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THEORETICAL AND EMPIRICAL STUDIES IN STRATEGIC PRICING
By

Pedro A. Almoguera

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Economics

2006

ABSTRACT
THEORETICAL AND EMPIRICAL STUDIES IN STRATEGIC PRICING

By
Pedro A. Almoguera

This dissertation provides two theoretical studies of a firm maximizing profits
under demand uncertainty, these studies are preceded by aliterature review on similar
research; and lastly, one empirical essay where oil prices are used to measure the
efficiency of OPEC as a cartel.

Chapter 1 presents a literature review comparing the models that will be devel-
oped in Chapters 2 and 3. The first half of the chapter reviews models that involve
a firm’s choices from amongst its price level, production output, or both variables
simultaneously. The review is then extended in the second half, to incorporate the
models developed in Chapter 3, to include research on storable goods and network
effects.

Chapter 2 studies a two-period model in which a firm faces uncertainty in the
slope of the demand function and is able to choose either the price or the production
level of its good. The model is solved in two different scenarios: one in which the
firm sells a storable good, hence, second period demand decreases when first period
demand is high; and the other in which consumers experience a network effect. The
main result is that welfare increases when the firm sets prices instead of quantity in
either scenario. Also, the myopic solution generates greater first period welfare than
with the dynamic outcome.

Chapter 3 examines a single firm that maximizes expected profits. Two models
are analyzed, one in the spirit of the “newsboy problem” where the firm must commit
to the production level before realizing the state of demand, and the other where a
price or quantity combination, or a commitment to both variables must be chosen

before realizing the true demand.

Conditions for the optimal price and quantity combination are found to depend on
the probability of the states of demand and the marginal cost. The main result is that
pre—committing to only the production level gives higher expected profits than when
there is pre—commitment in the two variables, however, production levels are lower. In
the model with pre-cornmitment in the two variables, the optimal price is constrained
by the monopoly outcome of the two possible states of demand under certainty while
on the other hand, the optimal quantity is constrained by the monopoly outcome
of the low demand. However, under certain conditions the optimal quantity can be
greater than the monopoly quantity with the high demand.

Chapter 4 provides a test for the cooperative behavior hypothesis for OPEC during
the period 1974 to 2004. A modification of the Green and Porter model allows non-
OPEC producers to be treated as a competitive fringe. The proposition of whether oil
price fluctuations are a consequence of noncooperative behavior within the cartel or
only shocks in the demand for oil is examined. A Three Stage Least Square estimation
and a simplified version of the EM algorithm are implemented after constructing a
cooperative behavior variable. The main finding is that, overall, OPEC has not been
effective in keeping price and quantity above competitive levels. Moreover, oil prices

are significantly higher in periods of collusion among OPEC members.

Contents
List of Figures vi
List of Tables vii

1 Firm Strategies: Setting Prices and Setting Output — a Short Liter-

ature Review 1
1 . 1 Introduction ................................ 1
1.2 Description of Literature ......................... 2

2 Demand Uncertainty with Storable Goods and Network Effects 10

2.1 Introduction ................................ 10
2.2 Uncertainty with Uniform Demands ................... 12
2.2.1 Storable Goods .......................... 15
2.2.2 Network Effect .......................... 18

2.3 Uncertainty in the Saturation Quantity ................. 21
2.3.1 Storable Good ........................... 25
2.3.2 Network Effect .......................... 30

2.4 Uncertainty in the Reservation Price .................. 33
2.4.1 Model with Production Decision ................. 34
2.4.2 Static Model with Pricing Decision ............... 37

2.5 Conclusion ................................. 39
2.6 Appendix ................................. 45
3 Price-Quantity Commitment Under Demand Uncertainty 47
3.1 Introduction ................................ 47
3.2 Uncertainty with Uniform Demands ................... 49
3.2.1 Price and Quantity Commitment ................ 49

iv

3.2.2 Price-Quantity Commitment ................... 49

3.2.3 Comparative Statics ....................... 57

3.3 Uncertainty in the Saturation Quantity ................. 59
3.3.1 Price versus Quantity Commitment ............... 59
3.3.2 Price-Quantity Commitment ................... 60
3.3.3 Comparative Statics ....................... 63

3.4 Uncertainty in the Reservation Price .................. 64
3.4.1 Price versus Production Commitment .............. 64
3.4.2 Price—Quantity Commitment ................... 65
3.4.3 Comparative Statics ....................... 67

3.5 A Two-Period Model ........................... 68
3.6 The Model with a Continuous Distribution ............... 73
3.7 Conclusion ................................. 77
A Study of OPEC Cartel Stability 89
4.1 Introduction ................................ 89
4.2 The Model ................................. 92
4.3 The Data ................................. 100
4.4 Empirical Results ............................. 104
4.5 Conclusion ................................. 108
4.6 Appendix ................................. 109
References 122

2.1
2.2
2.3
2.4
3.1
3.2
3.3
3.4
3.5
3.6

3.7
3.8
3.9
3.10
4.1

4.2

4.3

4.4

4.5

List of Figures

Uncertainty in the Intercept with Parallel Demands .......... 41
States of Demand with Uncertainty in the Saturation Quantity . . . . 42
States of Demand with Uncertainty in the Reservation Price ..... 43
Regions for Profits with Unknown Horizontal Intercept ........ 44
High and Low States of Demand ..................... 79
Profits as a Function of Produced Quantity ............... 80
Regions for the Profit Functions ..................... 81
Optimal Pricing Path ........................... 82
Myopic versus High State Profits .................... 83
Intervals for the Profit Functions with Uncertainty in the Reservation

Price .................................... 84
Profit Rinctions for the Static and Dynamic Model .......... 85
Optimal versus Monopoly Price ..................... 86
Optimal versus Monopoly Quantity ................... 87
Optimal versus Monopoly Profits .................... 88
Oil Production (thousands of barrels per day) ............. 117
Real Prices and C00perative Period Indicator ............. 118
OPEC’S Herfindahl Index ........................ 119
Market Shares .............................. 120
Combined Share for Iran, Iraq, Kuwait and Saudi Arabia ....... 121

4.1
4.2
4.3
4.4
4.5

List of Tables

OPEC Members Production (thousands of barrels per day). ..... 112
Descriptive Statistics ........................... 113
Herfindahl Index for OPEC Members for Related Years ........ 114
Elliott and Jansson Unit Root Test ................... 115
SEM Estimates .............................. 116

vii

1 Firm Strategies: Setting Prices and Setting Out-

put — a Short Literature Review

1. 1 Introduction

The idea that a firm can adjust its pricing and production strategy due to changes
in the market, number and quality of consumers and rivals, or new information gives
more flexibility to models. When the firm is restricted to make some of these decisions
prior to entering the market, either in terms of production or pricing, the problem
becomes more complicated due to the possibility of overproduction or underproduc-
tion if the firm chooses output, or over and underpricing with a price-setting firm.
In the quantity-setting case overproduction leads to unnecessary extra costs, while
the underproduction case represents an inefficient allocation. Whereas when the firm
chooses prices, overpricing leads to zero profits since it is charging a price that ex-
ceeds the reservation price of any consumer, surpassing the willingness to pay for any
type of consumer; and underpricing leads to an inefficient rationing because it is not
maximizing revenues.

The following two chapters study these limitations on firms, particularly focusing
on the case in which the firm faces uncertainty in the demand for its product. Firms
having to make production and/ or pricing decisions under conditions of market un-
certainty are found in almost all industries. Uncertainty in this study is modelled in
three frameworks where the difference lies in the assumptions placed on the uncer-
tainty of the demand function. All models assume linear demand curves. The first
model considers a possible uniformly higher demand, the second model addresses
uncertainty in the saturation quantity, and the third considers uncertainty in the
reservation price.

Chapters 2 and 3 solve each of the three frameworks when the firm chooses either

prices or quantities for its product, and each addresses a different question. In Chapter

1

2, two cases are differentiated. First, we consider goods that are storable, and second,
we suppose that the good exhibits network effects. Furthermore, a static benchmark
is compared to a two-period model, in order to give insight into how the introduction
of dynamics in the model affects the firm’s decisions. Chapter 3, following a similar
procedure, compares models when the firm can choose either prices or quantities, or
is restricted to commit to a price-quantity combination before realizing demand. The
benchmark for this case follows the classic "newsvendor problem".

The following section briefly describes the main works in the existing literature

related to Chapters 2 and 3.

1.2 Description of Literature

A firm setting prices or production levels presents different issues to all par-
ties involved. From the firm’s perspective, it imposes a different approach on how
to maximize profits, since dealing with overpricing has different consequences than
overproduction. As it is shown in Chapter 2, overpricing can lead to zero profits
if the price charged exceeds the reservation price of every consumer, whereas with
overproduction the firm can still earn profits but has an associated larger cost of pro-
duction due to the unnecessary leftover produced. Ftom the consumers’ perspective,
consumer surplus changes if the market does not clear and it depends on who gets the
sold units (e.g. with efficient rationing the good is only sold to the consumers with
the highest valuation). Finally, the government’s perspective also changes since the
regulation policies have different meanings; they can be imposed in terms of quotas
for a quantity-setting firm or in terms of price floors or ceilings to a price-setting
firm; and depending on the industry’s characteristics, this could result in distinctly
different outcomes. The standard consensus is that if a firm chooses prices, then the
efficient output will always be chosen as its profit maximization outcome (Bertrand

competition). However, this remark holds depending on the type and number of con-

sumers to whom the product is being sold or how profits are defined, as is explained
in the following Chapters.

Weitzman (1974) finds that there is no difference between a price- or quantity-
setting firm when the same information is needed for choosing one of the two vari-
ables. In Miller and Pazgal (2001), with a two-stage, differentiated-product, oligopoly
model, assuming linear demand and constant marginal cost, the final outcome, when
prices are chosen versus quantities, is the same. Addressing the issue of prices versus
quantities in specific industries, in Menanteau, Finon and Lamy (2003), a quantity-
based regulation policy for promoting the development of renewable energy is found
to be more efficient since it is based on a quota system allowing better control of
social costs. If the renewable resources are complements, as in Gaudet and Moreaux
(1990), then a price-decision strategy dominates instead. This result is also substan-
tiated in Singh and Vives (1984) and Cheng (1985). Hence, the study of price- versus
quantity-setting models is extremely sensitive to the type of assumptions applied to
the industry and consumers, and this produces ambiguous results.

In Chapter 2, demand is modelled for a storable good or a network effect good.
With a storable good, each consumer buys the product only once; that is, if a con-
sumer purchases the good in period 1, she will leave the market at the beginning of
period 2. Classic examples for durable goods include the purchase of houses, boats,
appliances or furniture which have a life span long enough to be considered lifetime
purchases. However, storable goods are a much broader group, also including goods
that can be bought and stored for a period of time, making unnecessary its purchase
in more than one period; among durable goods examples are cleaning and hygienic
products (e.g. razor blades, dishwashing detergent, paper towels). The amount of
sales in period 1 decreases demand for period 2, therefore, the firm must evaluate
the trade-off of having high profits in one period versus how that reduces demand in

the next. The study of storable goods has been of great interest over the last several

decades, not only for academic purposes but from the industry side as well. As stated
in Waldman (2003), durable goods accounted for 60% of aggregate production for the
manufacturing sector in the United States, in the year 2000.

From the theoretical perspective, there have been different approaches to solving
models with storable goods. One direction is with the durability of the good, as in
Coase (1972), where the issue is the competition and market trade-off between new
and used goods in a multi-period framework. Another direction is with asymmetric
information, as described in Akerlof (1970) for the ‘lemons’ market. A different
approach for firms dealing with storable goods was first addressed in Bulow (1982),
where the firm’s decision to sell or lease its good and the trade-off between these
options is developed. Furthermore, Bhaskaran and Gilbert (2003) find conditions
when a firm chooses to combine a leasing and selling strategy versus choosing one
or the other, following the model first implemented in Bulow (1982). Another area
of study deals with the firm having the option of learning about demand over time
through experimentation, as developed in Mirman, Samuelson and Urbano (1993),
where different incentives are presented to a price- or quantity-setter monopolist.
Experimentation can also be useful for learning how consumers vary their tastes over
time depending on their experiences, as analyzed in Conlisk, Gerstner and Sobel
(1984) and Paredes (2006). More recently, Gowrisankaran and Rysman (2005) have
estimated dynamic consumer preferences in the context of storable goods. In Bagnoli,
Salant and Swierzbinski (1989), assuming a finite number of consumers instead of a
continuum, gives an opposite result from the Coase conjecture, where the monopolist
does not lose its monopoly power in a dynamic setup. Before turning to network
effects, it is worth mentioning that durable goods can be produced in advance and
stored in order to be sold in a later period, hence, they are different than perishable
goods which have a shorter life duration.

With a network effect, the more sold in the initial period, the more is demanded in

the next. This indicates that overproducing is the best strategy for the firm. However,
overproduction implies lower prices, hence, it might compromise profits. Therefore,
the firm must once again compare the trade-off between producing at the myopic
solution and deviating from it.

Goods that present network effects or network externalities, have the special char-
acteristic of attracting more consumers if everyone else in their environment are using
them. A classic example for goods with network externalities can be found in the
telecommunication industry, where the greater the number of people that are part of
a network, the cheaper the calls are since they are originating from the same carrier.
This is taken to the limit with most carriers within the wireless industry now-a-days;
for example, if a consumer originates and terminates a call with another customer of
the same carrier, it is completely free for both users. The strategy involved is that
each user encourages everyone around him to belong to the same carrier in order to
have a greater common satisfaction for the good. Other examples include the Band—
wagon effect, which is interpreted as the desire to be in style or fashion; the use of the
same operating system in personal computer, or the use of emails at the workplace.

The pioneer in studying network effects on consumers’ expectations and produc-
tion decisions for the firm is the seminal model in Katz and Shapiro (1985). In Mason
(1999), network effects are related to learning-by-doing, confirming the perception of
network externalities as economies of scale on the demand side, due to its incentive to
increase future demand. Lemley and McGowand (1998) describe the legal perspective
of network effects, and address the question of in which cases and contexts should
laws involving network effects be modified. Notice that because network externalities
can be interpreted as economies of scales, then a one-product firm might have an
advantage that would not be fair in other markets since the network effect is not
incorporated in the antitrust laws. In Spiegel, Ben-Zion and Tavor (2005), conditions

are found where a network effect monOpolist achieves the social welfare maximum,

making any type of regulation unnecessary. One of the most recent studies is Econo-
mides, Mitchell and Skrzypacz (2005), where it is estimated that in a duopoly a
small network effect will help the higher quality firm set higher prices, whereas, a
strong network effect gives the advantage to the firm with the largest market share.
In Cabral, Salant and Woroch (1999), the intuition of the Coase conjecture is over—
turned with network externalities since prices rise over time. This happens when the
network effect is strong enough to make the firm underprice in order to earn higher
profits in the future. The empirical study of network effects can be best described
in the telephone and internet industries, as addressed in Werden (2001), Madden,
Coble-Neal and Dalzell (2004), and Birke and Swarm (2005).

For Chapter 3, the classic "newsvendor problem" is used as the static benchmark
in order to compare it to a less flexible model when the firm has to commit to prices
and production levels for its good, before knowing the true demand. The "newsven-
dor problem" or "newsboy problem" considers a newspaper vendor that has to choose
the amount of newspaper copies to have in stock to be sold the next morning. The
size of the stock has to be chosen without knowing how many people are going to
stop and purchase a newspaper. Consequently, a firm must find the optimal quan-
tity to produce that maximizes expected profits without knowing the exact quantity
demanded. Therefore, in the general setting of the "newsvendor problem", prices are
not a control variable, rather they are determined by the market depending on the
chosen production quantity.

The first to solve a version of the "newsvendor problem" was Whitin (1955). He
assumes that the firm simultaneously chooses the price of the good and the stocking
quantity. The solution to his model consists in finding the optimal stocking output
and then finding the corresponding optimal price, however, in this model demand is
known. A similar approach is used in Dada and Petruzzi (1999), where their single

period additive uncertainty resembles our model except for two main points. First,

they assume the distribution of the stochastic term depends on the price strategy
of the firm, as in Whitin (1955). Second, they do not allow for price-quantity com-
binations that result in zero profits if the chosen price is too high and the realized
demand is low. Khouja (1999) provides an comprehensive review of the "newsvendor
problem" with possible extensions in terms of a multi-period framework, managing
inventories and firms choosing price-quantity combinations.

