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git; o:
llBRARY
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dissertation entitled

THREE ESSAYS ON UNCERTAINTY AND LEARNING BY ECONOMIC AGENTS

presented by

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THREE ESSAYS ON UNCERTAINTY AND LEARNING BY ECONOMIC AGENTS
By

Hilde Patron

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Economics

2001

ABSTRACT

THREE ESSAYS ON UNCERTAINTY AND LEARNING BY ECONOMIC AGENTS
By

Hilde Patron

Based on the assumption that agent’s decisions affect their understanding of their
environments I introduce Bayesian updating of beliefs in three different economic models
to study how the agent’s ability to learn affects the decisions of rational agents.

In Chapter 1 I study a two-period model with an incumbent firm threatened with
entry. Demand is unknown and stochastic, and prices contain statistical information
about demand. The incumbent’s first period decision affects the informativeness of the
price level and through it, the probability of entry. Hence the incumbent can manipulate
its quantity to discourage entry. In equilibrium, unless the possible demand functions
differ by a constant, the incumbent always manipulates first period output to reduce the
probability of entry, i.e. limit prices. In particular, if given its prior information the
entrant is currently not entering, then the incumbent limit prices by concealing
information from the entrant; if at current beliefs the entrant is entering, then the
incumbent limit prices by revealing information.

In Chapter 2 I study a model in which fiscal policy determines that for a short
period of time the government must rely exclusively on the income from issuing money,
but it is uncertain about the way in which monetary policy influences the public’s

demand for money. I assume that at any point in time the government uses all the

information available to it to make the best possible decision, and that as new
observations become available the government updates its beliefs about the demand for
money using Bayes’ rule. The three results of the model are: First, the government
values more information about the demand for money as this allows it to make a more
accurate decision. Second, if the government can affect the informational content of the
demand for money (that is, unless the possible demand functions differ by a constant) it
will adjust the rate of growth of the money supply to increase information. Third, under
some parameter specifications the government induces a hyperinflation to learn about
demand.

In Chapter 3 I study the design of incentive contracts for central bankers when the
government and the private sector are imperfectly informed about the central banker’s
preferences. Since the contract affects the inflation rate set by the bankers, which in turn
contains statistical information about the banker’s loss function, the government can
manipulate the contract to increase information. However, bankers have incentives to
manipulate the government’s and the public’s perception of their preferences and hence
they manipulate the inflation rate accordingly, which affects the informational content of
each contract and the expected variability of the inflation rate. The interaction of these
two effects determines the gains and losses from inducing more transparent monetary

policies, and whether bankers should be appointed to one or two periods.

Copyright by
HILDE PATRON
2001

ACKNOWLEDGEMENTS

I would like to thank Rowena Pecchenino, Carl Davidson and Thomas Jeitschko

for their guidance and help in writing this dissertation.

I also benefited from comments from and discussions with Chris Curran, Ana

Maria Herrera, Leonard Mirman, Elena Pesavento and Ed Schlee.

Finally I would like to acknowledge my parents, my brothers, my husband,

Laureano, and my beautiful, sweet daughter Andrea.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. viii
LIST OF FIGURES ................................................................................. ix
INTRODUCTION .................................................................................... 1
CHAPTER 1
MONOPOLY EXPERIMENTATION AND ENTRY DETERRENCE:
LIMIT PRICING THROUGH LEARNING ...................................................... 4
1.1 Introduction .............................................................................. 4
1.2 Related Literature on Limit Pricing ................................................... 7
1.3 The Model ............................................................................... 10
1.4 The Equilibrium ........................................................................ 13
1.4.1 The Second Period Equilibrium ........................................ 13
1.4.2 The First Period Equilibrium ........................................... 16
1.5 An Example ............................................................................ 17
1.6 If the Incumbent Could, Would it Try to Discourage Entry? ................... 22
1.6.1 General Discussion ...................................................... 22
1.6.2 Conditions Under which the Incumbent Limit Prices ............... 24
1.7 Conclusions ............................................................................. 33
CHAPTER 2
UNCERTAINTY, LEARNING, AND SEIGNORAGE MAXIMZING
GOVERNMENTS .................................................................................. 36
2.1 Introduction ............................................................................. 36
2.2 Literature Review ...................................................................... 41
2.3 Model and Equilibrium ............................................................... 44
2.3.1 The Basic Model ......................................................... 44
2.3.2 The Equilibrium .......................................................... 51
2.3.3 The Myopic Equilibrium ................................................ 52
2.4 The Value of Information ............................................................ 53
2.5 Effects of Experimentation in High Inflation Economies ........................ 55
2.6 Solution for Uniformly Distributed Shocks and Linear Demand F unctions...65
2.7 Some Intuition on the Model ......................................................... 76
2.8 Case Study: The Bolivian Hyperinflation of 1984-1985 ......................... 77
2.9 Conclusions ............................................................................... 80
CHAPTER 3
INCENTIVES CONTRACTS FOR CENTRAL BANKERS: INFORMATIONAL
IMPLICATIONS OF LINEAR CONTRACTS ................................................. 82
3.1 Introduction ............................................................................. 82
3.2 The Model .............................................................................. 89

vi

3.3 Delegation with Learning ........................................................... 102

3.3.1 The Model ............................................................... 102
3.3.2 The Equilibrium ......................................................... 105
3.3.3 The Value of “More Transparent” Monetary Policy (The Value of
Information) ............................................................. l l 1
3.3.4 Central Banker’s Incentives to Manipulate Information .......... 124
3.3.5 Government’s Incentives to Induce More Informative Inflation
Rates (Incentives to Experiment) ..................................... 130
3.4 Conclusions ............................................................................ 143
CONCLUSIONS ................................................................................... 147
APPENDICES ..................................................................................... 150
APPENDIX A
Appendix to Chapter 1 ................................................................... 151
APPENDIX B
Appendix to Chapter 2 ................................................................... 161
APPENDIX C
Appendix to Chapter 3 ................................................................... 171
REFERENCES .................................................................................... 183

vii

LIST OF TABLES

Table 2.1 Summary of Example ............................................................. 72
Table 2.2 Minimum and Maximum Inflation Rates ...................................... 74
Table 2.3 Gains in Seignorage from the Experiment ...................................... 75

viii

Figure 1.1
Figure 1.2
Figure 1.3
Figure 1.4
Figure 2.1
Figure 2.2

Figure 2.3

Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

LIST OF FIGURES

Expected Demand Functions for the Example ................................. 18
Increasing Quantity Increases Information .................................... 27
Increasing Quantity Decreases Information .................................... 28
Expected Demand Functions: Case ((1) ......................................... 33
Seignorage Function with Laffer Curve Property ............................ 48
Informativeness of the Inflation Rate .......................................... 55

Conditions under which the Non-Myopic Inflation Rate is Higher the

Myopic Rate ........................................................................ 56
Money Growth Paths ............................................................. 59
Hyperinflationary Money Growth Paths ....................................... 61
Spread of the Possible Demand Functions ..................................... 64
Intuitive Representation of the Model .......................................... 77
Value of Information: Restricted vs. Unrestricted Models ................ 113
How Bankers Signal their Type ................................................ 117
Value to the Central Banker of Signaling its Type .......................... 123
The Central Banker does not Manipulate Information ..................... 126
The Central Banker Increases Information ................................... 127
The Central Banker Decreases Information .................................. 128
One Type of Central Banker Decreases Information, One Type Increases

Information ....................................................................... 129
Myopic reaction functions ...................................................... 134

Given the Strategic Behavior of the Bankers, Increasing the Punishment
Increases Information ............................................................ 136

ix

Figure 3.10 The strategic behavior of the banker induces more transparent monetary
policy .............................................................................. 141

Figure 3.11 Static Loss Function of the Government ..................................... 142

INTRODUCTION

When rational, optimizing agents have to make decisions without fully knowing
their environments they must use all the information available to them at any point in
time to make the best decision possible, they must revise their beliefs as new information
becomes available and they must adjust their actions according to what they learn. Based
on these premises I study how the agent’s ability to learn, and firrthermore to generate
information affect the decisions of rational optimizing agents in three different models.

In Chapter 1 I present a two period model of two rational, risk neutral, profit
maximizing firms, the incumbent and the entrant, that face an unknown and stochastic
demand function. There are two possible states of demand but firms do not know which
is the true state. In the first period the incumbent sets a maximizing output, and a price
and output are realized. Firms use this information to update their beliefs. If given these
beliefs the expected profits of the entrant in the second period are positive the entrant
enters and both firms compete in quantities, if they are negative the entrant does not enter
and the incumbent stays a monopolist.

Since the incumbent’s first period output affects its posterior beliefs, the
incumbent can induce more informative outputs. With more information it can make a
better informed decision in the second period and increase expected profits. However,
the entrant starts out with the same priors as the incumbent does, observes the same
variables, and updates beliefs in the same way, thus whatever the incumbent learns, the

entrant learns as well. More informative outputs then also affect the expected

profitability of the entrant. The incumbent can use this to its advantage to try to
discourage entry.

I find that the expected profits of the incumbent firm are decreasing in the
probability of entry, and thus that unless the incumbent cannot manipulate the
information of the entrant (that is unless each output level is equally informative), it
manipulates the first period quantity to deter entry.

In Chapter 2 I study a model in which the government has accumulated large
deficits or has large spending needs, but it can only finance itself with the income from
money creation (seignorage), for example because of the inefficiency of the tax system
combined with low credit ratings.

The government is uncertain about the parameters of the money demand function,
and can learn about them in a Bayesian fashion. Knowing the demand function
accurately is important because this helps the government determine what proportion of
money issue turns into inflation and what into real seignorage revenues.

I find that the government always gains fi‘om better knowledge of the demand
function and thus it seeks to learn and it adapts to new information. Constant money
growth rates are thus suboptimal (unless the possible demand functions differ only by a
constant). Furthermore, I find that for some parameter values the government creates a
hyperinflation in order to learn about demand.

In Chapter 3 I study the design of linear contracts for central bankers when the
central banker’s preferences for output over inflation stabilization are not fully known.

Bankers are hired for one or two periods and given a contract that stipulates a punishment

if the inflation rate is above the socially desirable rate. If they are hired for two periods
the contract is revised after the first period.

After the bankers are hired, the private sector forms expectations of inflation for
the period and sets wages accordingly. A supply shock is then realized and observed by
the bankers who then set the inflation rate.

Based on the inflation rate, which is a noisy observation of the banker’s
preferences, the government updates its beliefs and adjusts the second period contract (or
punishment) accordingly. The private sector also updates its expectations of inflation and
wages after observing the inflation rate.

The three results of the model are: First, there exists a parameter space for which
the government benefits from more transparent monetary policy (more information). For
these parameters however, the bankers might not benefit from more transparency.
Moreover, as long as the banker can affect the beliefs of the government it will do so, that
is, bankers act strategically. In particular if the banker’s value of signaling its type is
positive (negative) it will try to increase (decrease) the government’s amount of
information. Second, the strategic behavior of the central banker has two effects on
welfare. On the one hand it increases or reduces the informativeness of the inflation rate,
and on the other hand it increases or reduces the variability of the inflation rate (and of
output). Third, the interaction of these two effects determines whether the government
sets contracts that increase information, and whether bankers should be hired to serve one

or two periods.

CHAPTER 1

MONOPOLY EXPERIMENTATION AND ENTRY DETERRENCE:

LIMIT PRICING THROUGH LEARNING

1.1 Introduction

Firms often face uncertainty about their environments. For example, they may
not fully know their cost structure, the quality of their product, or the demand they face.
If so, they might take deliberate measures to increase their information, thereby
sacrificing current period profits for more information and thus future higher profits.

When there is more than one firm and when information is a public good the first
period decision becomes a complicated issue, as it can be used not only to learn
(experimentation) but also to alter the rival firms’ perception of profits (limit pricing or
predatory pricing). Thus considering two fnms introduces all sorts of dynamics into the
decision making process in the presence of uncertainty. One particular case is a
monopolist faced with the threat of entry, which is the topic of this paper.

I present a two period model of two rational, risk neutral, profit maximizing firms,
the incumbent and the entrant, that face an unknown and stochastic demand fimction.
There are two possible states of demand but firms do not know which is the true state.

In the first period the incumbent sets a profit maximizing output, and a price and
output are realized. Firms use this information to update their beliefs. If given these

beliefs the expected profits of the entrant in the second period are positive, the entrant

enters and both firms compete in quantities, if they are negative, the entrant does not
enter and the incumbent stays a monopolist.

Since the incumbent’s first period output affects its posterior beliefs, the
incumbent can induce more informative outputs. With more information it can make a
better informed decision in the second period and increase expected profits. However,
the entrant starts out with the same priors as the incumbent does, observes the same
information, and updates beliefs in the same way, thus whatever the incumbent learns, the
entrant learns as well. More informative outputs then also affect the expected
profitability of the entrant. The incumbent can use this to its advantage to try to
discourage entry.

I find that unless the established firm is not able to affect the informativeness of
the price level, it limit prices, that is, it manipulates output to reduce the probability of
entry. This result is similar to Fudenberg and Tirole (1986), who found that a firm might
price below monopoly to reduce the probability of other firms joining the market. In my
paper however, limit pricing can mean increasing or decreasing quantity.

Under some circumstances entry deterrence takes the form of concealing
information (less precise beliefs), and under other circumstances it takes the form of
revealing information (increasing the chance of bad news). The intuition of this result is
as follows: the entry decision is based on what the entrant learns after the first period. If
for example the entrant is currently considering entry, new information might reveal
some bad news with more precise beliefs. Thus the incumbent can deter entry by

increasing the entrant’s information.

The form of uncertainty and the information generation process I study in this
paper have already been studied in monopoly and duopoly situations. Among the many
papers in the literature, two that are particularly important to my study are Mirman,
Samuelson and Urbano (1993) and Mirman, Samuelson and Schlee (1994)

Mirman, Samuelson and Urbano (1993) study the conditions under which a
monopolist faced with an uncertain and stochastic demand function experiments. They
find that as long as the monopolist can affect the informativeness of the price level it
experiments in the direction that increases information, except for the case in which the
value of information is zero'.

Mirman, Samuelson and Schlee (1994) study the same problem for a duopolistic
market, and find that since the value of information for the duopolist could be negative,
information could take the direction of reducing information. Otherwise, firms
experiment so as to increase information.

In this paper I study a firm that can be either a monopolist as in Mirman,
Samuelson and Urbano (1993) or a duopolist as in Mirman, Samuelson and Schlee
(1994). The firm has a belief concerning whether it will be one or the other but does not
know for sure which one. The formulation of my model is similar to their models,
however I study how the decision to increase or decrease information depends on how the
probability of entry is affected with more or less information.

The rest of the paper is organized as follows. In Section 1.2 I review the limit
pricing literature. In Section 1.3 I set up the model, and in Section 1.4 describe the

sequential equilibrium of the game. In Section 1.5 I solve for the equilibrium of an

 

‘ One such case is a zero marginal cost monopolist that faces two possible linear demand functions that
cross at the quantity axis.

example. In Section 1.6 I study the conditions that determine whether the incumbent

limit prices or not, and in Section 1.7 I conclude and discuss several extensions of this

paper.

1.2 Related Literature on Limit Pricing

In a seminal paper Bain (1949) suggested that prices in an industry could reveal
future pricing policies or the character of the industry. Subsequent papers on limit
pricing can be divided into one of these two groups. Papers that study limit pricing as a
credible threat (e.g. Bain (1949), Modigliani (1958), Friedman (1979), Salop (1979) and
Spulber (1981)), and papers that study limit pricing as revealing something about an
industry characteristic known to the established firm but unknown to the entrant.

Among the latter, the preeminent example is Milgrom and Roberts (1982). Other
papers are Matthews and Mirman (1983), Harrington (1986) and Jain, Jeitschko and
Mirman (2000). My paper is among this second group and for this reason I concentrate
on reviewing this strand of the literature.

Milgrom and Roberts (1983) study a signaling model with two firms. Each firm
is uncertain about its rival’s unit costs, which can be low or high. Moreover, the price set
by the incumbent contains information about its costs. Thus a price below the monopoly
price could be a signal of low costs. Since the incumbent firm wants the entrant to think
it is low cost, limit pricing occurs in equilibrium. Entry however is not necessarily
deterred. For example, in a separating equilibrium the entrant is not fooled and entry is

not deterred.

Matthews and Mirman (1983) study a similar model of asymmetric information.
They assume that firms know all cost functions, but that there is a persistent industry
profitability parameter known to the established firm but unknown to the entrant.
Moreover, there are unobservable random shocks to demand, which occur after the
established firm has made its decision, and which prevent the price from perfectly
revealing the industry characteristic. They find that limit pricing always occurs in
equilibrium.

Harrington (1986), also in a model of asymmetric information, assumes that the
incumbent is uncertain about the entrant’s costs and the entrant is uncertain about the
costs of both firms. If the two cost functions are correlated, the first period price conveys
information about both cost structures. For example, a high price signals that the
incumbent frrrn has high costs and that the entrant is likely to have high costs as well.
Decreasing price thus does not necessarily deter entry.

For Milgrom and Roberts (1983) and Matthews and Mirman (1983) limit pricing
means charging a price below the short run monopoly price. Unlike them, Harrington
finds that if the incumbent’s and the entrant’s cost are sufficiently correlated, which is
reasonable if both firms have access to the same technology, limit pricing can take the
form of charging a price above monopoly price.

Following Harrington’s result, I will define limit pricing as any action taken by
the incumbent firm with the intent to reduce the probably of entry, whether it means
increasing or decreasing prices.

Jain, Jeitschko and Mirman (2000) study an incumbent firm threatened with entry,

but unlike my paper the incumbent firm must contract with a bank for outside funds. In

their model uncertainty arises because the bank and the entrant are not sure about the
incurnbent’s type (costs) but observe a noisy signal of it. They study the design of
contracts between the bank and the incumbent firm, and how these are affected when the
incumbent is threatened with entry. They show that the bank structures the first period
contract so as to reduce the probability of why, because deterring entry increases the
profits of the incumbent and thus the payments that the bank can extractz.

My model is different from theirs because I do not model contracts between
financial institutions and firms, and thus I do not concentrate on learning in principal-
agent models. Instead I concentrate on the pure entry deterrence model among producing
firms.

In the above papers, as well as in other models of limit pricing, the results depend
crucially on how information is modeled. In particular, it is the incentive to preserve its
informational advantage which determines the incumbent’s incentives to limit price. In
my model I relax the assumption of asymmetric information of any kind. I relax this
assumption to study the more realistic question of limit pricing when the incumbent is
still learning about its environment (demand). This adds richness to the problem because
now the incumbent faces a real dilemma: given the public nature of information, if it

wants its rival to learn less it has to learn less as well.

 

2 Iain, Jeitschko and Mirman (2000) is a modified version of the paper by Jeitschko and Mirman (2000)
who study short term contracting in a principal-agent model where the principal does not know the agent’s
type. They however do not consider the threat of entry.

I find that even in the absence of an informational advantage, as long as firms do
not fully know the characteristics of the market demand, limit pricing almost always

occurs in equilibrium3.

1.3 The Model

I analyze a two period game. There are two rational, profit maximizing, risk
neutral firms. In period one there is only a single firm, an incumbent ( I ) that can
produce at a marginal cost k, . The incumbent sets an output and observes the resulting
price. There is a second firm, the entrant ( E ) that observes the incumbent’s output and
the resulting price.

In period two the entrant can choose to produce an identical product with
marginal cost kg. The entrant incurs a small entry cost § if it chooses to produce. If
expected profits are negative, the entrant will not enter and the incumbent remains a
monopolist. If it enters, each firm independently and simultaneously sets output. Given
the total output, a market clearing price results.

In each period the inverse demand function is given by p = g(Q, 7) + a , where p
is the market price, Q is the total quantity produced, and a is the realization of a random
variable, whose distribution is characterized by the density function f (a ). I assume that

the expected value of a is zero, that f (5) is continuous and positive and that it satisfies

the monotone likelihood ratio property (MLRP), i.e. [ff—6% is strictly decreasing in a.
s

 

3 Only when the incumbent firm cannot affect the informativeness of the price level, that is when the
possible demand functions only differ by a constant, limit pricing does not occur in equilibrium

10

y is an unknown parameter that can take on two possible values, 7 e {7, z},
where the upper bar means the “good” demand and the lower, the “bad” demand, i.e, \7’Q

g(Q,?) 2 g(Q, z). The function g is non-increasing in Q, and for convenience, twice
continuously differentiable in Q. Finally assume that there exist numbers Q and Q with

0 < Q < Q such that g(Q, y) > g(Q, y) 2 min(k, ,kE) , that is assume that there is positive

production in equilibrium.
The firms do not know which is the true state of demand. However, in period one

both firms have common beliefs about demand. That is, for a given output, both firms
expect the same price. In particular, they have a belief p° that ytakes on the value ; .
In period one, firm 1 chooses an output Q,. After the first period quantity is

chosen, a random shock is realized (although not observed) and hence a price is realized.

Firm [’3 expected profit in period one is,
2r,(Q,.p°> = I g(Q,,?)p° +g(Q,,z)(1- p") —k,IQ, .
Assume that there are no fixed costs, that 7r, (Q1. p°) is strictly quasiconcave, and

that the fimction g is such that Q g’ + 2g is strictly decreasing in Q. These last two
assumptions are technical assumptions that are needed in order to ensure the existence of
an equilibrium (the proof is in Appendix A).

Both firms observe p and Q, but not a. With these commonly observed variables
and common priors, firms update their beliefs about demand according to Bayes rule and

have the same posterior. Let p denote the posterior belief that y takes on the value ; ,

then, by Bayes rule,

11

_ p°7 ’
p°f +(l-p°)[_

 

p(p:Q1) :

where 7 = f(p-§) = f(p —g<Q,,?» and 1 = f(p —_g;) = f(p-g(Q,,z)).

In period two, after observing the quantity chosen in period one (Q) and the
resulting price, firms update their beliefs and the entrant decides whether to enter or not.
If it enters, both firms compete in quantities; if it does not enter, the incumbent sets the
quantity that maximizes monopoly profits.

The timing of the model can be summarized in the following diagram.

Time Line

t=l —""‘ Firms start out with prior beliefs p0.
__ Firm 1 sets Q, .
__ a and p are realized

(but a is not observed).

__ I and E observe Q,, p,7r, and update beliefs (p).

 

YQTCidCS whether to enter or not.
E Enters E does not enter

__ Firms compete in quantities.____ Firm 1 sets the monopoly
maximizing quantity.

__ a and p are realized __ a and p are realized
(but a is not observed). (but a is not observed).

 

 

—— Firms get profits. — Firm 1 gets profits.

12

1.4 The equilibrium
I am interested in a sequential equilibrium and thus I solve for the second period

equilibrium first.

1.4.1 The Second Period Equilibrium
Afier the first period, having observed price and quantity, firms update their

beliefs and the entrant makes a decision of whether to enter the market or not. This

decision is made by forecasting the duopoly equilibrium (q19q5)9 and calculating
expected profits (719,275). Formally, each firm solves the following maximization

problem,

Max ”.(qlggp) M q,- =argmax rum/gap), i=1,E, (1.1)
I q]

where
7r, (611,615.10) = [g(q, + «JED/0 + g(q, + C115,1)(1 - p) -k,]q,,
”Anya/9) = [g(q, + (15,?» + g(q, + 45,1)(1- p) -k5]qs -€ -

Let q, *( p) and q5 * (p) denote the outputs, as a function of beliefs that
maximize the above profits. Finally let total quantity be denoted by
(NP) 5 q, *(p)+q.- *0?)

Let V5 (p) denote the resulting value function of the entrant, and let V, (p) denote

the resulting value function of the incumbent (if entry occurs), i.e.

mm = [g(q*(p),;)p + g(q *0», 1x1 — p)—k.—Iq. *(p) -5.

W) = [g(q *(p),?)p+ g(q *(p),1)(1-p)-k,]q,*(p)-

l3

If the value function of the entrant is positive, i.e. if V5 (p) 2 0, E will enter and
the two firms will compete in quantities. If V5 (p) < 0 the entrant will not enter and the
incumbent will remain a monopolist. I.e. for some beliefs, q E * (p) may be zero or close
to zero so that profits would be strictly negative, as the entrant must pay an entry cost 5 .

Since the value function of the entrant is non-decreasing in prices (Lemma 1.1
below), the entry decision of the incumbent can be written in relation to a minimum price
and a minimum associated posterior belief.

Lemma 1.1: The value function of the entrant is non-decreasing in prices,

mm
dp '—

Proof: Differentiating the value function of the entrant with respect to first period

price yields,

dVE(p) _ d<Ig(q*(p),7)p+ g(q*(p),z)(1- p)—k.]q. *(p)—:)
dp — 51.0

 

 

9

dVE (P) =

a; — _ III III d_p.
dp [(g(q (pm g(q 0)).qu rm] dp.

equals

By assumption (g (q *(P),;) ‘8 (q *(P), 1)) 2 0 , then the sign of ——dV§ (p )

0_
the Sign of 1'2. To find 1’2 remember that by Bayes rule p(p,Q,) = p f

dp dp p°7+(l-p°)[ ’

 

and thus taking the derivative with respect to p yields,

13 _ p°7'(p°7+(1—p°)1)-p°7(p°7'+ (1-p°)f)
dp ’ (p°7+(1-p°)_f_)’
= p°(1—p°)(71—7p
(p°7+(1—p°)1>2

 

 

l4

which is non-negative given the monotone likelihood ratio property (i.e. according to the

likelihood ratio property % is strictly decreasing in a , and thus

>

\ll‘fl
I". I“:

97,671“). Thus fl[1440120.

Cl

Since entry happens for high enough prices (Lemma 1.1), there is a minimum
price w (Q ,) such that if the first period price is higher than this price, the entrant enters,

if it is lower, the entrant does not enter. The minimum price that induces entry is

determined implicitly through Bayes rule as follows,

,0*(p Q)E pof£W(QI)—g(Qp;»
’ p°f(W(Q,) — g(Q, . 7)) + (1 — p°)f(v’(Q,) — g(Q,, z»

(1.2)
where p“(p,Q,) is the minimum value of the posterior belief needed for the entrant to
enter, i.e. p*( p, Q,) is defined by VE(p*) a 0. Alternatively, we can define the
minimum price that induces entry W(Q,) by the identity V5 ( p(l//(Q, ), Q, )) E 0.

When first period price is not high enough, the entrant does not enter and the
incumbent remains a monopolist. Let V," (p) denote the value function of the incumbent
when there is not entry, i.e.

mp) = [g(q'"(p),7)p+g(q'"(p).z_)(1— p)—k,1q"'(p),

where q’" (p) is the quantity that maximizes monopoly profits for the incumbent, i.e.

q”(p) e arg max [g(qiho + g(q, {XI - p) - kllq -

15

In summary the second period equilibrium is given by the quantity q’"( p) if

p< p*(p,Q,) , and by the quantities q, *(p) and qg *(p) if p 2 p*(p,Q,)-

1.4.2 The First Period Equilibrium

In period one the incumbent accounts for the effect of its first period decision on
the informativeness of the price level and thus on future profits. Thus in period one the
incumbent maximizes first period profits plus the expected discounted value of second
period profits

I have to take expectations of second period profits because in period one the
posterior belief p is a random variable whose distribution depends upon the incumbent’s
first period output and price. At the time of the maximization problem, p is random
variable and thus I have to take expectations of second period profits over all possible
prices.

Formally, the incumbent chooses Q, to maximize,

”1(ano)
H: I, o = 919;) °° 1'3
-ao V(QI)

where 5 is the discount factor, and h(p,Q,) is the posterior distribution of first period
price implied by posterior beliefs and by the shock a.

To find h( p, Q,) let E( p, Q,) be the value of a that must result if the period one
price is p , the first period quantity is Q, , and the true state of demand is ;; and let

g( p, Q,) be the value of a that must result if the period one price is p , the first period

16

quantity is Q, , and the true state of demand is Z . The density of p can be written as

h (AQI) = .00 7 + (1- 10°);-
Intuitively, the incumbent firm maximizes first period profits plus the discounted
expected value of second period profits. Expected second period profits are expected

duopoly profits conditional on entry (i.e. conditional on p being at least l/I(Q,)), plus

expected monopoly profits conditional on no entry (i.e. conditional on p being less than

V’(QI ) )

Let Q, *(p°) be the resulting first period equilibrium quantity. The existence of

equilibrium is proven in Lemma 1.2 and Proposition 1.1. For continuity of exposition I
prove these statements in Appendix A.
Lemma 1.2: If a period two equilibrium exists, then that equilibrium is unique and

at least one firm produces a positive quantity’.

Proposition 1.1: Let g(Q,y) be defined and continuous on 1R+ for 7 e {7, z}and let

there be Q such that g(Q,7)=0 for 76 {7%}. Then a (possibly mixed strategy)
sequential equilibrium existss.
In the next section I develop an example to illustrate the model and the

equilibrium.

1.5 An Example
For illustration, the following linear demand example is used throughout the

paper. Consider the following two linear demand functions:

 

‘ Mirman, Samuelson and Schlee (1994), Lemma 2, page 368.
5 Mirman, Samuelson and Schlee (1994), Proposition 1, page 370.

17

p=20—Q+£
p=20—1.5Q+£'

Since for all Q>0 and all 3, 20—Q+£220—l.5Q+£, then p=20—Q+£ is

the “good demand”. Graphically these two demand fimctions are shown in Figure 1.1.

Assume that a is uniformly distributed in the interval [-15,]5]. The density

 

 

function of a is thus given by f (a) = T5——(L—-1_5) = 310 , and the expected value of a by
" a 52 ‘5 (15)2 - (-15)2 6
E(£)= j—da= = =0.
_,5 30 60 _,5 60
P
A
20

 

 

40/3 20 Q

 

 

 

Figure 1.1. Expected Demand Functions of the Example

¥

6 In general, if s is uniformly distributed in the interval [0, b], its density function is given by f (s) = b;
— a

 

 

b 2 b 2 2
and its expected value by 5(5) = Ide5 = 2“: )I = (12(1)- (a)) = (b :20: a) = a: b .
._ a — a - a - a

18

This example violates two of the assumptions of the model: first, that the shock
has unbounded support, and second that its density function is differentiable. Although
these assumptions are violated, the example retains their general implications, namely
that there is no learning with probability one after the first period, and that higher prices
increase the beliefs that demand is the good one (that is that beliefs are monotonic).

Finally assume that 621, k, =4, k5 =8, 6:55, and p” =0.5.

To find the equilibrium I first find the second period equilibrium. If there is no

entry, the incumbent chooses q’” to maximize monopoly profits, which are given by
It," (p) = [(20 - q’" ) p + (20 —1.5q"')(l — p) - 4] q'" , which gives maximizing quantity

64

,and rofits V’" =____.
p ’ (p) 1.5-0.5p

8
* :—
q’" (p) 1.5—0.5p

If the entrant enters, firms compete in quantities, and thus solve (1.1), which in

Max [9(20 - q, ‘45) + (1 - p)(20 - 15(9; + ‘112)) - 414;

4!

the example is given by ,
Agax Iprzo—q, ~q.)+(1-p>(20—1.5(q, +q.»—8Iq. -55

 

 

 

20
which ives the Coumot e uilibrium uantities * = ,
S q q q, (1)) 3,1545,»
... 8 . . 400
qE (p) = , and value functions for the Incumbent, V,( p) = ,
3(1.5 — 0.5p) 9(1.5 — 0.5 p)
64

 

and for the entrant V = — 5.5.
50’) 9(1.5—0.5p)

To find the incumbent’s expected second period value function I need to integrate
second period profits over all possible posterior beliefs. With a uniform distribution,

there are only three possible posterior beliefs, zero, the same prior, or one, as follows. If

19

the true demand is given by p = 20—1.5Q, +8, then the first period price is within the
following interval pe[20—1.5Q, —15,20—1.5Q, +15]; If the true demand is given by
p = 20—Q, +5 , then the first period price is within the following interval
pe[20—Q, -15,20—Q, +15]. If the first period price is above 20—1.5Q, —15 but
strictly below 20—Q, —15 , then the firms learn that the true demand is the “bad”
demand. If the first period price is strictly above 20-1.5Q, +15 , then the firms learn

that the true demand is the “good” one. Formally posterior beliefs are given as follows,

0 if 20-1.5Q,-15$p<20—Q,—15
p: p°=o.5 if 20—Q,—15 spszo-I.5Q,+15,
1 if 20—1.5Q,+15<p$20—Q,+15

and each posterior happens according to the following distribution function7,

f 0

’3’ if 20—1.5Q, —lSSp<20—Q, —15

 

h(p,q)=< if 20—Q,—15$p$20—1.5Q,+15 .

if 20—1.5Q, +15<pSZO-Q, +15

 

U) U) H
Olbc OIF‘ be

Since the value fimction of the entrant V5 (p) is only positive for values of p

above 0.41, if the entrant believes that p is below this value, he will not enter. That is

p“ = 0.41 and the minimum price that induces entry is given by,

 

7 To find the density function note that h(p,q)= p°7+(l—p°)£ = 0.5(7+£), and that for a uniform

. - l 1 l 1
distribution function = =———=—, and thus I: , =0.5 —+— . Finall , from Ba es’
f [- (15-(—15)) 30 (p q) [30 30) y y

0-
' P f o“ .
rule, if p=0 then 0: _ <:>p f=0; and if p=l, then
p°f+(l-p°)[
1_ p°f

_ p°7+(1—p°)/”(l-po)£=0'

20

i/I(Q,) = 20 — Q, - 15 = 5 - Q, . Therefore the entrant’s decision can be represented with

the following rule,

- 0
Entry Rule {Entry If p e {p ’ 1} ,
No Entry if p = 0

or by,

Ent’y if pZS-Q,

Entry Rule .
No Entry If Otherwise

For prices above S—Q, the incumbent expects duopoly profits V,( p) , and for

prices below 5 - Q, , it expects monopoly profits V,"' (p). Second period expected profits

are thus given by,

 

ZO—Q, +15 2 0—1.5Q, +15
1,, ,Q V, (p =1)h(p = 1)dp + [M V, (p = 0.5)hrp = one
W (Q )= ' ' '
1 20-Q,-Is m ’
I,,_,_,Q,_,,V, (p=0)h(p=0)dp
thus,
35-9, 400 1 35-159, 400 1
law-so:9(1.5-0.5(1))5dp + 5-9, 9(1.5-0.5(0.5))§dp
W(Q,)= =35.6+0.13Q,.

 

+ J-S-Q, 64 1 d
s-I.SQ, 1.5—0.5(0) 60 p

Expected second period profits are an increasing function of first period quantity,
intuitively because higher quantities both increase information and reduce entry. (More

on this in the next section.)

21

To obtain the sequential equilibrium the incumbent chooses quantity Q, to
maximize [0.5(20—Q,)+0.5(20—1.5Q,)—4]Q, +W(Q,), which yields the first period

equilibrium quantity Q, "‘ =6.45.

1.6 If the Incumbent Could, Would it try to Discourage Entry?

So far I described the model and equilibrium, and illustrated it with an example.
Now I address the following natural questions. Given that this is a model with a threat of
entry, how does the incumbent react to the threat of entry? That is, does the incumbent
limit price? What does limit pricing look like? Does the incumbent limit price by
concealing information from the entrant? Or instead, by increasing the entrant’s

knowledge of the industry? And exactly, what is the definition of limit pricing?

1.6.1 General Discussion
In the first period the incumbent sets a quantity Q, , which affects the function
g(Q,,y) , and through it the price level p = g(Q,,y) + 8. Although the incumbent or the

entrant does not observe the shock a , the quantity Q, affects the possible distribution

functions of prices 7(e)=f(p-g(Q,i» and 1(a)=f(p—g(Q,._r_». and through

them the minimum price that induces entry w(Q, ) , implicitly defined by,

p°f(v/(Q,)-g(Qp;))

” (”’Q’) E p°f(w(Q,)-g(Q,.?»+(1—p°>f(w(Q,)-g(Q,. _y_»’

posterior beliefs p ,

p°7

,0(PIQ1)= p77:- (1_p0)£ ,

22

their distribution,
h (PeQI) = P0 7 + (1- POM,
and therefore the probability of entry, G , formally given by G(Q,) = f, 9,) h( p, Q, )dp .

If the posterior distribution of beliefs is riskier in the Rothschild-Stiglitz sense,
then information increasess. Thus if changes in Q, increase information (lead to more
accurately beliefs) and through more information deter entry (reduce the probability of
entry), then the incumbent limit prices by enhancing the entrant’s information. If
changes in Q, decrease information and through less information deter entry, then the
incumbent limit prices by concealing information from the entrant.

In the example of Section 1.5, entry occurs for p2 p*= 0.41, or prices above

i/I(Q,) = 5 — Q, . Thus the probability of entry is given by,

_ 20-1.5Q,+15 1 20—Q,+15 p0 _ I
G(Q’)— Jio-Q,-15 26- I + «go-1.59”” 25 p_l-fiQ"

Since the entrant enters for prices above w(Q,) = S—Q, , an increase of one in

first period quantity decreases the minimum price that induces entry by one. All else
constant, there are more price observations that induce entry, and thus one would expect

an increase in the probability of entry. The probability of entry however goes from

G(Q,) = l—éé-Q, to G(Q, +1) =l-$(Q, +1) , which means that G decreases by i,

 

8 This comes from the definition of a more informative experiment as defined by Blackwell (1951) and
(1953). An experiment is a pair of measures [ f (p-g(Q,, 1)), f (p—g(Q,,;))] , one for each state of
nature that gives the distribution of the first period price. An experiment
21 = [f ( p - g(Q, ', 1)), f (p — g(Q, ',;))] is more informative than an experiment
2. = min-g(Q, '.z)).f(p- g(Q, 2?)». if lame, ".piihrpe, )dp 2 Jame, '.p»h(p.Q, )dp for every
continuous, convex function z(p(Q, , p)) : [0,1] —> 91 .

23

not increases by one. In this example, if the incumbent wants to limit price it must
increase quantity.
Note that not all changes in Q, that affect information necessarily affect the

probability of entry, hence, when studying limit pricing I must abstract from everything

other than changes in 7 and i that translate into changes in the probability of entry. In

particular I will say that if the incumbent sets a quantity Q, "‘ so as to reduce the

probability of entry, then the incumbent limit prices.

Traditionally, the standard definition in the literature is that the incumbent limit
prices if it charges a price below the short run monopoly price with the intent to
discourage other firms from joining the market (as in Bain (1949), Matthews and Mirman
(1983), or Harrington (1986) for example). I do not want to use this definition however,
because as explained before the difference of my model and the short run monopoly
might include manipulation of information that does not affect the likelihood of other
firms joining the market. That is, since information may be of value to the incumbent
firm, the firm might manipulate information to make a better informed decision even if
this does not affect the entrant’s decision to enter or not. It seems natural thus to define

limit pricing by its purpose, by any action aimed at reducing the likelihood of entry.

1.6.2 Conditions Under which the Incumbent Limit Prices
To study the incentives of the incumbent to reduce the probability of entry I
compare its maximization problem with a hypothetical incumbent that takes the

probability of entry as constant.

24

An incumbent firm that takes the probability entry as given chooses Q, to maximize

equation (1.3) subject to Egg-Q12 = 0. Using Leibniz rule to differentiate G(Q,) with

I

respect to Q, yieldsg,

dG(Q,)=_ . h °° dh(p.Q,)d
“Te, we» (w(Q.).Q,)+V(£)——dQI p.

Hence the incumbent that takes entry as given solves the following problem,

V(QI) to

Agax e,(Q,,p°)+6 I W(ptp,Q,))h<p,Q,)dp+ I Krprp,Q,))h(p,Q,)dp
I "'3 V(QI)
dG(Ql) t w dh(p, Q1)
. —=o , h ,, , = ——d
sh dc, Oil/(Q) (W(Q)Q) mi) are, 12

That is, an incumbent that takes the probability of entry as given maximizes first

period profits plus the discounted expected value of second period profits, subject to a
fixed probability of entry”). Let Q ,F E be the solution to this problem.

In general, given some assumptions that introduced next I can show that the

incumbent threatened with entry will always limit price. This result is presented in

 

(X) (I)
9 According to Leibniz’ rule if y(x) = Em z(x,t)dt , % = idx'B-zbc, ,B(x)) -%z(x,a(x)) + If“) gz—xdt .

10 The probability of entry could change in two ways, through the minimum price that induces entry and
through the distribution of prices above this minimum price. When assuming that the probability of entry

is fixed, the correct assumption is that VI'(Q,)h(U/(Q,),Q,) = I flE-ip’&ldp , but not necessarily that

W(Q!) I

Q

w

W'(Q1)=0 and j
V(QI)

dh(ps Q1 )
sz

dp = 0. That is, the incumbent limit prices if it engages in any behavior

such that VI'(Ql)h(W(Q1)e Q1) ii I
W(Q!)

dh(P,Q]) dp .
(Q:

25

Lemma 1.2, and the proof, in Appendix A. Before I present the lemma however, I will
present and discuss the following assumptions.

1 will assume for the remainder of the paper that if at current beliefs the entrant is
entering, i.e. if p° > p * , then increases in information reduce the probability of entry; if
at current beliefs the entrant is not entering, i.e. if p° < p * , then increases in information

increase the probability of entry1 1. Formally these assumptions imply the following:

Assumption A1 . 1: In any of the following two scenarios the probability of entry is

non-increasing in quantity, i.e. d—Zfl s 0.

1
ALL] If increasing quantity decreases information, (Eng) <0 , and if at
current beliefs the entrant is not entering, p° < p*.
A1.1.2 If increasing quantity increases information, (g'—g') > 0, and if at

current beliefs the entrant is entering, p° > p *.

Assumption A1.2: In any of the following two scenarios the probability of entry is

5199.220,

non-decreasing in quantity, i.e.
sz

A1.2.1 If increasing quantity decreases information, (Q— g') < 0 , and if at

current beliefs the entrant is entering, p° > p *.

 

” This is an assumption and not a result of the model. For example consider the example from Section 1.5.

If prior beliefs had been assumed to be given by p° = 0.1 then at prior beliefs the entrant would not enter

and thus increases in information should increase the probability of entry. However the probability of entry

dG(Q,)
d

1
construct models that violate thia assumption. Since these cases do not make intuitive sense I will rule
them out.

in this case is given by G(Q,)=1—%:—Q, , and hence < 0. Hence mathematically it is possible to

26

A1.2.2 If increasing quantity increases information, (E'— g') > 0 , and if at
current beliefs the entrant is not entering, p° < p * .

The term (E'- g') gives the distance between the means of the two possible
distribution functions of prices, and it is thus a measure of the amount of information. If
(E'— g') is positive, it means that as quantity increases, the distance between the means

of the two possible distribution functions of prices increases, and thus information

increases (as in Figure 1.2).

 

 

 

’9

Figure 1.2. Increasing Quantity Increases Information

 

 

 

If (33') is negative, as quantity increases the two possible posterior

distributions of prices move closer together and become more undistinguishable (Figure

1.3). Increasing quantity thus decreases information.

27

 

 

 

(0'

 

 

 

Figure 1.3. Increasing Quantity Decreases Information

The role of (Q— g') in the assumptions (and in Lemma 1.3) is twofold. First, it
determines whether limit pricing means increasing or decreasing quantity. Second it
establishes a relation between the entry decision at priors and at posterior beliefs as

follows. For example, if (g'—g_') s O (as in Figure 1.3), increases in quantity decrease

information. If at prior beliefs the entrant does not enter (i.e. p° < p*), increases in

quantity do not reveal any new information, and thus should not change the decision of
the entrant. Decreasing quantity however might reveal some good news with more

precise beliefs and increase the profitability of the entrant. One would thus expect the

dG(Q,)<O
d _ .

l

probability of entry to increase as quantity decreases, i.e.

Now assume that at current beliefs the entrant would enter (i.e. p° > p*).

Decreasing quantity, which increases information, might reveal some bad news for the

28

entrant with more precise beliefs. One would expect the probability of entry to decrease

as quantity decreases, 2%(91—1 2 0.

1

These assumptions, although arbitrary, have intuitive appeal. This is a model
where the only reason why an entrant’s profitability might increase or decrease is through
the beliefs of the entrant. Thus the only reason why an entrant might change its mind
about entering or not entering is if it learns something hurtful or helpful. If given priors

the entrant is entering, no news should not change its mind; bad news might.

Lemma 1.3: Given assumptions AH and A12, an incumbent firm threatened

with entry limit prices, that is G(Q, *) s G(Q,” ). In particular,

(a) If reductions in quantity increase information, and through more
information increase the probability of entry
[(3— g') < 0 and ii—(QQL) s O i, the incumbent sets a quantity

1

higher than the incumbent that takes entry as given, i.e. Q,* 2 Q,” .

Since this reduces the probability of entry and decreases
information, then the incumbent limit prices by concealing
information from the entrant.

(b) If increases in quantity increase information, and through more

information increase the probability of entry

((§'—§y>o and d—(jg’lzo J, then the incumbent sets a

I

quantity lower than the incumbent that takes entry as given, i.e.

29

(C)

(d)

Q,*s Q,"E . Since this reduces the probability of entry and
decreases information, then the incumbent limit prices by
concealing information from the entrant.

If increases in quantity increase information, and through more

information reduce the probability of entry

[(g'-g')>0 and (Egg—030], then the incumbent sets a

1
quantity higher than the incumbent that takes entry as given, i.e.
Q,*2 Q,” . Since this reduces the probability of entry and

increases information, then the incumbent limit prices by revealing
information to the entrant.
If decreases in quantity increase information, and through more

information reduce the probability of entry

((3:8) < 0 and dig!) 2 0 j, the incumbent sets a quantity

I
lower than the incumbent that takes entry as given, i.e. Q,* s Q,”.

Since this reduces the probability of entry and increases
information, then the incumbent limit prices by revealing

information to the entrant.

Proof: In Appendix A.

30

Lemma 1.3 states that the incumbent threatened with entry manipulates first
period quantity to reduce the probability of entry. The only case in which the incumbent
does not limit price is when the demand functions differ only by a constant because in
this case the incumbent cannot affect the informativeness of the price level. The lemma
also characterizes what limit pricing looks like. In particular, if the incumbent can reduce

the probability of entry by concealing information from the entrant

dG(Q1)

[i.e. ('g'i-£v)_7Q__20], the incumbent limit prices by decreasing information; If
I

enhancing the entrant’s knowledge of demand reduces the probability of entry

dG(Q,)

[i.e. (Eng—s O], the incumbent limit prices by increasing the entrant’s

dQ:

information.

Of cases a through d of Lemma 1.3, case a is the traditional result of the limit
pricing literature. That is, in case a the incumbent limit prices by increasing quantity,
which conceals information from the entrant. Unlike the previous literature however, this
implies that the incumbent’s information is reduced as well.

In case b the incumbent limit prices by reducing quantity, but also by concealing
information. Harrington’s (1986) result could be adapted to fit this case. For Harrington,
if the entrant and the incumbent’s costs are sufficiently correlated, higher quantities
might reveal good news to the entrant, namely that it is likely to be the low cost type.
The incumbent wants to limit price by setting a lower quantity and thus concealing this
information from the entrant. Once again, unlike Harrington’s result, reducing the

entrant’s information also reduces the incumbent’s.

31

More surprisingly however is the result that limit pricing can mean revealing
information, cases c and d. Since the incumbent uses information as a strategic tool, and
since more information can mean bad news to the entrant with more precise beliefs, limit
pricing can very well mean enhancing the entrant’s knowledge of demand.

The example of section 1.5 corresponds to case c (Figure 1.1): increasing quantity

decreases the probability of entry iciQi: -—1— < 0 , and thus by Lemma 1.3 the

dQ, 120
incumbent limit prices by increasing quantity, i.e. Q,* > QfE. Since increasing quantity
increases information, i.e. since (E'— g ') = —1 — (—1.5) = 0.5 > O , then the incumbent limit
prices by increasing the entrant’s information about demand. The intuition of this result

is as follows. At the current beliefs the entrant enters, i.e. p° =O.5 > p"‘= 0.41. By

increasing output the incumbent increases the chance of bad news for the entrant, and
thus the probability of entry goes down.

As an example of case d, assume that the two possible demand functions are

given by p = 30— Q+£ and p = 20—§Q + a (Figure 1.4). Assume that a is unifome

distributed in the interval [-20,20], that 5 =1, k, =4, k,. =8, 5 =20, and p° =0.7,
which yields the first period sequential equilibrium is Q“ = 10.09. In this example the
minimum posterior that induces entry is p* = 0.4 , thus at prior beliefs the entrant is
entering. The probability of entry is given by G(Q,) = 0.23 +.01Q, , and thus a reduction

in quantity, which increases information, reduces the probability of entry. According to

Lemma 1.3 then the incumbent wants to increase information to increase the chance of

32

bad news, QFE > Q“. Thus the incumbent limit prices by reducing quantity, which

increases information.

 

30

20

 

 

30 Q

 

 

 

Figure 1.4. Expected Demand Functions: Case ((1)

1 .7 Conclusions

Going back to the idea first proposed by Bain (1949) that prices embody
information about the industry I study limit pricing under a set up in which prices clearly
embody statistical information about the demand function. F inns do not know the
demand function and learn about it through the actions of an incumbent firm that can
increase or reduce the informational content of the price level to limit price.

The incumbent’s costs are fully known and so its actions only reveal information

about the market, both to itself and to others observing the market and contemplating

33

entry, which implies that if it wants to learn, its rival learns as well; if it wants its rival to
learn less, it learns less itself.

I find that in equilibrium unless the demand functions differ by a constant the
incumbent firm limit prices. More importantly however is the result that limit pricing can
mean revealing information to the entrant. In particular, if given current beliefs the
entrant is entering the market, more information might reveal some bad news with more
precise beliefs and thus reduce the probability of entry. In this case the incumbent limit
prices by increasing information.

It would be interesting to study the incentives to limit price in the presence of
symmetric information, such as in this model, but uncertainty about different things, such
as the cost functions of firms, or a general industry parameter. I would expect the results
to hold: even if information is symmetric, the incumbent can manipulate the probability
of good or bad news to try to deter entry.

I would expect the results to also hold in a model of information jamming such as
Fudenberg and Tiroles’s (1986). That is, if the entrant cannot observe the first period
quantity, only the price, then it is said that the incumbent limit prices by jamming its
rival’s information. Assuming that the entrant cannot observe quantity implies that the
entrant and the incumbent update beliefs in a different manner and thus learn differently.
The intuition behind the results should not change however. That is, even if the firms
learn in different ways, the incumbent has incentives to jam the entrant’s learning so as to
make entry appear less profitable.

Allowing for different initial priors (asymmetric information) should also extend

the results of this model, although once again, the intuition behind the results should not

34

change. That is, even if firms have different information to start with, as long as both the
incumbent and the entrant are not fully informed about demand, the incumbent still has
incentives to manipulate (or jam) the entrant’s learning so as to make entry appear less

profitable.

35

CHAPTER 2

UNCERTAINTY, LEARNING, AND SEIGNORAGE MAXIMIZING

GOVERNMENTS

2.1 Introduction

The relationship between the instruments of monetary policy such as the money
growth rate or the interest rate, and policy targets such as the inflation rate is not fully
certain. Governments or central bankers must then conduct monetary policy less than the
fully informed. To aid themselves, policy makers gather data, run econometric models,
and simulate responses of the economy to various policy shocks. Moreover, they update
these models regularly after new data becomes available, which means that they are
trying to learn or to update their beliefs about the economic environment.

This is a simultaneous control and estimation problem that affects the optimal
decisions of policy makers. The policy maker tries to reach its target as best as possible
given its current beliefs, but it must also revise its understanding about the relationship
between instruments and targets, and as it revises its beliefs it must adjust monetary
policy accordingly.

The relevance of learning in monetary economics has already been recognized in
the literature. Most papers however study the role of the private sector’s learning about
the economic environment, but there are not many papers that study the complementary
problem of learning by the monetary authority. I review the literature of learning by

monetary policy makers in Section 2.2 of this essay.

36

The policy maker’s need to learn about the economy is particularly important
during abnormal periods of time during which the government is very uncertain about the
reaction of the economy to different monetary actions. One such period of time is a
hyperinflation, or a high inflation episode. During episodes of high inflation the velocity
of money increases significantly, and eventually people substitute bad (inflating) money
for good, stable money (such as the dollar for example), or they simply turn to barter. In
order for the government to meet its targets it must learn quickly how the demand for
money will react to monetary policy. For example, if a significant portion of the
government’s finances must come from seignorage, knowing to what extent people will
actually use the national currency determines which portion of money issue will translate
into inflation and which portion into real seignorage revenues. This in turn determines
how much money the government should issue.

In this paper I study the impact of learning by the monetary authorities on the
optimal supply of money and the income from seignorage in one of these highly
inflationary economies. I assume that the government relies heavily on money issue to
finance its budget and thus its objective is to maximize seignorage.

The reason why a government comes to rely largely on income from money issue
can be explained using Sargent and Wallace’s (1981) unpleasant monetarist arithmetic
scenario. They showed that if fiscal policy precedes monetary policy, if there is a limit to
the debt that the government can issue, and if the government cannot raise taxes, then the
tighter monetary policy is today, the looser it will be in the future. If the government
accumulates a lot of debt and if tax collections are not significant sources of income then

the government will have to issue a lot of money.

37

The conditions that lead a government to accumulate debt and to run large deficits
can be as varied as the financing of a war or the inefficiency of the tax system.

Deficits can soar for example during periods of civil unrest during which the
government’s expenditure rises to combat the rebellious groups. Tax revenues also fall

as this portion of the population stops paying taxesll

. Moreover, wars in general (not
only civil wars) are very difficult to finance with traditional taxes. Hamilton (1977) for
example points to “the unwillingness of our political leaders in both parties to attempt to
pay the cost of the (Vietnam) war through taxation”12 as the cause of the US. inflation of
the 1970’s. He argues that “this method of payment would have revealed the true cost,
and thus ended the war.”13

Low reliance on tax income is not exclusive to wars or social unrest. Inefficient
or unsophisticated tax systems, very common in countries with high political instability
and polarization, or in countries with large rural areas and poor infrastructure, result in
too little resources invested in tax collection or in tax avoidance and thus in little tax
revenue”.

Low credit ratings and high transitory spending also increase the need for
seignorage”. Transitory spending is not usually financed with tax reforms, which are

meant to be permanent, but with domestic or foreign debt. If the government cannot get

any loans, it turns to printing money.

 

" Capie (1986).

‘2 Hamilton (1977), p. 17-18.
'3 Hamilton (1977), p. 17-18.
“ Cukierman et. al (1992).

‘5 Click (1993).

38

Finally, the inability to reach a policy decision can also lead to deteriorating

deficits and to seignorage as the major temporary source of income16

. As an example
consider France during the first half of the 1920’s. France needed to pay for the
reparations of the war, and it was clear that the Germans could not keep up with the
reparation payments imposed by the Versailles treaty. The French tried invading the
region of the Ruhr in Germany to collect their production as payment but the German
government promoted a policy of passive resistance during which German workers were
paid not to work”. The French effort proved unsuccessful and did not solve (nor help)
the budget’s problem. A tax reform was imminent but the Chamber of Deputies could
not reach an agreement for several years because of the opposition between
Conservatives and Socialists. This led to an 18-month period of complete fiscal inaction
during which the government relied on printing money to finance itself”.

All of the conditions that are likely to lead to the maximization of seignorage by
the government are in essence short-lived or transitory. For this reason I work with a
short horizon. In particular, I work with a two-period model. In the first period the
government chooses a growth rate of money given some prior beliefs about the demand
function. There are stochastic shocks to demand, a demand for money is realized, and the
government collects S1 in seignorage; the government then updates its beliefs according

to Bayes rule and sets a new growth of the money supply for the second period, and

collects S; in seignorage. A riskier distribution of posterior beliefs in the Rothschild-

 

16Alesina and Drazen (1991).

'7 A policy such as this also increases the government’s need for seignorage. In particular, the German
policy of passive resistance and the reparation payments imposed on Germany after the war are considered
to be the two leading causes of the German hyperinflation of the 1922-1923.

1’ Alesina and Drazen (1991) suggest France and other European countries as examples of this type of war
of attrition where the different political parties that made up the government could not agree on fiscal
reforms following the war.

39

Stiglitz sense means that the government learns or that information increases. Intuitively,
a riskier distribution of beliefs makes the possible demand functions easier to distinguish.
The description of this model and the definition of the equilibrium are presented in
Section 2.3.

In Sections 2.4 and 2.5 I present the three results from the model. I find that the
government can increase real seignorage collections by learning about the demand for
money (Proposition 2.1). Hence information is valuable. Intuitively, learning about the
demand for money allows the government to make more accurate and efficient decisions.

Given that the government gains from learning, the government actively seeks to
learn (Proposition 2.2), and thus constant money growth rates are not optimal, except
when the possible demand functions differ by a constant because in this case every
quantity of money demanded is equally informative.

Finally, for some parameter values the government induces a hyperinflation in
order to learn about the demand for money (Proposition 2.3). That is, for some parameter
values, if the government does not update beliefs the seignorage maximizing money
growth rate is not hyperinflationary, but if the government updates beliefs the money
growth rate generates a hyperinflation. If this is the case then the hyperinflation is the
result of a rational decision by the government, which is interesting because as most
people would agree hyperinflations should (if possible) be avoided. I should note
however that this is not the result of monetary conditions but instead of fiscal policy,
which determines the conduct of monetary policy.

In Section 2.6 I illustrate the model and the three results with an example. I show

how learning can more than triple the inflation rate and trigger a hyperinflation.

40

In Section 2.7 I present an intuitive explanation of our model, and in Section 2.8,

a short case study for the Bolivian hyperinflation of the 1980’s. At last I conclude.

2.2 Literature Review

In this section I review how the policy maker’s uncertainty and learning affect
certain monetary problems as studied previously in other papers.

The idea about learning and monetary policy goes back to the 1960’s: Brainard
(1967) studied the making of monetary policy when there is uncertainty about the effect
of the policy on the targets, and Poole (1970) studied the choice of policy tool under
uncertainty about output. Neither one of them however studied learning. The more
recent literature focus on the effects of the monetary authority’s and the public’s ability to
reduce their uncertainty about the economy, e.g. Sack (1999), Kasa (1999), Hardy (1997),
Caplin and Leahy (1996), Balvers and Cosimano (1994) and Bertocchi and Spagat (1993)
among many others”.