In Lau (1980), a variation of the “newsvendor problem” is solved with a firm facing
uncertainty and maximizing for the optimal probability of achieving a predetermined
level of expected profits. In this case, the control variable is the probability of the
states of demand instead of prices or quantities. In a dynamic setting, as in Lazear
(1986), a firm facing uncertainty in the consumer’s valuation and unit-demand, ad-
justs the price of the good from the myopic solution in the first period in order to use
the information obtained for the second-period pricing. If the good was not sold in
the first period, it is assumed that the price was too high, and the optimal approach is
to reduce it in the second period. In this framework, information is useful as the firm
is able to update its beliefs about the consumers’ valuation and get higher expected
profits than with the myopic solution. Wolinsky (1991) found conditions in a two-
period model where a monopolist should hold inventories for strategic considerations
under a continuum amount of consumers with unit demand and unknown reservation
values. However, in a static model learning and information have no value. This
raises the question of under what conditions the firm should overproduce.

The use of models where a firm has to commit to a price-quantity combination is
becoming more popular in the literature. Li and Atkins (2005) solve a model where
the headquarters of a firm is in charge of the pricing and replenishment strategy.
Their finding is that both price and service levels decrease with demand variability.
In contrast to the model solved in this study, they assume that the pricing decision

affects the level of uncertainty, implying a correlation with the realized demand.

Chen and Levy (2004) solve a finite horizon model for a price and quantity-setting
firm and find the optimal inventory policy based on the inventory and variable cost
of production. Khouja (2000) finds an optimal price strategy to sell the leftover
sequentially from the first period, in a typical "newsvendor problem" setting.

In contrast with the bulk of the literature, the dynamic model in this study is
solved assuming that high demand is realized with certainty in the second period.
In order to maximize expected profits, the firm must make a price and production
level decision simultaneously, before realizing the true demand and not solve for only
one of the variables assuming the other as given or to be determined by the market.
Hence, there is less strategy and flexibility involved from the firm’s standpoint where
none of the previous mainstream approaches can be used. For example, in this model
the firm does not have the option between leasing and selling its product, as in Bulow
(1986) and Tirole (2000), since the good is sold for one period. Price dispersion,
as described in Wilson (1988), is not optimal either because the product is offered
only once to the consumers available at that particular moment. Experimentation is
also of no value to the firm, as addressed in Mirman, Samuelson and Urbano (1993),
or to consumers, as in Riordan (1986) and Paredes (2006), since neither firm nor
consumers gain profits or utility from static outcome deviations in order to learn
about the future. Also, clearance sales are not an option for the firm as detailed in
Nocke and Peitz (2005), since there is certainty in the second period regardless of
production or pricing decisions. If a firm can sell its product with information on
the valuation of its consumers, a price discrimination approach as described in Segal
(2003) can be used. Finally, Mills (1959) find that for a one-period model, a price
setter firm facing uncertainty in the demand for its product with a constant marginal
cost, charges a lower price compare to the case with certainty.

Subsequently, Chapter 2 presents conditions for each of the three models assuming

uncertainty in different intercepts of the demand function: the first scenario examines

a uniformly higher demand, while the other two scenarios analyze uncertainty in one
of the two intercepts. The latter scenarios are interpreted as uncertainty in the
saturation production (vertical intercept) or consumers’ reservation price (horizontal
intercept). Each model is then solved in a static and dynamic setting for either a
storable good or network effect. Chapter 3 solves the same three models, but assumes
a price-quantity setting firm rather than a firm that can only choose one of the two

variables without the inclusion of storable goods or network effects.

2 Demand Uncertainty with Storable Goods and

Network Effects

2.1 Introduction

A firm faces crucial decisions regarding the pricing and output level of its product
before introducing it to the market. The more information about consumers available,
the more accurate these decisions will be. However, perfect information about demand
is not always feasible or available. If the firm’s decisions are going to affect the number
of their total potential consumers, then a different approach might be necessary where
the main objective is not only to maximize present profits, but to maximize future
profits as well.

This study examines a firm’s choices when it faces uncertainty in the demand
for the good it sells. The firm must choose either a price or quantity to be produced
before realizing the true demand. After making the production or pricing decisions for
the first period, the firm realizes the actual demand. At the beginning of the second
period, the realized demand might fluctuate but by now the firm has full information
about its consumers.

The models developed address the following questions for storable goods and net-
work effect goods, independently. By how much should the firm deviate from the
myopic solution in a static setting? Is it different if the firm commits to prices or
quantities in a static versus a dynamic setting? In a dynamic setting, is the firm better
off by choosing prices or quantities? Finally, what are the welfare implications? The
purpose of this chapter is not only to compare a myopic equilibrium to its dynamic
counterpart, but also, to estimate welfare changes when the firm can choose either
prices or the production level, in the presence of uncertainty and varying demand

across time. As discussed in the previous section, in the case of a storable good, the

10

number of second period consumers is reduced if sales are high in the first period,
while with a network effect, the number of consumers increases in the second period
if sales are high in the first.

The contribution of this study is to compare firm decisions and welfare implications
when consumer preferences are modeled for storable goods and network effects, in a
static and dynamic setting, using three different settings for the demand function.
This provides an intuition for in which settings the firm and/ or consumers are better
off when the firm chooses either prices or quantities, and how assuming a storable
good demand or a network effect changes the results.

The chapter is divided into three sections. Each section covers a different model
with every model addressing both a static and dynamic setting. For the dynamic
models there are two possible assumptions for the demand function; it is modelled
either as a network effect or a storable good. The purpose of modelling demand with
either of the two previous assumptions is to provide insights on how the firm and/ or
consumers are better off when the firm chooses either prices or quantities in different
industries. For example, consider that when a firm sells a storable good, such as
a car, or a network effect good, such as a cellular phone, the resulting profits the
firm earns and satisfaction the consumers receive might not be the same if the firm
sets prices or production level since it depends on the type of product demand. The
type of demands to be considered are addressed in this chapter as follows: Section
2.2 assumes a model where there is uncertainty with uniform demands. Section 2.3
assumes uncertainty in the saturation quantity. Section 2.4 considers uncertainty in
the vertical intercept with different slopes and finally, Section 2.5 summarizes the

main results.

11

2.2 Uncertainty with Uniform Demands

The model solved in this section is used as a benchmark for the static model in
the remainder of this section and the first section of Chapter 3.

Assume a profit maximizing firm chooses either the price or production level of
the good it sells. Following this decision, in a traditional framework with full infor-
mation the market determines the remaining variable and clears. But if the firm faces
uncertainty in the demand for its good in a static setting, its optimal strategy might
deviate from the full information model. It can be expected that the firm decides not
take any risks due to the uncertainty and produce at the full information level of the
low demand if it believes that the low demand is more likely. Nonetheless, with a
possible higher uniform demand, there is the chance that the firm can sell more at a
higher price.

If the firm is instead assumed to sell for two periods, then by the beginning of
the second period it will have full information about the demand curve. Hence,
in contrast with models where experimentation or learning by doing is involved, it
will be assumed that there is known demand at the beginning of the second period,
regardless of the firm’s output in the first period. However, consumers change their
tastes depending on the previous demand for the good. That is, the demand of the
second period might adjust depending on how many consumers bought the good in
the first period. But, by that point, the firm is aware and has full knowledge of
the adjustment. Consider a firm maximizing expected profits where the demand is
assumed to be linear in the form q = a - p. For simplicity, it is assumed that the
uncertainty lies in the intercept of the demand function, and the slope of the two
possible demand curves is normalized. The demand function can be q = 2 — p (a
good, or high state of demand) with probability 7, or q = 1 — p (a bad, or low state

of demand) with probability 1 — 7. Assume a marginal cost 0 6 (0,1) and no fixed

12

cost.l Figure (2.1) presents the two possible demands and the marginal cost. Due to
the symmetry of the change in both intercepts, it can be expected that there is no
difference between maximizing with respect to quantities or prices.2

It is assumed that the firm chooses the quantity to be produced before realizing
the true state of demand and then the market determines the price. The uncertainty
has a Bernoulli distribution, and the loss in profits associated with underproduction

is used as a penalty instead of an explicit shortage penalty parameter.3

The firm maximizes expected profits as follow:

E( )3 = { 7(2—q—C)<I+(1-7)(1-q-C)q fqu [0,1] (2.1)
7(2-(1)q-cq zfqe[1,2].

The first branch of Equation (2.1) restricts the production decision to less than or
equal to 1. With probability 7, the firm gets the profits of the high state of demand
and with probability 1 — 7, profits from the low demand. The second branch is
never going to be an optimal decision since it loses all the profits from the low state
of demand (7 = 0); generating lower revenues and generating a cost of c per unit
produced regardless of if it is sold. In order to have profits in the event of the low state
of demand, the quantity to be produced must lie in the interval [0,1]. Maximizing
the first branch of Equation (2.1) with respect to quantity yields:

,_1—c+7

2 (2.2)

q

The optimal quantity, q‘, lies between [0,1] since the probability and marginal cost

are also restricted by the same interval. It is also greater than or equal to the optimal

 

1The assumption of no fixed cost is only for simplicity as it would not affect any of the results.

21h fact, if the axis names are switched, the same model is obtained.

3With a shortage penalty parameter, if there are D consumers and prodution is P < D then
there will be a penalty imposed to the firm because D — P consumers were not able to purchase the
good. The profit function would then have an extra term of E(D - P) where E is the penalty for
each consumer not satisfied.

13

output of the low demand under certainty. If the firm knows that it is facing the
low demand (7 = 0) then the optimal output is q“ = %. If the firm knows that
the high demand is realized (7 = 1) then the firm produces at the optimal output of
the high state of demand under certainty (q‘ = gg—C). In Equation (2.3), the optimal
expected price for the static model, E(p"‘)’, is calculated as the weighted optimal
price associated with each state of demand when q“ is the production level. The high
demand has an associated price of p; = 2 - q‘ and the low demand of p; = 1 — q“.
Expected price is calculated as: 7p}, + (1 — 7) pf. It must be noticed that E (p*)“’ also

equals the optimal price of each state of demand when 7 equals 0 or 1.4

_1+7+c

Eur): , (23)
Equation (2.4) presents the static expected profits associated with the optimal

output q“. This is calculated by plugging in q" in the first branch of Equation (2.1).

 

(1-0)2 7 7’ vc_(1+7-C)2
2+4 2“ 4 (2'4)

The first term of E (7r"‘)'s is the monopoly profits when the low state of demand is
realized with certainty. The other three terms of the profit function can be interpreted
as by how much profits adjust with respect to the probabilities. If 7 equals 0, we
obtain profits associated with the low state of demand with certainty; if 7 equals
1, then 7r“ equals %(2 — c)2 which are the profits associated with the high state of
demand under certainty. Equality to the output with certainty only holds when 7

equals zero. Therefore, profits with production pre—commitment are defined in the

interval between the optimal profits under certainty for each state of demand.

 

4If the model is solved assuming the firm is instead committing to a price and quantity is deter-
mined by the market, the same results are obtained.

14

2.2.1 Storable Goods

This section solves the previous model in a dynamic setting rather than a static
framework. Second period demand is modelled as a storable good. The amount of
consumers in the second period are reduced depending on first period sales. Hence,
the second period intercept, a2, is defined as a2 = A(q)a1, where a1 is the first period
intercept (a1 Q {1,2}). The parameter /\(q) measures the change in the number
of consumers. For a storable good /\(q) 6 (0,1], in order to represent a reduction
in second period demand. The derivative of /\(q) with respect to first period sales,
measures the effect of the first period sales on second period demand. This derivative
is defined in the interval [—1,0) to represent a demand decrease of no more than one
unit in the second period per unit sold in the first period. By the beginning of the
second period, the firm has full information about demand (i.e. knows if it is high or
low), but the firm has full information about A(q) at all times. Second period profits
are defined as 7rg(a) = %A(q)2(a — c)2,where differentiating with respect to q equals
ég-Z‘Ma - c)2.

The problem solved by the firm is:

mamEUr);i = 7 [(2 - q - c)q + 67r2(2)l + (1 - 7) [(1 - q - Clq + 6W2(1)l (2-5)

‘1

where 6, the discount factor for second period demand, is assumed to be in the interval
(0, 1). If 6 equals 0, the dynamic problem is identical to the static model. Also, 6 is
assumed to be less than 1 to have a discount on second period profits. The first branch
represents the expected profits of the high state of demand plus the discounted profits
that the firm will earn in the second period. Similarly, the second branch represents

the expected profits with the low state of demand.

Lemma 2.1 With a storable good and a quantity-setting firm, expected profits in a

static setting are higher than the first period profits of a two-period model

15

Proof. Maximizing Equation ( 2. 5) with respect to quantity the following results are

obtained:5

—c (MBA 2 2
(1d = 5%‘l'751‘ (7(2-6) +(1‘7)(1—C)) (2-6)
1500).; = 53%15_%%(7(2—c)2+(1—7)(1—c)2), (2-7)

which evaluated in Equation ( 2. 5) results in associated profits of

—c 2 8 2 2
EM): = (—1+—74—-)— - % (5A5: (7(2 - C)2 + (1 - 7)(1- C) )) (2-8)

since the first term of E (n1): equals 7r‘, and the second term is always negative, then

7r‘ _>_ E(7rf)g. I

Lemma 2.1 shows that for a two—period model and demand for a storable good,
the firm will underproduce in the first period. In order to have higher second period
profits, the firm sacrifices production and first period profits. Hence, profits from
the static model are always higher than the first period profits from the dynamic
model. While this result is trivially true, because the myopic firm can always decide
to mimic the first-period production plan of the forward—looking dynamic firm, it is
not immediately clear how total welfare in the first period compares across models.

In order to measure the surplus consumers obtain, consumer surplus for each state
of demand is defined in the usual form: the maximum of the consumers’ willingness

to pay, the area below the demand curve, minus the price he actually pays, E(p‘)’.
q.

For the high and low demand, these areas are defined as q‘ f (2 — q — E (p*)) dq and
o

q

q" f (1 — q — E (p‘))dq, respectively. Expected consumer surplus is the consumer

o
surplus from the high state of demand weighted by 7, plus the consumer surplus

 

5See Appendix

16

corresponding to the low state of demand weighted by 1 — 7. For the benchmark

model expected consumer surplus is defined as:

(1+7-Cl2.

8 (2.9)

E(CS"); =

When 7 equals 0 or 1, E (CS *)3 is the consumer surplus from the low or high state
of demand, respectively. As in a classic microeconornics model, expected consumer
surplus equals half of the associated expected profits, therefore all the analysis from
Equation (2.4) follows through for E(CS‘);.

Expected welfare for the static model is defined as the sum of the static profit
function from Equation (2.4), plus the expected consumer surplus from Equation
(2.10):

EM); = 3n + 7 — C)? (2.10)

As with the associated profits and expected consumer surplus from Equations (2.9)
and (2.4), expected welfare is bound by the welfare under certainty with the low and
high state of demand and they are equal when 7 equals 0 or 1, respectively. To shed
some light on this, Lemma 2.2 compares the associate welfare from Equation (2.10)

and the first period welfare of the two—period model.

Lemma 2.2 With a storable good and a quantity-setting firm, welfare in the static
model is greater than first-period welfare from the dynamic model.
Proof. Expected consumer surplus from the dynamic model is:

1 — 2 1 — A

where A = i6)‘% (7(2 — c)2 + (1 - 7)(1- c)2).
The first term of E (05*): equals E (03‘); Since 53:— is contained in the interval

[—1,0), the first factor of the second term is always negative. Moreover, the second

17

factor is always positive, resulting in a negative second term for E (03“); Hence,
E(CS"); Z E(C’S*)f,’. The equality only holds when x\, 2—2, or 6 equal 0.

First period welfare in the dynamic model is defined as W: = E(7r‘1‘)g+ E (08")?
Then, W: — W; = A (q‘ — é). Since 68—: S 0 implies A S 0, then A (q’ — 13-) 5 0.

Therefore, welfare from the static model, IV is greater than first period welfare of

8
q)

the two—period model, “7:. I

Comparing a static versus a two-period model, it is obtained that profits and
consumer surplus are greater in the static setting; therefore, welfare is greater with
the myopic solution for the first period.