Bertocchi and Spagat (1993) consider a central bank that wants to minimize
output variability in an infinite horizon with unknown structure of the economy. They
find that Friedman’s argument (1968) for a fixed money supply rule is only optimal in
very specific circumstances, because learning is valuable, and it implies periodic
adjustments of the optimal monetary actions.

Balvers and Cosimano (1994) study the debate of gradualism in monetary policy

when the central bank minimizes the variability of inflation and is uncertain about the

 

'9 Learning in monetary economics has concentrated mostly on learning by the public, not by the
government, unlike my paper. The papers by Marcet and Nicollini (1998) and Marcet and Sargent (1987)
are for example two papers that study the dynamics of hyperinflations given different learning mechanisms
by the public, for example least squares learning.

41

parameters of the aggregate demand function. By assuming that inflation changes
stochastically with money growth, they develop an optimal way to learn about the
inflation process and find that gradual convergence of money growth is optimal, because,
although sharp reductions in money growth increase information about the economy, they
also increase its variability, which makes it harder to forecast future inflation.

Caplin and Leahy (1996) study a central bank that learns via the reactions of the
investors in a recession to the choice of monetary policy. They find that very gradual
reductions in interest rates are ineffectual whereas aggressive polices are more successful,
because when faced by small cuts in interest rates investors have new information that
implies that further cuts may be in order and thus wait to invest. Aggressive policies on
the other hand, reveal that there will probably be no further cuts and induce higher
investment.

Hardy (1997), by studying how different central banks intervene in different ways
in the financial market to keep interest rates near their targets, argues that the design of
instruments has an important effect on the informational content of market prices. Since
information is valuable in the determination of the operational target, by not intervening
the central bank allows fluctuations that give rise to valuable information. One
implication of Hardy’s result is that since information increases the value function of the
central bank, monetary policy actually reduces market efficiency.

Kasa (1999) considers a central bank that attempts to stabilize output and inflation
but is unsure about the relationship between these two variables. He derives conditions

on parameter values that lead the central bank to erroneously believe that there is an

42

exploitable relation between output and inflation, which provides a new explanation to
the inflation bias.

Finally, Sack (1999) sets up a model of monopoly experimentation when the
Federal Reserve sets an interest rate rule, is uncertain about the policy multiplier and
learns through the reaction of the economy to monetary policy. He finds that since the
Fed faces greater uncertainty about the impact of its policy as it moves away from the
previous interest rate level, the optimal monetary rule is gradual adjustments towards the
target.

The previous papers have in common that they study a central bank seeking to
maximize a social benefit over an infinite time horizon, whether it is a trade off between
output and inflation, the minimization of inflation variability, maximization of output,
etc, and they do it in the presence of uncertainty and learning. In my paper however, as a
result of fiscal policy the government needs to maximize seignorage. Since this is likely
a short term objective, I restrict my analysis to a short horizon.

This model can be used to determine the conditions (as in Bertocchi and Spagat
(1993)) under which Friedman’s argument is true in a scenario in which the government
needs to finance itself strictly with money creation, but can also be used to determine
what learning does to money growth and to seignorage, and the conditions under which

learning leads to hyperinflations, which has not been studied before.

43

2.3 Model and Equilibrium
2.3.1 The Basic Model

There are two periods. At time t=1 the state of the economy is as follows: the
government has a high level of desired spending, past inflation is high, people are not
buying bonds and the government cannot raise taxes, and thus to finance public
expenditures it must issue money. Given the stock of high—powered money, the

government selects a growth rate for the money supply for the period, gl , and collects
seignorage, 5,.
Seignorage in period t is given by the rate of growth of money times the supply of

real balances,

where M, is the nominal supply of money, 13 is the price level, and g, is the growth rate
of money supply in period t. The condition for equilibrimn in the money market is given

by,

M
—J'=L(ir!),r)r

!

where L(i Y) is the demand for real balances, i is the nominal interest rate and Y, is

(’1

real output. Assume a traditional money demand function, i.e. dL(11,__,_Y ’) < 0 and
I!

> 0. Also assume that the interest rate and the output level are functions of the

__(__dLil. Y)
dY

I

money growth rate.

According to Fisher’s theorem (1930), the nominal interest rate can be written as

follows,
(1+1}) = (1+r.)(1+flf),

where r

, is the real rate and 723‘ are the expectations of inflation, and where both 7; and

Irf are functions of g,. Thus the reduced form equation for the interest rate can be

written as,
i, =a+flg, (2-1)
or is a positive parameter, and ,Bcan be positive or negative. If an increase in the
money growth rate reduces the interest rate, which is known as the liquidity effect, then ,6
< 0, otherwise, 13> 0.

The traditional explanation given to the liquidity effect is that without flexible
prices, an increase in money growth requires a reduction in real rates. (Think of the IS-
LM model. An increase in money growth shifts the LM curve to the right along the
downward sloping IS curve.) Additionally, an increase in money growth increases
inflation expectations. If the reduction in the real rate more than offsets the increase in
expectations, we say that we have the liquidity effect.

In highly inflationary economies I would expect the effect of expectations of
inflation to dominate over the interest rate effect, and thus that ,6 > 0.

Output is also a fimction of money growth,
Y. = i’ + 45g, (22)

where 1" is the (known) level of output associated with full employment.

45

If an increase in the money growth rate increases output (i.e. (i > 0), I have the
Tobin effect; if it reduces output (i.e. ¢ < 0), the anti-Tobin effect.

The Tobin effect is the effect that an increase in the rate of growth of money
supply has on the demand for assets. If there is a Tobin effect, an increase in the rate of
growth of money, which increases the cost of holding cash balances decreases the
demand for real balances; individuals switch out of money and into equities, which
increases the demand for capital and thus increases output.

Increasing the money growth rate also reduces the demand for bank deposits,
which means that bank lending is reduced to firms that want to purchase capital goods.
The anti-Tobin effect results is the reduction in investment fi'om firms dominates over the
individuals’ increased demand for assets”.

In highly inflationary economies the effect of money growth on the real variables
is expected to be small compared to the nominal changes in the economy and thus that

¢z0.

Let m, = ' , then, based on equations (2.1) and (2.2) assume that the money

5|:

demand equation can be written as a function of the rate of growth of money as follows”,
m, = l(g,,Q)+£, (2.3)
where a, is a random shock to demand, distributed with a continuous, differentiable

distribution function f (8,) in — oo < s < oo , and with zero expectation. The shock a, is

 

2° Gale (1983), pages 8-9.
2' An extension of this model is to consider heteroskedastic shocks as follows, m, = l (g, ,Q) + g,s, .

46

meant to capture all the variables excluded from the analysis, as well as the stochastic
nature of the economyzz.
Q is a set of parameters unknown to the govermnent, which include among

others a,,6 and ¢.

l(g,,Q) is a function of the rate of growth of money such that

£54m=ltgnfn s O and l(g,,Q)+g,l'(g,,Q) is strictly decreasing in g,. That is,
g,
assume that the demand function is downward sloping”, and that marginal revenue from

seignorage is decreasing. The latter assumption is satisfied for example if l(g,,Q) is

concave or not too convex.
The reason why I want to introduce the latter assumption about the demand for

money (that l(g,,Q)+g,l'(g,,Q) is decreasing in g,) is that I want to have a Laffer-

curve such as in Figure 2.1, as I explain next. Remember that seignorage is given by the

demand for real balances times the rate of growth of money, S, = g,l(g,,Q). Since
demand is downward sloping, as g, increases, l(g,,Q) decreases, and hence the two

terms in the seignorage equation move in opposite directions, i.e.

dS
T'=I(g.,n)+g.1'(g,.n).
g

I

 

22 Since the distribution of a is independent of g, and since I assume that -00 < s < co , my formulation
allows the possibility that m<0. This assumption significantly simplifies the analysis, and it is
traditionally used in the learning literature given the transparency of the results. When dealing with
examples 1 will however assume that negative quantities are not allowed.

23 It is in principle possible to obtain positively sloped demand functions by assuming a Tobin effect and a
liquidity effect. This possibility however does not appear to be very attractive, specially during
hyperinflations when one would expect the effect of expectations of inflation to be large compared to
changes in the real side of the economy.

47

where the first term is positive, and the second term is negative. As g, approaches zero
g, l '(g,,Q) approaches zero (as long as l '(g,,Q) does not go to minus infinity too fast).
Since I(g,,Q) is positive, then for low values of g, increasing the rate of growth of
money increases seignorage. However, it is reasonable to assume that as g, becomes

very large, the second effect dominates the first. This means that it is reasonable to

assume that the seignorage function is concave in g, , or that demand function is such that

l(g,,Q) + g,l '(g, ,Q) is decreasing in g,.

 

 

 

’g

 

 

 

Figure 2.1. Seignorage Function with Laffer Curve Property

There are several possible representations for the money demand function, e. g.
Cagan’s (1956), Blanchard and Fischer’s ( 1993), and Lucas’ (1994).
Cagan (1956) suggested that a good representation for the demand for money
_ a—bi,+lnY,

during hyperinflations is given by m — e hence,

I 9

48

-b’,lnY,_ -b-b,li, -»
I(g,,§2)=e" '+ —e" a ”g +"‘ i“ ’ , where a and b are posrtlve parameters. Cagan’s

formulation is usually used for convenience, although it appears to resemble high
inflation economies very well.

Blanchard and Fischer (1993 — p. 513-517) suggest a more general demand

function for money of the form =0(r;+7rf), where 6'(-)<024. In this case,

.~< [_5

l(g,,Q) = 6(7; +7t,‘)Y,.
Another representation of the demand for money (although not necessarily a good

fit for a hyperinflation) is given by l(g,,Q) = Ai’”Y , where A is a constant, and where

both the nominal interest rate and real income can be functions of the money growth rate.
This function is used by Lucas (1994) to characterize the US. data”.

Under either Cagan’s (1956) or Blanchard and Fischer’s (1993) specifications, Y
and r are assumed to be constant. The argument traditionally given is that during
hyperinflations the changes in the real variables in the economy are either null or very
small compared to the large changes in money supply and prices. A notable exception to
this hypothesis has been suggested by Pickersgill (1968) for the hyperinflationary episode
in the Soviet Union during 1921-1926.

I will not follow any particular form for the demand for money, but instead work

with the general specification l(g,,Q).

 

e
“"bmflq ) and i, = r, + nf , Blanchard and Fisher’s specification becomes

2‘ If for example 90‘, +7rf) = e
Cagan’s.

2’ Of the three demand functions reviewed only Lucas’ specification satisfies the restrictions of this model,
given some additional restrictions on the signs of ,6 and d.

49

To introduce uncertainty, assume that the demand function can take on one of the

two following representations,
m, =1(g,,§)+a, , (2.4)
m, = l(g,,Q)+e, , (2.5)
and assume that Vg, I(g,,§) 21(g,,Q_). This means that one demand function is always

above the other, and thus I will refer to l(g,,-O) as the “high” demand curve and to
l(g,,_g) as the “low” demand curve. I explain the meaning of this assumption in a later
section.

In the first period, the govermnent has a prior belief p, that the demand for
money is equation (2.4), i.e. it believes that the correct parameters are T)- with prior
probability ,a, , and with prior probability 1- u, that they are _Q.

In period one, the government chooses a growth rate for the money supply

g,(,u,) , and a value of e and hence ml are realized. The government’s expected
seignorage in period one is then,
EMS: =[fll1(g.,5)+(1-#1)1(g.,9)]gl-
The government does not observe the realization of a, but a, prevents the

government from perfectly learning the parameters of the demand function after the first

period. The observation of ml , together with knowledge of g,(p,) , leads the

government to revise its beliefs about 0. Let p, be the posterior belief that the correct

parameters are 5 , then, through Bayes rule write ,u, as follows,

50

Am. —I(g. ,5»

”’ = (1-#1)f(ml —1(gpQ))+Mf(ml-l(gle5)) '

With the new beliefs, the government sets a growth rate of money for the second

period, g2 (,a,)and collects S2012) in seignorage.

2.3.2 The Equilibrium
Since posterior beliefs are a function of prior beliefs and of the first period money
growth rate, I solve for the second period equilibrium first. In period 2, the government

maximizes expected seignorage given posterior beliefs,
E..Sz =[#21(g2a§)+(1—#2)1(g2e9)]82,
with respect to g2 . The first order condition is given by,
[AItg.,fi)+(1—A)I(g.._a)]+[lhIrefi)+(1-h211'rg.,a)]g. =0,
and the second order condition by,
2[#21'(gze6)+(1-.U2)1'(82eQ)]+[#2("(g2,§)+(1-#2)1'(82e@]82 so.
Let g2 (11,) denote the second period optimal money grth rate. Given the
assumptions about l(g,,Q) , the second order condition is satisfied with strict inequality,

and thus g2 (#2) denotes a unique maximum.

Let S2 "' ([1,) denote the resulting value function of the government, that is,

S. *(112)=[#21(g2(,u2),6)+(1-#2)l(g2(#2),Q)]g2(/12).

In the first period the government chooses a rate of growth of money by

maximizing expected seignorage over the two periods. Since in period one the posterior

51

beliefs #2 are a random variable whose distribution function depends upon the first
period money growth rate and upon the distribution of ml implied by a, I take the

expectation of the second period seignorage over all possible beliefs and all possible
amounts of money demanded. Assuming that the time discount parameter is given by 6 ,

the government’s first period problem is as follows,
Max EMS, + a j 52 *(p,)h(m,g)dm,
81

where the distribution m implied by e is given by,

h(m.g) = (1— #1 )f(m —I(g.a» + #lf(m — I(g,§)).

Let gl (a) be the equilibrium money growth rate in period one.

2.3.3 The Myopic Equilibrium
In this dynamic problem periods one and two are connected by posterior beliefs.

Thus when the government chooses gl (,u,) it accounts for the effect of period one money
growth on posterior beliefs, and of posterior beliefs on second period seignorage. I now

want to compare gl (p,) with the money growth rate that the government would set if it
did not account for the effect of g,(,u,) on posterior beliefs. Therefore, I compare

g,(,u,) with the optimal myopic money growth rate, gmp,c(p,), which is chosen to

maximize first period expected seignorage, Max E A Sl .
81

(Note that gmpkml) and g,(p,) only differ in the value of ,u. The first and

second order conditions are also the same except for the value of p .)

52

If the government issues a rate g,(,u,) different than the myopic rate gmpkful),

then the government experiments; if this leads to a riskier distribution of beliefs, then it

increases information.

2.4 The Value of Information

In the previous section I defined the equilibrium money growth rate g,(,u,) and
the myopic money growth rate gmp,c(p,) , and defined the difference between the two as

the amount of experimentation. Thus experimentation is the deviation from myopic
policies in order to increase or reduce information, where more information means a
riskier distribution of posterior beliefs. In this section I study whether the government
benefits from more information.

Blackwell (1951, 1953) defined more informative experiments and the sufficient
conditions for a more informative experiment to be valuable. Intuitively, an experiment
2; is more informative than an experiment 22, if every distribution of beliefs that can be
generated from 22, can also be generated by z]. Formally, an experiment is a pair of
measures [f (m—l(g, Q», f (m—l(g,§))], one for each state of nature that gives the
distribution of the first period money demand.

Definition: An experiment 21 = [ f (m—1(g ", Q», f (m—l(g ",O))] is more
informative than an experiment 22 = [f (m — l(g ', 9)), f (m - l(g ', 5))1 , if

Iz<p.(g”.m»h(m.g>dg 2 jZ(#2(g'lm))h(meg)dm

for every continuous, convex function x(,u,(g",m)) : [0,1] —> ‘R .

53

In the spirit of Blackwell (1951, 1953), the second derivative of the second period

2 :1-
value function with respect to beliefs, %, is called the value of information, and

more information is said to be good if the second period value function is convex in
beliefs, which comes from the definition of a more informative experiment.
In my model, experiments are rates of growth of the money supply, observations

are money demanded, and more informative experiments are determined by the demand
functions as follows. Remember that Vg, , l(g,,§)21(g,,g), which implies that the two
demand functions look like one of the two panels in Figure 2.2 (although they do not
have to be straight lines). In Figure 2.2(a) the possible distribution functions of ml
implied by gl move further apart and overlap less as gl increases. Thus a greater gl
produces a more informative signal. On panel (b), the possible distributions of MI move
closer together and overlap more as gl increases, and thus a greater g1 produces a less
informative signal.

Intuitively, more informative money growth rates spread the means of the two
possible demand functions apart making them more distinguishable.

Proposition 2.1: The government that maximizes expected seignorage values

2 a1:
__dsz 91»...

information, i.e.
dflz

Proof: See Appendix B.

54

 

 

 

 

 

, \ 4g , , g
(a) (b)

Figure 2.2. Informativeness of the Inflation Rate

 

 

 

Proposition 2.1 shows that the government values more information.
Mathematically, since expected seignorage is a linear function of posterior beliefs, the
value function is the supremum of a collection of linear firnctions, and it is thus convex.
Intuitively, more information is valuable because it allows the government to make a

better informed decision.

2.5 Effects of Experimentation in High Inflation Economies

In this section I study the government’s incentives, created by the effect of period
one money growth rates on the informativeness of money demanded, for the government
to adjust the rate of growth of money supply away from the myopically optimal rates. To

do so, I compare the first order conditions of the myopic and non-myopic problems.

55

The non-myopic government maximizes 5.1.51 + 6 IS, "‘(p2 )h(m,g)dm , and

 

the myopic government, E ”I Sl , both with respect to g,. Therefore, if
d I S. *(#2)h(m,g)dm
"° d > (<)0, in order for the non-myopic first order condition to be
31

 

dE S
satisfied, d” l < (>)0. This means that the myopic objective function is decreasing
81

(increasing) at g, (a). Since expected seignorage is a concave function, it is decreasing

(increasing) only for values of gl above (below) gmp,c(,u,). Graphically this argument

 

 

 

 

is represented in Figure 2.3.
dEp, S1 (#1) < O
dg 1
Ep. S1 (#1)
4 W
gmpicUJl) 31011)

 

 

 

Figure 2.3. Conditions under which the Non-Myopic Inflation Rate is Higher than the

Myopic Rate

56

d I S. *(flz)h(m,g)dm
After some algebra (in Appendix B) ‘°° d can be written as a
31

 

function of the value of information and of the amount of information. In particular,

d IS. *(;12)h(m,g)dm

dgl

(125,1- all
1— 2 d .2.6
an? M “)dm£m( )

 

 

 

=(I'(g..5)—I'(g..e» I

The relation l'(g,,§) > (<)l'(g,,@ indicates whether increasing the money

growth rate increases (decreases) information. In Figure 2.2(a) l'(g,,E2-) >l'(g,,_g),

hence, increasing the rate of growth of money spreads the means of the two demand
functions apart so that they become more distinguishable. Higher money growth rates are
thus more informative. Since information is valuable (Proposition 2.1), one would expect

the government to increase the rate of growth of money above the myopic rate to increase
information. Alternatively, if l'(g,,5) <I'(g,,Q , e.g. as in Figure 2.2(b), lower money
growth rates are more informative, and thus one would expect the government to set
gl (#1) below the myopic rate. Formally, these arguments are summarized in the next
Proposition. Before presenting the proposition I state one assumption commonly used in

the information literature.

Assumption A2: Assume that f (5) satisfies the monotone likelihood ratio

property (MLRP): %£)2 is strictly decreasing in c.
a

57

An implication of the MLRP is that a higher demand for money leads to a higher
belief that the demand is the “high” one, i.e. % 2 0. (The proof to this statement is in
771

Appendix B.)
Proposition 2.2: Let the MLRP hold. Then the Optimal money growth rate gI (,u,)

is higher (lower) than the myopic rate gmp,c(p,) , if increasing the rate of growth of
money increases (decreases) information. That is, g,(,u,) 2(5) gm”, (,u,) if

I'(g.,fi)2(s>l'(g.,el.

Proof: See section Appendix B.

The relation between the myopic money growth rate gm”, (,u,) , the non-myOpic

money growth rate gl ([1,) , and the second period money growth rate g2 (#2), is
presented in Figure 2.4. In the graph, by assumption, there are three possible posterior
beliefs, ,u, > p, , ,u, = ,u, and p2 < ,u,. Figure 2.4(a) represents the three possible time
paths for the growth rate of money when l'(g,,O) 21'(g,,Q) . For presentation purposes
I assume that before period one the prevailing rate of growth is the myopic rate. In
period one, the growth rate jumps up to g,(,u,) , because this increases information. In

period two it can go either up or down again depending on posterior beliefs. For
example, if the posterior beliefs are the same as the prior beliefs, the second period

money growth rate is exactly the same myopic rate, i.e. g2 (,uI ) = gmp,c(p,) .

Figure 2.4(b) shows the possible paths when l'(g,,§)sl'(g,,Q), which

corresponds to Figure 2.2(b). In period one the rate is lower than the myopic rate because

58

this increases information, but in period two it can go up or down depending on posterior

 

 

 

 

 

 

 

 

 

 

 

 

 

beliefs.

8 g
A #2 >141 I

g I 112 >111

gmyopic H2 :11] gmyopic P2 :11]

112 <ill H2 <11!

i >

1 2 t 1 2 t
I'(g..fi)21'(g..s_2) I'tg..'6)s1'(g..a)
(a) (b)

 

 

 

Figure 2.4: Money Growth Paths

Finally, I can identify the conditions under which experimentation leads to a
hyperinflation, which I present formally in proposition 2.3. The definition of a
hyperinflation is arbitrary. Cagan, for example, who was the first to study
hyperinflations, defined a hyperinflationary episode as starting in the month the inflation
rate is above 50% and ending in the month when it has fallen below that rate and has
stayed there for at least a year. His definition is arbitrary, but it served the purpose of his
study of seven hyperinflations of the 1920’s and 1940’s. Other definitions are slacker, or

cover longer periods of time. I will not specify a particular inflation rate as

59

hyperinflationary, thus when I say that the money growth rate is hyperinflationary, I
mean that it leads to a very high inflation rate.

Definition: The money growth rate gl is hyperinflationary if it leads to an

inflation rate above I, where the inflation rate is given by,

”(gl)=(-——R§,g') ~1],

and the price level is given by,

(1+g,)M0 .
l(gl!Q)+£l

Pl (gt) =

To obtain a hyperinflation as the result of experimentation, the myopic rate must

not be hyperinflationary (i.e. it must lead to yearly inflation rates below I or below

8,000% if we were to follow Cagan’s definition), but the optimal amount of

experimentation must be large enough to generate a hyperinflation. This case is
represented in Figure 2.5 and in Proposition 2.3.

To establish Proposition 2.3 I compare the myopic and nonmyopic problems for

the cases where higher money growth rates are more informative“. I assume that

gm”, (,u,) is not conducive to a hyperinflation but that gl (#1) is, and I look at the

conditions that make the difference between these two rates, which is given by equation

(2.6), large (hyperinflationary).

 

2‘ I could apply Proposition 2.3 to hyperdeflations. In this case I would study cases where decreasing the
money growth rate increases information, such as Figure (2b).

60

 

git
T—L_

> Hyperinflationary

 

 

 

gmyopic

 

 

 

 

 

Figure 2.5. Hyperinflationary Money Growth Paths

Proposition 2.3: Assume that the myopic inflation rate is not hyperinflationary, i.e.

 

_ <1+g....,..(h.»Mo _ _ _
MW“’[Itgm...(hi.m+el 1]" "’

where n E R4, , and where R, is normalized to one. Then if the following two conditions

are true, hyperinflations are the optimal result of uncertainty and learning:

(3.1) Increasing the rate of growth of money increases information, i.e.

I'(g..fi)>1'(g.,t2), and

61

 

 

(l(gmp,c,Q) + 6,)n

* d”
[12 l-u, — £17712
2 ( )dm 1" M0

(3.2) 6(I'rg.,5)—I'(g.,a» I :52

,u
Proof:

Since I '(g,,(_2) > I '(g,,Q) then the non-myopic inflation rate is above the myopic

rate (Proposition 2.2), and thus given that l'(gl , Q) < 0 , then,

[ (1+gm...(a»Mo H (Harrow. J

l(gmyopic ([1,),0) + £1 — l(g1(/11):Q) + £1

and thus ”(Sample (#1)) 5 ”(gr (#1)) -

Condition 3.2 is a sufficient condition for the difference between these two rates

to be larger than n, that is,

(1+ g1 (.111 ))Mo _ (1+ gmyopic(/11))MO
«graham. Ilg....-.(A).n)+e'

 

”(81011))" ”(gmpic (.ul )) =

Note that since I '(gl ,9) < 0 and gm”, (11,) s g1 (11,) , then,

(1+gl(/«))Mo 2 (1+gl(#l))Mo
l(gl(/1l)an)+gl l(gmyopic(/11)9Q)+£l,

 

and thus that,

(1+g. (#1))M0 _ (1+gm...(M))Mo
(gm... (#1), 9) +6. “gm-4710.0) + a. '

 

 

7r(gl(rul))-”(gmyopic(#l)) 2]

Hence, if

(1+gl(M))M0 _ (1+gmyopic(#l))MO >
l(gnryopic(#l)ra)+£l l(gnryopic(lul)’Q)+£l-

 

9

which is equivalent to,

(l(gnryopic (#1 ), Q) + 51)”
Mo

 

(g. (#1 ) - awn-4711)) 2

9

62

then Mg. (h. )1 — hummus) 2 n, and thus Mg. (#1 » 2 1.

Now note that the non-myopic inflation rate equals the myopic inflation rate plus
the optimal amount of experimentation, where the latter is given by equation 2.6 (see

proof in Appendix B), thus,

 

 

d S "‘ d
2 #2(1-.U1) ”21m.

_ 6° 2
g.(u)—g...,..(h.)=S(I'(g..Q)—I'(g..a» I d”, dm

Thus, if

 

(l(gmyopic’a) + £1)"

— "’sz * dp,
61' ,Q -I' ,Q ———2— 1— -— d 2
((g. ) (gl_)) I M“ h.) dmi’" M.

then ate. (#1 )1 2 I.

For experimentation to lead to hyperinflations I need that higher money growth
rates be more informative, and that the incentives to experiment (equation (2.6)) be large.

Among other things, the larger the discount parameter 5 , the larger the value of

2 :1:
information d S’, , and the more spread out the demand functions, that is the larger

#2

 

l '(g,,5) - l '(g,,Q) , the larger the amount of experimentation. In particular, the more
spread apart the means of the two possible distribution functions of posterior beliefs, the
more there is to be learned, and thus the more likely is the government to hyperinflate.
That is, the probability that the non-myopic money growth rate is hyperinflationary

equals the following probability,

63

sz * d
2. lea—h.) ”2 1d»:
dp, dm
-l(gmyopic9n) 9

 

 

Moatl'rg..fi)—I'(g..a» I

n

Pr 3,5

 

2S2;
2

 

which given 20, increases as l'(g,,§—2)—l'(g,,Q) increases. Thus, the larger

l'(gl ,5) — l'(gl , Q) , the larger the probability that learning leads to a hyperinflation.
In Figure 2.6(b) for example, I '(g,,§)- I'(g,,_Q) is larger than in Figure 2.6(a),

thus a hyperinflation is more likely under Figure 2.6(b) than under Figure 2.6(a).

 

 

 

 

 

(a) (b)

 

 

 

Figure 2.6. Spread of the Possible Demand Functions

I should emphasize that although in this model the introduction of learning can
lead to large (hyperinflationary) money growth rates, this is the result of fiscal decisions

which precede monetary policy and which determine the need for large seignorage.

64

2.6 Solution for Uniformly Distributed Shocks and Linear Demand Functions

To illustrate the results from the model I solve for the simplest possible case,
linear demand functions and uniformly distributed random shocks. This example is meant
to illustrate how significant the effect of learning can be on the inflation rate and on the
seignorage level, and how a rational government can induce hyperinflations to learn
about demand.

Assume that the random shocks 5, are distributed uniformly on the interval

[-2.115,2.115]. The demand function is linear in the nominal interest rate and output
level, 771, = a-bi, + Y, +8, , which using equations (2.1) and (2.2) leads to
m,=(a-ba+Y)+(¢—b,6)g,+a,. That is 1(g,,O)=(a-ba+i')+(¢—bp)g, and
O: {a,b,a,fi,¢,Y}.

Assume that prior beliefs are given by p1=0.5, and that
Z—W=—o.os, g-gg=-o.l9, and Z-Ed: g—bg+Y=9. Graphically, these
types of demand functions correspond to Figure 2.2(a). Finally, assume starting values
go = g......(h.) and 1% =1-

1 am assuming that (¢—b,6) < 0 for both types of demand functions. This is a

sign restriction that must be imposed to satisfy the restrictions on money demand (I ' < 0
and 21'+ gl"<0). Assuming that (¢—b,6) < 0 is a reasonable assumption for highly
inflationary economies where it is expected that the effect of the money growth rate on
the real variables of the economy be negligible or at least small compared to the effects

on the monetary variables (the expectations of inflation). It would therefore be expected

that ,6 > 0 and It to be close to zero, and thus (¢ ——b,6) < 0.

65

Since 9 — 0.08g + 5 >9 — 0.19g + a Vg , then 9 — 0.0.8g + a is called the “high”
demand, and 9 -0.19g +8 , the “low” demand. #0 = 0.5 is then the prior belief that the

demand is the “high” one.

With a uniform distribution fimction there are only three possible posterior beliefs

as follows. Since 8 e[-2.115,2.115], then m, e (9—-0.l9gI -2.115,9—0.19gl +2.115) if
9—0.19gl +3 is the true demand, and ml 6 (9—0.08g, —2.115,9—0.08gI +2.115) if
9—0.08g| + e is the true demand function. If observed money demand is at least
9—0.19gl —2.1 15 but strictly less than 9—-0.08gl —2.1 15 , then only the “low” demand

could have generated this observation. In this case the government learns that the true

demand fimction is 9—0.19g,+£. The posterior belief is thus given by ,u2 =0. If
observed money demand is between 9 —0.l9gl + 2.115 and 9—0.08gl + 2.115 , then only
the “high” type of demand function could have generated the observation. The
government thus learns that this is the true demand, p, =1. Otherwise, the government
does not learn anything. In summary, the three possible posterior beliefs are,

0 if 9—0.19g,-2.1155m,<9—0.08g,-2.115
#2: ,u, if 9—0.08gl—2.115 sm,s9—o.l9g,+2.115.
1 if 9—0.19g,+2.115<m,s9-0.08g,+2.115

Posterior beliefs are a function of first period money demand. To find the
distribution of first period money demand implied by the random shock, let g(m, , g,) be

the value of e that must appear if the money demand is ml , the money growth rate in

period one is gl , and the true state of demand is 6; with g(m,,g,) being the value of e

that must appear if the money demand is ml , the money growth rate in period one is gI ,

66

and the true state of demand is Q. Then the density of m, is given by
h(m,g>=(l—h.)f(m-I(g.n»+h.f(m—I(g.fi».
Since by assumption p, =0.5, then the distribution of ml is given by

h(m, g) = 0.5 f (m -I(g,@) + 0.5 f (m —l(g,§)) . Note however that from Bayes’ rule if

the posterior belief is given by p, = 0 then,

0-5f(m1-I(gn§))

= 0.5f(m. -l(g.,Q))+0-5f(ml-1(g.,5)) Q 0 = O'Sflm' ”l(g"m)’

and for ,u2 =1,

0.5f(m. —I(g.,fi»

: 0.5/(m1—1(gp9_))+0.5f(m, _l(g1’§)) <3 0 :05f(ml -I(gl"_2)) ,

 

 

 

 

thus,
,
f(m 3&9» if 9—.19g,—2.115$m,<9—.08g,—2.115
h(m,g)=< f(m-l(g’g):f(m"l(g’a» if 9—.08g,-2.115$m,S9—.19g,+2.115
f(m';(g’9)) if 9-.19g,+2.115<m,S9—.08g,+2.115
Finally, since I assumed that the shocks are uniformly distributed in [—£‘,£‘],
.. l l
where £=2.115,then f(£)=—-;=-——. Hence,
26‘ 4.23
8—2-6 if 9—0.19g,—2.115$m,<9—0.08g,-2.115

h(m,g)=u4L23 if 9—0.08g,-2.115sm,s9-o.19g,+2.115.

8—il—6- if 9-0.19gl +2.115 < 7711 S9—0.08gl +2.115

 

67

Having defined all the preliminaries I now move on to find the equilibrium money
growth rates.
Given posterior beliefs and given that the expected value of a is zero, in the

second period the government maximizes,

52012) = ((9 - 0-0382)/12 + (9 - 0-19gz)(1- #2 ))ga
= [9-(0-19-0-11/12)g2]gz

9

with respect to g,. The first order condition is that 9—(0.38-—0.22,u,)g2 = 0 , which

9
0.38 -0.22p2 '

 

yields the second period equilibrium money growth rate g2 (p2) =

Plugging this rate in the seignorage function gives the second period value function as a

function of posterior beliefs,

 

 

 

 

9 9 9
s* = 9—008 +9—0.19 1—~ ,
2 (”2) [[ 0.38-0.22il,]”2 [ 0.38-0-22#2l( M)l°-38‘°-22/‘2
40.5
th 5* = .
“s 2 (”2) 0.38-0.22/1,

Given the three possible posterior beliefs and the probability of each occurring,

the expected value of S2 * (,u,) is given by,

 

(9-0.08g,—2.115 1 9—0.19gl +2.1 15 1 \
S,"'(,a2 =0)—dm + S,"‘(p2 =p,)—dm
9—0 19 -2115 8'46 9—008 -2115 4'23
ES*(!12)= -31- .g..
m 2 9-0.08g.+2.115 l
+ Sz *(/12 =1)—dm
\ 9-0.19g,+2.115 8'46 1
and thus,

68

 

/ 9—0.08g.—2.ll5 40.5 1 m \
9-0.19g.-2.115 0'38 - 022(0) 8-46

9—0.19g.+2.1 15 40.5 1
dm
9-0-0381-2-115 0.38 — 0.22(0.5) 4.23

9-0.083.+2.115 40.5 1 m
0.38-022(1) 8.46 ,

 

EMSZ I“(‘12) = +

 

= 150 + 0.78g,.

 

+
( 9—0.19gI +2.1 15

 

 

Since increasing the rate of growth of money increases (valuable) information,
second period expected seignorage increases as the rate of money growth increases.

In the first period the government maximizes expected first period seignorage
given prior beliefs p, = 0.5 , plus the discounted expected value of second period
seignorage,

((9 — 0.08gl )0.5 + (9 — 0.19g,)(0.5))gI + 5(150 + 0.78g,) ,
with respect to g,. Assuming that 6 =1 , the first order condition is that
9 -— 0.27gl + 0.78 = 0 , which yields gI (0.5) = 36.22 = 3,622% .

Total (undiscounted by assumption) expected seignorage over the two periods is
thus,

((9 — 0.08 * 36.22)0.5 + (9 — 0.19 * 36.22)(1- 0.5))(36.22) + 150 + 0.78(36.22) = 327 .

To find the maximizing myopic money growth rate and seignorage level I

differentiate myopic expected seignorage ((9—0.08g,)0.5+(9—0.19g, )(1-0.5))g, with
respect to g, in each period, which yields myopic money growth rate
= 33.33 = 3333%. Myopic seignorage over the two periods is thus,

gmyvpic

2 *[(9 — 0.08(33.33))0.5 + (9 - 0.19(3.33))0.5)](33.33) = 300.

69

Thus when the government accounts for the effect of the first period money
growth rate on posterior beliefs it sets a money growth rate of 3,622%, which is 8.7%
higher than the rate that it would set if it did not account for this effect (3,333%). This
increases seignorage by 9% from 300 to 327. Real seignorage then increases by more
than the increase in the rate of grth of money.

To calculate the effect of the government’s ability to learn on the inflation rate I

M0
9—019ch + a

 

first need to find the initial money stock, Mo. Since R, = or

M0
1% =
9 —0.08g,w,, + e

 

, and since Po=1, then Mo=9—0.19gmp,c+e or

M0 = 9-0.08gmp,c + a. A reasonable assumption is thus that the initial money stock is
an average of these two, i.e. Mo = 0.5(9 — 019ng,) + 0.5(9 - 0.08gmp,c) = 4.5 .

With the initial money stock and the rate of growth of money I can calculate the

stock of money in period one, M1 = (1+ g,)M0 , and the price level in period one. The

4.5(1+g,) or P: 4.5(1+g,)
9-0.19g,+eI ‘ 9—0.08g,+el

 

 

price level in period one will be P, = depending on

which is the true demand function for money. The government does not know which is
the true demand function, nor does it know the value of the shock a , but it can calculate

an expected price level. To find the expected price level, integrate over a and over
beliefs. Let f} be the expected price level in period one, then 13, is found according to

the following formula,

70

2"“ 4.5(1+g,) 0.5+ 4.5(l+g,)
9—0.19g, +5, 9—0.08g, +8,

2"“, 4.5(l+g,) 05+ 4.5(l+g,) 05) 1 d8

0.5) f(£)ds

9-0.19g,+£, ' 9-0.08g,+s, ' 4.23

-2.115

This integral can be solved to yield,

 

p _ o.5e4,5*(1+g,)[Liam—0.19g, +2.115)+Ln(9—0.08g, +2.115) ]
l_ .

4.23 —Ln(9-0.l9g, —2.115)-Ln(9—0.08g, —2.115)

Substituing the value of g, =36.22 and taking the natural logarithm gives

~

P, = 156.62. Doing the same for the myopic money growth rate gm,“ = 33.33 , yields

expected price lamp“ = 52.1. The expected inflation rates are thus

7r, =M100% =15,562% for the nonmyopic money growth rate, and

0

=(ijlmwpic_130)

0

ic
”(WP

100% = 5,110% for the myopic money growth rate. The expected

inflation rate increases by 205% when the government updates its beliefs about the
demand function. Moreover, according to Cagan’s definition of a hyperinflation, which
is roughly 8,000% a year, the myopic money growth rate is not hyperinflationary but the

nonmyopic (rational) money growth rate is.

The results from this example, and for different discount parameters are

summarized in Table 2.1.

In this example the government’s ability to learn implies an increase in the
expected inflation rate of about 205% for a discount parameter of one. As the discount

parameter decreases, the non-myopic maximization problem resembles more and more

71

the myopic problem, for example, if the discount parameter is 0.5 the increase in inflation

is of 28%; if the discount parameter is 0.1 the increase in inflation is only of 4%.

Table 2.1 - Summary of Example

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 g, :1- gmyapic % Increase Inflation Myopic %

Inflation Increase

1 3,622% 3,333% 8.7% 15,562% 5,110% 205%
0.9 3,592% 3,333% 7.7% 9,693% 5,110% 90%
0.8 3,563% 3,333% 6.9% 8,369% 5,110% 64%
0.7 3,535% 3,333% 6.1% 7,588% 5,110% 49%
0.6 3,506% 3,333% 5.2% 7,003% 5,110% 37%
0.5 3,477% 3,333% 4.3% 6,544% 5,110% 28%
0.4 3,448% 3,333% 3.5% 6,167% 5,110% 21%
0.3 3,420% 3,333% 2.6% 5,858% 5,110% 19%
0.2 3,391% 3,333% 1.7% 5,578% 5,110% 9%
0.1 3,362% 3,333% 0.9% 5,331% 5,110% 4%
0.0i 3,333% 3,333% 0% 5,110% 5,110% 0%

 

 

Table 2.1 showed the government’s expected inflation rates. These are however
not necessarily the true inflation rates. Depending on the value of the shocks and of the
true demand function the increase in prices can be higher or lower. Table 2.2 for

example presents the largest and smallest possible inflation rates under both myopic and

72

non-myopic behavior. To find these inflation rates I first find the largest and smallest

possible prices. The largest possible price (E) will result when the true demand function

is 9—0.19g, +8 and the shock is a =—2.115, and the lowest price (1”,), when the true

demand is 9—0.08g, +3 and the shock is a = 2.115 , that is,

Moa'l'gl)

E= .
9-0.l9g,-2.115

 

and

= Mo(l+g1)
—‘ 9—0.08g, +2.115’

 

 

with the lowest and highest prices under the myopic rate (R’W’i‘, Rm” ) being analogous

for the myopic money growth rate.

(ft-Po) and

The lowest and highest inflation rates are thus given by Z = P
0

P — P
12', = :42. This minimum and maximum inflation rates are shown in Table 2.2.

0
In the absence of discounting (6 =1), the increase in the inflation rate can be as
little as 12% and as high as 18,000%, thus depending on the correct model (i.e. on the
true demand function), learning can lead to drastic increases in the inflation rate.
Although the numbers on Table 2.2 can be rather large, they are not necessarily

unrealistic. In Hungary for example between March of 1923 and February of 1924 the

73

total increase in prices was 4,400%, and between August of 1945 and July of 1946 the

inflation rate was about 3.8lx1027%27.

Table 2.2 - Minimum and Maximum Inflation Rates28

 

 

 

 

 

 

 

 

 

 

 

 

 

5 Z W % increase ’1; ”row“ % increase

1 5x106% 27,871% 18,679% 1,938% 1,729% 12%
0.9 275,880% 27,871% 890% 1,916% 1,729% 11%.
0.8 142,862% 27,871% 413% 1,894% 1,729% 10%
0.7 96,977% 27,871% 248% 1,874% 1,729% 8%
0.6 72,472% 27,871% 160% 1,853% 1,729% 7%
0.5 57,656% 27,871% 107% 1,832% 1,729% 6%
0.4 47,731% 27,871% 71% 1,811% 1,729% 5%
0.3 40,830% 27,871% 46% 1,790% 1,729% 4%
0.2 35,434% 27,871% 27% 1,770% 1,729% 2%
0.1 31,233% 27,871% 12% 1.749% 1,729% 1%
0.0- 27,781% 27,871% 0% 1,729% 1,729% 0%

 

 

 

 

 

 

 

 

 

To complete the analysis of the impact of learning on the model I want to measure

the gains of learning on seignorage. In Table 2.3 I present the total collection of

 

27 Cagan (1956), Table 1, p. 26.
28 These numbers can vary significantly depending on how many decimal places 1 work with. For example

ifI work with g = 36.2222 , then It, = 6x10°%, ifI use instead g = 36.22 , then F, = 5x10°% .

74

seignorage over the two periods, and the ratio of the percentage increase in seignorage

over the percentage increase in the money growth rate. Formally the measure is given by

S, + 6S2
_ (1 + 6)Smp,c

 

85,8

_§I__1

gmyopic

seignorage to a 1% experiment, that is, 65,, measures the percentage increase in

seignorage from a 1% increase in the money growth rate, where both increases are

measured with respect to the myopic levels and rates.

Table 2.3 - Gains in Seignorage from the Experiment

This last measure can be interpreted as the elasticity of real

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 S, + 6S2 (1+ 6 )Smpk g, * gm,”c em

1 327 300: 3,622% 3,333% 1.04
0.9 309 285 3,592% 3,333% 1.09
0.8 291 270: 3,563% 3,333% 1.15
0.7 274 255 3,535% 3,333% 1.21
0.6 256 240 3,506% 3,333% 1.28
0.5 238 225 3,477% 3,333% 1.36,
0.4 221 210- 3,448% 3,333% 145'
0.3 203 195 3,420% 3,333% 156'
0.2 185 180- 3,391% 3,333% 1.68
0.1 168 165 3,362% 3,333% 1.83
0.0 1501 1501 3,333% 3,333% -

 

 

 

75

Increasing the money growth rate by 1% above the myopic rate increases total
real seignorage collections by at least 1.04%. Moreover, the lower the discount
parameter the higher the elasticity of real seignorage to the experiment. This happens
because for low discount parameters the myopic and non-myopic money grth rates are
more similar, and relatively low compared to the nonmyopic money growth rates that

result fiom high 6 ’5. Since marginal seignorage is decreasing in g, the higher the

money growth rate (i.e. the higher 6) the smaller the increase in seignorage from a 1%

increase in the money growth rate.

2.7 Some Intuition on the Model

All the results of the paper hinge on what is the relationship between the two
possible demand functions, whether they fan in or out. There is nothing in the theory that
says which way is the correct assumption. An alternative way to present the problem is
to assume that for a very long time the government has set a constant money growth rate,
and thus one point in the demand function is observed, e.g. mo, go. The two possible

demand functions then look like Figure 2.7.

Given this assumption, the government will always find it optimal to experiment
because it can always generate information. Since highly inflationary economies are
characterized by increasing money growth rates, it seems reasonable to assume that these
countries operate in region 11 of Figure 2.7. According to Proposition 2.3 then for these
highly inflationary economies, the more spread out the demand functions, the more likely

a hyperinflation.

76

 

"’4

 

 

 

 

g0

 

 

 

Figure 2.7. Intuitive Representation of the Model

In the next section I present a case study for Bolivia, and characterize the growing
uncertainty in this country at the time the hyperinflation process was getting started as a

possible application of my model.

2.8 Case Study: The Bolivian Hyperinflation of 1984-1985
Between April 1984 and September 1985, Bolivia experienced a 26,000 percent
inflation rate. Its immediate cause was the government’s loss of international
creditworthiness in the 1980’s. In this short case study however, I allege that Bolivia’s
hyperinflation could have been in part the result of, or at least could have been worsened
by, the government’s uncertainty about the economy during the first half of the 1980’s.
From 1964 to 1978 Bolivia was ruled under an uninterrupted military regime.

During this time the government financed itself mostly by accumulating debt. From 1978

77

to 1982 there was significant political instability, in particular, eleven heads of state in
four years, violent coups, one accidental death, interim governments, and fraudulent
elections. During this period, the various Bolivian governments relied heavily on foreign
borrowing to finance government expenditures. However, the combination of the
buildup of international debt, the poor macroeconomic management, weak tax system,
poor export prospects, and the preceding political instability, precluded the Bolivian
government fi'om obtaining new international loans after 1981. When the borrowing
stopped, instead of raising taxes or cutting expenditures, the government expanded
money supply rapidly, setting off the hyperinflationary process.

Proposition 2.3 shows that the degree of uncertainty about money demand, output
and interest rate determination affects the optimal money growth (and inflation) rate, and
since some specific aspects of the Bolivian economy make it reasonable to argue that the
knowledge that the government had of the economy was deteriorating precisely at the
time that the hyperinflation process started, I believe that uncertainty might have lead (at
least in part) to the Bolivian hyperinflation.

“The political chaos between 1978 and the end of 1982 had a paralyzing effect on
the economy. The uncertainties that arose from this situation delayed recognition of the
external disturbances that the national economy faced and obstructed the process of
decision-making needed to take appropriate action. Political antagonists attributed the
effects of external shocks to their political foes, instead of looking at the true causes...
Bolivia remained largely isolated from the ongoing discussion in academic and official
circles about the way to cope with the crisis. Some of the macroeconomic mistakes that

were made can be attributed to this isolation.” (Morales and Sachs (1990), p. 185.)

78

After 1982, the government was not as well informed as it predecessors: there is
in general a lack of fiscal data, and what exists is mostly for the central government, and
not the consolidated public sector. The lack of data can lead to misdiagnosis of the
economy, under or overestimations of policy effects, and to inadequate econometric
analysis.

In 1982 the coca market boomed and the share of coca exports increased, making
the predictions of exports and output, and thus money demand hard to estimate.

In 1983 wage indexation was introduced for the first time but eliminated two
years later, an indication that the policy did not achieve the expected outcome, and also
an indication of the government’s lack of understanding of the economy.

In 1984 an important black market for dollars is developed, smuggling becomes
significant, and a large portion of imports go unrecorded; the official demand for dollars
goes down dramatically and unexpectedly, and capital flight reached new highs, all of
which makes predictions of money demand more complicated.

It is also reasonable to assume that money demand shifted in this period in its
sensitivity to expected inflation, inflation uncertainty and the underlying monetary
disequilibria (Asilis et. a1. (1993)).

For all the reasons mentioned above, together with Propositions 2.1 through 2.3, it
is reasonable to argue that the uncertainty, which reached its peak in the first half of the

1980’s, had an important effect on the Bolivian hyperinflation of 1984-1985.

79

2.9 Conclusions

In this paper I assumed that the government maximizes seignorage in a two-
period model with uncertainty about the money demand function and about the
relationship between interest rates and money growth and, real output and money growth.
The model is aimed at resembling unstable governments or highly inflationary
economies.

The conditions under which the government experiments are similar to the
conditions of monopoly experimentation found before by Mirman, Samuelson and
Urbano (1993): As long as the value of information is non zero, if experimentation
increases information, the growth of money in period one is greater then the myopic rate;
if it decreases information, it is lower.

Proposition 2.2 implies that constant money growth rates are in general
suboptimal. They are only optimal in the very specific case in which the demand
functions differ only by a constant.

Proposition 2.3 implies that if increasing the rate of growth of money increases
information, the more spread out the possible demand functions, the higher the
probability that experimentation can lead to a hyperinflation.

Marcet and Nicolini (1998), in a study of hyperinflations under bounded

rationality have said that “. . .the reduction in seignorage that is needed to achieve an

(if

inflation equal to E [ the maximum tolerable] is ofien quite moderate, which raises the

issue of why governments have used ERR [exchange rate rules] instead of lowering the
fiscal deficit (and seignorage) sufficiently. One possible answer is that lowering

seignorage by the exact amount requires much more information: it can only be

80

implemented when the government knows exactly the model and all the parameter

values, including those that determine the (boundedly rational) expectations [of prices]

P‘

1+1: and all the shocks.” (Marcet and Nicolini (1998), p. 10). They have however,
provided no answer to this question. My paper is a first approximation as it shows that
the lack of information about the determinants of seignorage (including those that
determine the public’s expectations) can produce hyperinflations. An extension of my
model that includes exchange rate regimes should answer this question in more detail,
and should provide more insights in to the process of hyperinflations.

My paper is also similar to Marcet and Nicolini (1998) because both papers study
hyperinflations under learning. In their paper the public learns using a switching
mechanism that consists of least squares learning during low and stable seignorage levels,
and a “tracking” learning mechanism when seignorage becomes high (possibly because
of stochastic shocks). Since the rational expectations equilibrium is harder to learn under
high levels of seignorage, hyperinflations become more likely. In my paper I do not
explicitly model the learning of the public, instead I focus on the government’s learning.
Furthermore, I do not assume any particular learning mechanism for the public, only that
it be a function of monetary policy. An interesting question is how my results would

change if I explicitly model the public’s expectations. I leave these topics for further

research.

81

CHAPTER 3

MULTIPLE YEAR IN CENTIVE CONTRACTS FOR CENTRAL BANKERS:

INFORMATIONAL IMPLICATIONS OF LINEAR CONTRACTS

3.1 Introduction

Low and stable inflation is believed to help the economy function better, to
prevent fluctuations in employment and output, and erosions in income. However,
distortions such as taxes or unemployment benefits, or the existence of a monopolistic
competitive sector, which lead to an equilibrium output level that is too low, create
incentives for the government or the central bank to pursue an output target above the
natural rate, and hence to set inflation above the socially desirable rate.

In order to avoid or reduce the inflation bias two different paths of academic
research have emerged, which in turn have led to major structural changes in monetary
institutions. The two approaches are the legislative approach and the contracting or
targeting approach.

According to the legislative approach monetary policy should be delegated to a
central bank that is by law flee to conduct monetary policy without the interference of the
government (instrument independent) but that has the exclusive mandate of attaining
price stability (goal dependent). The most important academic contribution in this area is
Rogoff (1985) who showed that the government should choose a banker with a weight on
inflation stabilization that is higher than society’s although not infinite. If this is the case,

the bank’s reactions to supply shocks are not as radically distorted as when the banker

82

ll

only cares about inflation and supply shocks are passed entirely through output.
Moreover, the benefit of output stabilization outweighs the costs from excess inflation.

Within the same approach Lohmann (1992) showed that the government could do
even better than this by appointing a conservative central banker such as Rogoff’s but
overriding him when shocks were too large.

In the contracting or targeting approach the government imposes a target on the
banker or allows the banker to explicitly set his targets. The banker is then held
accountable for the success of monetary policy through linear contracts, inflation targets,
combinations of inflation targets and contracts, or dismissal rules. Walsh (1995) and
Svensson (1997) are pioneering studies in this area.

Walsh (1995) showed that if the central banker and the government’s preferences
were the same there exists a linear contract that leads to the precommitrnent outcome.
The precommitrnent outcome is found by assuming that the government can credibly
precommit to a zero inflation rate (or to the socially desirable rate), which yields the
lowest welfare losses of all possible outcomes, but is time inconsistent. Time
inconsistency means that after the public’s expectations are formed it is no longer optimal
for the government to implement the announced rate.

Svensson (1997) found that (in the absence of unemployment persistence)
inflation targets are equivalent to Walsh’s linear inflation contracts. They both eliminate
the inflation bias without a cost in output stabilization making inflation targeting (or
linear contracts) superior to Rogoff’s conservative banker.

The previous papers assume that both the government and the public know the

preferences of the central banker, however, there are at least two reasons why the

83

preferences of the head of the central bank may not be fully known. First, if the
government or society selects the central banker at random from the population (or from
a group of the population) and if there is disagreement within this group regarding the
costs of inflation, the preferences of the banker are a random variable. Second, if the
preferences of individuals are a function of the state of the economy (as proposed by
Cukierrnan and Meltzer (1986) and Ball (1995)), which is subject to random shocks, then
the preferences themselves follow stochastic processes and are hence not firlly known at
any point in time.

The unknown nature of the central banker’s preferences has been part of
academic, political and even judicial discussions. Goodfriend (1986) for example reports
on a lawsuit brought against the Federal Open Market Committee (F OMC) by a graduate
law student under the Freedom of Information Act of 1966 to make the minutes from the
meetings available as soon as these were over, whilst the directives of the Federal
Reserve argued for secrecy on the grounds that “such disclosure would be injurious to its
function and the nation’s monetary and economic status”29 and that “uncertainty in
monetary markets best serves (F OMC) needs.”3°’3 '

In Goodfiiend (1986) there is also a reference to a 1983 report released by the
House Banking Committee in which the Democratic majority’s opinion characterized the

Fed’s directives as “having a ‘near obsession’ with secrecy about its goals and actions”32.

 

29 This excerpt comes from the lawsuit itself as reported by Goodfriend (1986) p.67.

3° Goodfriend (1986) p. 68.

3' After several years and several appeals the Court eventually ruled in favor of the FOMC.
3’ Goodfriend (1986) p. 63.

84

More recently Alan Greenspan said in Congress, “If I’ve made myself clear, you’ve
misunderstood me”33.

Assuming that the preferences of the banker are unknown, Beetsma and Jensen
(1998), Muscatelli (1998a), (1998b), (1999) and Schaling, et. a1. (1998) address the
accountability problem previously studied by Rogoff, Walsh and Svensson under
conditions of certainty. The three main findings of these studies are, first that uncertainty
about the banker’s preferences affects the variability of output and inflation, and hence
welfare. Second, that under uncertain central banker’s preferences the trade-off between
the inflation bias and output stabilization reemerges resurrecting a possible role for
Rogoffs conservative central banker. Third, that the equivalence between Svensson’s
inflation targets and Walsh’s linear contracts breaks down. In particular inflation
contracts are superior to targets. For example in Beetsma and Jensen (1998) the reason is
that contracts help to stabilize the output and inflation variability induced by the
uncertainty, while targets only affect the mean inflation rate and output level.

Two correlated issues also arise from these models. On the one hand, the degree
of uncertainty about the banker’s preferences affects the welfare of society, the optimal
delegation agreement, and the losses of the central banker. On the other hand, the
inflation rate contains statistical information about the banker’s preferences, that is, for
any given delegation arrangement and for every set of preferences34 there is a
corresponding inflation rate. In the absence of supply shocks this would imply that after

the first period the government and the public fully learn the preferences of the banker.

In a stochastic model although learning is not complete (that is there is always a positive

 

33 As reported in David Smith’s (October 22, 2000) Sunday Times column “Dim Wim’s darkest hour”.
3‘ The set of preferences is determined by the relative weighs the banker sets on output and inflation
stabilization.

85

probability that the banker has a given set of preferences), the expected value and the
variance of the parameters that characterize the banker’s preferences changes. Hence the
degree of uncertainty about the banker’s preferences evolves over time. Hence a rational
government, private sector and central banker should account for the effect of the
inflation rate on their beliefs (i.e. on the expected value and variance of the parameters
that characterize the banker’s preferences) and on their expected losses. However, I do
not know of any paper that incorporates the government’s and the public’s updating of
beliefs in the study of central bank accountability. My paper is the first to study the
performance of linear contracts (or any other delegation agreement for that matter) when
the government and the public learn about the banker’s preferences based on the
observed inflation rate.

The paper is divided in two parts. The purpose of Section 3.2 is to introduce the
general model and related literature. The bulk of this section is not original research but
mostly a review of the literature that motivates the model. A textbook exposition can be
found in Walsh (2000), and also most of the mathematical derivations can be found in
Persson and Tabellini (1993). In this section I show that linear contracts that reproduce
the optimal precommitrnent outcome only exist contingent on the bank’s private
information. Contingent contracts are either very costly or impossible to enforce. I then
Show the best alternative contract that the government can design, the contract that
minimizes social welfare losses subject to the bank’s inflation rate and the public’s
expectations. This contract however might be inferior to the contract that accounts for

the government’s learning process.

86

In Section 3.3 I extend the model to add learning, which is the point of departure
of this paper. I assume that there is a limit to the number of periods that the bankers can
be in office, which seems to be corroborated by several real life examples, two of which
are the US. Federal Reserve Bank and the European Central Bank. In the US. the seven
members of the board of the Federal Reserve can serve up to a full term of 14 years. One
term begins every two years. A member who serves a firll term may not be reappointed,
but a member who completes an unexpired portion of a term may be reappointed”. The
directors of the European Central Bank are appointed to eight-year nonrenewable
contracts”.

For simplicity I will assume that the bankers can be hired for two periods at the
most. I assume that the government offers one period contracts to the central bankers37 —
the myopic, static, or one-shot government — or two period contracts, where the terms of
the contract are determined at the beginning of each period”. If bankers are offered one
period contracts, they cannot be reappointed.