Lemma 2.1 and Lemma 2.2 can be summarized as follows: since the firm is under-
producing in the two-period model, it will earn less profits than in the static setting.
The change in consumer surplus is explained in the same manner; a lower produc-
tion implies a higher price resulting in less welfare for consumers. Since profits and
consumer surplus are lower in the first period of the dynamic model, welfare is also

lower.

2.2.2 Network Effect

When the consumers demand is modelled with a network effect instead of a
storable good, the more sold, the more the good will be demanded. In particular,
when first period sales increase, it will result in an increase in second period demand.
In contrast with the previous model, second period demand faces an augmentation
instead of a reduction. The intercept of the second period is again defined as a2 =
/\(q)a1, but since second period demand increases, A(q) is defined in the interval [1, 2].
The parameter A(q) cannot be less than 1 since this would represent a reduction
instead of an augmentation. Restricting A(q) to be less than or equal to 2 lets the
second period demand be modelled assuming that the increase in demand will not be

more than double the first period demand. The effect of first period sales in second

18

period demand,g—:, has to be positive and less than 1, then each unit sold in the first

period will attract at most one more unit to be demanded in the second period.
The following Lemma compares the static versus the two-period welfare when

a network effect is assumed instead of a storable good demand as in the previous

Lemmas.

Lemma 2.3 With a network efiect and a quantity-setting firm, first period welfare
of the dynamic model is greater than welfare in the static model.
Proof. From Equation ( 2. 8) it is obtained that Lemma 2.1 also holds for a network
effect, even though the firm is producing more than in the static model. That is, the
firm earns higher profits in the static model versus in the first-period of the dynamic
model.

However, consumer surplus from the dynamic setting is much larger. With a
network effect 2% Z 0, from Equation (2.11) it is obtained that the second term is
always positive resulting in an opposite inequality for the consumer surpluses { in this

case E(CS"): S E(C'S'*)g).
Comparing the two welfares in question, the difierence W: — W; = A (q* — g) is

positive since A 2 0 and q" 2 523. I

Lemma 2.3 provides a result opposite of Lemma 2.2. When a network effect is
assumed instead of a storable good, welfare in the first period of the two-period model
is greater than welfare from the static model. Even though first period profits are
always smaller than profits from the static model, consumer surplus in the dynamic
model is greater. Moreover, the difference in consumer surplus is much larger than
the difference in profits, resulting in greater welfare in the dynamic model. The result
is explained by the tendency of the firm to overproduce with a network effect in order
to have higher sales in the second period. The overproduction also results in higher
consumer surplus for each period since it has an associated lower expected price which

benefits consumers.

19

The possibility of having an unknown uniformly higher demand before the firm
makes a marketing decision, makes the firm indifferent between choosing prices or

production levels. The following Lemma shows this comparison.

Lemma 2.4 With parallel demands and uncertainty in the intercept, a firm is indif—

ferent between choosing prices or quantity.

Proof. When the firm chooses prices in a static setting, it solves the following
problem:
mngm; = 7(2 —p><p— c)+<1— 7x1 —p)<p— c) (2.12)

after maximizing Equation (2.12) with respect to prices, p; is obtained

1+7+c

p; = ———2 (2.13)
t 3 1+ 7 _ C

As with the quantity-setting firm model, for this case after the optimal price has been
estimated, the expected production level is calculated as the weighted average of the
two state of demand optimal quantities which is 7 (2 — p;)+ (1 — 7) (1 — p3) . Expected
profits for the price-setting firm are calculated as 7(2—p;)(p;—c)+(1—7)(1—p;)(p;—c)
which simplifies to:

(1 + 7 -— c)2

EM"); = ——4— (2.15)

F innaly expected consumer surplus is calculated in the same manner as before, that

is as the weighted average of the consumer surplus from the high and low states of

demand
(1 + 7 — c)2

E(cs*); = 8

(2.16)

which is the same outcome associated with the static model and a quantity-setting firm
solved above.

For the dynamic model we obtain the same results since differentiating rrg(a) with

20

respect to p results in ég—g—g—gMa — c)2 = —§)\53(a — c)2. I

Consumers are also indifferent if they are purchasing the good from a price— or
quantity-setting firm. Due to the symmetry of the model, the firm’s price or pro-
duction level does not change depending on which of the two variables it chooses,
therefore, the benefit consumers obtain from buying the good also remains the same.

The following Lemma states this result.

Lemma 2.5 With parallel demands and uncertainty in the intercept, consumers are
indifl'erent between purchasing from a price- or quantity-setting firm.

Proof. The proof follows directly from comparing Equation (2.9) and Equation
(2.16). I

The simplicity of the proof arises from the fact that the symmetry in the model
allows the variables to switch on the axis without affecting the consumer surplus. The
next section shows how changing the assumption of the demand functions changes
the preferences of consumers and firms, when the firm chooses price or production

levels.

2.3 Uncertainty in the Saturation Quantity

This section assumes a common intercept but the uncertainty lies in the slopes,
giving two possible saturation quantities. The demand function for the good produced
by the firm is assumed to be linear, and for simplicity, with a normalized vertical
intercept. It can be either a high (good) state of demand, p = 1— bq, with probability
7, or a low (bad) state, p = 1 — bq, with probability 1 — 7, where b > (3. Hence,
the uncertainty lies in the slope of the demand function. The two states of demand

have an intercept in the vertical axis at 1; prices greater than 1 will always result in

1

no profits for the firm. The saturation quantities for each demand are 3 and % for

21

the high and low state, respectively. Figure (2.2) presents the two possible states of
demand with the constant marginal cost. If the output level is set above if and the
realized demand is low, then the firm will make profits up to where the price equals the
low demand function. If the output is set between i and 1,1; and the realized demand
is high, then the firm underproduced and will have lower profits than if it would have
chosen a price-quantity combination lying on the high demand. In addition, it will
be assumed that the firm has a constant marginal cost 0 6 (0,1). Without loss of
generality b is normalized to 1, and b E [.4, 1]. The assumption of having (_2 on the
defined interval is necessary for the Lemmas stated in this section.

The firm maximizes expected profit in a two-period framework. In the first period,
the firm has to choose its quantity or price before realizing actual demand but knowing
the possible states and their probabilities. At the beginning of the second period, the
firm learns the true demand function and is able to allocate the monopoly outcome.
However, the second period slope is the first period SIOpe parametrized by a factor,
A, that depends on the first period sales;6 hence, the second period slope is defined as
b2 = A1215. Profits in the second period are then defined as rr2(b) = £0 — c)2, where
b could be either the high or low state slope. The parameter A represents one of two
possible case scenarios:

A storable good: if a consumer buys the good in the first period he will not
consume it again in the second period; hence, demand in the second period is reduced
by first period sales.

A network effect: demand in the second period will be augmented from the
first period; for stationary purposes second period demand will be assumed to, at
most, double the first period demand. Also, with a network effect it is assumed

that an increase in first period consumption has a positive impact on second period

demand.

 

6When the market clears, the quantity produced equals sales so there is no leftover.

22

The derivation of the myopic optimal outcome when the firm chooses prices or
quantity in a static setting will be used as a benchmark. The expected profit maxi-

mization problem for the firm choosing quantity is:

maxE(7r)Z = 7(1 - 9(1) + (1 - 7)(1 - 5(1) - at (2-17)

‘1

which yields the following results:

 

 

q’ = 2 7b+(1-7)5 (2.18)
E02). = "2L6 1 (2.19)

The static quantity, qs, equals the monopoly outcome of maximizing the average of the
two possible states. The expected average price E(p), is calculated as the weighted
average of the optimal price for each state of demand. However, as is explained below,
the optimal price for each state of demand equals 1%. Expected profits are derived as
the sum of the profits obtained from each state of demand weighted by its respective

probability. Expected consumer surplus is calculated in a similar manner as Equation

 

 

(2.16):
s _ (1—c)2 1
Em” ‘ 4 (we—775) (2'20)
13(05):; = (1'6) 1 _. (2.21)

8 (79+ (1 - 7))

Maximizing expected profits with a price-setting firm we obtain:

P

maxEvr); = 7 (Lg—P) (p — c) + <1 — 7) (1%”) (p — c) (2.22)

23

resulting in:

p. = ’2,” (2.23)

_ 1—c 75+(1-7)b
E(q). — 2 ( 59 ) (234)

 

 

Horn Equation (2.23) it is obtained that prices are the same, independent of which
variable the firm chooses, but the expected production level is lower than the output

level from the previous model as the following Lemma states.

 

 

E(7r); = (11C) (%+L;—7> (2.25)
E(CS); = (1;6)2(%+-1—g—7). (2.26)

Expected profits and consumer surplus are calculated in the same manner as in Equa-
tions (2.20) and (2.21). The following Lemma compares the outcome between a price

setter and quantity setter firm in a static model.

Lemma 2.6 In a static setting, total welfare is greater when the firm sets prices
instead of quantity.
Proof. Welfare is calculated as the sum of profits and consumer surplus. Comparing

Equations (2.20) and (2.4) it can be concluded that profits and consumer surplus are

1

greater with a price-setting firm sznce m

< :3! + 1—;1 which can be interpreted
as by how much profits are reduced in a price- or quantity-setting model. Therefore,

welfare is greater with a price setter firm. I

The intuition of having greater welfare from a price setter firm, as stated in Lemma
2.1, is that the optimal price for the high and low state of demand are the same (Iii-C).
Hence, the average price E (p),, equals the optimal price p3. But expected quantity

E (q), is larger than the optimal solution for the quantity—setting problem, qs, since

24

the former is the quantity obtained after choosing the optimal price for either state
of demand. The static output, q,, is the optimal outcome for maximizing expected
profits but is not the optimal output level for either of the two states of demand.
This result resembles Mirman, Samuelson and Urbano (1993) where experimentation
is of no value in a similar framework, and the firm would prefer to set prices instead
of quantity for the same reason. The next section presents the two—period framework
assuming that the firm sells a storable good. This result matches Fudenberg and
Tirole (1983), where learning by doing increases first period output compared to
the myopic quantity in order to have higher first period profits and shift downward

constant marginal cost in the second period.

2.3.1 Storable Good

In this section, comparisons between dynamic and static models for price and
quantity setters are provided when a storable good assumption is used in the second
period demand. Since second period demand is reduced with a storable good, the
slope parameter A(q), will be defined in the interval (0,1]. Therefore, second period
demand is bound between the first period demand and no demand at all. Moreover,
an increase in first period sales has a negative effect on the second period demand
which can be represented as defining the marginal effect of first period sales on second

period demand, %’3, to the interval [—1,0) .
9

Lemma 2.7 With a storable good and a quantity-setting firm, welfare in the static
model is greater than first-period welfare in the dynamic model.7
Proof. The maximization problem in a dynamic setting is.

mngUrV 7 [[1 — Qq - C] q + 6r2(12)l + (1 - 7) [(1 - 5(1 - C)q + 572(5)] (2-27)

q:

 

7The proofs assume 5 normalized to 1, and b E [.4, 1]

25

resulting in

 

 

 

 

 

_ 1 _ C 072(12) — ' 8W2(5)
‘“ ‘ 2<)1+<1—7>B> (1 +’l” o‘q ”1 "’l 6g ll (2'28)
— 6 7n " —
E(P)1 = 126-5(7—8 62:9)+(1—7)d7;2;b)]. (2.29)

Second period profits are defined as n2(b) = fiA(q)(1 — c)2, and assumed to be dis-
counted by the factor 6. Using the chain rule with respect to first period quantity we

have
07W“) __ Qi
(Sq — dq 4b

 

(1 — c)2. (2.30)

The Optimal dynamic first period quantity, q], consists of two terms: the first is
the optimal quantity for the static model solved above, the second term represents
the discounted adjustment of the second period profits to changes in q]. Notice that

a_,\

q1 < q, since with a storable good the second term is negative because sq

is negative.
Also. q] = q, if second period profits are not taken into account for the maximization
problem (i.e. (5 = 0); or if there is no relationship between the slope parameter, A,
and the first period quantity (3% = 0) .

Similarly, the first period expected dynamic price, E (p)1, consists of the optimal

static price plus a dynamic term, that for the case of a storable good is positive,

resulting in a higher dynamic price compared to the myopic solution.

1 — c 2 (52 8r b (in b 2 1
E(7T1)Z= ( ) _ __ 7 2(_) +(1_7) 2( ) . _,
4(712 + (1 - 7)b) 4 09 5C1 712+ (1 - 7)b

 

 

 

 

(2.31)

where E(rr1)g is the expected profits in the first period. E(7r1)f, > E(7r)g since the

second term of Equation {2.31) is always negative.

26

Expected consumer surplus in the dynamic model is:

d _ (l-c)2 6(1—c)
Ems”: ‘ sewn—7767+ 4 l’

6M2)
6q

 

 

 

52 622(9) 197.2(5) ’ 1
’2? 7 6q 0—7) 6C1 l 72+(1-7)5'

 

 

Comparing the expected consumer surplus of the one-period versus the two-period

model we obtain:

E(csl)g - 19(05):; =

 

 

 

= “lidl’aflflmm’” 6a

£552 [76??? ( ‘ ”87?;le 71.2 + (i - 775
1;c) [7812(2) +(1_7)§1:§] +

+§83 [79%. +(1— 7)87r;(1(5)]2 712+ (l ‘ ”1’

8r; (5)] +

 

 

 

5(

 

 

 

 

 

 

 

 

 

 

 

6 1 -— c 3 8A 1— 7
= __<__16>_[%+<_ 7],
582 8_qA2(1__)—c 7 1
1[b(157)2] 7b+(1-7)5
: 6(1—6c)3a_/\6[1+<1_v)][1+§Q<1—c)4[1+(1:7>]2 1 _]
16 Sq b b 8 Sq 16 b b 713+ (1 — 7)b
where 1+ 7383—:(1—Cl4 [1121 + (1E7)]2 79+(l—7)5 > 0 since the second term is less than

1 in absolute value.
Hence, the same result is obtained for first period expected consumer surplus as
in profits: E (031): < E (CS )3 Therefore, first period welfare for the static setting is

larger than welfare in the dynamic model. I

In Lemma 2.6 it is demonstrated that the firm in the dynamic setting underpro-

duces and, as a consequence, charges a higher price compared to the myopic solution.

27

In order to get higher profits in the second period, the firm’s underproduction leads
to lower profits in the first period. In addition, with a reduction in consumer surplus
in the dynamic model, we obtain lower first period welfare when the. firm chooses
the optimal output of the two-period maximization problem over the myopic output
level. This is a similar result to Lemma 2.2 in the previous Section. The following

Lemma proves that this result also holds with a price-setting firm.

Lemma 2.8 With a storable good and a price-setting firm, welfare in the static model

is greater than first-period welfare in the dynamic model.

Proof. In this case the firm solves the following maximization problem:

mngvr): = 7 K?) (p — c) + we] +<1— 7) [(1—g3) (p — c) + ma]

differentiating with respect to prices and making the derivative equal to zero, the

optimal price is defined as:

 

 

 

 

 

p. = 1,6 g] 87?]f’)+(1—7>a’:,jfb’], (233)
Ed). = ’;C(V”+(.,jb‘”9)-§[7—6”;If’-’)+(1—v)a’;jf”]. (2.34)

The expected production level was calculated as in the previous models, as the weighted
average of the optimal quantity for each state of demand. Using the chain rule on

second period profits this time with respect to price gives

 

6p _ 6A 6q6p—6q 4b

 

07r2(b) _ 87r2(b)0A§_q_ _ 6A(1-—— c)2 ( 1)
b

28

Then first period expected profit and consumer surplus are calculated as in Equation

(2.30) and Equation {2.31):9

 

 

 

 

 

 

 

 

 

d_(1—c)2 7 :7 _f 672(2) _ 072(5) 2 5;.
ET)“ 4 (3+ 2 l 4 l’y 6p +0 7’ 67» l 75+<1—7)1’
(2.35)
.1 7 7 62 672(2) 672(5) 2 51>
Ewsl)”: f1C)C(§+ +(Tb))+ 8 l7 6p +(1—7) 6p l 7b+(1—7)Q—
Ml ‘ “ll—“aflilbq - (236)

First period expected profits with a price setter firm present a similar structure as
a quantity setter firm. The first term is the static price setter profits, whereas, the
second term represents the discounted profits obtained from the second period. In the
case of a storable good, the second term is negative, resulting in lower first period
dynamic profits.