If the government offers one period contracts and hires a new banker in the
second period the government does not care about learning about the banker’s

preferences. However, since the inflation rate has information about the bankers’

 

3’ The chairman of the board of governors is chosen from among the members of the board to serve for
four-year terms. This appointment may be renewed several times.

3‘ According to Article 11 of the Protocol of the Statute of the European System of Central Banks and the
European Central Bank (ECB), the term of office of the Executive Board of the ECB (the president, vice
president and the other four members) shall be eight years, and shall not be renewable. The initial
appointment of the Executive Board (Article 50) shall be for eight years for the president, four years for the
vice president, and fixed terms between five and eight years for the other four members. None of these
terms of office are renewable.

37 Note that a period can be interpreted as one year, 18 months or two years, depending on how long it takes
monetary policy to take effect.

38 An example of multiple year contracts in which the terms of the contract are negotiated yearly is New
Zealand where the inflation targets (the basis for dismissing the governor) are negotiated between the
government and the bank while the governor is under contract. A simpler example is a wage increase set
by the government at the beginning of each year.

87

preferences the government can induce more informative inflation rates than the myopic
rates. The government can thus offer two-period contracts and use the new knowledge
about the banker’s preferences in future decisions. In particular I assume that the
government learns using a Bayes rule mechanism.

Finally, I assume that there are only two types of bankers; a simplifying
assumption that allows me to study the bank’s incentives to manipulate information and
the government’s to reply accurately, and the effect of learning on the optimal number of
years that the banker should be appointed to.

For some parameter values I find that the government, which I assume is the
faithful agent of society, and thus society benefits from more information about the
banker’s preferences. Given that it is said that monetary policy is transparent if there is
little uncertainty about the banker’s preferences”, transparency of monetary policy can
thus increase the welfare of society.

For the parameter space for which the government values information, the banker
does not necessarily benefit from a better informational state of the government.
Moreover, as long as the banker can affect the beliefs of the government it will do so.
That is, the central banker acts strategically by increasing or reducing the likelihood as
perceived by the government of its type being revealed.

Since the government forecasts the behavior of the bank when it designs the
contract, the optimal contract accounts for the bank’s expected manipulation of

information.

 

’9 There are actually two aspects to monetary transparency: transparency of policy goals (which I study)
and transparency of policy apparatus and procedures.

88

The degree of transparency that can be achieved through multiple year contracts is
also a contributing factor in the optimal length of time of the contract. Studies about the
optimal length of time to which bankers are appointed have focused on the relation
between central bank independence, inflation, and turnover rate of central bankers. It is
believed that the optimal length should be long enough to insulate the bank from political
pressures, timed rightly so as to reduce the uncertainty fi'om the turnover of political
leadership, but not long enough to lock in bad policies. See for example O’Flaherty
(1990), Waller (1992), Waller and Walsh (1996) and Garcia de Paso (2000).

In my model the results are driven by the government’s and the central banker’s
value of more transparent monetary policy, and by the excess variability in the inflation
rate induced by the banker’s manipulation of information, which are the two effects of
the strategic behavior of the banker on the government’s loss function.

In Section 3.4 of the paper I summarize the most important results and discuss

several possible extensions of the model.

3.2. The Model
The notation and general set up of the model follows Beetsma and Jensen (1998).

The government has preferences over output and inflation according to the

function,

LG1

. =3[h(y. Jr +(u. -1? )2]. (3.1)

where (y, — 3) are deviations of output around a target level, and (tr, —1T) are deviations

of inflation around a target rate. These targets are the government’s preferred values for

output and inflation.

89

The parameter 7. determines the government’s preferences for output over
inflation stabilization.

Output is determined according to the following supply equation,
yr =1I,—1t,¢+8,, (32)
where n ,e are the public’s expectations of inflation, and thus (it , —n,') is unexpected

inflation. c, are serially uncorrelated supply shocks with zero mean, variance 0 ,2 , and

density function f (a) .

Equation (3.2) can be motivated as arising from the presence of nominal wage
contracts that are signed prior to the setting of the money growth rate based on the
public’s expectations of the inflation rate. If actual inflation is higher than expected, real
wages decrease, and fnms demand for labor, and hence output, increase. If actual
inflation falls short of the expected rate, real wages increase and firms reduce
employment.

The public’s expectations are formed before a is observed. However, before the
central bank (CB) sets inflation it observes the shock a . The role of private information
is to emphasize that the government often has problems knowing what central bankers
know. For example, a monetary expansion could be the central bank’s answer to its
forecast about the velocity of money; or it could be a desire to expand output. The
government however, has no means to tell one reason from the other. This assumption is
discussed in more detail in Canzoneri (1985) and in Garfinkle and Oh (1995).

An implication of equation (3.2) is that the natural rate of unemployment is zero

(a convenient normalization). This formulation also implies that the government’s target

for output is above the natural output level, 3 >0. This assumption is crucial in the

90

literature as the existence of this difference gives rise to the government’s (and the
bank’s) incentives to inflate above the inflation target.

Barro and Gordon (1983) and Calvo (1978) suggest that in the presence of labor
market distortions such as unemployment compensation and income taxation, the natural
output level will tend to be inefficiently low. Another motivation (Walsh (2000)) can be
provided by the presence of a monopolistic competitive sector that leads to an
equilibrium output level that is too low. If this is the case the government might pursue

an output level above the natural level.
The value of E is determined from the government’s fiscal side optimization
problem. I will not address this problem in this paper. Instead I will assume that

monetary policy is carried out given a predetermined value for E.

Precommitment

I first present the precomrnitment outcome, when the government is able to
commit to the announced inflation rate. Since in this case welfare losses are at a
minimum it serves as the benchmark case.

The sequence of events in the model is as follows. First the public sets

expectations Tl: ,' and based on these expectations the public sets nominal wages. Then

the supply shock a, is realized. The government, who functions as CB, observes a, and

sets inflation. Hence output is realized.
The government sets inflation by minimizing the expected present discounted

value of (3.1) subject to (3.2) and subject to the restriction that average inflation equals

91

announced inflation on average (the precommitrnent assumption). The resulting inflation

rate is given by“),

n'precommitment =;___8 . (33)

The optimal response of inflation to a supply shock is negatively proportional to
a . A negative aggregate supply shock lowers output, and to stabilize output, inflation

should be increased. The more the government cares about output stabilization relative to

inflation stabilization, i.e. the larger A , the larger l—k—A, and hence the larger the
+

adjustment in inflation.
Expected losses for the government under precommitrnent are given by

E(LG.Prewmmitment) = k 0,: + £332 .
2(1+ 7t) 2

It is well known that this policy is time inconsistent and therefore not credible.

Once expectations are formed according to the policy rule given by (3.3) and nominal

wages set, the government has incentives to deviate from (3.3) to stimulate output.

 

4" Given that the government acts as the bank, it knows all the variables at the time of its minimization
problem. Thus it minimizes (l) with respect to it , and n,‘ , subject to the restriction that expected inflation
equals announced inflation on average. 1 thus minimize the Lagrangian,
%[A(1t,-1t,' +c, -})’ +(1r,-1T)2]+00[7r,' -E(n,)],

with respect to it, , n,‘ and a) , where to is the Lagrangian multiplier associated with the constraint that
n,‘ equal the mathematical expectation of it, , conditional on the public’s information.

dE(1t,) _

l

—A.(1t, -1t,' +8, —;)+0) = 0 and (iii) n,‘ -E(n,) = 0. Taking the expectation of (ii) and combining it

The first order conditions are (1) Mn, -1t,' +8, —})+ (it, —1T)-to 0 , (ii)

with (iii) yields a) = 4.}. Taking the expectation of (i) and combining with (D = 46 and 1t,‘ = 1? yields

dig.) =,, Subs,,,u,,ng ‘%’H_)=1 and m :4} in (i) yields equation (3.3). More detailed

explanations of this precommitrnent scenario can be found in Persson and Tabellini (1993).

92

Therefore it is more realistic to assume that the government is not able to precommit to

an announced inflation rate.

Discretion without delegation

If the government is not able to bind itself to its announcements, the optimal

inflation rate is found by minimizing (3.1) with respect to n, , subject to (3.2) and taking

expectations of inflation as given. The first order condition of the government yields,

'scretionw o e ation A. “e +—"'8 +7?
’d' / (“98 = ( t y ) . (34)
(1+7t)

 

fl

Since the public is aware that the government sets inflation according to the rule

(3.4) expectations of inflation are given by,

A(1tf+;—e)+1?
(1+7e)

 

TE: = E(1t:iiscretion w/odelegalion) : E[ ]:fl: = x;+;, (35)

where E represents the expectation given the public’s information.
Expected inflation exceeds the optimal target 1? by it; because of the incentives

to create an inflation surprise in order to stimulate output.
Substituting (3.5) in (3 .4) gives the discretion equilibrium inflation rate in the

absence of delegation,

. . . — 7t -
7t tducrehon w/odelegatlon : 1t ——8 + A
1+ k .
_ precommitment —
_ “II, + 1y

The government sets an inflation rate above the optimal precommitrnent rate by

A; in order to bring output closer to the target.

93

Expected losses of the government under discretion without delegation are

E (Lo'd'“""°""” °"""g"”°”) -—L 2 $93-97, which are higher than expected losses

_ O a
20 + A) 2
under precommitrnent. The government could thus try to delegate monetary policy to an

independent central bank to try to reduce these losses.

Delegation with discretion
Assume now that monetary policy is conducted by an instrument-independent
central bank, whose preferences are uncertain at the moment of delegation. Assume that

the CB’s loss function is given by,
cs __ 1 - 2 - 2
L. —3[0» —a) (y, —y) +(1+a)(n. —n) ]. (3.6)

where or e (—l,?») is a stochastic parameter unobserved by the government and the

private sector.

Assume that in the first period the government and the public have some beliefs
about a , such that the expected value of a is given by E (0t), and its variance by 0:.
Finally, assume that or is independent of e , hence E (as) = 0 4'.

Note that if at = —l , the banker is an “employment nutter” — according to Mervyn

King’s (1997) jargon - a banker that only cares about output; if a = A , the bank is an

 

" The way uncertainty is modeled in Beetsma and Jensen (1998) is referred to as “pure” uncertainty. It is
called pure uncertainty because the sum of the coefficients of the banker’s function equals the sum of the
coefficients of the government’s function, i.e. (A -a)+(l +ct) = 1+ 1 , which implies that the preferences

of the banker on average coincide with the preferences of the government (or society), and that monetary
policy will on average coincide with monetary policy in the absence of uncertainty about the banker’s
preferences. A way to relax this assumption is to assume the following objective function for the banker,

L,“ = a}. (y, -;)z + (l + x)(1t, —1?)2] , where x is a stochastic parameter.

94

“inflation nutter” because it only cares about inflation; if a = O the banker’s and the
government’s preferences are the same.

The central bank chooses inflation by minimizing the discounted value of (3.6)
subject to (3.2) and taking the expectations of inflation as given, which results in an

discretion

inflation rate, 1t , given by,

discretion _ (A. _a)(n’€ +;-8)+(1+a);t-
' (1+;t) '

 

1t (3-7)

Since the public is aware that inflation is set to satisfy (3.7), inflation expectations

are given by,

TI: =E(n’discretion)=E[(x —a)(7tf+;-8)+(l+a)7?]
(HA)

2 (Mn: = (A —E(a))(n: +})+ (1+ 5(a))? .
Canceling terms and rearranging yields,

a: 44.93301)". (3.8)
(1+E(a))

Substituting (3.8) in (3.7) gives the equilibrium inflation rate when monetary

policy is delegated to the CB,

nfiscretion _;_()" ’U.) + (x _a) 3;
1+)» 1+E(ct)

(3.9)

= n'precommitment + (A. ’a) 3; a 8
1+E(0t) 1+),

95

 

The term represents the distortions on the reactions to supply shocks

as
(1+)t

(7» ~00 -

relative to what is socially optimal. The term y represents the bank’s incentives

to create output surprises.

A negative aggregate supply shock, which lowers output, induces an increase in

e , which is less than

 

(iv-<1)
+

inflation to stabilize output. The CB increases inflation by —

the increase that the government would like for positive values of a , and more than the
government would like for negative values of or .

Finally, the larger or , i.e. the less the central banker cares about output, the
smaller the incentives of the CB to create output surprises.

The expected losses of the government under delegation are given by,

zlflflcuawmwnr)?‘

EU“ ...a....) 2(1+>.) ‘ 2(1+E(ot))

 

Since delegation of monetary

policy to the central bank does not bring inflation down to the precommitrnent level, nor
does it improve on welfare, the government can try to design contracts that give
incentives to the CB to set the precommitment inflation rate. Walsh (1995) showed that
as long as the government and the bank have the same preferences such a contract exists.
We now study whether it also exists when the bank does not necessarily share the
government’s preferences over output and inflation, and furthermore, the government is

unsure about these preferences.

96

Linear contracts
Assume that the government wants to design a contract to try to induce the CB to

set the precommitment inflation rate. Let the new loss function of the CB be given by,

L,CB+m,(1t, —1?), (3.10)
where m, defines the contract: If actual inflation is above the target the banker incurs a
penalty of size m, for each additional unit of inflation above the target.

The purpose of the contract is to increase the marginal cost of inflation of the

banker (by m, ). The government therefore wants to artificially generate conservative

central bankers (such as Rogoff’s) without inducing distortions in the reactions to supply
shocks (if possible).

Note that the contract is not meant to reflect the preferences of the government
over inflation stabilization. That is, the government dislikes inflation above or below 1? ,

but bankers are not punished for setting inflation below 1? . The purpose of the contract is

to increase the banker’s marginal cost of inflation to induce lower inflation rates (if
. . . . . - I.
pOSSlble to induce the precomrmtment lnflatlon rate, 1: —fi8 )42.
+

In order to set the contract m, the government forecasts the bank’s reaction

function. The central bank minimizes the discounted expected value of (3.10) subject to

(3.2) with respect to 1!, , and taking the expectations of inflation as given, which is

 

‘2 This issue also arises when contracts are defined as is traditionally done in the literature, as "1,1: , . In
this case, bankers are punished by setting any positive inflation rate, even the socially desirable inflation

rate 1: , or the precommitment inflation rate. The relevant point however is that the marginal cost of
inflation also increases by m,. Furthermore, if the bankers and the government have the same preferences,

then the government can manipulate the banker’s marginal cost of inflation, inducing the precommitrnent
inflation rate without inducing distortions in the stabilization of supply shocks (Walsh (1995)).

97

equivalent to solving a sequence of single period decision problems”. The first order

condition of this problem is given by,
(it—0t)(1t,—1t,‘+8,-;)+(1+0t)(1t,-1-t-)+m, =0.
Collecting 1t, ,

n _(x—ot)(n;—e,+"y‘)+(l+ot)it'—m,
" (1+7.) '

 

The public is aware of CB’s reaction function and thus the expectations of

inflation are given by,

n, =E[(x -a)(1tf-8,+;)+(l+0t)1-t——m,].
(1+)t)

Since the expected value of and as, is zero,

 

n. = (x -E(a»(n: +})+(1+E(a»v?—m.
' 0+2.) '

Solving for 1t,‘ ,

, - A—E(a)- 1
=1r+ y— m,.
1+E(0t) 1+E(a)

 

Plugging 1r f in the CB’s reaction firnction yields the CB’s inflation rate for each

contract m,

=1-t_+ (A-a) -_(}t—a)e _(l+}t-a+E(0t))
' 1+E(Ct) (1+1)’ (1+A)(1+E(ot)) ‘
a............+(x-a)- a _(1+x-a+E(a»m'

’ 1+E(a)y (1+x)’ (l+2.)(1+E(ot)) '

 

(3.11)

 

(:11

 

‘3 Since time periods are independent of each other the term of office to which bankers are appointed is
irrelevant so far.

98

The govermnent would like to set a contract that induces the precommitrnent

contract precommitmem

outcome, i.e. choose m, so that 1:, =1t, , which is given by

m’=[(X-a)-+ a 8% (1+7v)(l+E(a))

y , , but this contract is a function on the
1+E(0t) (1+7t.) (1+)t—0t+E(a))

bank’s private information about supply shocks 8 and on the unknown parameter a , but

neither of these is verifiable. If contingent contracts are not enforceable the

government’s best alternative (so far) is to minimize the discounted expected value of
Lf—m,(1r, —1r) subject to (3.2) and (3.11) and subject to the public’s expectations.

F orrnally the government solves the following problem,

Afr: EBptot,—a;+a,—§)2+(n,—E)2]—m,(a,—1?)]

_ 0» -a)(n: —e. +})+(1+a)tT—m.
" 0+2.)
, —+ h—E(ot)-_ 1

& rt, =1: y m,
1+E(0t) 1+E(a)

 

s.t. 1t (3.12)

The approach I use to solve this problem is as follows. I first square the terms
inside the brackets and then take the expectation. Finally I differentiate.

Squaring the terms in brackets,

l[x(n’2_2nlnl¢+2n’gl—21t’;+1t:2—2nf8,+2n:;+8,2—28,;+;2)]
E 2
l 2 "' —2 _
+301, —21t,7t +7; )—ml(nl—n)

Taking expectation using the facts that Es2 =0,2 and En, =1t,‘,

[g(Enf _nf +2E(n,g,)+o:+;2)+-;-(E1t,2 —2n:E+EZ)—m,(nf-1?)],

99

(2 -a)<n: —e. +;)+(1+a);_m, 8 ]=_(x-E(a»o3

where E(2t,e,) = E[
(1+}t) (1+7t)

The government’s objective function can be written as follows,

 

 

 

 

 

 

 

 

 

 

 

 

2.(-a:’+o3+7) 416+? _ _ 2
515an ]+[ ]-m,[nf—it]—m “an“, (3.13)
2 2 2 0+2.)
where
En.’ :E[(2. —a)(1tf-8,+;)+(l+a);t--ml]2,
(1+1)
((k-a)’(1tf-8.+})’+2(k-a)(1+a)(nf-8.+3); ‘
:>E1t2=E (1+1):
' +-2(2.-ot)(1tf—e,43):»,+(l+ot)21?2-2(1+ot)i?m,+m,2 ’
\ (1+-’02 J
((2.2-2AE(a)+E(0t2))(1tf2+21tfy+of+;2) \
(1+?t)2
+2(x-Eia)+xE(a)-E(a2»(n:G);
25a}: (1+2) _2 44.
+ —2(2. — E(a ))(n f + y)m, + (1+ 2E((1) + £01 2 ))1t
(1+}t)2
+ -2(1+ E(ot ))E' m, + m,’
( (1+2)2 )

 

Differentiating the government’s loss function (equation (3.13)) with respect to

 

m, ,
2 e _
liliEft._[2m;+a +m,]d“' -nf+1t :0, (3.14)
2 dm, 61m,

 

“ When taking the expectation Err,2 I use the assumptions E8 = 0 and E (as) = 0.

100

2t—E(ot)-_ 1 fl:_ 1

which, after substituting n: :1? + y m,, _ ——, and
1+E(ot) 1+E(0t) dm, 1+E(0t)

 

 

 

_ 2 ;_20+93—E(a)-kE(a)+E(a2)_E(a)2)_
6115“.” (1+E(a)) (1+2t)(l+15(ot))2 y .
= yields the
0"": +2(1+ZA+X’+E(a2)—E(a)2)
(1+2.)’(1+13(ot))2 '

following first order condition,

(iii—)1?— (7"E(O‘»(1+M+E(a’)-E(a)’ -+(1+2)’ +E(a’)-E(a)’ m)
(1+E(a)) 0+E(ot))2 (l+2.)(1+E(a»2 .
[l[_ l-E(a)- m, J — ]
1! + y- +1t +m,
1+E(Cl) 1+E(a) _;_l-E(a)-+ m! 1:
\ 1+E(a) 1+E(0t) 1+E(a) )

 

 

Simplifying and solving for m, yields the optimal punishment,

m : (1+2.)(22.-2E(a)+AE(a)-2E(a)z+5012»— (315)
' 3+32. +15(ot2)-E(0t)2 +2(1+7~)E(a) ° .

Plugging the optimal contract in equation (3.13) yields the expected (myopic) loss
function of the government.

Up to this point this paper has mostly been a review of the state of the literature of
linear contracts. In the following section I expand this literature to include the natural
possibility of the government and the public learning about the bankers. Since the
observed inflation rate is a function of a , and since different contracts (different m’s)
result in different inflation rates, the government can improve upon the contract mmp"
by learning about the parameter a . In particular, if the government sets a contract that
induces a riskier distribution of posterior beliefs we say that information increases. In the

next section I study how the government’s learning affects the delegation problem.

101

3.3 Delegation with Learning
3.3.1 The Model

Assume now that the government hires the bankers for two periods in order to
improve its information about their preferences and thus be able to design more accurate
incentives. For simplicity I will work with a simplified model as follows. The central
bank and the government contract over the inflation rate, which is unknown at the time
the contracts are set. Assume that in general the inflation rate has two components: a

deterministic component 1: (x ,a) and a stochastic component g(ct,l< )e, , where the

government is uncertain about a and 8,. K is a set of commonly known parameters that

include 113,13: f and m,. Formally,
2t,(0t,e,) =1r,(0t,lc)+g(0t,l<)e,.
The parameter at determines the preferences of the banker, and the function
g(a,1c ) , which is a function of the preferences of the bank determines by how much the

inflation rate is adjusted in the presence of a supply shock. This function comes from the

central banker’s minimization problem and is given by g(a ,K ) = -)l‘—:—:i.

I assume that there are only two types of central bankers, thus at can take on two
values a e {auafl} , Where a, <0t,,. That is, the banker of type H cares more about
inflation stabilization than the type L banker.

The first time that an individual is hired no is the prior belief that the individual is

type L, i.e. thatct =0LL; l—po is the prior that at =0t”.

102

At the beginning of period one, and given prior beliefs no , the government hires a
banker and sets the contract ml . Given this contract the banker sets the first period
inflation rate 1tl . At the outset of the second period the CB and the government know the

first period contract and the first period inflation rate. The government and the public use

this information to update beliefs about the CB’s preferences using Bayes’ rule. Let )1,

denote the posterior belief that a = a, , then by Bayes rule,

 

= “°f'- 3.16
11) “oft. +(l-Flo)fu , ( )

where fl. =f[nl—nl(aL9K)] and f” =f[nl —fll(aH’K)].
g(aan) g(am'c)

 

 

Finally, the random shock 8, is distributed according to the density function
f(8) on —n <8 <11 , where n 6 R and assume that E(8) = 0, and that e, is distributed
independently from a and thus E (set) = 0. Three final characteristics of the random
shocks are that 11 must be “large enough” with little weight on the tails of f (8) to avoid
learning of the banker’ preferences in the myopic model, but that 11 must not be infinite,
i.e. n < oo . Otherwise the losses of central banking would be unbounded. I explain these

assumptions in the next paragraphs.
Since the inflation rate is a stochastic variable, if I assume that the random shocks
are appropriately large and that there is little weight on the tails of the density function,

that is if f (-n)= f (n)50, then in equilibrium the type of the banker is not fully

9

revealed after the first period. To illustrate what I mean by “appropriately large’

consider the static (myopic) problem of section 3.2 (Beetsma and Jensen’s (1998) model),

103

 

and assume the following parameters, no =05). =1,0to =0.l,0t,,, =02; =0 and

; = 0.1 . Using equations (3.11) and (3.15) the inflation rate set in the static model by the
L type is given by 1t,(0to,8,)= 0.0289- 0.458l , and the inflation set by H type is given by
1t,(0t,,,8,) =0.0226-0.4e,. If the random shock is distributed in the interval [—n,n ],
then if the banker is the L type the inflation rate is within the interval
I L =[0.0289 —0.45n,0.0289 + 0.4511 ] and if the banker is the H type then the inflation rate
is within the interval 1,, =[0.0226—0.4n,0.0226+0.4n ]. Now assume that 1] =0.001,
then IL =[0.028,0.029] and I” =[0.022,0.023]. Since the interception between these
intervals is empty, i.e. since I o HI” = Q , then the type of the banker is fully revealed in
the first period with probability one. Shocks are therefore not appropriately large, and
my model reduces to Beetsma and Jensen’s. Now assume that 11 = 0.056. In this case
I L = [0.004, 0.054] and I H = [0000,0045]. Then only if the first period inflation rate is

very low (below 0.4%) or high (above 4.5%) the government learns that the banker is the
L type. If additionally the probability of observing the very high or very low rates (i.e. if

there is little weight on the tails of f (8)) then the probability that the type of banker is

revealed is low. If this probability goes to zero then it is safe to assume that the banker’s
type is not fully revealed in the first period in the myopic model. In this case shocks are
appropriately large making this a model of learning.

The last assumption regarding the random shocks is that these must be only
appropriately large but no larger, and definitely, not infinite. This assumption is
necessary in order to avoid unbounded central bankers’ losses, which would prevent

individuals from accepting the job of central banking. In the example, if 11 = 0.056 static

104

losses for the bankers are no higher than 0.02 (for either banker). If however 1’] = 00 ,
these losses are infinite. A maximum size of the shock of n = 0.056 therefore seems
appropriate in this example as it satisfies both assumptions: bankers’ losses are bounded
and there is no full learning in the myopic model.

Note that this model could not be applied to an environment in which to avoid
learning shocks must be infinite.

This last assumption also implies that the model is only applicable to stable
environments or to economics not subject to extremely large disturbances. For example,

this model is not applicable to hyperinflations.

3.3.2 The Equilibrium

Since the decision in period two depends on the expected value of a given
posterior beliefs and since posterior beliefs are a function of the first period inflation rate,
and hence of the first period contract, the government must take into account when
choosing the contract the effect of mI on its future beliefs u,. The government’s
objective is thus to minimize expected discounted losses over the two periods. Therefore
the model must be solved through backward induction beginning with the second period
equilibrium, and within each period beginning with the problem of the central banker.

I will assume that within each period the individual rationality constraint of the
central banker is satisfied. This implies that whatever losses the central banker gets in

each period are lower than its reservation losses, that is I assume first, that there exists a

number 5 6 IR such that if m, > Z the bankers do not accept the contract, and second,

that the equilibrium contract in every period is lower than the limit contract 3 , that is

105

m,* <5. With this assumption I rule out (or I do not consider my model to be
applicable to) cases in which equilibrium punishments are too large (such as m, * = 00 ).

Assuming that the participation constraint of bankers is satisfied has been called
previously in the literature the “ego rents” of central bankers. This is a sensible
assumption in this model given that I have assumed that shocks are only appropriately
large to avoid learning but not infinite, and therefore that the losses of bankers are

bounded as explained in the previous subsection.

The second period
Given the posterior beliefs the government designs the second period contract.

To do so, the government first forecasts the central bank’s behavior.

The central bank
The central bank of type i (i=L,H) sets inflation by minimizing L?" + m2(1r2 a?)
with respect to 2t 2 , where these losses are defined by,
L?" = 10‘. —ot,.)(2t2 —2t2‘ +82 —;)2 +-1-(1+0t,.)(1t2 -1?)2 ,
2 2
for i=L,H. As a result the i-type’s first order condition is given by,
(A -0t,.)(1t2 —1t§+82 —;)+ (1+0t,.)(1t2 —2-t.)+m2 = 0.

Solving for 1t2 yields the reaction function of the banker of type i. In particular,

the type-L banker sets inflation according to,

106

(1+at);+0~ —a.)(n; +})—m. _ (x ““0.
0+2.) 0+2.) 2
(7" “(11.)

0+2.)

"2(au32)=
, (3.17)
=rt2L(m2,1t2‘)—

and the H-type according to,

8 )=(1+au)n +0” —0t,,)(1r2'+y)—m2 _ (A ‘0”)8

’ 0+2.) 0+2.) 2
(A ’au) °
(1+2)

1t,(ct,,,
(3.18)

 

=1tf(m2,1tze)— 32

Since the public knows the reaction function of each banker, it forms expectations
rationally forecasting equations (3.17) and (3.18),
“2‘ = “1E“2(a1.282)+(1_ P1)E7t2(am€2)

= u,n2‘(m2,1t2')+(1—fllhzflonztnze)

Solving for it; ,

e _- (k-PlaL—(l-P'lhfl);_m2
n2(m2,pl)_n+ (1+“laL+(l'—plhll) . (3.19)

Note that the public forms its expectations already knowing the punishment m2 ,
and thus expectations are a function of the contract designed by the government. Since

the CB responds in an optimal way to inflation expectations, we can write the reaction

functions of the bankers as functions of the optimal contract as follows,

0" —al‘)8

“2(auez) =1tzl‘(m2,1tze(m2, Pl))' (1+)” 2

9

=1t21‘(m2, “|)+gL82

and

107

 

(A ’0'”)

“2(aH982)=1t2H(m2’n2e(m2’ul))— (1+1) 82

=n§'(m2.lxi)+g”82

The government
Given 2:2(0to,82) =1t2L(m2, 0,) + g‘t:2 and 1C2(C1H,82)=1t;{(m2,p1)+g”82 the
government minimizes its expected losses G2 with respect to m2 , where
02011.. u.) = #1050122. #1) +(1 - ui)Gz”(m2, +1.),

mezl “1): Lg'i(m2,p,)—m2[1t;(m2,ttl)—1-t-],

and
G" x i i e -2 1 i ,- -2
L. (m2,u.)=E 3102(m2.u.)+g82-1t2(m2.lu,)+ez-y] +3ln2(m2.lu.)+g82-nl .

for i = H, L.
To eliminate the expectation’s operator I square the expressions inside the

brackets and take the expected value with respect to 8 , setting E8 = O , E (as) = 0 , and

2
a,

E82=6

1+}. ,. e _ _
+-(—2—)1t§(m2.ui)2-1tz(m2.ui)[?mz(m2.ut)+>~y+nl
-'= .2 _, . (3.20)
l(g’+1)2+g' 2 M§(mz.ui)2 . - Ky
+ 2 o, +——2—+}.1t2(m2,u,)y+—2—

1—2
-1t
Cl 2

 

Let "12(lll) be the contract that minimizes 6,. (Note that the optimal contract

m2 (pl) corresponds to the static optimal contract found in the previous section, equation

(3.15). with E(a) = ma. +(1-u.)aa and Eel) = ma.’ +<1—u.)a..’).

Finally, let V601,) denote the resulting value function of the government, i.e.

108

V0041) = A'gNGAmz, 111)

= G2(m2(p't)2pl) ,
= ’4le ("72010: 111) + (1 _ 11062” ("720-10, ill)

and let V,”’ ( 11,) denote the value fimction of the central bank of type i, i.e.

VFW.) = Li”"(m2(ul). l1.)+ m2(u.)(n§(m2(l1.). it.) + 8382 -17 ) -

Period 1
The CB ’s problem

The banker’s second period decision (1:2) is a function of the second period
contract m2(u,) , which is a fimction of the first period inflation rate, and hence the
banker’s second period losses (V,”(ug) are a function of the government’s posterior
beliefs, and thus in period one the central bank of type i chooses it, to minimize the
discounted value of the two-period losses,

If“ +m.(n. —1?)+ rimmed. (3.21)
taking the expectations of inflation as given, where p is the discount parameter,
L?” =%(}. —0t,.)(1tl —1t,' +8l —;)2 +-%(l+a,)(1tl —1T)2,
and
Etc-“(ml = IKC”(ui)f(8)de .

That is, I must take expectations of the banker’s second period losses with respect

to 82 , which is unknown to the CB in period one. Since at the time of its decision the

109

bank

hteo

0P6

react

mu
p61

PC

lit

banker knows all the variables that determine the posterior beliefs 1;, , I do not need to
integrate with respect to posterior beliefs.

Let 1t,(a,,8,)=1t:(m,)+gi8, be the resulting reaction function of the banker of
type i. When choosing the first period contract ml the government forecasts these
reaction functions and minimizes the discounted expected value of the two-period losses,
GI (m1, no) + pEVG(u,) , where

(7.072.410) = qui‘Onn no)+(1- ua)G.”(mi, 110),

G.‘(mouo)=LtG"(mnuo)-mi[1tf(mnuo)4?].
G,i _ i» i i _ e _‘2 1 i i _‘2
L1 (mlaP0)—E 3[nl(ml’“0)+g81 “l(ml!“0)+81 Y] +'2‘[nl(mlallo)+gel 7‘] -

and
EVG(P1)=IVG(Pl)[llofL +(1_Po)ffl]dn 2

that is, when I take the expectation of the government’s second period losses V“( 111) I
must integrate with respect to u, , which is unknown to the government in the first
period. I know from Bayes’ rule that for a given value of mI , it, is a function of the first
period inflation rate, which is a random variable in period one (see equation (3.16)). To
find the density of this random variable, let 8L(1t,,m,) be the value of 8 that must appear
if the first period inflation rate is 1r, , the contract set in the first period is ml , and the
banker is type L; with 8” (1r,,m,) being analogous for the H type, then the posterior
distribution of 2t, is given by

Pof(8L(“laml»+(l‘H0)f(3H(7tlaml))-

110

Let m, * be the resulting equilibrium contract in period one.

I now want to study how the model with learning differs from the static model. In
particular I want to study what the government’s ability to learn implies in terms of the
behavior of the banker, how the government reacts to the banker’s behavior, and for how

many periods should the banker be appointed to office.

3.3.3 The Value of “More Transparent” Monetary Policy (The Value of Information)
At the outset of the second period the government knows the first period inflation
rate and contract (but not the random shock) and updates beliefs using Bayes rule. If the
distribution of beliefs in the second period is riskier in the Rothschild-Stiglitz sense then
information increases”. If the expected losses of the government are reduced with more
information i.e. if the value function of the government is concave in posterior beliefs,

then the value of information to the government is positive.

 

‘5 The definition of more informative experiments comes from Blackwell (1951) and (1953). F orrnally,

 

 

an experiment is a pair of measures [f(m -1r,L(0t,,x (m‘ »],f[n' -1t,’(0t,,,l<(m,)))] , one for each state
8 (“L’K) g (antic)

of nature that gives the distribution of the first period inflation rate. An experiment

”[1,, -n. (unison. 0)] f(m -n. («Mm '»

L ,, J] is more informative than an experiment
8 (aux) g (aunt)

 

[[ni-ni(ai.x(nh'))]’f[n.-1I.(a.,.l<(m.'))

L H J, if for every continuous, convex function
g (aL’K) g (ain't)

X(l4l(mn7tl)) 3 [0:1] _’ SR a then IXQJlO’h '91It1))h(’nl"’rl)617t ->- IX(Pl(mI 'tnl))h(’nl ,1t,)d1t 2 Where h(mlrnl)

is the distribution 0 f 1r, implied by m, and 8,.

111

This is a complex model, and showing that the value function is concave is not
straightforward. The reason for this is that information enters into the government’s
expected loss function in several ways: directly through the inflation rate, but also
through the public’s expectations of inflation, which in turn feed into the inflation rate
themselves. The value of information to the government depends on the interaction
between all of these channels. More information for example allows the government to
make a better informed decision, reducing its net expected losses (the losses due to the
variance of the inflation rate and output level, plus the transfer to bankers - or minus the
transfer from bankers“).

More information also allows the public to make more accurate predictions,
reducing surprise inflation and therefore the effectiveness of monetary policy.

The difficulty of the flow of information in the model arises from the public’s
expectations of inflation. To study the complexity of the value of information consider
the example depicted in Figure 3.1. In the Figure I show how the value of information
varies between the restricted model in which the public does not update beliefs and the
unrestricted model in which it does. The thin curve in the figure shows the second
derivative of the government’s value function with respect to u, . The thick curve shows
the same derivative for the restricted model in which the public cannot update beliefs.

For both models and for any value of the posterior beliefs the value of information
to the government is positive, that is the second derivative of the value function is

negative.

 

‘6 This is a model in which the government cares about the transfer per so, which implies that the
government may be willing to trade off the socially optimal inflation rate for costly rents to bankers. These
rents are affected by the flows of information as well.

112

For n, < 0.57 the value of information under the restricted model is higher, but
for p, > 0.57 the ordering is reversed. An intuitive explanation is as follows. Starting at

u, = 0.57 , assume that it, increases. As the probability that the banker is the L type

increases, then the expected losses due to higher variance of the inflation rate around 2?
increase, and the rents to bankers are reduced (or alternatively, their punishment
increased). Since information is always valuable, the gains are higher than the losses.
However, if the expectations of inflation are fixed, according to the thick curve, as u,
increases the gains from higher rents do not increase as rapidly as the losses, that is, as
the government learns that the banker is the low type its value of information is reduced

(it gets closer and closer to zero).

 

 

0.05

 

 

-0.45 -
-0.95 4
-1.45 ~
-1.95 -
-2.45 4

 

 

-2.95

 

Posterior Belief

no =0.9,u, =-0.99, (1,, =0.99, i=1, E=Oand }=0.1.

 

 

 

Figure 3.1. Value of Information: Restricted vs. Unrestricted Models

113

When the public updates beliefs (thin curve), as it, increases, the expectations of
inflation increase, which leads bankers to increase the inflation rate. This increases the
variability of the inflation rate (further than in the restricted model) but at the same time it
also increases the rents to the government. Since according Figure 3.1 the value of

information increases as it, increases, then rents increase more rapidly.
The reverse argument would hold for low values of p, .

What is most important to grasp from this example is that when information
affects the expectations of the public, as the government and the public learn that the
banker is the low type, the losses due to more variability in the inflation rate, and the
gains from higher rents from bankers intensify, but the gains increase more rapidly,
increasing the value of information. When they learn that the banker is the high type, the
gains from less variability in the economy and the losses from lower rents intensify as
well, but the losses grow more rapidly reducing the value of information to the
government.

Although information flows in this model are complex, it is possible to show that
for some (reasonable) parameter values the value of information to the govermnent is still
positive, as we show in the next proposition.

Proposition 3.1: There exists a (sensible) parameter space for which the value

filnction of the government V G( 0,) is concave in posterior beliefs.

Proof: The value function of the government is given by

V6011) = 512171020?!” l-ll) : “le(m2(P'l)2 Pl) +0" 14002” ("HO-‘1), “1) Where "12041) is the

114

optimal contract. To show that this function is concave in u, I first show that for some
parameter values,

VOW.) S HiG2L(m2(fi.). EH (1- ui)G§'(m2(lii)al1.).
for all )1, at u,. In a second step I show that since V G(u,) is the lower envelope of a set

of linear functions it must be concave.

Step 1: There exists a sensible parameter space for which
Vanni) s n.G.‘(m.(li.). ti.)+ (1— anti.” ("2.010. ii.) for an ti. c 11.-

Proof:

By contradiction assume that Mn.) > 1+in (cam. ). ti.)+ (1 — n00.” 02.01.). it.)
for some ii, it 1.1,. Then, provided that there exists an admissible it that is a solution to
the following quadratic equation,

#in (r71, u.)+(1- +1002” (fit. it.) E 11in (M2010, fi.)+(l-u.)Ga” ("1.011). in).
there exists a contract, call it fil( 1.1,, fi,), which yields strictly lower losses than m,(u,).
Since 211201,) is the loss minimizing contract this is a contradiction.

The parameter space for which the value filnction is concave is determined by the

existence of a real-numbered solution to the quadratic equation.
Step 2: Given step 1, 0 V601, ')+ (1 -0 )VG(p., ') S VG(0p., '+ (1 —0)p., ") for
it. til. " it +1..
Proof:
Since Vth.) s p.05 (M2011), ti.) + <1 — n00.” (nation) . for an ii. a n. . then
9 Vow. 2 se in. 'G.‘ ("12010. ti.) + (1 - n. 20.” (M2011), ti. )1

and

115

(1 -9 )VGUH ") S(1401111 "Ga" ("12010, 110+ (1 - #1902" ("1.011). 11.)]
for any two beliefs [1, ', u, " at 11, and 9 6 (0,1).
Since this is true for all 11,, it is true for ti, =0u,'+(1—0)p.,", in which case
adding these two inequalities yields,
9 VG(Hi')+(1-9)VG(11. ") S VG(9u. '+ (1 -9)u. ')+

hence V001,) is concave in 0,.

Note that Proposition 3.1 does not necessarily apply for all possible values of the
parameters, i.e. it is possible that for some parameter values the quadratic equation may
not have a real solution. In this case Proposition 3.1 cannot be applied.

Given that monetary policy is transparent if there is little uncertainty about the
central banker’s preferences, and for the parameter space for which the government’s
(society’s) expected losses decrease as the information about the bankers’ preferences
increases, then more transparent monetary policies are better in terms of welfare. The
bankers however do not necessarily benefit from a more transparent monetary policy as
shown next.

The definition of the value of information to the CB is a little different from the
government’s, in particular, the value of information to the central banker is positive if its
second period losses decrease as the central banker differentiates itself from the other

type ofbanker. For example, if 1t,(a,,8,)>1t,(0t,, ,8,) for all m, , then the L type banker

distinguishes itself from the H type by increasing n,; the H type distinguishes itself by

116

decreasing 1:, (see Figure 3.2a)“. The reverse would be true if 1t,((1L,8,)<1t,((1H,8,) as
seen in Figure 3.2b. Finally, if 1t,(aL,8,) >1t,(a”,8,) for some values of m, and
1r,(0t,,8,) <1t,(0l,,,8,) for other values of m, , then the bankers differentiate themselves

from each other by rotating their reaction functions as shown in Figure 3.2c.

 

 

 

 

 

 

 

 

 

Figure 3.2. How Bankers Signal their Type

 

‘7 In figures la through 1c, the reaction functions of the central bankers are negatively sloped. To show
why this is so note that the reaction functions are found by minimizing If“ + m,[1r, —1r]+ pEV,“(u,) with

CBJ CB
d; fl/‘—(-”‘—)=0. To find the

 

respect to n, . Hence the first order condition is given by +m, + p

slope of the reaction function, differentiate the first order condition with respect to m, and solve for % ,

 

which yields fl 1 , which is negative if If“ +m,[1r, —1?]+ pE V,“ (11,) is a

m - dZECBJ + p dzEVIC'(l.ll)
dir.’ dir.’
convex function. The same reasoning holds for the non-myopic reaction functions.

117

The correct relation (or ordering) between 1t,(aL,8,) and 1r,(a,,,8,) depends on

the size of the shocks 8,. To see this consider first the second period reaction functions
given by equations (3.17) and (3.18). Subtracting (3.18) fi'om (3.17) it can be shown that

1t,(0t,,82)>2t,(a,,,82) only if

(at, —ot,,)(i't'-it,‘—}+e,)
(1+2)

 

20c>n-n{—y+~82$0©825m+n2'+y.

Substituting for “II; from equation (3.19),

 

82 S-fl:+1t2e+;c§_2;+[;+(x_llla£ —(l—p,hfl)y—m2]+;
(”Hana—um.)
©82< (l+h.)y—m2

 

— 1+Pla1. +0-14le

m m2

> , where Z is the maximum
1+PlaL+(1"P~l)au 1+Plat+(l’11l)an

 

 

Since

(”2.5—'5
1+ Plat. +0-11le

 

punishment that bankers accept, then assuming that 82 .<. yields that

for all m , 1:2(cto,82) >1t2(aH,82).
However, if shocks to the economy are “very large” it is possible that
1t,(0t,,82) <1t2(0t,,,8,). Intuitively, since the banker of type L cares more about output

than the banker of type H, its reaction to supply shocks is more distorted, i.e. 'ng > lg” I.

The larger the shocks, the larger the distortion, and thus the larger the possibility that

48
“2(au82)<7‘2(am32) '

 

(1 +2.)}—7n'
1+ poor, +(1— uo)a,,
of the type L banker is always above that the type H’s.

then myopic reaction function

 

’8 A similar argument can be used to show that if 8, S

118

Now consider the strategic reaction fimctions in period 1. The reaction function
of each banker is found by minimizing L?” + m, (a, —1?)+ pEV,“ (12,) with respect to it, ,
which yields the following first order condition for the banker of type i,

dEViCBOJI) dill
dill d"II

 

FOC’ = (2. -a,.)(7[, -n,‘ +a, —'y')+(1+ot,)(a, —1?)+m, + p = 0.

Note that the first order condition of banker of type L can be expressed in terms of
type H’s,
Foc” +(a.. —a.)(n. —nr +c. -'y')+ (a. —c..)(n. —t?)

L _
FOC — +p[dEV.C”<n.)_dEV:”(n.)]dtt.
dill d“! dfll

II
C

 

Therefore the first period reaction function of banker of type L is above that of

banker of type H only if

dEVtC”(nt)_dEV§”(ui)Jdut $049

- _— ‘+ —_+
((1,, (12X1t “l 81 Y) P[ dill dill d“!

which once again depends on the size of the shock”. In footnote 49 I show the difficulty

in actually pinning down the exact restrictions that 8 must satisfy in order for

 

 

C8 ('8
“EV‘ (“‘)- “EV" (“‘)]&s 0 then it must be the case that
d u, du, dit,

FOC” >0 otherwise the first order condition of the type L banker would not be satisfied. This means that
the loss function of the type H banker evaluated at the L inflation rate has positive slope. Since the loss
function of the bankers is convex, this means that the reaction function of banker of type L is above that of
banker of type H.

5° It is very difficult to clearly identify the restrictions on 8 implied by the non-strategic reaction functions

‘9 If (a.-a.>(tT-n:+c.—})+p[

for two reasons: I do not know the explicit expression for the function %”'— , and it is very hard to find a
l

dEVLCB("l)_dEVHCB(Hl)
dui du.

 

neat expression for [ J. For example assume that a good approximation for

-d— is given by fl= bu, , where b is a real number. This allows me to find an expression for the
1!, l

inflation expectations by taking the expectations of the first order condition and solving for 1r,‘ ,

119

1t,((1L,8,)>1t,((1”,8,), or vice versa. However, it is important to note that either Figure
3.2(a) or Figure 3.2(b) are feasible depending on the size of the shocks. However, I will
assume throughout the rest of the paper that 1r,(0t,,8,)>1t,(0t,, ,8,) (as in Figure 3.2(a)),
that is that shocks are not too large. The reason I do so is two-fold. First, as I explained
before in the paper, I want the losses of the bankers to be bounded, and second, because
this allows me to assume that beliefs are monotonic. Formally my assumptions are,

Assumption A31. 8, is not too large, in particular since —'q s 8, Sn , assume that

n is such that 1r,(0to,n)>1r,(a,,,n) and 2t,(aL,—n)>rt,(a,,,—n) for all t and all m,.

 

WWI)”

(2.—amt. ‘7‘:+31‘;)+(1+al)(7‘l —t't')+m. + c an.

=0- 5(a))(ni-n -y)+<1+£(a»<n:-n)+nt+p[n .i’Ej—Qi'l+(i-u.)w]tbnn=o.
u. du.

0- -E(c»§+(i+£(a»t?— m.

[1+E(a)+bp[u ww- wflttC—"(L‘QD
d“, d“,

:>2t,'=

dEVLC.(ul)_ dEV:'(tr,)\dp,
dul d”! )dnl
( \

S 0 is equivalent to,

 

The“ ((1,, —aL)(1T—nl¢+el-;)+ p[

p { rial/Put.) _ dEV:’(u.))dtt.

 

 

((1,, ”aflk dilly du, )dn,
8, S— + 'a a .
[1+E(a)+b:[f,m+(l_u1)@%]]

dEd du,

[1+E(a)+bp[p,flfiflz+(l_m)d531m(fll)]]
J

 

 

K

120

Assumption A3.2. Higher inflation rates lead to higher beliefs that the banker is

theL type, %205'.

7!,
Putting assumptions Al and A2 together leads to the following definition
regarding the value to the central bankers of signaling their type.
Definition: The banker of type L benefits (is banned) by signaling its type if its

second period expected losses decrease (increase) as the first period inflation rate it,

dEVLCB(“l) = dEVLCB(Hl) d“! I
dnl d“! dnl

 

increases (decreases). Hence, given that and given

Assumption A3.2, the type L-banker benefits (is harmed) by signaling its type if

dE VLCB (Pl)
d 111

< (>)O.
Definition: The banker of type H benefits (is harmed) by signaling its type if its

second period expected losses decrease (increase) as the first period inflation rate 11:,

decreases (increases). Hence, the type H—banker benefits (is banned) by signaling its

CB CB
dbl/Hf (”1) = dEV” (p ‘) du, > (<)0 , which given Assumption A3.2 is equivalent
dn, d“! dnl

 

type if
C8

to m > (<)0 _
dill

Unlike the government who gains from more transparent monetary policy the
banker does not necessarily benefit from signaling its type as we show in the next

example.

 

5' Assumption A3.2 is the result of Assumption A3.1 together with the maximum likelihood property
according to which f '(8)/ f (8) is strictly decreasing in 8 .

121

Example: central bank’s value of signaling is {we

Assume that ao=0.1, (1,, =0.9, 1:1, ir—=0.03 and 33:1. In Figure 3.31

dEV.-CB(111)

present the value of
dill

for both types of bankers (using the assumption that

E8 = 0 ), which is a function of the posterior belief u, .

dEVLCB (“1)

Pl

For the L type > 0, and thus its losses are increased by signaling its

type. Since this function reaches a maximum at )4, =1, then the value of information to
the banker of type L is minimized when its type is fully revealed.

dEVIEBOJl)

“I

For the H type <0 for most values of 11,, and thus its losses are

increased by signaling its type as well. Unlike for the other banker, the type H banker’s
value of information is positive and maximized when its type is revealed 0, =0.

For these parameter values, the value of information to the government is positive
as seen in the third panel of Figure 3.3. Thus the government and the banker might act at
cross purposes. This leads to the question of whether the government can induce more
informative inflation rates, and if so, at what cost. The answers to these questions depend
on the type of the bankers, on whether they benefit from revealing their type, and on what
this implies in terms of the variability of the inflation rate around the target. I address

these questions in the next sections.

122

 

TYPE? L
0.6

0.55:
0.5
0.45»
0.4.

0.35:

 

 

0.2 0.4 0.6 0.8 1 “1

0.04»
0.03'
0.02~
o_01.

 

. . - . . . . “1
0.2 0.4 0.6 0.8 1
-0.01>
-0.02

 

Vfi—fy

dV_G will

 

#1

H l-

 

 

 

 

 

Figure 3.3. Value to the Central Banker of Signaling its Type

123

3.3.4 Central Banker’s Incentives to Manipulate Information

Since the second period losses of the central banker are a function of the
government’s posterior beliefs then the banker has incentives to manipulate the
government’s beliefs. In this section I show that if the value to the banker of revealing its
type is positive, then the banker acts strategically to signal its type. The reverse holds if
the value to the banker of signaling its type is negative. Before I prove this statement I
formalize a series of definitions.

Definition (strategic banker): Assume that the banker is appointed to two periods.
If the banker accounts for the effect of the first period inflation rate on the government’s
posterior beliefs and thus on its second period losses, then the banker acts strategically.

A strategic banker minimizes (3.21) with respect to n,. The resulting reaction function is
given by 2t,(0t,.,8,) =1t,’(m,) +g‘8,.

Definition (myopic banker): Assume that the banker is appointed to two periods.
If the banker minimizes L,“ + m,(1t,—1?) with respect to a, without regards to the effect
of the inflation rate on posterior beliefs and thus on its second period losses, then the
banker acts myopically or non-strategically. Let 1r,"”"”‘c(0t,,8,) =1t,""”"’”’° + g’8, be the

solution to the myopic problem.
Definition (information manipulation): If when appointed to two periods the

strategic central banker sets an inflation rate that differs from the myopic inflation rate,

that is if n, (a,,8,) at 1t,’"’°”" (a,,8,) , then the banker manipulates information.

124

[l

To study the CB’s incentives to manipulate information I compare the first order

conditions of the CB’s problem when it minimizes equation (3.21), and when it acts

myopically and minimizes L,“ + m, (it, a?) .

Proposition 3.2: Given Assumption A3.2, a strategic central banker will adjust the

inflation rate away from the myopic rate in order to influence the government’s posterior

beliefs if the banker’s value of signaling its type is not zero, i.e. if

particular,

3.2.1

3.2.2

dEViCBO‘l)
dill

$0. In

If the value to the central banker of signaling its type is positive (negative)
and the banker is the L type, then the banker increases (decreases) the

inflation rate to increase (decrease) the likelihood of its type being the true

dEVLCBUJl)

141

type: I.e. if <(>)0 then Vm, 1t,(0t,,8,)>(<)1r,’"”°”“(0t,,8,).

If the value to the central banker of signaling its type is positive (negative)
and the banker is the H type, then the banker decreases (increases) the

inflation rate to increase (decrease) the likelihood of its type being the true

dEV:B(p,)

Fl

type: I.e. if > (<)0 then Vm, 1t,(aH,8,) <(>)1r,"""""(a,,,8,).

Proof: In Appendix C.

According to Proposition 3.2 if the banker’s losses are reduced through

transparency, then the banker wants to increase information or to signal its type. To

signal its type each banker separates its signal from the other type’s signal. Proposition

3.2 can be represented graphically in the next set of figures. In Figure 3.4 the reaction

125

function of each type of banker is the same whether the banker acts myopically or

strategically. This is the result of a zero value of information.

 

“'T

   
  

“popic(aLa31) =1!,((XL,8,)

“(W‘mmsa =fli(a,,,ei)

 

 

 

 

 

Figure 3.4. The Central Banker does not Manipulate Information

In Figure 3.5 the strategic behavior of the bankers implies that their reaction

functions are more spread out than the myopic reaction functions. In this example, for

126

any given contract m, the strategic behavior of the bankers increases information.

Intuitively the reaction functions are more spread and are easier to distinguish”.

 

 

 

 

 

 

 

Figure 3.5. The Central Banker Increases Information.

In Figure 3.6 the strategic behavior of the bankers implies that their reaction

functions move closer together, which means that each inflation rate is now less

 

’2 The movement of the reaction functions is not a parallel movement. In particular, the myopic reaction

 

 

LWIC
function has slope gj—dm,_= --d-,lLl5,-,- and the strategic reaction function has slope
chi?
its. _ 1
dm. c121.“ d’EV.“(u.) '

 

+9

da,’ da,’

127

informative. This happens because the value to the bankers of signaling their type is

negative.

 

 

 

 

 

 

 

Figure 3.6. The Central Banker Decreases Information.

There are finally two other cases in which one type of banker’s value of signaling
its type is positive and the other type’s is negative. These cases are presented in Figure
3.7. In either of these cases it is not clear a priori whether the strategic inflation rate is
more or less informative.

Note that the assumptions imposed on the random shocks, which lead to
Assumption A3.1, limit my model to the cases presented in Figures 3.4 through Figure
3.7. That is, the two reaction functions at most converge to the same reaction function

but it is never the case that 1r,(01,,,8,)>1t,(a,,8,).

128

 

filmmmmei)

 

 

 

 

 

m, "'1

 

 

 

Figure 3.7: One Type of Central Banker Decreases Information, One Type Increases

Information

Since the banker, regardless of its type has incentives to manipulate the
government’s informational state by increasing or decreasing the inflation rate I now
study the government’s reaction to the strategic behavior of the banker. Before going on
note however that the strategic behavior of the banker, which is aimed exclusively at
manipulating the posterior beliefs of the government, has two effects on the
government’s optimization problem. The first effect, which was discussed in detail in
Proposition 3.2 and in Figures 3.4 through 3.7, is that the strategic behavior of the banker

increases or reduces the informativeness of each inflation rate by spreading the two

129

reaction functions apart or moving them closer together. The second effect of the

strategic behavior of the banker is to move the inflation rates closer to, or away from the
target 1?. For example, in Figure 3.5 1t,((1L,8,)-7T increases, and 1t,(aH,8,)-7-t—

decreases. Depending on the magnitude of these changes and on prior beliefs uo , this

implies that the expected variability of the inflation rate around 2? , which is given by
po(n,(aL,8,)—;)2 +(l-po)(1r,(a”,8,)—Tt-)z, increases or decreases. Thus even if the

strategic behavior of the central banker increases information, it might also increase (or
decrease) the variability of the inflation rate and of output, which affects the
government’s first period total and marginal expected losses. Hence, to study the
government’s reaction to the banker’s strategic behavior I will study the interaction of

these two effects.

3.3.5 Government’s Incentives to Induce More Informative Inflation Rates (Incentives
to Experiment)

In this section I study the government’s incentives to increase information, and
the government’s reaction to the manipulation of information by the central banker. I
will do so in two parts. I first assume that the banker does not act strategically. This will
allow me to study the pure gains from learning that the government can generate when
bankers do not manipulate information, and it will help me differentiate the government’s

response to the two effects of the strategic behavior of the banker discussed previously.

130

Incentives to experiment when the banker does not act strategically
If the banker does not act strategically the government’s problem is summarized

as follows,

Min G,+pEVG(tl,) s.t. 1t,""’°”’c(a,,8,) & 1t,’"’""“(0t,,,8,). (3.22)
”'1

Note that the reaction functions 1t,’””°”’" (0t,,8,) correspond to

“(WWW-,8,) =;+ (2. —0t,.) —_ (1+1 +E,0t —0t,.) ml _ (2t —0t,.)
(1+E,0t) (1+E,0t)(1+7t) (1+2)

 

8,, which is the result of

the static minimization problem given prior beliefs.
Let m,"°"””""g’° be the solution to (3.22).
If the punishment (contract) that minimizes the discounted two-period expected

losses of the government m,"°""”""g" differs from the contract that minimizes first period

expected losses, then the government experiments. The latter contract, call it m,”"’°”" ,
comes from solving the following optimization problem,

Min G, st. 2t,""“””"(0to,8,) & 1t,""’°”’c(0t,,,8,).
"’1

Thus m,""’°”" is set according to the first order condition,

dG, da,“"""”" + (10, da,"""’"”“
dnf dm, dn," dm,

 

-[n n ”PPP'P +(1—n )n ”"PPPP -1? ]=0. (3.23)
01 0 l

which are the first period marginal expected losses of the government.

-strategic

The contract m, is set according to the first order condition

dG, da,"""'°”" + dG, dn,”'""’°”'°
dn,‘ dm, 011:," dm,

 

dEVG(lll)]=O’

-[uoit.“""°”” + (1 - 11m.”'”""”" -1-r_ 1+ 9[
dm,

131

which are the first period marginal expected losses plus the marginal gains or losses in
the second period due to more or less information.