Comparing the expected consumer surpluses of the static and dynamic model:

14.70503, —- 13(05):, =

 

 

 

 

 

 

 

 

_ g 672(9) _ 572(5) ’ bl_) _6(1—c) 672(9) _ 672(5)
’ Sl’ 6:9 +0 ” 6p l 75+(1—7)z_2 4 l7 62 ”I 7’ 62 l
_ {EU-c)“ 1 (1-7) [ 5(1-C)Q _7_ (1-7) 512

‘ 168g 16 (122+ '52 l” 8 6q(_2+ 52 )7E+(1—7)12l

 

 

("145(18- C)g_:1\(§75 +0-7)) 512

 

therefore E(CSl)g < E(CS);.
Expected consumer surplus in the dynamic setting is also smaller, hence, welfare

in the static model is greater than first period welfare in the dynamic setting. I

 

9See Appendix for proof of Equations (2.30) and (2.31)

29

Lemma 2.7 presents a result similar to Lemma 2.2; first period expected profit
and consumer surplus in the dynamic setting, are smaller than in the static model.
Hence, welfare in the myopic case is greater than first period welfare in the dynamic
model. Therefore, with a storable good, regardless of whether the firm sets quantity
or prices, welfare is greater in the static case versus welfare in the first period of the

dynamic model.

2.3.2 Network Effect

The following two Lemmas make the same comparisons as the previous section
but instead use a network effect. With a network effect, second period demand in-
creases compared to the first period. In the model this assumption is included by
assuming the slope parameter, A(q), is defined in the interval [1,2] so that the aug-
mentation in second period demand is no more than double the period one demand.
Consequently, an increase in first period consumption has a positive effect on sec-
ond period demand; it is assumed that the demand will not increase by more than
a one-to-one ratio, meaning that each unit purchased in the first period will attract,
at most, one more consumer which is represented as the marginal effect of the first

period sales on second period demand, 2—2, being bound in the interval [0, 1].

Lemma 2.9 With a network effect and a quantity-setting firm, first period welfare
in a dynamic model is greater than the static model welfare.
Proof. From Equation (2.28) the dynamic quantity, ql , is larger than the static
optimal output q,, because with a network efi'ect, since 23—: is positive, the second term
of ql, is also positive.

First period profits in the dynamic setting (E(7r1)§,") are smaller than in the static
model, but in contrast with Lemma 2.2, with a network effect first period consumer

surplus from the dynamic setting is greater than consumer surplus from the static

model. Moreover, E(CS,): — E(CS,); > E(rr,); - E(rr,):. Thus, first period welfare

30

increases in the dynamic model. I

Since a network effect implies that the more consumed in the first period the
greater second period demand will be, the firm overproduces compared to the myopic
case in order to increase second period demand. This increase in output implies a
lower first period price resulting in greater expected consumer surplus. The gain in
consumer surplus is greater than the losses in profits implying greater first period

welfare in the dynamic scenario.

Lemma 2.10 With a network effect and a price-setting firm, first period welfare in
a dynamic model is greater than in a static model.

Proof. Using Equation (2.33), the dynamic price is lower than the myopic price
since 2—2 is positive. Also, first period profits of the dynamic model are lower than

the static expected profits. However, consumer surplus is greater and E(CS,): —

E(CS1 )f, > E(rr,); — E(rr,)g, so welfare increases with a network effect. I

As with an output setter firm, a network effect also implies greater welfare in the
dynamic case with a price setter firm because of the increase in expected consumer
surplus.

A network effect gives a result opposite of a storable good: first period welfare
for the dynamic setting is greater than the myopic profits regardless if the firm sets
prices or quantity. However, all the previous comparisons have been between dynamic
and static models. The next Lemma examines the best decision for the firm between

setting price or quantity in a dynamic framework.

Lemma 2.11 Welfare is greater when the firm sets prices instead of quantities in a
two-period model.
Proof. Comparing Equations (2.28) and (2.33), the expected quantity obtained

from the price-setting problem, E (q)1, is always larger than the optimal output for the

31

quantity setter problem regardless of whether it is a storable good or network efi'ect.

However, prices do not have the same behavior. With a storable good, the expected

price obtained from the quantity setter problem is larger than the optimal price from

the price-setting model.

With a network effect the result is reversed. In terms of

expected profits, the firm is better ofi choosing prices instead of quantity:

(1-6)2 1 7 (1-7)

4 l7Q+(1-7)5—(§+ B ll+
526A2(1--c)4
I??? 16

 

 

 

 

 

which with the assumed slope values is always positive.

1 (1—7) 2 512 _[7 (1---7)2 1
{(22+ 52 )7b+(1-7)b (2+ 5 l7b+(1-7)5

 

Comparing the expected consumer surplus of the price versus quantity setter firm

we get:

15(05):,i — 113(05):,l =

 

 

 

 

 

 

(1:)2l7b+(i-7)5-(f+(157))l+
6(11—(56)3g—:[% (1%?) 32+(15-2’7)]+

__ 2
$330”) Hi“1 bfll 7b+<i—7>b‘ (bl

 

The first and third terms are negative and the second is positive, however, the

difference is always positive for a storable good, furthermore, if g0 — dig-2 > .14, the

second term is big enough for the difference to still be positive.

The comparison of expected consumer surplus gives a similar result except for

a new constraint imposed in some of the parameters: E(CS): > E (CS): with a

storable good; when a network effect is assumed, the inequality is also true when

32

l

6(1 - dag—2 > .28. The constraint on the network effect case can be interpreted as the
combination of a small marginal cost and discount factor, and a relative large marginal
effect of the first period production on the second period slope parameter. Therefore,
welfare increases with a price setter monopolist regardless of with a storable good or

network effect. I

Lemma 2.10 corroborates the finding in Lemma 2.1, regardless of whether it is
a storable good or network effect, the firm and consumers are better off when the
firm sets prices instead of quantities. This result holds in either a static or dynamic
framework. As previously discussed, if the firm faces uncertainty only in the satu-
ration quantity then when it maximizes expected profits with respect to prices, the
obtained price is also the optimal price for each state of demand when the firm knows
the realized demand. On the other hand, when the firm chooses quantity, the opti-
mal output that maximizes expected profits is neither of the optimal outputs with the
states of demand, resulting in lower expected profits. This analysis explains why first-
period profits are always higher with a price-setting firm. Consumer surplus presents
a similar behavior, with a storable good consumers are always better off with a price-
setting firm since it is associated with a lower price and a larger expected output
than in a quantity-setting firm. With a network effect the same inequality holds but
it is restricted to having a large effect on second period sales and/ or discount factor
compared to the marginal cost. This restriction is explained by the consumer sur-
plus increase associated with a network effect that compensates for the asymmetric

relation of prices and quantities in the possible states of demand.

2.4 Uncertainty in the Reservation Price

The model in the previous section solved for expected profits when the slope

of the possible demand curve was unknown with the same vertical intercept, thereby

33

affecting only the saturation quantity. This section solves a similar model but assumes
uncertainty in the vertical intercept. This uncertainty is interpreted as an unknown
reservation price, or the price at which consumers will no longer be interested in
purchasing the good since that price would be greater than the ‘happiness’ generated
by the good. The high demand is defined as p = 2 — 2q, with probability 7, and
the low demand as p = 1 — q, with probability 1 — 7. The marginal cost remains
constant between 0 and 1. Figure (2.3) presents the defined states of demand with
the marginal cost. The vertical axis is similar to the model solved in Section 2.2 but
for this scenario the firm will never produce more than 1. It also resembles the model

from Section 2.3, if prices and quantities are shifted on the axis.

2.4.1 Model with Production Decision

Equation (2.37) presents the problem that the firm faces when there is uncer-

tainty in the vertical intercept and the firm chooses its production level in a static

setting.
mngbr):=7’[2-2q-c}q+(1—7)[1-q-c}q (2.37)
which results in:
1—c+7
8 —— 2.38
q 2(7 + 1) ( )
1+c+
E(p), = ——é——l. (2.39)

For this model, the optimal quantity depends on the probability of the states of

demand. Notice that q, and E (p), equal the optimal output level and price when the

34

firm faces the low demand with certainty (7 = 0):

(1-c+7)2

BUT); = W (2.40)
s _ (1 — c -l- 7)2

In a dynamic setting, as in the previous sections, once the firm has made its
optimization decision and the market clears, the firm has full information about the
demand function its product faces. For the second period the demand function is
p = a2 —bq, where the second period intercept is the first period intercept parametrized

by A(q), a2 = A(q)a1 with the same slope (b). Hence, second period profits are defined

as r2(a) = .4.qu - or.

Lemma 2.12 With a storable good, welfare in the static model is greater than first
period welfare of the two-period model.

Proof. The dynamic optimization solved by the firm is:
mngbr): = 7 [l2 - 241 - C] q + 672(2)] + (1 - 7) [(1 - 141 - C)q + 672(1)] (2-42)

resulting in the following outcome:

 

 

 

 

 

 

_ 1—C+’)’ 6 67TQ(2) 87(2(1)

qd ‘ 2(7+1>+2(7+1)l7 6g +0") 6q l (2'43)
1+c+7 6 67r2(2) 67r2(1)

Em 2(7+1)‘2<7+1>l 641 ”1'” 6q l (2'44)

with the following associated expected profits and consumer surplus:

(1—c+7)2 _

_ (1 — 0+7)2 A
E(CS): -— w‘f'fl m+1—c+7 (2.46)

35

I

where A: 6 7%22 +(1—7)

8W2(1)
5Q

 

As in the previous sections, profits from the static model are greater than first
period profits from the two-period model. Since A is negative with a storable good
and the term in parenthesis is positive then consumer surplus from the static model
is greater than from the two-period model. Since profits and consumer surplus are

greater in the static model, it follows that welfare is also greater. I

Once more, a storable good implies lower first period welfare for the dynamic
model. With a storable good the marginal effect of second period demand on first
period sales is negative, hence, the static quantity is equal or greater than the dynamic
output. This results in a greater expected dynamic price than its static counterpart.
Therefore, the decrease in second period demand makes the firm produce at a lower
output in the first period in order to sell more in the second period. As a consequence,
consumers are worse off since they are paying a higher price which results in overall
lower welfare.

In the case of a network effect, as before, it is obtained that first period welfare
of the two-period model is greater than welfare in the static model. The following

Lemma proves this finding for the model with uncertainty in the reservation price.

Lemma 2.13 With a network efiect first period welfare of the two-period model is
greater than welfare in the static model.

Proof. It follows from the results of the previous Lemma that profits from the
dynamic model are also going to be less than the associated profits of the static model.
The change in consumer surplus, however, is not the same. Since 2% Z O, A is
positive resulting in a larger consumer surplus for the dynamic model. Moreover, the
difference in welfare Wd — W, = A {—87% + 1 — c + 7] is positive which results in

greater welfare for the dynamic model. I

36

When a network effect is assumed, similar to previous sections, the first period
welfare associated with the dynamic model is greater than the welfare from the static
model. Even though profits are always going to be lower in the first period of the
dynamic model, as Equation (2.31) shows, with a network effect the optimal dynamic
output is greater than the static output, having an associated lower expected price

which increases the consumer surplus in the first-period dynamic model.

2.4.2 Static Model with Pricing Decision

When the firm chooses prices instead of quantity, it runs the risk of pricing itself
out; specifically, if the firm chooses a price greater than 1 and the realized demand is
low there are no consumer willing to pay that much. The problem to be solve by the

firm for this setting is:

maxE(7r );zyby—c](—2—2 p)+(1—7)[p—c]max{0,1—p}. (2.47)

There are two possible solutions for this model, depending on if the firm charges a

price greater or less than 1. If p < 1, then Equation (2.47) is:

mng() ();1=7[P- c](——— gp)+<1—v)(p—c>(1—p). (2.48)

When Equation (2.48) is maximized with respect to prices, the following pricing level

is obtained
p1‘(2+2c_7c)
’ 2(2 -- r)

 

(2.49)

37

With a price of pi, the associated expected production level and expected profits are

calculated as the weighted average used in the previous models, this results in

1 2-26+2’)’C

 

E(q). = ———4 (2.50)
— c 'c2
19(2):} = (2 a: :77) l . (2.51)

If p > 1, then Equation (2.47) reduces to maximizing only in the high demand since

the price exceeds the reservation price:

mlaxE(7r); = 7(1) — c) (2—1’) . (2.52)

Therefore, the firm only earns profits in the case of realizing the high state of demand.

The optimal outcome for this scenario is:

 

 

P3 = 2:6 (2-53)
Ea): = “24“” (2.54)

the associated expected profits and expected consumer surplus are defined as:

E(7T)32 =

p

 

(2.55)

13(05);2

 

(2.56)

As was previously stated, the firm only earns profits if the high demand is realized.
Moreover, this result also extends to consumers, since a price greater than one is too
expensive for them, their optimal decision is to not purchase the good, producing no
benefit or surplus. The following Lemma compares the firms earnings when it sets

either prices or production levels.

Lemma 2.14 With an unknown vertical intercept the firm earns more profits when

38

choosing production levels instead of prices if c < Ti? (2 + 27 — \/ 27 + 272).

Proof. Comparing the associated profit functions for each case we get that E (7r);

is always equal or greater than E(7r);’,1 for any values of c and 7. Comparing ea:-

pected profits associated with the two pricing options, it is obtained that E (7r);1 >

E(rr);2 (=2 e < 5—}7 (2 — 7 - V27 — 72) = c1. Finally, comparing the profits from
the quantity-setting model with the profits obtained with a price greater than 1, yields
E(7r): > EM);2 4:) c < -2+1—7 (2 + 27 — \/27+ 272) = c2. But clis less than c2, as
shown in Figure (2.4), so profits for the quantity-setting firm are always higher when

the marginal cost is less than c2. I

The preceding Lemma compares profits between a price and quantity-setting firm.
In contrast with Lemmas 2.4 and 2.5 from Section 2.3, switching the uncertainty from
the reservation price to the saturation quantity does not reflect the same change in
terms of choosing prices versus quantities. That is, when the uncertainty lies in the
vertical axis, the firm is better off choosing prices instead of the production level.
But when the firm faces uncertainty in the horizontal intercept of the demand for
its good, it is not always better off choosing quantity instead of prices. With high
marginal costs (approximately .8) and almost any probability (7 > .2 ), the firm is

better off charging a price greater than 1 rather than choosing quantities.

2.5 Conclusion

This study provides comparisons amongst three different models where the firm
can choose either price or production level in a two—period model. It is assumed that
demand in the second period is related to sales in the first period. This connection

comes about in one of two ways: either there are network externalities in consumption

39

across periods, or the good is storable so that consumers can purchase in either period,
regardless of when exactly they wish to benefit from the consumption of the good.

In addition to the previous assumption, the firm has uncertainty about the demand
for its good. This uncertainty is modelled in three different cases. When the firm
faces two possible parallel demand functions, the firm and consumers are indifferent
if the firm chooses prices or quantities. With a storable good, welfare from the static
model is greater than first period welfare of the two-period model. If a network effect
is assumed instead of a storable good, first period welfare from the dynamic model is
greater than welfare from the static model since the firm will overproduce in order to
increase second period sales.

A different setting presents the model when there is uncertainty in the saturation
quantity (i.e. unknown horizontal intercept). The same result is obtained as in the
first model except that the firm and consumers are better off when the firm chooses
prices instead of quantities. This is because the optimal price for a price setter firm,
under uncertainty, is the optimal price for each of the possible states of demand.
Whereas, the optimal quantity that maximizes expected profits is not the optimal
output for any of the possible demand functions. Consumers are also better off when
the firm sets prices, and the gain in consumer surplus is even greater with the presence
of a network effect.

Finally, a third model is solved where the uncertainty lies in the reservation price
allowing for the possibility that the firm be priced out of the market. This uncertainty
is modelled with an unknown vertical intercept. In this case, the firm is better
off choosing quantity instead of price except with high marginal costs, since the

repercussions for overproducing are not as severe.