If the CB does not act strategically the government will experiment to influence
the degree of transparency of monetary policy (information) if the first order conditions

of the two problems differ, that is if,

G
flip—”$0,

dm,
To find this expression first note that,

dEVPm) dEVPut.) du."”P"P"‘ dEVPot ) dtt ”"PPP’P
= l = + l l .

Q L H
dm, dn, dm, dn, dm,

 

 

Differentiating E V G(u,) = IVG(u,)[uof, + (1— no)f,, ]d1t inside the integral with

respect to “it," yields,

 

 

w— ”6de _ a dln.f.+<1-n.)f..l
da,’ —I dill d1tli[“0fl.+(l Ho)fH]dK+IV ((1,) d“: d‘lt.

dEVGQ’a)

L
1

After some algebraic manipulation (in Appendix C), can be written as,

dEVGw.) = (d’VGUii) dill
dnlL J 511412 dn,

 

 

“l(l ' 1-10)flild7r i

dEVG(tt,)
H

l

and can be written as,

 

dEVG(}.l,) =_J‘d2VG(P'l) dtt,
2

1- d .
dtt,” dill d1!!!“ Po)fu 7‘

132

dEVGUli) and dEVGUt.)

is as
drt,” dn,‘

The reason for the asymmetry on the signs of

follows. If 1t,‘L increases, then information increases which reduces the government’s
expected losses. However, if 1:,” increases, then information decreases which increases

the government’s expected losses.
The expression 0 can therefore be written as a function of the govermnent’s

value of information and of the expected amount of information as follows,

 

d1! Lnryopic dn ”.mpic dZVG d
Q—( l _ l J (111) “I 111(1-llo)fnd71o

_ dml dml 511412 dfll

1.. ' H . '
d1! ‘ "WP“ d7! 1 "001710

 

The term [ ] is a measure of the expected amount of

dm, dm,

information as it gives the distance between the means of the two possible distributions of

 

L.myopic H ,myopic
d2: drt 0t —0t .
— 1 = L ” < 0 , then decreasrng the

bel'efs and . S'nce '
l f‘ f” 1 [ dm, dm, 1+2

punishment increases the expected amount of information. This can be seen as f, and
f” move further apart and overlap less as m, increases (as in Figure 3.8).

Since decreasing m, increases information and since the government’s value of

information is positive, then the government decreases the punishment to the banker to
induce learning. This result is formally stated in Proposition 3.3.

Proposition 3.3: Given Assumption A3.2, if the central banker does not act
strategically then if the government’s value of information is positive, it will decrease the

punishment compared to the myopic punishment to increase the degree of transparency of

monetary policy, that is, m,"°”""""g" < m,""""“.

133

Proof: See Appendix C.

 

ic,L
7! {'WP

7t {utopia}!

 

 

 

 

 

Figure 3.8. Myopic reaction functions

Not surprisingly, if the government’s expected losses decrease with more
information or more transparent monetary policy, the government will increase the
degree of transparency of monetary policy by decreasing the monetary punishment to the
banker. The government of course will only deviate from the myopic contract if the

gains from information overcome the losses from experimentation.

134

Proposition 3.4: If the central banker does not act strategically then the
government is better off or at least not worse off hiring the bankers for two periods
instead of one.

Proof:

If m,”"’°”" is a feasible contract i.e. if it satisfies the individual rationality

non- strategic

constraint but the government chooses m, , then it must be that the expected two

non-strategic

period losses are lower under m, than under m,"""’”’°.

Effect of the strategic behavior of the central banker

The central banker does not necessarily act myopically. In particular, if

dEViCBOll)

d at O the banker will act strategically by deviating from the myopic reaction
Fl

1. H
functions. If for example [ii—f4] < 0 , the strategic behavior of the banker
ml "’1

implies that the reaction functions move outwards as in Figure 3.5, or inwards, as in

L H
ESL-11L] > 0 , then the strategic behavior of the bank implies more

Figure 3.6. If [
dm, m,

complicated movements, such as in Figure 3.9 (full lines represent myopic reaction

functions, dashed lines, strategic reaction functions).

135

 

 

 

 

 

 

 

 

 

Figure 3.9: Given the Strategic Behavior of the Bankers, Increasing the Punishment

Increases Information

For simplicity, I will present our results for the case in which decreasing the

punishment increases information, although the intuition of the results that follow

L H
remains the same when [git—‘— _fl] > 0.
dm, dm,

Assumption A3.3: Given the strategic behavior of the bank, decreasing the unit

 

. . . . dfl,L dn ,”
pumshment moreases lnformatron, dm, — — < 0.

dmi

The government’s optimization problem when the banker acts strategically, which

we described in detail previously, can be summarized as follows,

Min G,+pEVG(u,) st. 1t,((xL,8,) & 1t,(0l,,,8,). (3.24)
"'1

136

Remember that I called the equilibrium m, *.

To study how the optimal contract and the degree of transparency of monetary
policy are affected when the banker acts strategically I study the two effects of the
strategic behavior of the central bank on the government's expected losses. First, I study
how the strategic behavior of the central banker spreads the reaction functions apart or
moves them closer together. Second, how it increases or reduces the distance between
each signal and the inflation target. The first effect increases or reduces the

informativeness of the inflation rate, and the second effect increases or reduces the

variability of the inflation rate around 1? .
The effect of the strategic behavior of the central bank on the optimal contract

non—strategic

m, * - m, can be decomposed as follows,
* non—strategic _ * ? non-strategic myopic ? myopic
m, -m, — (m, —m, )— (m, — m, )+ (m, — m, ) ,

where mf is the contract that minimizes the government’s first period expected losses

subject to the strategic behavior of the central bank, that is m,’ is found by minimizing

Min G, s1. 1t,(aL,8,) & 1t,(0t,,,8,), (3.25)
"'1

with respect to m,. Hence (m: —m,'"”°”") shows the deviation in the period-one loss-

minimizing contract due to the higher or lower variability of the inflation rate around its

target implied by the strategic behavior of the bank.
non-strategic

The term (m, —m,”""”’c) shows the optimal amount of experimentation

when the bankers act myopically (Proposition 3.3), and the term (m, * —m,? ) shows the

deviation from the first period loss minimizing punishment given that the bankers use the

137

strategic reaction functions (i.e. the net incentives to increase information)”. These
terms are explained in Propositions 3.6 and 3.7.
Proposition 3.5: The punishment that minimizes the government’s first period

expected losses is lower (higher) when the banker acts strategically (m: < (>)m,""’°”" ), if

the first period marginal losses of the government are higher (lower) when the banker

acts strategically. That is,

 

 

dG, da,‘ _danP-PPP‘P + dG, dn,” _da,”-P'-PPP"P
If 9 = da,‘ dm, dm, dn,” dm, dm, > (<)0 :> m,? < (>)m,’"-’°”".

“[1400! 1L -1t ILMWP‘C) '1' (1 - FOX“ I” '7‘ [H.ntvopic )]

Proof: For the proof see Appendix C.

Proposition 3.6: When the central banker acts strategically, if the government’s
value of information is positive, then the government deviates from the punishment that
minirrrizes its first period losses in order to increase information. In particular, given
Assumption A3.3, m, * < m,’ .

Proof: See Appendix C.

By comparing how the strategic behavior of the central bank affects the first
period minimum through higher or lower variance of the inflation rate around the target
(Proposition 3.5) and the gains from information (Propositions 3.3 and 3.6) I can thus

study the effect of the strategic behavior of the central bank on the optimal punishment

 

’3 Alternatively I could say that the effect of the strategic behavior of the central bank is equal to the
difference between the incentives to experiment when the banker act strategically and when it does not, i.e.

m, "—m,""""""""‘ = (m, ‘-m,""””")—(m,"°"""""‘ -nr,"’°”"). The incentives to experiment when the banker
acts strategically can then be decomposed in the gains from more information and from less variability in
the inflation rate, m, "' -m,"-"””‘ = (mI ‘—m,7)- (m,7 — m,””°”") .

138

non-strategic

m, * —m, , and on the degree of transparency of monetary policy. In particular I

define the effect of the strategic behavior of the banker on the degree of transparency of

monetary policy as follows,

non—strategic

Definition: If m, * is more informative than m, , that is if
m, "' < m,"°"“”"’eg" , then the strategic behavior of the central bank implies more transparent

monetary policy. Alternatively, if m, * <m,"°”""""g" the government’s reaction to the

strategic behavior of the central bank is to induce a more transparent monetary policy.
Proposition 3.7: Given Assumption A3.3 then if the government’s value of

information is positive, the strategic behavior of the central banker implies more
non-strategic

transparent monetary policy (i.e. m,* < m, ), if any of the following conditions are

true,

i) The gains from information to the government are higher when the banker
acts strategically, (m, "‘ -m,? < m,"°"""""g" —m{"’°”“) , and the strategic
behavior of the central bank implies higher first period marginal losses
due to more variability of the inflation rate around its target (m: < m,’""°”") .

ii) The gains from information to the government are higher when the banker
acts strategically, (m, *—m,? < m,”°"“”’°"g" —m,"""’”"), the strategic behavior
of the central bank implies lower first period marginal losses due to less
variability of the inflation rate around its target (m,? > m,”"’°”’c) , but the
extra gains from information compensate for higher first period losses, i.e.

(”11 * _ml?) __ (”Iron—strategic _ mptyopiC) < mlmyopic _ m: .

139

iii) The gains from information to the government are lower when the banker
acts strategically, (m, *—m,? > m,"°"””""g" —m,"”°”") but the strategic
behavior of the banker implies much higher first period marginal losses

due to more variability in the inflation rate, i.e. m,’ < m,""’°"" , but

non-strategic

(m.*—m;’)—(m. —m."PPP’P)<m."PPPP-mf.
Otherwise the strategic behavior of the bank implies less transparent monetary
non-strategic

policy, i.e. m,* > m,

Proof: For the algebraic proof see Appendix C.

An intuitive explanation of Proposition 3.7 can be given in Figure 3.10. In this
figure the value to the bankers of signaling their type is positive, hence the reaction
functions are more spread out than the myopic firnctions, and the incentives to

experiment are higher when bankers act strategically, i.e.

non—strat ic ic
ml 98 _ mlmyop

 

 

d7! L d“ H dnlelyopic danJnyopic
_.1_ _ _1_ _ _
dm, dm, dm, dm,

]<0. Thus lm, *—m,?|>

and m] * _ml? < minon-strategic _ mlmyopic .
The strategic behavior of the bank affects the marginal losses in period one as

well through more or less variability in the inflation rate, in particular according to Figure

L L,myopic H H,myopic .
3.10, dn‘ _dn, <0, dn, —dn' >0, n,‘ >a,"""’°p'c and
dm, dm, dm, dm,

 

 

H .myopic

2t,” <1t, . Hence depending on priors no and on the set of parameters 1c , the first

period marginal losses implied by the strategic behavior of the bank may be higher or

lower than the expected marginal losses implied by the banker that does not act

140

strategically. Thus both m: > m,’"’"’"" and m,? < m,’"”°”“ are feasible. If m,’ < m,""’°”" (parti

non — strategic

of PrOposition 3.7), then m,. < m, i.e. the strategic behavior of the bank implies

more transparent monetary policy. If however 221,? > m,’"’"”ic the relation between m,. and

non-

m, "mg" depends then magnitude of the gains and losses implied by the strategic

behavior of the bank.

 

“l4 1tl k

   
  

1t?””’(aui<) “qr. (aux)
\

n;””"(amiC)

> ‘D
m {nelstrategic m leropic j I {I

 

 

 

 

 

 

 

Figure 3.10. The strategic behavior of the banker induces more transparent monetary

policy.

When the central banker acts strategically, even if the banker benefits from
signaling its type, it might be in the best interest of the government not to induce more

transparent monetary policies, unlike when the banker acts myopically.

141

It is also possible (part iii of Proposition 3.7) that even if the value to the bankers
of signaling their type is negative, their behavior implies more transparent monetary
policy. This will happen if the strategic behavior increases the first period marginal
losses of the government significantly.

The combination of these two effects also determines the optimal length to which
bankers should be appointed. That is, Proposition 3.4 does not necessarily hold when
bankers act strategically. For example, if the strategic behavior of the central banker

implies that for every punishment m, the static loss function of the government is always

above the government’s loss function given the myopic reaction functions of the bankers,

then one-year appointments are not necessarily inferior any longer.

 

 

' Lmvpic ”.mpic
G, given 1:, ,n,

 

 

- l
mi"’°”'°=m{’ m.

 

 

 

Figure 3.11. Static loss function of the government

142

In the hypothetical situation of Figure 3.11, the period-one loss-minimizing

contract is the same under both reaction functions, m,"”'°”" = m,’ , but the static losses of

the government are lower under m,’"’°”’°. In this case, two-year appointments will only

reduce the government’s losses below one-year appointments if the government is able to
induce large gains from information. This of course depends on the parameters of the

model.

3.4 Conclusions

Based on the assumption that distortions such as taxes or unemployment benefits
make the natural rate of unemployment inefficiently high, the government has incentives
to create surprise inflation to stimulate output. Some of the institutional solutions to this
problem deal with the type of individual appointed to the central bank (Rogoff (1985),
Lohman (1992)), or with the proper incentives that bankers should face, such as linear
contracts (Walsh (1995), Beetsma and Jensen (1998)), inflation targets (Svensson (1997),
Beetsma and Jensen (1998)) or dismissal rules (Walsh (1999)).

In this paper I studied the performance of linear contracts in the case in which the
government is uncertain about the banker’s preferences but can learn about them in time.
When the government is uncertain about the preferences of the central bank, the
government, based on the observed inflation rate, which is a noisy observation of the
banker’s type, can design contracts to enhance its information in order to reduce welfare
losses. The banker also has incentives to act strategically to affect the government’s

perception of its type being the true type.

143

Neither the strategic behavior of the banker nor the reaction of the government
has been studied before in the literature. I found that as long as the banker can affect the
beliefs of the government it will do so, that is, bankers act strategically. In particular if
the banker’s value of signaling its type is positive (negative) it will try to increase
(decrease) the government’s amount of information. Since the government forecasts the
behavior of the bank when it designs the contract, the optimal contract accounts for the
bank’s manipulation of information.

For a given parameter space, the government, unlike the bankers, values
information about the bankers’ preferences, and thus expected losses are reduced with a
better informational state. Transparency of monetary policy, defined as little uncertainty
about the banker’s preferences can thus increase the welfare of society. If the bankers do
not act strategically then two-year terms are superior to one-year terms because the
government can always induce more transparency of monetary policy.

If bankers act strategically the latter result is not necessarily true. In particular the
optimal length of time in office will depend on the effect of the strategic behavior of the
bank on the losses due to excess variability in the economy, and on the extra amount of
information that can result from the strategic behavior of the bankers.

Studies about the optimal length of time to which bankers are appointed have
focused on the relation between central bank independence, inflation, and turnover rate of
central bankers. I add to this list the degree of transparency of monetary policy that can
be achieved through multiple year contracts, and the cost of transparency in terms of the
variance of the inflation rate around its target. In particular, if the strategic behavior of

the central bank implies more transparency of monetary policy and less variability in the

144

expected inflation rate, two-year appointments are superior to one-year appointments.
However if it also implies higher losses due to excess variability in the inflation rate, then
two-year appointments are not necessarily optimal.

There are several possible extensions to this paper. For example, in this paper I
only considered two types of bankers. The model should be generalized to a continuum
of types.

I must also study how uncertainty and learning about the bankers’ preferences
affects the design and performance of inflation targets and of more sophisticated
arrangements such as dismissal rules and nonlinear contracts.

My model should also be extended to allow for the possibility of reappointment as
a function of learning. If bankers are always appointed to one-year terms and can serve
up to two years, then reappointment in the second period will be a function of what the
government learns. This adds dynamics to the model because now the banker needs to
think not only of minimizing future losses but also of getting reappointed.

Another extension of the model is to assume that bankers are appointed for a fixed
number of years but that their appointments can be renewed. This model would then
capture better the US system in which the chairman of the Fed is appointed for four year
renewable terms.

Finally, virtually any contractual arrangement between the government and the
central bank is subject to learning of some kind, not necessarily of the banker’s
preferences but also of economic phenomena such as the existence of a Phillips curve for
examples". Particular studies that would be enriched by including learning are for

example Svensson’s (1997) and Lockwood’s (1997) papers of inflation contracts and

 

5‘ Patron (2000) presents a review of the literature of learning and monetary economics.

145

targets with persistent unemployment in which the dynamic dimension they emphasize

naturally gives rise to the possibility of learning.

146

CONCLUSIONS

Based on the assumption that agent’s decisions affect their understanding of their
environment I introduce Bayesian learning in three different economic models.

In Chapter 1 I study a two-period model with an incumbent firm threatened with
entry. Demand is unknown and stochastic, and prices contain statistical information
about demand. I find that, unless the possible demand functions differ by a constant, the
incumbent firm always manipulates the entrant’s information to discourage entry.

The model of Chapter 1 could be generalized first by assuming asymmetric
information in the form of different initial priors for the incumbent and entrant, and
second, by studying a model of information jamming, in which the entrant cannot
observe the first period quantity, only the price level. In both cases the intuition behind
the results should not change. That is, even if firms have different information to start
with or learn differently, as long as the incumbent and the entrant are not fully informed
about demand, the incumbent has incentives to manipulate or jam the entrant’s learning
so as to make entry appear less profitable.

In Chapter 2 I study a government that maximizes seignorage over two periods
but is uncertain about the money demand function and about the relationship between
interest rates and money growth and, real output and money growth. This model is aimed
at resembling unstable governments or highly inflationary economies.

I find that unless the possible demand functions differ by a constant, a constant
money supply rule is suboptimal. That is, the government should seek to increase its

information about the demand function and should adapt to new information. Moreover,

147

if the two possible demand functions are very spread apart, the value to the government
of more information, and hence the incentives to experiment, are large, making
hyperinflations more likely.

This model leads naturally to the question of how learning affects the end of a
hyperinflation, and in particular to the role that learning plays in the reduction of
seignorage in the presence of uncertainty.

Another extension is to allow for different learning mechanisms of the public,
such as adaptive expectations or least squares learning for example, which would enrich
the dynamics of the model by allowing us to study the interaction of learning of the
monetary authority and the public.

In Chapter 3 I study the design of linear contracts for central bankers in a two-
period model when the government and the public are uncertain about the bankers
preferences and about the stochastic variables that constrain their choices, given that they
can learn about the bankers in time. I find that there exists a sensible parameter space for
which the government values information about the bankers preferences but for which the
bankers might not. Moreover the bankers will act strategically to increase or reduce the
government’s information depending on their own value of signaling their type. The
strategic behavior of the central banker has two effects. It increases or reduces
information but it might also increase or reduce the variability of the inflation rate around
its target. The combination of these effects determines whether two-period contracts are
superior to one-period appointments, and whether the government’s response to the

strategic behavior of the banker implies more or less transparent monetary policies.

148

This model should be extended to study how uncertainty and learning about the
bankers’ preferences affects the design and performance of inflation targets and of more
sophisticated arrangements such as dismissal rules.

Finally, another extension of the model is to assume that bankers are appointed
for a fixed number of years but that their appointments can be renewed. This model
would then capture better the US system in which the chairman of the Fed is appointed

for four year renewable terms.

149

APPENDICES

150

APPENDIX A

APPENDIX TO CHAPTER 1

151

Proof of existence of equilibrium.

To show that in general that an equilibrium exists, I use a Lemma 1.2 and
Proposition 1.1 established in Mirman, Samuelson and Schlee (1994).

Lemma 1.2: If a period two equilibrium exists, then that equilibrium is unique
and at least one firm produces a positive quantity.

Proof:

I first show that if the entrant enters and thus if the firms compete in quantities,
then if an equilibrium exists, then that equilibrium is unique and at least one firm
produces a positive quantity“.

Let

§(q,p)Epg(q,?)+(1-p)g(q.z)- (21.1)

In equilibrium, q; must satisfy the first order condition:

8141 +q5’p)qt' +§(ql +qE’p)—ki 5 0° (A.2)

However, I assumed that there was positive production in equilibrium, then, the
first order condition must be satisfied with equality for at least one frrrn. If it is satisfied

with equality for both firms, adding the two first order conditions yields,

8141 +q£ap)(ql +q£)+2§(ql +q5,p)-(k, +195): 0 (A3)

One of my assumptions was that qg' + 2g is strictly decreasing in q, then at most
one q = q; + qg satisfies (A.3).

There are then two possible equilibria, one in which both firms satisfy their first
order condition with equality and one in which only the low cost firm produces. There

cannot be an equilibrium in which only the high cost firm produces. If this were true, the

 

5” Mirman, Samuelson and Schlee (1994), Lemma 2, page 368.

152

price would exceed the marginal cost of the high cost firm and also the marginal cost of
the low cost fum, but then, production of zero is not optimal for the low cost firm.
If firm I is the low cost firm, the first order condition for the equilibrium in
which only the low cost firm produces are,
§'(qicp)qi +2§(qf.p)-k, = 0 (A4)

8017,20) -ka < 0 (A.5)

The first order conditions for the equilibrium in which the first order conditions of
both firms are satisfied with equality are:
s '(q.’ + ql’op)q.’ + g‘(q.’ + qécp) -k. = 0 (A6)
s“ '(q.’ + qétpMé + g(q,’ +ql’s. 20) -ka = 0 (A7)
Since g'(q,' + q I, , p) < 0 conditions (A5) and (A7) can hold only if
qf>q§+q£-
Using §'< 0 , equations (A.4) and (A6) give
§'(q.’ +qé.p)(q.’ +qé)+§(q,’ +4242) 5 é'(q.’ +qé.p)q,’ +801} + qécp) (A 8)

=10 = §'(q§,p)qi +801? .20)
The first and last term in equation (A8) are the marginal revenue of firm I. The

inequality of these two terms given q,‘ > q ,’ + q f, , contradicts quasiconcavity of expected
profits. Thus there is a unique aggregate output in equilibrium. Individual outputs are
then determined by the first order conditions.

If there is no entry and the incumbent sets the monopoly quantity, then by the

assumption of strict quasiconcavity, the monopoly-profits maximizing quantity is unique.

(Q.E.D.)

153

Proposition 1.1: Let g(Q,y) be defined and continuous on HR, for 7 e {7, Z}and

let there be Q such that g(Q,y) = 0 for 7 e {7, z}. Then a (possibly mixed strategy)
sequential equilibrium exists.

Proof (sketch): For a complete proof, see Mirman, Samuelson and Schlee (1994).
As a sketch consider the following proof.

First, restrict the strategy space to the closed interval [0, Q]. By the assumptions

of quasiconcavity and continuity of It}, there exists an equilibrium of the game played in
the second period, (q,‘(p), qg.(p)). This equilibrium (these two reaction functions) is

continuous in p. The functions h( p, q) , V, (p) and V,"' (p) are continuous in p as well,

which is continuous in Q], thus the entire equation 7:, (Q, , p°) + 6 [W (Q, )] , where
W(Qi) to
W(Q.) = I V,"'(pm Q. ))h(p. Q. )dp + I V.(p(p. Q. ))h(p.Q. )dp .
—co V(QI)

is continuous in Q1. Applying Glicksberg’s (1952) extension of Kakuthani’s fixed—point

theorem completes the proof. Then a solution to the entire game exists.

(Q.E.D)

Lemma 1.3: Given assumptions AU and A12, an incumbent firm threatened
with entry always limit prices, that is G(Q, *) S G(Q,” ). In particular,

(a) If reductions in quantity increase information, and through more information

increase the probability of entry ((E'—g')$0 and g(EQ’lSO], the

l

incumbent sets a quantity higher than the incumbent that takes entry as given,

154

(b)

(C)

(d)

i.e. Q,*2Q,FE . Since this reduces the probability of entry and decreases
information, then the incumbent limit prices by concealing information from

the entrant.

If increases in quantity increase information, and through more information

increase the probability of entry [(g'— g') 2 0 and %QQ 2 0 J, then the

l
incumbent sets a quantity lower than the incumbent that takes entry as given,
i.e. Q,* s Q”. Since this reduces the probability of entry and decreases
information, then the incumbent limit prices by concealing information from

the entrant.

If increases in quantity increase information, and through more information

reduce the probability of entry [(E—g') 2 0 and d%(QQ’—)$0], then the

I
incumbent sets a quantity higher than the incumbent that takes entry as given,
i.e. Q,*2 Q,” . Since this reduces the probability of entry and increases
information, then the incumbent limit prices by revealing information to the

entrant.

If decreases in quantity increase information, and through more information

reduce the probability of entry [(33030 and igLQQ’JZO], the

l
incumbent sets a quantity lower than the incumbent that takes entry as given,

i.e. Q,*s Q,“ . Since this reduces the probability of entry and increases

155

information, then the incumbent limit prices by revealing information to the

entrant.

Proof:

If the incumbent sets a quantity above or below the quantity of a monopolist that
takes the probability of entry as fixed (QfE ) so as to reduce the probability of entry then
the incumbent limit prices.

To study the incumbent’s incentives to deviate from Q,” I look at how the first
order conditions of the two problems differ. I first look at the maximization problem of

the incumbent that does not take entry as fixed, which maximizes 7r,(Q,, p°) + 6 W (Q,),

where
W(Q!)
W(Q.)- — I V,"'(p(p.Qi))h(p.Qi)dp+ I V.(p(p.Q.»h(p.Q,)dp.
W(QI)
0
The first order condition is given bygiggfol+6igégfi= 0, which using
I l

Leibniz’ rule and looks like,

 

-w'(Q.)h(W(Q,).Q,)[V,(p(v/(Q,) Qi))-V."'(p(w(Q,). Q. 2)]+
dfll(QI’po) 6 '0 V11 _h d V(QI)VIM' —ph 50.
————dQI + “I”; in) do (no) p+ I (p) dQ, (p.Q.)dp+
W(Q!)
_le.) —Q—_d, dQI ,

 

156

w

Since G(Q,) = I h(p,Q,)dp , then using Leibniz rule,
W(Q!)

dG
dQI

 

= —l// '(Q,)h(1/’(QI)’ Q1) + I

w dh(p, Q!)
dQ, 01” ’

W(QI)

then the first order condition can be written as follows,

d_g__
dQI

iv,

d”l(Ql’po) +6

dh(p, Q1 )
dQ, d”

W(QI)

'(p) —h(p. Q. )dp +

W(Q!)

[V,(p(W(QI).QI))-V,'"(p(w(Qi),Qi))]+

V,"'(p)—— d” h(p.Q.)dp+

III
C

dQI

W(Q!)

dQI

1 o.

W(Q!)

dh(p’Ql i Vlm(p)—

h (P a Q___1 )
are. d”

)
—d
”L do

I V,(p)—

_V(QI)

 

 

(11.9)

The incumbent that takes entry as given (or that naively thinks that it cannot affect

the probability of entry) solves the same problem but assumes that :—G = 0.

1

Alternatively, assume that the incumbent that takes may as given maximizes

7:, (Q, , p°) + 5 W (Q,) subject to the restriction that SQE = 0 . This firm then maximizes,

1

d0
”l(Qppo )+5W(Qz)+4;é—

I

. . d
where 21 is the Lagrange multiplier associated with the restriction that d—g— =0. The
1

two first order conditions are thus given by,

157

d”I(QItpo)

 

 

 

     

do
79. —WII)— do. p][ ..—(p(w(Q.)Q.)) (PW/(Q1) Q.»]+
co W(QI)
+6 '(p)——h(p.Q.)dp+ V,"'(p)d ph(p.Q.)dp+
Vi£I)V dQ, I dQI
d__h(p.Q.) "PP .. chine.)
V(p)—— dp+ V,(p)—61p
34;), dQI —cc dQI _I
(120
mi
and
d_G=0
dQ, ’ '

Plugging the second equation in the first gives,

d”! (QIipo

dQ,

r— Q
[ V(QI)

I V,

W(QI)

+6

 

W(QI)
de
dQ.’

 

+2

I V,(p)—

)

dQ,

'(p)—h(p.Q.)dp+

dQI

dQI

dh _(_p’Ql)

q

:Il:Vl (PO/(91 )a Q1» — VIM (PO/(Q, )a Q1 ))] +

W(Q!)

l

V,"'(p)——dh(dZQ’)dp

V'" (mafia Q0611? +

v(QI)
dp + I

 

Since this hypothetical incumbent assumes that the probability of entry is a

constant, then it must also be true that

differentiated to yield

2

 

I

2
d C: = 0. (Alternatively, since 10— 5 0
I

 

l

2 = 0 .) Its first order condition thus reduces to,

158

it can be

 

 

—I d—’1———’(”’QP—dp [V (p(w(Q.).Q.)) V"'(p(iV(Q.) Q.»]+
W(QI) dQ,
0 W(Q!)
girl—$512+5 IK'(P)%h(PcQI)dP+ IV“'(p)-§—g-h(p,Q,)dp+ 20
I W(QI) I l
dhth.) "PP .. d_h___(p.Q.)