4O

Figure 2.1: Uncertainty in the Intercept with Parallel Demands

A
P

 

 

V

41

Figure 2.2: States of Demand with Uncertainty in the Saturation Quantity

 

 

1/13 1/5 C:

42

Figure 2.3: States of Demand with Uncertainty in the Reservation Price

 

 

 

43

Figure 2.4: Regions for Profits with Unknown Horizontal Intercept

 

 

 

 

1
0.81: 7?,
0.53
c ytq
0.43
. f?
0.2: q
0 ""072' ' ' 11'4' ' Pals ' 'Pofsfi I '1

gamma

44

2.6 Appendix

Derivation of Equation 2.6-2.8. Differentiating Equation (2.5) with respect

to output and setting it equal to 0, it is obtained that:

3%»; [<2 — q — c)q + 6r2(2)l+(1- r)[(1— q — c)q + 672(1)] =

 

 

aiqi’[(2-q-C)q+6Mq) 9—6) ] +(1-7) [(1-q—C)q+6

3A(2—c)2
7[2—2q—c+53 2

 

 

+](1—7)[1—2q—c+6%3(lgc) ] =0.

After some algebra, qd is obtained. Equation (2.7) is defined as 7 (2 — qd) + (1 -— 7)(1—

qd). Finally, expected profits from Equation (2.8) are calculated as:

 

 

Mari? — er] A(qdru — or] _

+(1‘7) [(1—Qd—C)Qd+5 4

E003 = 7 [(2 - qd — C)qd + 6

Derivation of Equations 2.28-2.31. Expected profits in Equation (2.27) are
defined as:

maber)“ = r[[1 -12q - C] q + 6r2(b)] + (1 - r) [(1- 59 - C)q + 6r2(5)].

q q
The first order condition after differentiating with respect to quantity becomes:

71—flJq—c-l-6g7r2—(Q) +(1—7)1—2bq-c+66L(b) =0.
6q 5‘1

After rearranging terms, ql is obtained. Following the same approach as before, the

expected price is calculated as a weighted average of the form: E (p)1 = 7(1 - (2(11) +

(1“ 7)(1 — 591)-

The associated expected profits from Equation (2.30) are calculated in the same

45

manner as for the static model of Equation (2.20). That is, in this case evaluating ql
in Equation (2.27).
Expected consumer surplus for the dynamic model is defined as the sum of the

consumer surplus from each state of demand weighted by its associated probability::

1 — - 9+(1— )5
'2‘ oq [1 - (1 — bqflqldq + “71’ fo‘“[1 — (1 — bq1)q]dq = (47—7243

46

3 Price-Quantity Commitment Under Demand Un-

certainty

3. 1 Introduction

A firm facing uncertainty in the demand for its good might prefer to choose
either quantities or prices in order to earn higher profits. But in some cases they
have to commit to both variables in advance which brings about different issues for
the firm to consider. These issues, or limitations, arise from the flexibility that is lost
by having to choose a price-quantity combination; making it less likely to achieve a
market clearance for its goods, as is the case when only one of the two variables is
chosen.

Consider a firm launching a new advertising campaign in order to attract more
consumers; if the campaign is successful there will be higher demand for the product,
if the campaign fails current demand for the product will be retained. However, in
many instances, the firm must choose the price and production level of the product
before realizing the effect of its advertising campaign. The limitation of having to
commit to production and price before entering the market can be explained by
various factors: the firm might have to include the price and production data in
the advertising campaign or market studies; the dealers of the product may have to
make an inventory decision that depends on the price (due to budget constraints)
before they realize the demand for the product; or, the firm might have to submit a
forecast of assets or future sales before getting the results of the advertising campaign.
As such, it faces the risk of overproducing if the campaign fails, or underproducing
and then losing profits if the campaign is successful but the selected price-quantity
combination was not adequate.

Another example is a car dealer that offers a new model and is not sure if it is

going to attract consumers with high or low valuation for the car. The dealer has

47

information about the tastes of consumers and their willingness to pay for the car,
including how likely each state of demand is, but it must choose the price and quantity
of cars to be kept at the dealership before knowing what kind of consumers are actually
going to purchase it. After pricing and ordering the number of cars, consumers will go
to see it, and the dealer will learn which of the two types of consumers are purchasing
the vehicle.

Following the structure presented in Chapter 2, the same three models are going
to be used. The first model presents a uniform case where the possibility of higher
demand exists. In this case, consumers could purchase more of the good even at
higher prices. The second model presents uncertainty in the reservation price that
consumers are willing to pay. The third model’s uncertainty lies in the saturation
quantity, so that the firm does not know the amount of consumers willing to buy its
product at a specific price. Instead of assuming a storable good or network effect
in the second period, as the previous chapter, the models are solved when the firm
has to choose a price and production level combination before knowing the realized
demand. Section 3.2 solves the model where the two possible demands are parallel.
Section 3.3 assumes uncertainty in the reservation price and solves the two-variable
model providing comparative statics with its one-variable model counterpart. Section
3.4 assumes instead uncertainty in the saturation quantity and follows the same pro—
cedure. Section 3.5 solves the model in a two-period framework and compares it with
Section 3.3. The main finding is that in the two-period framework, overproduction is
more likely since there is the possibility of selling the leftover in the second period.
In Section 3.6 a different variation is presented in which the static model is solved
assuming the uncertainty is an additive term with an uniform distribution. Section

3.7 summarizes the main results.

48

3.2 Uncertainty with Uniform Demands
3.2.1 Price and Quantity Commitment

As was defined in Chapter 2, when the uncertainty lies in the intercept of the
demand function, a firm has to choose either its production output or price for its
product under a high or low demand as presented in Figure (2.1).

For the "newsvendor problem" from Equations (2.2) and (2.3) from Chapter 2,

the optimal outcome is:

 

. 1—c+7
q — 2 (3.1)
, 1+c+7
p 2 (3.2)
which yields profits of
. _ (1 - C)2 7 72 7c
7r — 4 + 2 + 4 2 . (3.3)

As was explained in Section 2.2, the result from Equation (3.1) is indifferent if the
firm chooses prices or quantities, since there is no change if the variables are switched
to the other axis. Using Equation (3.1) as a benchmark, the next section solves the

same model when the firm commits, in advance, to a price-quantity combination.

3.2.2 Price-Quantity Commitment

The problem of committing to a price-quantity combination, as presented above,
is solved in two steps. First, maximizing expected profits with respect to quantity
q, assuming a given price p. Subsequently, in the second step, solving to find the
optimal price.

Figure (3.1) shows a graphic representation of the high and low state of demand.
If the firm chooses the price-quantity combination (pA, qA), the firm gains no profits

if the low state is the realized demand; that is, because the associated price pA, is

49

greater than 1, it is too high for any consumer to purchase the product. If instead
the high state is the realized demand, then the firm earns revenues of 1),; * qA and

profits of (PA - c)qA.
Expected profits are defined as:

(p—dq qsl-p
EM): 7pq+(1-r)pmaX{0.1-p}-cq if 1-p<q<2-p
rp(2-p)+(1-r)pmaX{0.1-p}—cq (122—p

(3.4)

Initially, the firm could choose any price-quantity combination in the space, either
in-between or outside of the two demand functions since there are no restrictions for
the quantity or price. Each branch of Equation (3.4) represents the expected profits
for the firm if it chooses a quantity within the defined intervals. The first branch is
the profit that the firm obtains if it chooses a quantity less than or equal to the low
demand. Since expected profits are linear and increasing in q, as shown in Equation
(3.5), not producing at its greatest possible value results in a shortage of production.
As is shown in Lemma 3.1, this branch reduces to the problem solved by a firm under
certainty with the low state of demand. Similarly, the third branch is also linear but
decreasing in q, in which case, the optimal choice of quantity is q = 2 — p. The firm
cannot sell more than q = 2 — p, since producing more than that increases costs but
not revenues.

The first term of the second branch of Equation (3.4) is the expected revenue
of choosing a quantity between the two demands, if the realized state is the high
demand which happens with probability 7. If the realized state is the low demand
with probability 1— 7, implying that the firm overproduced, then there are two cases:
first, the price associated with the quantity chosen is less than 1, resulting in revenues
of p(1 -— p); or second, that the price associated with the quantity is greater than 1,

resulting in zero revenues since no quantity is sold at that price. In either case the

50

total cost is the quantity produced times the marginal cost. Notice that in the first
case the firm ends up with a leftover of q — (1 — p), and in the second case since
nothing was sold the leftover is q (all that was produced). Because this is a static
model, the leftover is disregarded for now (see Section 2.5). The following first order

conditions with respect to the quantity to be produced are:

P-C (Isl—P

arr

EI- 7p—c if 1—p<q<2—p (3-5)
—c 922-17-

The second branch of Equation (3.5) could be positive or negative depending on
if 7p — c > 0. Figure (3.2) gives an interpretation of each case. If 7p — c > 0 (Case
A), the maximum is attained at q = 2 — p. If 7p — c < 0 (Case B), the maximum is
attained at q = 1 — p. Because of the linearity in quantity, the maximum is attained
at one of the two states of demand. Hence, the firm is not going to choose any
price-quantity combination off one of the two demand functions.

The model has three possible solutions: one under Case A, and two under Case B
depending on if the given price is greater or less than 1, as implied in Equation (3.4).
As previously mentioned, Case A is the regular monopoly solution under certainty
with low demand. The following three Lemmas give the optimal solution for each of
the three cases.

Lemma 3.1 states the condition necessary for the monopoly outcome of the low
state of demand with certainty (fim, 21‘...) to be the Optimal price-quantity combination.
Notice that this is identical to the problem solved by a firm with the low state of

demand and certainty.

Lemma 3.1 The outcome of the low state of demand with certainty is the optimal
price-quantity allocation that maximizes expected profits if c > £3-

Proof. As shown in Figure (3.2), if 7p — c < 0, then q = 1 — p. Maximizing the

51

first branch of Equation (3.4), the point (13..., a...) is obtained which gives profits of
7r... = i-(l - C)”. For 1’3... to be a maximum, the condition 7p — c < 0 must hold which

is true with the assumption c > 53:. I

For Case B, and a price less than 1, we get the following result from the high state

of demand.

Lemma 3.2 The price-quantity combination p1 2 141—3411, 21] = -3_%1 maximizes
expected profits if W > c when a price less than 1 is chosen

Proof. Rewriting the second branch of Equation (3.4) we get: 7p(2 — p) + (1 —
7)pmax{0,1 - p} — c(2 — p). Assuming 1 — p > 0 it reduces to: 7p(2 — p) + (1 ——
7)p(1 — p) — c(2 — p). Maximizing with respect to prices and evaluating at the high
demand the combination 131,51 is obtained. For 51,61 to be a maximum, the following
two conditions must be satisfied: 7 p, — c > 0 (=> ¥jfi > c. The condition 351%?)
is increasing and convex in 7. Notice that $1 S 1 => 0 + 7 S 1.

The expected profits associated with this price-quantity combination are if, =

il(1-c)2+rgl+%(1+C)—C- '

The price-quantity combination (fibril), obtained in Lemma 3.2, is only feasible
when the marginal cost is relatively small compared to the probability of the high
demand. Since “+312 is increasing and convex then with a high probability of the
high demand, (51,61) is the optimal outcome with any marginal cost.

Notice that p, is bound by the price of each of the states of demand under certainty.
On the contrary, a, is greater (or equal) than the output when the high demand is
realized with certainty. This is because a high price implies zero profits in case of
realizing the low demand, whereas a large output would only imply a surplus without
penalizing the expected profits of the firm.

The first term of the expected profits, fil, is the monopoly profits with certainty

from the low demand. Hence, as is shown below, when the probability of the good

52

demand is high enough, if] is always greater than if"...
The remaining possibility in Case B, assumes a price greater than 1. Now the

firm is taking the risk of selling nothing if the realized state is the low demand.

Lemma 3.3 The price-quantity combination p} = 212?, (72 = big, is the optimal
outcome that maximizes expected profits if 27 > c and a price greater than 1 is
chosen.

Proof. From 7p(2 — p) + (1 — 7)pmax{0,1 — p} — c(2 — p) using p > 1 and
q = 2 — p the second branch can be rewritten as 7r = (7p — c)(2 — p). Hence, ii} =
arg max {(7p — c)(2 — p)} . For 1’52 to be a maximum, the condition 7 fig — c > 0 must
be satisfied, which holds with the assumption 27 > c. Assuming 27 > c with 7 > .5,
for example, does not impose any condition on c but for other values it requires a

small c for a small 7.

For this outcome the associated profits are ffg = 7 + 31,778 —— c. I

The price in} obtained in Lemma 3.3 requires a strictly positive 7 (there is at least
a small chance of lying in the high demand) and is upper bound by price of the good
state of demand with certainty.

The price-quantity combination with a price less than 1 equals (52, a.) when 7 = 1,
but since )3, _<_ 1 and 1’52 2 1, this case is only feasible when c = 0.

Hence, depending on which of the two assumptions holds, a different solution
can be found. Nevertheless, the regions that these assumptions represent overlap
each other so the profit functions must be compared in order to determine the global

maximum.

53

Lemma 3.4 The relevant intervals for each profit function are:

A 2 . .
7n => c6 (OS—(273%)) 2f 7S7'andc€(0.-7+\/7) 2f727‘ (3-6)

7T2 => 0 E (—7 + J7, min{c1, 1}) and 7 Z 7"

, 2
7(7_+__),1) if 737'“ anch(cl,1)if 727'

3?", => cE
(2-7

 

where7‘ = arg {—7 + \/— = @3722} = .6—4x/2 z .34 andc1= Ti? (7 —- \/7 — 472 +473).
Proof. Comparing each of the profit function with each other we get the following

conditions:

7n > fig=>c<—7+\/7

7(7+2)

2-7

— —4’~’ 43 + —42+43

7 W1 7+7 andc>Cg=7 \/71 3 7.
—’Y _

7T1 > ffm=>C<

 

 

 

 

7T1> ’Tfm=>C<CI=

The result is obtained comparing the conditions (3.27}:2 > c and 1 — 7 > c) for
each price-quantity combination to be feasible, with the three conditions obtained

above. I

From Lemma 3.4 the global maximum can be calculated depending on the values
of the marginal cost and the probability 7, as illustrated in Figure (3.3). For costs
greater than approximately .22 and any 7, the choice is between the profits from the
low state of demand (fim), realized with certainty, and the profits obtained with a
price greater than 1 (fig). The constant 7‘ is the threshold between the different
profit expressions. Notice that at 7 = 7" and c = .24 the three profit functions are
equal

When 7 < 7*, ”rig is not an option, this is because the good state is not likely

enough for the firm to take the risk of charging a price greater than 1, not even with

54

costs close to 1. For 7 between 7‘ and .5 all three profit functions are feasible and the
choice depends on the marginal cost: low costs (less than approximately .22) imply
it], middle costs imply ffg (between .22 and c1) and high costs result in the profits of
the low demand with certainty. For values of 7 above .5 (the high state of demand is
more likely), the choice is only between ffg and 7’71: with low costs it is optimal to use
a price less than 1 (1’51), for large costs it is optimal to charge a price greater than 1
(fig). ’ng is always greater than ifm, making the difference nearly irrelevant when the
marginal cost is between .45 and .55 and probabilities near 7“.

Finally, a price equal to 1 is never an optimal solution. If the realized demand
is low the firm has zero profits and if the realized demand is high the firm makes
profits but less than the profits of the high demand with certainty. Hence, the result
from maximizing profits when the high demand is realized with certainty, gives higher
expected profits than a price equal to one, regardless of the probabilities of the states
of demand.

In order to compare in which scenario consumers are better off, consumer surplus
is calculated in two different ways. Since the market does not clear when the firm
chooses a price-quantity combination there are two possible cases. If the firm over-
produces then consumer surplus has the usual triangle area formula; but if there is
underproduction, then only some consumers will get the good. Under efficient ra-
tioning, consumers with the highest willingness to pay will be able to purchase the
good. Under least efficient rationing, consumers with the lowest valuation will have
access to purchase the product. Hence, efficient rationing consumer surplus (fl) ,
is always greater than least efficient rationing consumer surplus (Q51) , since there is
more satisfaction for the consumers with the highest valuation.

The following Lemma compares the expected consumer surplus associated with

each of the price-quantity combinations found above.

Lemma 3.5 Expected consumer surplus is the greatest when the firm chooses the

55

price-quantity combination (1’51, (’11) .

Proof. Expected consumer surplus is calculated as the consumer surplus of the asso-
ciated price-quantity combination in the high state multiplied by 7, plus the associated
consumer surplus of the low state multiplied by 1 — 7.

The consumer surplus associated with the price-quantity combination (fim, 21‘...) is
E(cs)... = gfo”m(2 — a. — q)dq + 9—3120 — 1’5me = 3,-(1 — c)2 + g (g + 5c —-,1-c2).
For the combination (pl, (7,) , the associated expected consumer surplus is E (Eh =
2(2-2) swam—21> 21::(1—c>2+1%1+i(1—§).

The first term of E(CS)1 and E(CS)m are the same; however, for E(CS)1 to be
feasible, c must be less than 152%. For this interval E (3)1 2 E(CS)m.

For the price-quantity combination with a price greater than 1 (fig, fig), the asso-
ciated expected consumer surplus is E (CS)g = %(2 — figfig = fiQEC-fi which is always
less than E(CS)1. I

When the firm chooses (1’)] , 61), it implies a price less than 1 and a production level
above any of the other possible outcomes, making it the best one for consumers. If
least efficient rationing is assumed instead, it would only make the difference between
E(Cfih and Egg)... even larger, since E(fi)... Z E(Q§)m. Moreover, E(fih =
E (CS )1 since all consumers are able to purchase the good in both scenarios, making
the consumer surplus under efficient and least efficient rationing identical. Expected
consumer surplus behaves similarly when a price greater than 1 is assumed instead,
since all consumers are able to purchase the good only in the case of realizing the low
demand.

Lemma 3.6 compares the results when the firm chooses a price-quantity combi-
nation versus when the firm commits in advance to only one of the two variables as

solved in the previous two sections.

Lemma 3.6 When a firm commits only to production levels under demand uncer-

tainty and prices are determined by the market, profits are always higher than when

56

the firm commits to a price-quantity combination before realizing the state of demand
except when 7 6 {0,1} or c = 0.

Proof. Since the first term of 1r‘ is if", = §(1 — c)2 and the sum of the other three
terms is always positive, 7r“ is always larger than the profits of the low state of demand
with certainty except when 7 = O { then they are equal).

Comparing 77‘ with frl we obtain:

A 1—2
f_m=<;)

 

fi_r_£:fl31 V r
4 2 2

'7
+ — +
2
which is always positive except when 7 = 1 and c = 0.
Finally, rr‘ is always greater than fig, except when c = 7+ \/7 but by the definition

of 7f2 that is not feasible, hence they are equal only when 7 = 1. I

This Lemma follows the intuition of an optimization problem of choosing one
variable versus choosing two, that is, since the firm has to commit only to one variable,
it has more flexibility to adjust to the uncertainty and the risk is not penalized in
profits as in the two-variable model. Therefore, expected profits in the "newsvendor
problem" setting gives higher profits than when the firm commits to a price-quantity

combination.

3.2.3 Comparative Statics

For the static model solved in the previous sections, we obtain the following

limits:

 

 

1i A 1+6 A
m = = m
7—+ 1 2

3—c
lim" = =1+Am
WfiOQI 2 q

57

When 7 approaches 0 (the low state of demand is extremely likely), if, goes to
p)... However, a, is the output when the low demand is realized with certainty plus 1
because it comes from the monopoly price of the low demand with certainty evaluated
on the high demand. Hence, when the firm knows that the low demand is going to be
realized, its best strategy is to produce at the certainty price as it would with perfect
information. The overproduction is feasible since there is no leftover cost, therefore,
if the high demand is going to be realized, the firm has an output level high enough
to satisfy consumers, but if demand is low the firm wastes resources at no extra cost.

When the high state of demand is most likely (7 ——2 1), p, and fig converge to
the price under certainty from the high demand (13%,). Also, 31} and Fig converge to the

high demand quantity with certainty ((33,).

 

 

2 + c
1. A = : . A : /\2
712111)1 2 712311)2 m
. A 2 — c . A /9
315.11% _ 2 — 71.1.1232 _ m

where 13?”, (7,2,, is the optimization result from the high demand under certainty. For
this scenario, if the firm knows that the high demand is going to be realized with
certainty, then the best outcome is to produce at the maximization outcome of the
high state of demand with certainty. This generates a level of profits as if there was
no uncertainty.

Figure (3.4) presents the results obtained above. Prices are bound by the price of
each demand function with certainty. Quantity is bound between the output of the
high state of demand with certainty and the output level evaluating the monopoly
price of the low demand in the high demand. Figure (3.4) shows this behavior where
M1, Mg are the outcomes with certainty for the low and high demand, respectively.

With low costs and a high probability of the high demand, the firm produces even

above the optimal output of the high demand with certainty. Producing at 1 + 21‘...

58

gives the firm the chance of making profits even in the case of realizing that the ‘true’

demand is low.

3.3 Uncertainty in the Saturation Quantity

Assume a profit maximizing firm chooses either the price or production level of
the good it sells. In a traditional framework with full information, following this
decision, the market determines the remaining variable and clears.

The demand function for the good produced by the firm is assumed to be linear,
and for simplicity in this model with a normalized slope. It can be either a high (good)
state of demand, p = 1 - %q, with probability 7, or a low (bad) state, p = 1 -— q, with
probability 1 - 7. Hence, uncertainty lies in the slope of the demand function. This is
the same model that was solved in Section 2.3, but here it is evaluated at b = 1, (_i = %.
As in the previous model, it is assumed that the firm has a constant marginal cost
c 6 (0,1). Comparing it to the previous model, notice that unlike before the price is
never going to be greater than 1; that is because the firm knows the reservation price
(no consumer would buy with a price greater than 1). The quantity axis resembles

the q axis from the model of the previous section, that was presented in Figure (2.2).

3.3.1 Price versus Quantity Commitment

If the firm sets quantity, then its maximization problem is the following:

mnge): = 7(1-%)+(1—7)(1-q)- cq. (3.7)

As before, the market will clear since the firm is only choosing quantities and the

price will be determined depending which of the two states of demand is realized.

59

The optimal quantity for the problem defined in Equation (3.7) is:

l-c

2 _ 7. (3.8)

 

(1b:

Notice that Equation (3.8) has the same solution as Equation (2.13) evaluated at
b = 1, b = 5 For this output the associated expected price is E(pb) = l—g—C. It is
worth recalling that the optimal expected price does not depend on the probability of
the states of demand. That is because the optimal price associated with q, for each
of the states of demand is the same, hence, its expected value is itself. This result
is further corroborated when the problem is solved for a price-setting firm, simplified

as follows:

maxEUr); = 72(1—p)(p - C) + (1 - 7)(1- p)(p - C) = (1+ 7)(1 - P)(P - C) (39)

P

where the optimal price is pg, = l—Jg—C— and the associated expected output is E(qb) =
l—g—‘(l + 7). Again, the expected quantity, E(qb), is greater than the Optimal output,
qb, from the quantity-setting problem. This is because, as was previously explained,
it is associated with the optimal price from either state of demand, while the optimal
output is not the optimal quantity level for either of the states of demand but for
maximizing expected profits. Lemma 2.5 from Section 2.3 still holds since the only
difference is that values have been assigned to the slope parameters. Hence, E (7r); =
%(1 — c)2(2 — 7) Z E(rr)2. The results obtained are used as a benchmark in order to

compare a more flexible model where the firm can choose one of the variables versus

a. model where the firm must choose both variables before realizing the true demand.

3.3.2 Price-Quantity Commitment

In this section the firm has to set a price-quantity combination before knowing

the true demand. Equation (3.10) presents the associated expected profits with this

60

optimization problem.

w-dq qSI—p
1E7W): 7pq+(1-7)p(1-p)-cq z'f 1-p<q<2(1-p)
7p2(1-p)+(1-7)p(1-p)-cq q 2 2(1—10)

(3.10)

In contrast to Equation (3.4), the firm will never charge a price greater than 1
and does not have the risk of ending with zero profits. The first branch of Equation
(3.10) is still the same as maximizing the low state of demand under certainty. The
second branch, represents the profits that the firm would earn if it chose an output
level between the two possible demand functions; that is, with probability 7, the firm
earns pq, and with probability 1 — 7, the realized state is low so that the firm earns
revenues up to the point where the market clears in the low demand. With either
state of demand the firm has cost, c, for each unit produced.

The first order conditions of Equation (3.10) are identical to Equation (3.5), hence,

the firm has two possible solutions, one when 7p—c < O, and the other when 7p—c > 0.

Lemma 3.7 If e > %(7 + 1), the firm achieves maximum expected profits at the
optimal outcome with certainty of the low state of demand.

Proof. When 7p — c < O, the firm achieves a maximum at q = 1 — p, hence,
the first and second branch of Equation (3.10) are identical providing the outcome of
the low demand under certainty (i.e. qf = 1%‘5, p‘{ = 3%? and r1 = i (1 - 0)?) . When

7P — C > 0, the optimal production level must lie in the high state of demand (q =

2(1 — p) ), resulting in q; = 12%:Ylfizf, p; = 3127—7131112; with associated expected profits of

.. 2
W22H§1+—1)(7+1-20)'
After comparing the two expected profit expressions it is found that 7r; 2 7r; 4:)

c2%(7+1). I

When the marginal cost is less than %(7 + 1), the firm earns higher expected

61

profits when choosing the optimal outcome of the low state of demand under certainty.
Therefore, when marginal costs are high enough, the firm will choose the outcome
associated with the high state of demand (p;,q§), regardless of the probability of
realizing it. This is because with a marginal cost close to 1 and producing in the low
state of demand, with probability 7, it is going to earn significantly lower profits since
the price is too close to the cost, and with probability 1 — 7, it is underproducing,
thereby wasting possible revenues. Whereas, if the firm produces in the high state
of demand, and the realized state is low it will earn low profits but with probability
1 — 7, it will earn greater profits.

For the allocation (p‘l‘,q{) the associate consumer surpluses under efficient and

least efficient rationing are:

 

 

 

 

_ _ . . q; _C 2
E(0—51) = (1 7X; pm‘ +7/(1-qi‘ -pi)dq= (“73:51 ) (3-11)
0
t t t 2
E(QSJ) = (1 -7)(;-P1)q1 +%(1- (131 —pi)(2- 2p} -qi') = (1;C)(3-12)

Since the firm produces in the low state of demand, with probability 1 — 7, the
market clears. With probability 7, the first qf are sold under efficient rationing,
and with least efficient rationing, the the last q; units are sold, which are defined as
(2 — 2p; — qf). E (El) implies higher expected surplus, as explained above.

When the firm chooses the combination on the high state of demand (p5, qg) then
E (fig) = E (QSQ) = E(CSg), because the firm is producing on the high state
of demand, therefore, there is no underproduction in either case (all consumers can
purchase the good), but there is the possibility of the firm ending with a leftover

stock.

(1 —p;><2 —p;)+ (1‘ 7)(1—pa)? = (7+ 1 ‘ 26) (3.13)

E(CSg) = 8(7 + 1)

 

 

NIQ

62

Notice that E (CSg) equals half of its associated profits as is the usual case under
certainty.

Comparing consumer surplus we obtain the same relationships as comparing ex-
pected profits. If e S %(7 + 1), consumers earn greater consumer surplus when the

firm chooses the price-quantity combination in the high state of demand.

3.3.3 Comparative Statics

In this section comparative statics are provided with the myopic solution and the
two possible outcomes when the firm has to commit, in advance, to a price-quantity
combination. Comparing outputs, we show that the myopic solution equals q; and
is less than q; for any marginal cost (and equal with c = O). This corroborates the
understanding that when the low state of demand is most likely, the firm will produce
in the low state of demand, and q; approaches the myOpic output as the low demand
becomes more likely.

l-c g>1, , 1—2c
= 1m =
q1—7 0q2 2

 

 

lim qb =
7-—20

When the high state of demand is going to be realized almost with certainty, the
myopic output is greater than the associated output when the firm has to commit
to both variables. Therefore, when the firm has to commit to both variables, it will
underproduce when the high state is most likely because of the loss of flexibility in
the model.

1. = — > . t: 1'
Wgnlqb 1 c _ 7115,1032 <11

Subsequently, prices follow a similar but much simpler behavior when the firm sets a

price on the high demand.

 

. ,, 1+2c

7_, 2 2 Pb = 191 = 71135111);

63

When '7 —-» 0, the price associated with the high demand (p5) is greater, however,
this price has lower associated profits, as was demonstrated in the previous Lemma.
Notice that when the high state of demand is most likely all prices are the same, in
which case, the highest output is associated with the highest profits and, as should

be expected, it is the benchmark output.

3.4 Uncertainty in the Reservation Price

This section uses the same model introduced in Section 2.4 where there is a high
state of demand, p = 2 — 2q, with probability 7, and a low state of demand, p = 1—q,
with probability 1 — 7. As before, if the firm overpricas it risks earning zero profits.
In particular, with probability 1 — 7, the firm earns no profits if the price is greater
than 1 since the low state of demand is realized. A representation of this model is
shown in Figure (2.3). In contrast to Section 2.4, the model is solved when the firm
commits to prices and production levels before knowing the realized demand instead

of assuming a dynamic model with storable good or network effects.

3.4.1 Price versus Production Commitment

As in the previous sections, the model where the firm chooses only prices or
production levels is used as a benchmark for comparative statics. If the firm chooses
production, then the optimal outcome is qb = é—Ffi, pb = 113;? When the firm sets
prices instead, there are two possible solutions, depending on if the price is greater
or less than 1. The solutions are p}, = W if the price is less than 1 and pi = 2—35
if the price is greater than 1. Since a price-setting firm generates greater welfare, the

prices pg and p3 are going to be the benchmark.

64

3.4.2 Price-Quantity Commitment

This section solves the model that the firm faces when committing in advance
to both variables. It is first solved finding the optimal output in terms of prices and
then finding the optimal combination. For this model expected profits are defined as
follows:

E(vr)= (p-C)q if qSl-p (3.14)

7pq+(1-7)pmaX{0,1-p}-cq 1—p<q<Q;—@-

Since the first order conditions are the same as Equation (3.5), there are three possible
solutions: one when 7p — c < 0, with associated profits, 7r“, and two with 7p — c > 0
since the price can be less (NI) or greater (a3) than 1. The following Lemma provides

the conditions for each case.

Lemma 3.8 The relevant intervals for each profit function are:

7r 4:) c527

,7
7r‘ (:1) CE 27,—)
1 ( 2-7
.a ’7’
4:) >——.
m 6—2—7

Proof. From the first order conditions in Equation (3.5), if 7p — c < 0 then
maximum profits are achieved at q = 1 — p, in which case the first and second branch

of Equation (3.14) are identical. Maaimizing (p — c)q, with respect to prices subject

to q = 1 —p, yields p" = 3%, q‘ = lg—C, and associated profits ofvr“ = %(1- c)2.

If instead 7p — c > 0, then the maximum is achieved at q = 2—3—3. If the price

is greater than 1, the second branch of Equation (3.14) becomes (7p - C) (2—3‘?) ; an

expression that is maximized at p} = 273%) with the following associated output, qf =
_ __C i. 2 . .
273%? and profits 0f7r1= {@1773 (2 — c) —2%Yc(1 — 7) . If7p—c > 0, and the price is

65

less than 1, the problem to be solved becomes 7p (2—23)+ +(1—7)p (1 — p)—c (2—33) , which

21—c #

. t -. C * 2 . . ,t
results in p2 = age, q2 = 47 , r2 = 31; (27 — ). For the condition 7121 — c > O to
hold, the restriction c < '2‘}, is needed. Also, for 7p; — c > 0 to hold, it must be true
that c < 27. But 7r; 2 7r; 2 7r‘; hence the conditions are determined depending on

in which interval each profit function is relevant. The proposed results are obtained

' , _Y_
sznce 27 2 2_7. I

This Lemma defines the interval for each of the possible profit functions, presented
in Figure (3.6). Notice that with low probabilities and high marginal costs, the firm
is better off choosing the outcome of the low state of demand with certainty. As the
low state of demand is more likely and there are high costs, the firm is better off not
taking the risk of overproducing. When the probability of the high state of demand
is high, the firm tends to produce in the high state of demand with a price less than
1. Nonetheless, for certain values in-between the firm is better off risking production
in the high state of demand with a price greater than 1, since the expected profits
with the high demand overcome the risk of no profits if the low demand is realized.

In order to measure the changes in consumer surplus, the same procedure as in

Section 2.5 will be used. The consumer surpluses associated with (p‘, q“) are:

 

 

 

E(CS*) = (1.. —p)2+7/(2—2q—c)d =(1-7)8(1—C)2+7(1;62)' (3.15)
0

With a probability of 1 — 7, consumers receive the usual consumer surplus from
the low state of demand. With probability of 7, they earn less than the consumer

surplus of the high demand since the good is not sold to all consumers.