V.(p)—dp+ V. (p)—— dp

1(1) dQI I dQ, _

(A.10)

The difference between the first order conditions of the two incumbents is thus
given by (A9) minus (A.10),

5a}.
dQI

Since V.(p) is decreasing in q *. then [V.(p(w(Q.).Q.)) - V."(p(i//(Q. ). Q. ))I S

[V.(p(iV(Q.).Q.))-V.’"(p(iI/(Q.).Q.))] (A11)

Thus equation (All) is positive (negative) if 5365 (2)0. If 3305 (>)0, then for the
l I

incumbent’s first order condition (equation (A9» to be satisfied, it must be true that

 

-I d———”’(”’QP ———dp [V.(p(v(Q.).Q—.» V.“(p(lI/(Q.) Q.»]+

W(QI) dQ,

0 V(QI)
flfi—Qé—flw I h(m,—Q phtp.Q.)dp+ I 16"(p)Eh(p.Q.)dp+ sew.
I W(QI) I I

.. dh(p,Q,) "Q” .. d__h(p.Q.)

V,(p)—_dp+ V. (p)— dp
LW((JQ‘I) dQ’ i dQ, _

 

which is the incumbent’s that takes entry as given first order condition. This means that

Q,‘ evaluated at the incumbent’s that takes entry as given objective function yields

negative (positive) slope, and thus if this is a concave function, it must be the case that

Q.* 2 (5)2”-

159

 

To characterize the lemma, pair up the possible cases of (E'— _g_ ') and d0 . For

I

example if (§'—g')>0 and p°>p*, then 5350. Hence according to (A.11),

l

9‘29”; “(F-55W and p° <p*,then £20. Hence 2.29.”.

l

(Q.E.D.)

160

APPENDIX B

APPENDIX TO CHAPTER 2

161

Proof of Proposition 2.1
Proposition 2.1: The government that maximizes expected seignorage values

2
M2,

information, i.e.
dflz

Proof: This proof originally comes fiom Mirman, Samuelson and Urbano (1993)

(p. 555).

The proof is a variation of an envelope argument. Let g2 ((12) be the optimal

period-two money growth rate as a function of #2. Fix I32. Define:

17012)=#2l(gz(fiz),fi)gz(fi2)+(l‘fl2)l(gz(fl2)ag)gz(fi2)- (3°12)
Then F(,u2) is linear in It, and S,*(,u2)2F(p,). Since S2*(,u2) is the

supremum of a set of linear functions, it must be convex (Rockafeller (1970) Theorem
5.5, p. 35)).
Q.E.D.

Proof of Proposition 2.2
Proposition 2.2: Let the MLRP hold. Then the optimal money growth rate

g,(p,) is higher (lower) than the myopic rate gmpic (p1) , if increasing the rate of growth

of money increases (decreases) information. That is, g,(,u,) 2(3) gmp:c(#n) if

I'(g.,§)2(5)1'<g.,o).
Proof:

First note that the first order condition of the non-myopic problem is given by

flsx<a>+dwm

= 0 , while the myopic first order condition is given by
dgl dgt

 

162

dE S *-
M = . If M _>_ (3)0 then, in order for the non-myopic first order
dgl dgl
. . . . dEA S] (”I ) . .
condition to be satisfied, it must be true that —d—— s (2)0. Given that E...SI(#1) IS a
g1

concave function, this implies that g(pl) 2 (s)gmyop,.c(p,). Graphically this argument is

represented in Figure 2.3.

-1?!

I now show that, j

d I S. *(Itz)h(m,g)dm
dgl

 

 

=(I'(g..n)— l'(g.,_)) I“: i.e(l- u.) fiejzdm.

Since EmS2 * (,u,) = I52 * (,uz)h(m, g)dm , I differentiate inside the integral,

-co

dEmS2*(,u2)_ dS2 *d;12 * dh(m, g)
————dgl.. I[— dpz dg —h(m, g)+s2 (pg— dg, Id m (13.13)

-co

Let 7=f(m.—I'(g.,§)g.) and 1=f(M.-l'(g..@g.). and let

i'=1'(gl,§) and ['=l'(g,,Q) then, since h(m,g)=(1-p,)£+.ul7,

dh(m, g)

d =-#.7'7'-(1-I1.)fl'- (B.14)
g!

Plugging equation (B.14) in (8.13),

W: IId—S— Id”: -h(m g)— S *(flz)[.u.f' I'+(1- A)f'11]dm (BIS)
dgl dy, dg

Integrating the second term in the rhs of equation (B. 15) by parts as follows,

S ’“(qu)(#.fl'-+(1 I1.)fl' )ll-

IS (#2)[#If'l'+(1- #.)_Jfl']dm= ..

I if: 1&[M7I'+(l-It.)_f_l']dm ’
I“: dm

(B.16)

-¢o

163

and since S2 *(p,72'+ (1—p,)£[')|:° =0 because 7(00) =£(oo) =7(—oo) =£(—oo) =0,

plugging (B.16) in (B.15) leads to,

* °° # * _—
fl’flifi: I fl-‘flMgmni’SZ—iW—ztmfleawn/1'1 dm.(B.17)
dgl _Q dy, dgI dp, dm ‘-

Now I want to write j—flz— as a function of % to plug in (8.17). To do so, I
g1 m

need a few manipulations. First of all, taking the derivative of h(m, g) with respect to m,

 

 

 

 

 

dh —
—=#If'+(1-#I)f' (8.18)
dm —
and combining it with (B. 14),
dh - dh -
-—=-1'--(1-#I)f'(l'-1')- (B-19)
dgl dm -
I also need the following manipulations. From Bayes rule,
#17 .1117 -
.112: _= ,thusph(m,g)=pf. Then,
(l-It.)_f_+#.f h(mug) 2 '
— dh
_I fllf(—)
dfl2 = #If _ 2d," (820)
dm h h
- dh dh
_ #11121?) #2 (I?)
but since p, f = p, h(m, g) then h, = h , thus equation (8.20) becomes,
- dh
#If "/12 (_J
d” = ‘1’" (13.21)
dm h

Using #17 = ,u2 h(m, g) once more,

— dh —,-, dh
-,-. #If[—] "#If l-Pz [—]
dflz __ [11f I _ dgl _ dgl (B 22)
__ 2 _ .
dgl h h h

 

164

I now can write equation (B22) as a function of 52—511 using equations (B.14),
m

(3.18), (3.19) and (3.21). First, plugging (3.14) in (3.22),

 

 

die _ -e.7'i'_ e(-V.7'?'-<1—Vr>rz')
dg, — h h ’

collecting I' ,

3

£&=_i' ”l7'_#2fll7' +fl2(l_#l)£_'.1.'
dg, h h h

Plea-mg
h

 

adding and subtracting

9

 

dfl2 _ -0 ”17' p2 —! I 1 ”(l—”')i'l'
T21-—1[T—7(fllf+(1-”1)£_(1—”l)[_)]+ h ,

using (3.18) to replace p,7'+ (1 - ,u, ) f ‘ above with %- ,
- m

 

 

d__2_,u_ -i g7__' Md}. #:(1-#.)f’#2(1-#.)fl'
h hdm+ h

dg, __— h ’
using (B.21) to replace ,u,f -#2 dh above with d__,uz,
h h dm dm

ifla=4ldfl2 _ h(l-fll)£'i'+ #2(1-#l)fl'

 

dg, dm h h ’
1 _ '
and collecting @L—hM—)-f—, 1 finally arrive at,

d_/1_ __l,dp,+ #:(1 #i)_f_'(.'-1')
dgl dm h

 

(3.23)

Plugging (B23) in equation (B.17),

165

dEmSZ * (#2) _

dgl

dS,*
Id-j;

[4.

die
dm

m(1-#1)_f_'(.'-1')

h ]h(g, m)dm +

 

I %‘Efl‘2—[M7i'+(l—M)fi'ldm
,u, dm

-oo

I now replace h(g,m) by (l — ,ul ) f + #17 in (3.24).

dEInSZ ‘ (#2)_

I f” I-I'%”I(<1—y.)f+/ )dm+
dg fl: (#2(1-#.)f'(_'-I'))dm+
I ‘2: “SIM +(1 #.)_f_11dm

 

K-co

 

 

I

 

(3.24)

Canceling the terms shown,

dEmSZ ‘ (#2)_

and collecting (r—i'),
dEmSZ * (#2)_

dgl

 

II%‘

S,
#2

(1:,
1%.

\
——[-7'%’f)«1—m)f)dm+

U

("#20 tom'— i))dm+

d_.._,_(1_ Ila/1d"?
dm

 

=a-I'>[If:,2d—”—,;(1—p.mdm+I%m(1-y.>1WI(B.2S)

Integrating by parts the second term in the rhs of (B25),

( \
dS“ «o
d—Zuea—mml;
#2
”dS,* ”dZS*dp
— 1- 'dm= —2———2- 1— dm—
Idle“ It); I d”: d... m It);
dS,"' d
I— ”(I- #1.);de
K-co dflz dm

 

 

Since 10») = £(—oo) = 0 , plugging (B26) in (B25) gives,

166

(3.26)

H

I—‘f’ 1"; mzdm —
_a m

U

dEmS2*(/12)=(lu_-iv) cidzsz‘fll;
' (#1,2 dm

1 -co

#2(1_/11)£dm‘

 

 

‘° t
I _fz 1’2 ,u,)£dm
_-°° m J

and canceling terms, 5‘

 

dE S,*(p,) - ” sz * dp
A—z l'—l 2 1— -—2 d 8.27
d (3)!”me mdmgm ( )

gl

2 at 1-
Given that d 52, 20 (Proposition 2.1), if 5&2 0, then Muse
dy, dm dgl

 

whenever (I'— 1') 2 (3)0. Thus, given % 2 O , gI 01,) 2(5) gm” (pl) if (i'—[ ') 2 (5)0.

I now show that the MLRP assumption implies $511 2 0.

By Bayes rule, p, = (1 3"; 7 , thus
" .ut _ + #1

 

 

d”: = #17'[(1—#1)1+fi]-M7[(1—M)1'+W]_ ,u,(1—p,)(£7'—7_f_').

_ 2 — 2
6"" [<1—p,)_f_+p,f] [awaym]
The sign of j—flz— is thus equal to the sign of (f7'—-f-f'). From the MLRP I
m _ _

f!

know that — is decreasing in a , where a = m —I(g,Q). Since by assumption 721 , then

m—ISm-l , and thus Cross multiplying gives us, 7121?. Hence

\II‘N
IV
I\ Il‘tg

(f7'—7f')20, and d—"Zzo. Q.E.D.
— — dm

167

Algebraic solution for the uniformly distributed shocks on linear demand functions.
In this appendix I find the non-myopic and myopic money growth rates under

linear demand functions of the type used in the example of section 2.6. For notational
brevity, assume that the two possible demand functions are m, = x+;g, +5, and

m, = x+ g, + a, , that is, assume that only the slope of the demand function is unknown.

By letting x = a —ba + l7 and z z ¢ — bfl , we get back the model of section 2.6. Assume

 

 

starting values g0 = gmp, , P0 2 land ,u, = 0.5.
Note that since R, = _ M0 or R, = M0 , and since 8, =1, then
x + 28mm + a x + 58mm + 3

M0 = x+zgwic +3 or M0 = x+ggmyopic +8, then it is reasonable to assume an initial

money stock of 0.5(x + Egmpic) + 0.5(x + 28mm) .

The random shocks 8, are distributed uniformly on the interval {—335} , and thus
the three possible posterior beliefs are,

0 if x+gg,—éSml<x+;g,—é
,u,= 0.5 if x+zg,—£‘ Sm,$x+gg,+é.

1 if x+§gl+§<m15x+zgl+é

The distribution of first period money demand implied by the random shock is

 

r1 . A - ..
—. If x+gg,-£Sml<x+zg,—£
4s
1 . - .. .
h(m,g)=<T 1f x+zg,—£ SmISx-tgglhem
8
1 . A - A
T If x+gg,+s<m,$x+zgl+6
( 5

Having defined all the preliminaries, I find the equilibrium money growth rates.

168

 

 

Given posterior beliefs and given that the expected value of a is zero, in the
second period the government maximizes,
«x +Zg2m +(x +zg2)(1- #2))82.
with respect to g, . The first order condition is that x+2(;y, +g(1—p,))g, = 0 , which

—x

2(2):. +£(l-flz))°

 

yields the second period equilibrium money growth rate g, (,u,) =

Plugging this rate in the seignorage function gives the second period value function as a

fimction of posterior beliefs,

 

 

#2[x+;[ - -x )1 .. -x ]+
2(2/12 + 2(1 " #2 )) 2(4‘2 + .Z.(1 ' #2 )) —x2

[ [ _, m _x I =4(Zu.+z(1—xe»
(1" #2) x + Z - ..
_ 2(2flz + z(1 - #2)) 2(zx12 + 2.0- #2))

d

 

Sz I"($12) =

 

 

 

 

2

To find EMS,*(,u,) note that if ,u,=0, then S,*(,u,)=-:i£-, if ,u, =p, =0.5,
E

._ 2 — 2
then S, I"(,1.l,)=—_L—, and if y,=l, then S, *(p,)=—£_—. Then, given h(m,g),
2(z+g) 4z

x+;g -c‘ 2 x+ggl+£ _ 2 x+§g +£‘_ 2
EMS,*(,u,)=[ Jl iidm+ x idm+ f éidm],

 

42 45‘ j . 2(;+g) 2t? 42 45

x+;g, -§ — x+zg, -€ x413, +5

thus,

—x21 - . . -x2 1 . - .
TTIX+th—£'x-Zgi+€]+2(— )E7IX+Zgi+£“x—Zgi+5]
z 8 2+2 6'

EmS2*(.u2)= _ - i

2
-x 1 — A A
+—=--:[x+zgt+8-x-_zg.-6]
42 4e

 

which, afier simplifying looks like,

—x2 (E — g)3 x2

 

EmSZ *(fl2) _

_16;;(;+g)é '-2(z+_z_)'

169

In the first period the government maximizes expected first period seignorage

given prior beliefs ,u, = 0.5 , plus the discounted expected value of second period

 

 

 

seignorage,
_ x2 (;_£)3 x2
0.5 x+z +0.5 x+z —-5 -5 + - 8.28
( ( g1) ( _gl))gl [l6gz(z+g)£‘ 1 2(z+£) ( )
__ 2 ‘_ 3
with respect to g,. The first order condition is that x+(z+_z_)g, — 6x_(z_ g) . = ,
l6gz(z+_z_)£

x + 6‘x2(z—g)3
(2+5) 1652(E+;)25’

 

which yields gl = - which after plugging in (B.28) yields total

seignorage.

To find the myopic values I first maximize the myopic seignorage function with

respect to g,, where the myopic seignorage function is (0.5(x+;g,)+0.5(x+ggl))g,,

which yields gm”, =-(T£-_—). Plugging back in the seignorage function, yields the
2 +2

(1 i; (5)152

myopic seignorage level over the two periods is — .
2(z + g)

To find the non-myopic average inflation rate I plug

 

 

2 - _ 3 _
g: = _ _ x + 6x_ (.2 '2‘): . in the following formula, 7: = 0.5(x + g) , where
(2+5) l6§z(z+§) a

;=M_1and ”Mn

 

 

 

_ . To find the average myopic inflation rate I
x + zg, x + £81
_ — 1+ . M
plug gm“, = — .. in formula,7r = 0.5(7t+7_r) , where I: =( gTP") ° -1 and
(Z + g) x + zgmyopic

1+ . M

g = ( 8mm) 0 —1. The formula for M0 is given by
x + zgmyopoc

M0 = 0.5(x + ng,c) + 0.5(x + ggmk) .

170

APPENDIX C

APPENDIX TO CHAPTER 3

 

171

 

Proof of Proposition 3.2

Proposition 3.2: Given Assumption A3.2, a strategic central banker will adjust

the inflation rate away from the myopic rate in order to influence the government’s

posterior beliefs if the banker’s value of signaling its type is not zero,

particular,

Proof:

3.2.1

3.2.2

CB
M1330, In
dfli

If the value to the central banker of signaling its type is positive
(negative) and the banker is the L type, then the banker increases

(decreases) the inflation rate to increase (decrease) the likelihood

dE VLCB (#1)

l

of its type being the true type: I.e. if < (>)0 then ‘v’ml

7r,(a,_,£,) > (<)7r,"”°”"(a,,e,).
If the value to the central banker of signaling its type is positive
(negative) and the banker is the H type, then the banker decreases

(increases) the inflation rate to increase (decrease) the likelihood of

dEV:B(/1l)

its type being the true type: I.e. if >(<)0 then Vm,

”l(aflagl)< (>)7r1my0p:'C(aH,£l).

If the banker acts strategically it chooses the inflation rate Itl according to the

following first order condition,

reaction function of the myopic banker is given by

dllf’ + m] (n. -?r')1+ p dEVC’w.)

= O , whereas the
dz!l dn',

CB _-
d[’“‘ ”W“ “1:0. Thus if
dz,

172

CB CB

> (<)0, then for the CB’s first order condition to be
dfll dfll

duf” + "wt. —Z)1
(171',

 

satisfied it must be the case that < (>)0. Since I,” + m1 (7:1 —-7;) is a

convex function of It, , then 7:: < (>)7r{‘”""”’°.

 

 

Proofof
dE__V__G(/J1)_d2VG(2.u1)dP1
—d‘[ :u1(1—1“o)fhld7r
d”! dzlli 51”]
and
dEVG(/11) =_J‘d2VG(#1)d.ul #(1_'u )f (171'
dz,” cl,u,2 drrl ' 0 H
Proof:

Throughout the proof I integrate over the support (-7],77), and I use the

assumption that f (—r)) = f (77) = 0.

First note that,

d__EVG(#.)_ _IdVdGUII) dfl.

d I.won+(1—yo)f,,)dn—Moo“? dz: (C29)
”1d/11 d”

 

Integrating the second term by parts,

IVG(#1)#°g—de”=W Idyjfimjfuofm,

and plugging in (C29),

 

173

 

 

 

P (ii/601061111 ‘
+ 1— )dz
m: I d” ”5 (yon ( ”om (C30)
dz‘ dVG d .
1 +J‘ (1011) ”Illa/I‘d”
#1 ‘17’1 .1

 

 

.uofL

Now note that from Bayes’ rule ,u, = where D = ,uof, +(l— ,uo)f,,,

L

fL =f[”l—Lfll ] and fI-I =f[”l-:lflj’then
g g

drul zflofL'_PofL dD zflofL'_fl_d£
dz, gLD 02 dz, gLD Ddz,’

EA__#_oftl_&£?_=_ia_a(£+ 010]

dzf— gLD Ddz,‘ dz, D dz," 217::

dB _ #ofL' (1-#o)fn'
- l. + H ’
dirt g g

 

dD __IuofL ' __ dD _,_ (1—#0)fl-l'

 

 

 

 

 

 

d? g‘ dzl g” ’
and thus,
fl=_ d#1_ ”1(1‘flo)fn'.
dz,‘ dz, g”D
Plugging ,d_pl = _ dial _ #10. Half” in (C30),
z, dz, g”
’ ,dVG(p.)[_ an. _p.(1-uo)f,,_'],,, f +,,_ ,1 , f >an
dEVG(/ll) = d”, d”, gHD 0 L 0 H
dz‘ dVG d
l +J' (#1) ,U, #Ofid”
\ dfll d7“ /

 

Canceling terms,

174

 

 

 

 

 

\

 

 

 

 

 

 

 

 

 

I_ cit/“(mam _
I d”, dzxM +(1 aware
mow.) = _IdVGoo [1:41—me Id”
dflf dflt g”
+ arr/“(11.) %
\ M771 I
thus,
(_ dVGUh) dfl1 1_ d \
mom): I d dz.“ yam”
dfln‘ _ 161V“ (#1) (.111 (1-#0)fu 'Iam
K d/‘I 8” I
,megmmg ,dVG(u.)(a(1—fio)fn']dfl by pans,
61% g
I o
L W wavy
draw.) #1(1-#o)fy')d z _ d’V“(u.)du. ,_ d
I tip, ( g” 7: I dfllz dfl,'ul( .uo)fu 7t 9
_ dVG(M)d#1 _
I du. d”, (1 yomdn
K
and plugging it in (C31),
I dVG(/1.)d#. ‘
- I—T—dm wow
dEVG(#i)_ d2V°(#1)d/11 _
T- +I dlhz dnlflia flo)fudfl:
dVG(/’1)dl’1 _
\‘I' I—dfl, #o)ffld” J
thus,

 

 

 

 

175

 

 

(c.31)

owl/“(111): ,dzvooz.)

(1
din,” #1#1(1‘#o)fud7r-

dz,

G 2 G
Similarly I now show that i%—§—;fi-)-=—Id V (:1,)d,u, po(l—,u,)f,dz. I first
”1 d”! d”!

differentiate EV G 01,) inside the integral with respect to z,” ,

_d___EV"‘(u,) _,dV“(M> d”; (nor. +(1—1Uo)fu )dvr- I WWW“ “132)
dz,” dp, dz, g

Using again Bayes rule p, = 52154— where D = ,uof, +(1— ,uO )f” , then

d/11_ _flofL ”_ofl. dD_/1of1.'_fli_£12
dzl g‘D D2 dz, g‘D Ddz,’

d,u, __l-‘ofL dD __IL1 dD

dz,” D2 dz,”= D dz,” ’

 

d0 = pof.'+(1-po)fn'

 

 

dfll 3L 8” ,
dD __(l-flo)fn'___d£ floft'
H ‘ H - + L ’
61”1 8 dz] 8
and thus,
dfll =_d _,_-/10(1 #1)fL .
dz,” dz,+ g‘D

- (#11 d.“ #1 (l-fli)f ' .
P1 ———=— l+ 0 L 111 C32,
uggmg dz,” dz, "D ( )

(IdVG(#1)[dfli you-13v. ‘

1 d
w: dtul dfl',+ gLD )(flofL'H golf”) 7!

dz” dVG d
I 4'}. (A) ”I (l-flo)fnd7’
( dp, dz,

 

 

Canceling terms,

176

 

\

 

dVG (#1)th
LI dfl] d”, (,uofL‘l'W)d7r
dEVG(y.)_ ,dVGw.) [flo(1‘/’1)fi')d,,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

din” d/l1 g ,
+IMdy, —#o)fnd”
K 1 ”I I
thus
I_ dV"(/11)d#i
0 L d”
w_ I d” ‘1’“ 01f) (c.33)
dz,” ,dVG(M)[#o(1- ”My,
K 8
Integrating ,dVG(,u,) [#0 (1 - 51)}; 'P” by parts,
dy, g
IdVG< ) N
—"i “#01;
dVGUIi) #o(1- yoft' _ _ ”OWN/’1 —
I 61/11 [ g" - I dll,2 dz1'u°(1 fill/Id” ,
J'dVG(/‘1)d/‘1 ode”
dial d”! J
K
plugging in (C33),
I_ ,dVGmoflAfl/dfl \
M”! 0 L
dEVG(p,)_ _ d’VG(#1)dfli ,_ d
dz," ‘ I dy.2 dfll#0( ”')fL fl,
dVG(#1)M
K+I/dfl1/d”1‘uo L It
thus,

177

 

dE V0011) d2VG(/11)dfli
# = —- 1— d .
dz,” I dfllz d”! #o( #1)f1. 7!

Finally, since from Bayes rule ,ulD = pofL <:> ,u,(,u0fL +(1—,uo)f,,) = ,uofL, then

anal/“(m z _ IdZVGm.)

d
dfllu ’Ul fl1(l—/lo)f,,d7t '

dfll d”!

 

”l(1_/‘l0)fH = flo(1"#l)fl. ’ and hence

Proof of Proposition 3.3
Proposition 3.3: Given Assumption A3.2, if the central banker does not act
strategically then if the government’s value of information is positive, it will decrease the

punishment compared to the myopic punishment to increase the degree of transparency of

monetary policy, that is, ml "mm“ < mfm’".

Proof:

2 G
—(—p‘—)-<0] thesign

Given Assumption A3.2 {-35— > 0] and proposition one [ d 2
.111

7’1

L,myopic H ,myqpic
dz, _ dz,

dm, dm1

 

of 0 equals the negative of the sign of [ ]. Since

< O , then (2 > O and thus in order for the first order

 

 

L.myopic H.myopic
dirl _ drt, = aL —a,,
1+ 1

condition to be satisfied, it must be true that

 

dG, dzrf'mpk + dG, dufm“

L.my0pic H ,myopic '—
— 7: +1— 7: —7r <0. Hence the
diff d d”: H dml [#0 1 ( #0) 1 1

non

myopic function G' evaluated at ml "M‘s" has negative slope. Since G1 is convex in

non—strategic

ml this implies that ml < mf'wp".

178

-1.

tr:

Proposition 3.5: The punishment that minimizes the government’s first period
expected losses is lower (higher) when the banker acts strategically (m,? < (>)m,""°”" ), if

the first period marginal losses of the government are higher (lower) when the banker

acts strategically. That is,

 

 

 

dG, dz,‘ _ dzf'mp‘c + d0, ‘17:,” _ “KW“
If e) = dn,‘ dm, dm, dz,” dm, dm, > (<)o:> m: < (>)m;W'c.

"“100“,. - ”IL'WPIC) + (1 " floXfl.” " ”limit )1

Proof:

m]? is found by minimizing equation (3.25) with respect to ml , which yields the
following first order condition,

as, da,‘ + dG, dz,”
dz,‘ dm, dz,” dm,

 

-[uorr.‘ +(1-xzo)2r.” J] = 0. (C34)
Subtracting the first order condition of the government given the myopic behavior
of the banker equation (3.23) from (C34) yields 9. Given convexity of 0,, if

(9 > (<)O :> m: < (>)m,’"”°”’°.

Proof of Proposition 3.6

Proposition 3.6: When the central banker acts strategically, if the government’s
value of information is positive, then the government deviates from the punishment that
minimizes its first period losses in order to increase information. In particular, given

Assumption A3.3, m,* < mf .

179

cur - - und-
,‘3‘

 

Proof: The difference between the first order conditions from equation (3.24) and
equation (3.25) implicitly determines ml * —m,7 . Using the algebra from Appendix C, this

difference (call it (b) is given by,

 

 

(1th dz” szG d
¢=£dni - d"; )1 dflfz’UI) d:l#1(l—#o)ffld”'
l l l

2 0
Given Assumption A3.2 [% > 0] and Proposition 3.1 {-d—Z—(z'lfl < O] the sign
”1 #1

 

. . drrlL dz,” . dzrll‘ dz,”
of (D equals the negative of the Slgn of —-—— . Hence if ——--— <0
dm1 dm, ml dml

then <D>0 andthus m,*<m,?.

Proof of Proposition 3.7
Proposition 3.7: Given Assumption A3.3 then if the government’s value of

information is positive, the strategic behavior of the central banker implies more
transparent monetary policy (i.e. m,‘ < mf’"“""‘g’c ), if any of the following conditions are

true,

i) The gains from information to the government are higher when the banker
acts strategically, (ml *-mf < mf°”""""g’c —m{’"°”’°), and the strategic
behavior of the central bank implies higher first period marginal losses

due to more variability of the inflation rate around its target (ml? < mfw’“) .

180

ii) The gains from information to the government are higher when the banker

acts strategically, (m, *—m,-’ < mron-strwegic

—-m,""°”"), the strategic behavior
of the central bank implies lower first period marginal losses due to less
variability of the inflation rate around its target (m: > m,""’°”’°) , but the

extra gains from information compensate for higher first period losses, i.e.

(m, * —mf) — (ms-””8": — W“) < mar“ - m3.

 

iii) The gains from information to the government are lower when the banker
acts strategically, (m, *-m," > m,”"""""'eg" -m,'""’”") but the strategic
behavior of the banker implies much higher first period marginal losses

due to more variability in the inflation rate, i.e. m,’ < m,"”’°”", but

("1, g. _m;?) _ (milieu—strategic _ mlmwpiC) < mlmyopic _ mi? .

Otherwise the strategic behavior of the bank implies less transparent monetary
policy, i.e. m,* > m,"°"“"""g’°.
Proof:

The difference between the first order conditions of the government’s problem

given that the banker acts strategically (problem 3.24) and we it acts non-strategically

 

 

 

 

 

 

 

(problem 3.22) is given by,
( dG, dn,‘ _ dnf-‘W‘c + dG, dz,” _ dz,”‘""”’”" ‘
6171'," dm, dm, dz,” dm, dm,
1‘: {pow-n.""”°”")+(1-no)(n.”-m”'"”"”">]
der dz” dnL-"W" dz ”W" szG d
l.» -——.» H :1 - 2,. ll
\ ml m, m, l dill 7’: J

If r > 0, then m, * < m:°"-“'°'eg"0.

181

The sign of Y depends on the one hand on the sign of

dG, cm," _dzrf‘mpk + (16, dz," _dir,”’""”pic
drtf dm, dm, dzr,” dm, dm, , which in turn determines the sign

’[flo (”1L _ ”1L'mpic)+ (1" #0 )(fllfl " ”lflmwopic )]

and magnitude of m,’"”°”"° —m,? , and on the other hand on the sign of

L H L,myopic ”.mpic
iii—175L- - dfl‘ — dfl‘ , which determines the sign and magnitude of
dm, dm, dm, dm,

(m1 * __ m1”) _ (”Iron—strategic _ mrryopiC) .

182

 

 

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183

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