 

 

2-2q' 2 2
E(Q‘)= (1—7)(21—p) +7 / (2—2q—c)dq= (1—7)8(1—c) _7(11_(;c) +143

(3.16)

66

As expected, consumer surplus with least efficient rationing is less than the con-
sumer surplus associated with efficient rationing. This is due to an associated smaller
area representing surplus as the good is sold to consumers with the lowest valuation
in the case of high demand.

For the two solutions in the high state of demand, there is no need of rationing as
all consumers are able to purchase the product. Expected consumer surplus is defined

as:

 

(1-7)(1-p'{)2 v<2—p1)2__(2_-_9& 19:32 (3.17)

E(CSI‘) = 2 + 4 —16(2 —7) + 2(2 -7)
E(ng) = 7(2 2195)2 = (”lg—76V. (3.18)

The first expression represents consumer surplus when the firm sets a price less
than 1 in the high demand, hence, there is a positive surplus for each state of demand.
On the contrary, when the firm sets a price above 1, it will only sell its good if the

realized demand is high, which happens only when the marginal cost is large enough

(c2737).

3.4.3 Comparative Statics

This section presents comparisons for the prices and quantities obtained when
the probability of the statas of demand approach the limits. When the high state is
going to be realized with certainty, the price in the high state of demand is greater

than the low demand price which equals its benchmark equivalent.

. ,, 2+c 1+c .
11m 1: 4 _<_ =P‘ = 111101);

7——' 2 7

 

As shown in Figure (3.6), when 7 ——» 0 the firm is better off choosing the allocation

with p‘, except when c = 0, due to all profits being equal. If the probability ap—

67

proaches 1, making the high state of demand more likely, then the firm is indifferent
as the optimal price of each state of demand is the same, even when the firm only

chooses a price.

However, the firm earns higher profits with a price less than 1 in the high state of
demand, as was proven in Lemma 2.8. For this case, expected profits are closest to
the associated profits when the firm only chooses a price less than 1 and quantity is

determined by the market.

3.5 A Two-Period Model

This section extends the model presented in Section 3.3 to a two-period model
where the firm is able to sell the leftover production from the first period in the
second. For simplicity, demand in the second period is assumed to be the high state
with certainty. The leftover is realized at the beginning of the second period, hence,
the firm solves the following problem in period 2: mpazc pq -— c(q — e) subject to
q = 2 — p, where e is the leftover from the first period and p,q are the price and
quantity demanded in the second period. Since e units have already been produced
in the first period, the firm only has to produce q — e units. The optimal outcome for

this problem is:

 

 

 

if? = 2 (3.19)
2 _
q? = C (3.20)
2
which results in associated profits of
A 2 — 2
7T2(e) = ( 40) +08. (3.21)

68

The present discounted value of the inter-temporal profit function is:

 

a—cm+ar%m iqul—p
7[pq+6fi2(0)]+ if 1—p<q<2—p
1r = < +(1— 7') [pmax{0,1—p} +6 fi2(q — max{0,1—p”) — cq (322)
rnmz—m+vr%ml+ iquZ—p
+0-7meumJ-rl+5fim-nWMQ1-mfl-wq

where 6 is a discount factor for the profits of the second period. Following the same
approach as before, assuming a given price, the profit function is linear in quantity.

The first order condition with respect to q is:

p-c qsl—p

arr .

8_q= 7(p—c)+6c—76c if 1—p<q<2—p (3-23)
76c+6c—76c—c q22—p.

As in the previous section, the first branch of Equation (3.23) is always positive and
the third is always negative. For the second branch to be positive, the following

condition is needed:

7p—c+6c—76c>O®————7p—-—>c. (3.24)

76+1—6

Case A (producing in the low demand) from Section 3.2 now is defined when
Equation (3.5) does not hold. The following Lemma provides the optimal price and

production decision when the output decision lies in the low demand.

Lemma 3.9 The price-quantity combination fin, = 1319, gm = 17-6, is the optimal

allocation that maximizes first-period ezpected profits if > c.

__L_
267-1-2—26—7

69

Pmof. Under Case A, the optimal choice of quantity is q = 1 — p. The result
is obtained maximizing the first branch of Equation {3.22) with respect to the price,
similar to Lemma 2.1 from the previous chapter.

To assure that it is a maximum, the condition of Equation {3.24) must not hold
and that is true with the assumption of 27577273267 > c.

Resulting profits are the first period profits with the low demand with certainty
plus the discounted high demand profits with certainty as well: 5?," = i (1 — c)2 +

(2—c)2. I

.5101

Case B represents when Equation (3.24) holds, resulting in a quantity of q = 2 — p.
Again there are two possible solutions for Case B, depending if the price is greater
or less than 1. Lemma 3.10 shows the optimal combination decision restricting the

optimal price to less than 1.

Lemma 3.10 If the optimal price is less than 1, then the price-quantity combination
,3, = 113—+5, q] = 3;}: is the local maximum expected profits if % > c.
Proof. Assuming the condition in Equation (3.24), the optimal quantity is attained

at q = 2 — p. The second branch of Equation (3.22) can be written as:

7p(2-p)+(1—7) max{0,1 - p} + 5 C((2 - p) - max{0,1- p})l-C(2-P)+g (2-6)2-
(3.25)

Assuming, 1 — p > 0, Equation (3.25) reduces to:

7p(2-P)+(1-7)[P(1-p)+56[(2-p)-(1-p)l-C(2-P)+ (2—c)2. (3-26)

“>401

Maximizing with respect to the price, {51 is obtained, however, for 1’51 to be a maxi-
mum Equation (3.24) is required which holds with the condition will: > c. The

associated profits are it} = i [(1 — c)2 + 72] + %(1 + c) — 6 7c. I

70

In order to compare the outcome when a price greater or less than 1 is chosen,
Lemma 3.11 solves for the second possibility of Case B. That is, a price greater than

1.

Lemma 3.11 If the optimal price is greater than 1, then the price-quantity combi-

nation p} = W, {1‘2 = M3 is the local maximum expected profits if

__27_
67+1—6 < C.

Proof. Using the condition of a price greater than 1, Equation (3.22) reduces to:

7p(2—p>+<1—~y>6 c(2—p) —c<2—p)+§ (2-.)2.

Maximizing with respect to prices, 1’52 is obtained. For 1’52, fig to be a maximum, Equa-

tion (3.24) must hold which is true with the assumption of 5%}: < c. The associated

 

 

profits are:
52c2 62c2 6c2 76262 37602 c2
A = —- — ———- — —— — — (5 — 6.
7T2 7+ 47 2 27+ 4 4 +47 7c c+
I

Before giving an explicit solution for this model, three examples are considered
with different values of 6.

o 6 = O is the one-period model case solved in the previous section. In this
case, the firm gains nothing from trying to sell the leftover in the second period.

0 6 = 1 is the case in which there is no discount factor. The solution is:10

”#1 => c E (O, —7 + ([7) and ’7?2 => c E (—7 + fl, 1). Notice that fim is never
chosen, even though 7 could be quite small making the good state most unlikely.
However, the leftover can always be sold in the second period and will give as much

profit as in the first period since there is no discount factor.

 

10The intervals are found comparing the two profit functions.

71

o = 1 / 2. The solution for this case is:

2 4 —2 2 2 3
iil => c6 (0,min{7+7—, 7 \/7 +7 +7}) (3.27)

 

 

2 7—1

 

 

4 —2 2 2 3
’7?2 => c6 ( 7 «7 1+7 +7,min{c3,1}) and727‘ (3.28)
7.—

,2
if", => c6 (7+%,1) if7S7' anch(C3,1) if 27‘ (3.29)

 

where 7" is the solution to 7 + %72 = 7:7 (47 — 2\/272 + 73 + 7), and equals ap-
proximately .284.

Figure (3.7) shows these results where the dotted line is the solution for the model
just solved, and the solid line is the graph from Figure 3.3 (static model). As before,
the firm never chooses ifg (profits with the price greater than 1) if the probability
of the high state is less than 7‘. it} is the optimal choice when the marginal cost is
small enough. Furthermore, this latter result is almost independent of the choice of
7. Notice that 7" is .284 and in the static mode it was .34. Now the firm is more
likely to choose the high state of demand with a price greater than 1.

Another observation emerging from Figure 5 is that for the static model when the
probability of the good demand was greater than 1 / 2, the decision was only between
$1 and fig. Here, with a discount factor of 1 / 2, 7 = 1 / 3 is the threshold for the profits
of the low demand (ifm). In other words, with the static model until the probability of
the high state was 1 / 2, the outcome of the low demand with certainty was attainable.
In the dynamic model, the low state of demand is feasible up to 7 = 1 / 3.

It has been showed that the possibility of selling the leftover in the second period

increases the “willingness” of the firm to choose the high state of demand.

72

The solution for the general model is:

A . 7(r+2)
7n CE(O’mm{2(267+2—26—7)’A})

c E (A,min {C4, 1}) and 7 2 7"

 

l

l

l

 

A 76+?) . . . ..
7r CE<2(267+2—26—7) )17 7an 0 (c4 )17 7

7— \/7573 —2(5272 +7+2672 -—267+673
67—1+26—62

 

 

where A =

 

_ 2672 + 27 — 276 - \/—4736 + 673 — 472 + 6726 — 2726 + 627 + 7 + 473 — 267
— 6—2627+1+7262+267—7—26

7(7 + 2) 7 — V673 — 26272 + 7 + 2672 — 267 + 673}

 

C4

 

 

an“ =argm3x{2(257+2-25—7)’ 67~1+26—62

3.6 The Model with a Continuous Distribution

As a last extension, the model from Section 3.3 is solved using an uniform distri-
bution instead of a Bernoulli distribution. Consider the following demand function:
q‘D = g — p + 6 with e m U —%, é]. Profits are maximized in the region between
the same two demand functions from Section 3.2. Since for this section a continuous
distribution is used instead, there is not only a high or low demand, but rather the
realized demand could be any curve with a normalized slope in the interval between

the high and low states of demand.

Profits are defined as:
7r 2 max {pmin {qD,qS} — cqs} = max {pmin {% —p + e,qS} — cqs} (3.30)
p,q Puq

where qD is the quantity demanded and q5 is the quantity supplied or produced.

Assume that qs must be chosen before realizing 6. Hence, if qD > qS, the firm is

73

underproducing, which means that it could have had higher profits by increasing the
quantity supplied. If qD < qS, the firm is overproducing and ends up with a leftover
of q5 - qD. As before, the model is solved for the production level assuming a given
price, and then solving for the optimal price.

Since the stochastic term is distributed with an uniform distribution in the interval
[—%, é], the smallest possible value for the quantity demanded qD is in the low state
of demand 1 - p; therefore, if qS < 1 — p, then 1r = pqs - cqs = (p -— c)qS.

If we take the largest possible value of 6, then qD is 2 - p. Now, if qs > 2 — p,

then
3
7r=qu-cq5=p(§—p+e)-cqs- (331)

Hence, expected profits are defined as:

(17-0qu qSSl-p
Ed"): pqSPr(qS<qD)+p(%-p)Pr(qS>qD)-cqs z'f 1—p<qS<2-p
p(%-p)—cq3 (1522-10
(3.32)

where Pr(q5 < qD) = 2—q3 —p, Pr(qS > qD)=1—Pr(qs < qD) = qS+p— 1.

After some algebra we can rewrite Equation (3.32) as:

(p-C)q3 (1531—19
3 .
Ed”): %p+%pqS-p(qs)2—2p2qs+%—p3-oq5 Zf 1-p<qs<2—p
196-10-ch (1522-1)

(3.33)

As in the discrete case, the first branch of Equation (3.33) is an increasing line
with slope of p -- c. The third branch is a decreasing line with slope of -c. The
second branch could go either way so it must be maximized with respect to q to find

the optimal output. The first and second derivatives with respect to the quantity

74

produced are:

 

0E€(7r) 7 s 2 02E€(7r)
—— = -— — — —- _ = —2 -
BqS 2p 2pq 2p 0, (92q5 12 < 0 (3 34)
0E€(7r) A %p — 2p2 — c
= => = —. 3035

From the second derivative we know that this turning point is a maximum. However,
it must be verified that this maximum is achieved in the interval (1 — p, 2 — p). The
upper bound 6‘ g 2 — p always holds. The lower bound has the following restriction:
l—psreecsgn

To get the optimal price, (7 must be replaced on the second branch of the profit

function, hence, profits depend on price and cost from above:

 

_ 25p2 — 16p3 — 28pc + 16p2c + 4c2

3.36
mp ( )

6(1))

Taking the first order derivative of this new profit function we obtain:

87r(p) _ 25p2 — 32p3 + 16p2c - 4c2

=0¢=>2 2—23162—4‘~’= .7
(9p 16p2 5p 3p+ p0 c 0(33)

 

 

Two of the three roots of the polynomial are ruled out for been imaginary numbers.

The remaining third root is:

A A2/3 + 625 + 8000 + 256a2 + 25A1/3 + 16CA1/3
p = A1/3

 

(3.38)

where A = —3609602 + 15625 + 300000 + 409603-1-

 

+192\/-3c2(15625 + 300000 -- 844802 + 409603).
The Optimal price with certainty is defined as pM = 1*5—29, q“ = 35525 and 1r“ =
%(3 — 2c)2. Figure (3.8) shows the price obtained above compared to the monopoly

price under certainty. Even though the obtained optimal price if, has a complicated

75

expression, it resembles a linear function that intersects the price with certainty ex-
actly at c = .5. Another important observation is that ii is greater than pM for small
values of 0. In other words, the obtained pricing path charges a price greater than
p“ when the cost is less than .5.

Evaluating {6‘ using EQuation (3.38), the resulting quantity is:

if = (—205977602 + 484375 + 6800000 + 409600c3 + 7552M + 193750”3 +
14800cC1/3 + 1356801/3c2 + 77502/3 —14400C2/3 — 6553664 — 25602/3c2 — 2048M? —
4096c301/3 — 6401/3x/3B)/D

B = ~02(15625 + 300000 — 844802 + 409603)

C = —3609602 -+- 15625 + 300000 + 409603 + 193V3B

D = 3201/3(02/3 + 625 + 8006 + 256a2 + 2501/3 + 16001/3).

Figure (3.9) shows the obtained quantity plotted when there is certainty. The
same behavior is observed for quantity as for prices, as was shown in Figure (3.8).
Choosing to produce above the output with certainty when the cost is sufficiently
small and determining a threshold intersection at c = .5.

Evaluating the price from Equation (3.36), we attain the profit function in terms
of the marginal cost. Figure (3.10) shows this profit function, if, and the profit
function under certainty. ’7? is always higher than the profits under certainty making
the difference diminutive when 0 is between .45 and .55.

As previously mentioned, with low marginal costs the firm should produce and
charge a price greater than the price under certainty, taking the risk to achieve higher
expected profits. For example at c = 0, the firm should have a price of .78 this is
greater than pM (.75) and a quantity of .96 (qM is .75). This will lead to expected
profits of .61 which is also greater than the associated profits with certainty (.56).
However, if an arbitrarily large 0 such as .8 is chosen instead, from the graphs we can

tell that the price and quantity are lower than the monOpoly outcomes, but expected

76

profits are still going to be higher than 7r“ .

3.7 Conclusion

When a firm faces uncertainty in the demand curve and must commit to a price-
quantity combination before realizing the true state of demand, expected profits
are higher when the pre-commitment involves only the production level, as in the
“newsvendor problem”, assuming a possible uniformly higher demand. The excep—
tion to this is when one of the possible states is going to be realized with certainty
or there is no marginal cost. In the pre—commitment case, expected profits are al-
ways greater than what a firm would earn with the low state of demand and certainty.
Moreover, the optimal price lies between the monopoly prices of each state of demand
under certainty. The optimal production level lies above the production level with
certainty of the low state of demand, but can be much greater than the output of the
high state of demand.

When the uncertainty lies instead in the reservation price or saturation output,
conditions are found where it is best to choose the optimal outcome of the low state
of demand with certainty, or to choose a combination in the high state of demand
with the risk of overproducing and, in some cases, zero profits due to overpricing.

Dynamic and continuous distribution models are considered as extensions for the
case of a possible uniformly higher demand. Compared to the discrete case, the choice
of choosing a price-quantity combination in the high state of demand increases when
a dynamic model is considered instead of a static one. To be more precise, in the
dynamic game, the low demand optimal outcome with certainty is less likely to be
chosen since the firm can better bear the risk of overproduction by selling the leftover
in the second period, making it more likely to charge a price above 1. The static model
shows an expected result: when the probability of the high demand is greater than

.5, the solution lies on the high state of demand which converges to the high demand

77

outcome with certainty as the probability of high demand increases. However, in the
dynamic model a probability greater than 1 / 3 is enough for the same shift when a
discount factor of 1 / 2 is used. The continuous static model shows also that the firm
is always worse off choosing the outcome under certainty of the low state of demand

rather than the Optimal combination of the static model.

78

Figure 3.1: High and Low States of Demand

P4)

 

 

 

/
/

V

79

Figure 3.2: Profits as a Function of Produced Quantity

 

 

 

 

Case A: 7p-c>0 Case B: yp-c<0
1c 1:
A A
E 5 \. E
: E ‘\\\ : :\\\
E : \_ i i \ 1
H) 2-p q 1-p 2-p q

80

Figure 3.3: Regions for the Profit Functions

 

 

 

 

 

81

 

M1

 

Figure 3.4: Optimal Pricing Path

 

 

82

Figure 3.5: Myopic versus High State Profits
1 -(

0.6%

0.6%
0.4-:

0.23

 

 

U 7 f1 v I v v 7 [it Tfiv f1 7 v r v 1

0'2 04 0.6 0.8
gamma

83

dd

Figure 3.6: Intervals for the Profit Functions with Uncertainty in the Reservation Price

 

  
  
      
 

 

 

11
0.8-
: 72m
0.6f
0.4-:
. 7T:
0.2:
0 ' 0'2' "074' 075” 0‘6 '7

gam

84

Figure 3.7: Profit Functions for the Static and Dynamic Model

 

 

 

 

85

Figure 3.8: Optimal versus Monopoly Price

 

o ' 0:2 ' '54 ' 0.6

86

d.

Figure 3.9: Optimal versus Monopoly Quantity

AAA
I

0 .9-1

    

vlvffv'vu

0.4-

0 .3«

'V'VYYV1TY'

 

87

Figure 3.10: Optimal versus Monopoly Profits

 

 

 

88

4 A Study of OPEC Cartel Stability

4. 1 Introduction

The Organization of Petroleum Exporting Countries (OPEC) was formed in 1960
at the Baghdad Conference.11 One of the stated objectives of OPEC is to "coordinate
and unify petroleum policies among Member Countries, in order to secure fair and
stable prices for petroleum producers,"l2 which has been taken to mean maximize
OPEC countries’ profits. In spite of this goal, significant price fluctuations over the
history of the organization suggest to some a competitive market perhaps abetted
by occasional attempts to restrict output that invariably leads individual countries
defections. On the other hand, it is well-known that in an environments with signif-
icant unanticipated demand fluctuations, the optimal strategr for a cartel results in
periods of price wars that help enforce the cartel in the long run (Green and Porter
1984). In particular, using data for the Joint Executive Committee railroad cartel,
Porter (1983) has tested and rejected the null hypothesis that “no switch took place,
so that price and quantity movements were solely attributable to exogenous shifts in
the demand and cost functions” ( pp.301). The question then is which better ex-
plains OPEC: whether such price fluctuations reflect switches between collusive and
non-cooperative periods in the cartel or whether the price behavior is indicative of
OPEC’s overall inability to affect the world’s oil prices. The goal of this paper is to
test for significant switches from collusive to non-cooperative behavior by OPEC.

Not surprisingly, there has been a good deal of work testing for collusion behavior
in OPEC. What is surprising is that with notable exceptions, previous work has used
a reduced form approach and have focused on individual country behavior rather

than looking at OPEC as a whole. That is, previous work does not use a measure

 

11The five founding members are Saudi Arabia, Iran, Iraq, Kuwait and Venezuela. The current
members also include Algeria, Indonesia, Nigeria, Libya, Qatar and the United Arab Emirates.
12 See http: / / www.0pec.org / aboutus/ history / history. htm

89

of collusion for OPEC members, nor does it focus on the overall behavior of OPEC.
In contrast, this paper follows Porter (1983), that is, we test for a structural model
based on an optimal mechanism to enforce a cartel that faces unobserved demand
fluctuations. In addition, by modifying the model to reflect how OPEC differs
for the cartel examined by Porter (1983), this paper makes a second contribution.
Specifically, we allow for non-OPEC producers to be treated as a competitive fringe.
Because OPEC’s production accounts for only 40% of the world’s supply of crude,
this modification is key in testing for switches in cartel behavior. Specifically, while a
cartel might keep to a quota until enough periods have passed to switch to a regime of
punishment, the fringe is not so constrained. Thus, an OPEC member in considering
whether to deviate from the collusive outcome must take into account that the fringe
will respond sooner to the resulting price decrease.

A novel result of this paper is our finding of significant switches between collusive
and noncooperative behavior between 1974 and 2004. We estimate that the proba-
bility of being in a cooperative period was 35%, with periods of collusion resulting in
28% higher oil prices relative to periods of quantity competition. It is worth noting
that the magnitude of this price increase is about half of that estimated to result from
wars involving any OPEC member (60%). Euthermore, we find statistical evidence
that OPEC does not behave as an effective cartel but as a non-cooperative oligopoly.
Our estimates are consistent with OPEC behaving as in Cournot competition in the
presence of a competitive fringe.

Regarding the overall market structure, our results are in line with the extant
literature’s findings that OPEC members produce at levels between quantity compe-
tition and the perfect collusive outcome. Although a number of studies in the 19805
and early 19903 suggest that OPEC behaves as a collusive cartel, recent work seems
to point towards a different market structure. The seminal paper of Griffin (1985)

tests individual countries’ behavior among both OPEC and non-OPEC members for

90

the period 1971-1983. He finds that most OPEC members behave as if they were part
of a collusive cartel, while non-OPEC countries behave as if in Bertrand competition.
Jones (1990) uses the same model for the 1983—1988 period with analogous findings.
Using an ARIMA specification, Loderer (1984) finds that OPEC did not influence
oil prices during the 1974-1980 period, but did so between 1981 and 1983. His re-
sults provide some evidence supporting the hypothesis that OPEC was a dominant
producer during the second subsample.

In contrast, Spilimbergo (2001) uses a dynamic framework and rejects the hy-
pothesis that OPEC was a market sharing cartel during the 1983-1991. Similar
results have been found by other authors using different techniques. For instance,
Gfilen (1996) examines whether OPEC is a cartel whose members agree on their role
assigned by the organization in an output rationing framework, and whether OPEC
has the power to affect the market price of oil by adjusting its production. Using
cointegration tests for the period 1965-1993, he finds that overall OPEC is not a co-
hesive organization. Griffin and Xiong (1997) calculate joint-profit-maximizing price
paths and find evidence that —for some countries— it is more profitable to cheat on the
assigned quota, despite the possibility of punishment if caught. Alhajji (2001) tests
the dominant producer hypothesis for: (a) OPEC overall, (b) the OPEC core and
(0) Saudi Arabia when non-OPEC oil producers are treated as a competitive fringe.
All but the final model are rejected. More recently Smith (2002) concluded that
OPEC’s market structure lies between a non-c00perative oligopoly and a cartel. In
the work closest to ours, Yang (2004) also uses Green and Porter (1984) as a starting
point. However, in Yang (2004) individual supply functions for each member country
are calculated so that instead of having a collusive variable indicator for OPEC as
a whole, the switches between regimes are based on specific characteristics of each
member and the total production of the rest of the organization, as in Griffin (1985).

While not considering perfectly competitive market outcomes, his main finding is that

91

each OPEC member produces between the collusive and Cournot outcomes.

In contrast, we remain closer to Green and Porter, by considering OPEC as a
single entity, rather than a collection of individual members. There are two com-
pelling reasons for doing so. First, there is ample evidence that individual members
frequently respond directly to other members’ outputs in such a way that overall
OPEC behavior vis-a-vis the world market remains unchanged. And second, to the
extent that oil prices are calculated as the average of the members’ crude oil streams,
a constant elasticity of variable cost appears to be a reasonable assumption.13 Hence,
there is an incentive to share technology among members to reduce costs since they
are the oil producers with the lowest cost as explained in Adelman ( 1993).

A second difference with respect to Yang (2004) is that our specification allows
us to test across many potential market structures — including perfect competition
and alternative Specifications of imperfectly competitive market structures with and
without competitive fringes.

The remainder of this paper is organized as follows. Section 4.2 presents the
model and the hypotheses that are tested. In Section 4.3 the data used is described.
In Section 4.4, the results and interpretations obtained are discussed and Section 4.5

concludes.

4.2 The Model

Using a model first proposed by Green and Porter (1984), Porter (1983) tests the
cooperative behavior hypothesis on the Joint Executive Committee of railroads -—
a cartel in operation from 1880 to 1886. His methodology estimates the level of
competition in a specific industry or market where each of the cartel members assumes

that a drop in demand could be explained either as an exogenous shock to the demand

 

13As of June 2005, OPEC’s reference basket now consists of eleven crude streams representing the
main export crudes of all member countries.

92

function or as a member’s deviation from the collusive output, which may trigger a
punishment stage.”

While the Joint Executive Committee controlled virtually all of the freight ship—
ments during the 1880’s in the US, OPEC has a significantly lower market share
(around 42%), that has been decreasing since the late 1980’s due to increased pro—
duction from non-OPEC producers. To account for the role of non-OPEC countries
in the oil market, we modify Porter’s model. Specifically, Porter’s supply function
is modified so as to allow non-OPEC members to be treated as a competitive fringe.
This modification also allows us to test the fringe’s statistical significance. Using a
constant elasticity of demand model, we assume that demand for OPEC oil is given

by a loglinear function of price in addition to a set of demand shifters:

ln opect = a0 + 01 ln pt + agOECDt + 03 In nopect -+- a4dummiest + U”, (4.1)

where 1n opect is the logarithm of OPEC oil supplied, ln pt is the logarithm of world
oil prices, OECD, is the first difference of the logarithm of GDP for OECD countries,
In nopect is the logarithm of non-OPEC major producers and dummiest is a vector
of seasonal dummies. U1, is an error term assumed to be i.i.d. normal with mean
zero and variance of.

The coefficient (11 is the price elasticity of demand for OPEC oil, which is expected
to be negative and greater than 1 in absolute value if OPEC indeed maximizes profits
as a dominant producer operating in the elastic segment of the demand curve.15
OECDt is intended to proxy for world income.16 The sign of (12 could be positive or
negative since OECDt is measured as a growth rate. Thus, a negative sign merely

indicates that the (positive) rate at which demand for OPEC oil grows decreases

 

1“See Tirole (1988) pp. 262 for a brief description of the model.

15see Alhajji (2001).

16Although world GDP would have been a better measure for world income, data is not available
at a quarterly frequency.

93

when the income for OECD countries increases; nonetheless oil is assumed to be a
normal good. The coefficient on In nopect, 03, is expected to be negative, since it is
the substitute good for OPEC oil.

For the supply equation, it is assumed that OPEC chooses the quantity to be
produced (which is consistent with the stated policy at a majority of their meetings).
Then the world price is set and the competitive fringe produces where the world price
equals their marginal cost. World demand, Qw, is given as the sum of non-OPEC
output, 62””, and OPEC output, Q", where OPEC output is the sum of each OPEC

country’s production:

ow = Q0 + or, where Q" = Z q.,. (4.2)

Each of OPEC’s member countries, indexed by i, maximize the usual profit func-
tion,

7T1 = ptqit — Ci((1it)- (43)

Let C,(q,-t) = a,q§’, + F, be the cost function for OPEC member i, 6 is the constant
elasticity of variable costs with respect to output, a,- is a firm-specific shift parameter,
and F.- is the fixed cost of firm i.

Hence Equation (4.5) is obtained following Porter’s derivation and including the
presence of the competitive fringe as in Church and Ware (1999). The market share
for OPEC is defined as s" = Qo/Q‘” and the market share of country i within the

organization as s, = qi/Q". The price elasticity of world oil demand is defined as:

s
Q)
<0
s
*e

(4.4)

 

0p 5275'

In line with Porter’s derivation, the parameter 19,-) is included in Equation (4.5) and

94

defines the possible five market structure cases as follows: 17

 

p, [1+ 6" x S J = M0,, (4.5)

611)

o if OPEC members exhibit noncooperative behavior and they price at marginal
cost, then we have evidence of Bertrand competition or, what is the same, a

perfectly competitive industry. In this case, 0“ = 0,Vi, t;

e if OPEC members maximize joint profits or, what is the same, follow efficient
cartel behavior using the collusive outcome in presence of a competitive fringe,

then 6,, = 1;18
e for an efficient cartel without a fringe, 6,: = 1/so;

o for Cournot competition with a competitive fringe, 0,) = 3,; and

for Cournot behavior without a fringe, 0“ = s,/s°,\7’i, t.

We assume that the oil produced by each country is of similar (i.e., equivalent)
quality; therefore, in equilibrium, each OPEC country obtains the same price.
After multiplying Equation (4.5) by each country’s market share and summing

across countries, we obtain:

 

 

6,- 0 0 0
Es.p.[1+ 1:3]:16 [1+8 X 1:23.140... (46>

6w

1 1'

With at = 2 83'6“.

On the other hand, the marginal cost for firm i can be rewritten as

Z s.Mc.(q.-,) = DQJ-l. (4.7)

17A detailed explanation of Equation (4.5) is on the Appendix.
lz’See Church and Ware (1999) for a detailed explanation of a dominant firm with a competitive
fringe model.

 

95

Notice that the left hand side of (4.7) equals the right hand side of Equation (4.6),

thus,

 

6w

0
p, [1+ at x 8 J 2 0625-1, (4.8)

where D is a function of the country-specific shift parameter and the constant elas-
ticity of demand.

Taking the logarithm of Equation (4.8), the aggregate supply relationship is

lnpt = 50 + flth + [3227 ‘l‘ 53L + U2t1 (49)

where I, is a dummy variable that equals 1 when the industry is in a cooperative
period and equals 0 when it is in a reversionary period. The error term U2; is
independent of time and normally distributed with zero mean and variance 0%. The
variable Zt is included to control for other factors of the supply function that would
otherwise be part of the error term, therefore it minimizes the omitted variable bias.

The parameters 50, B 1, ,83 are given by

 

flo =1nD,
181: 6— 11
0
53 = —ln (1+ 3 6706‘). (4.10)

Although, the market structure for an individual OPEC member could be esti-
mated from Equation (4.5), we opt instead to estimate and test a model of overall
OPEC collusive behavior. The motivation for this choice is twofold. First, there is
evidence that some countries have adjusted their quotas for a short period of time in
order to compensate for a shortage of production from another member country, thus
keeping the production level of the organization unchanged. This behavior would

give a wrong signal of overproduction for some countries in cases where the increased

96

production was clearly aimed at stabilizing OPEC’s total output. Second, the ob jec-
tive of this paper is to test OPEC’s collusive behavior and the market structure as a
whole, focusing on possible switches between collusive and non-cooperative behavior

over time.

Hypotesis 4.1 Given the assumption that 6, = 6 for all t, 63 reduces to — 1n (1 + 306/6”).
Letting H = Z: 3,2 denote the Herfindahl index we can test the market structure as

follows:

1. 6 = 0 for perfectly competitive behavior,

2. 6 = H z .15 for Cournot behavior with a fringe,

3. 6 = H / s0 z .3642 for Cournot behavior without a fringe,
4. 6 = 1 for a perfectly collusive cartel with a fringe, and

5. 6 2 Us0 m 2.428 for an efilcient monopolistic cartel.

In Porter (1985), where there is no competitive fringe, the parameter 6 only con-
trols for Bertrand, Cournot or perfectly collusive firms. In the present model the
market structure parameter allows us to test not only for these three market struc-
tures, but also for the significance of the competitive fringe represented by non-OPEC
producers within the previous market structures. Thus, we have five possible cases.

In the OPEC framework, we a priori expect the competitive fringe to be significant
due to the size of non-OPEC production. Unless, however, the market is perfectly
competitive with OPEC and non-OPEC countries behaving as in Bertrand competi-
tion. Moreover, if there is no fringe and OPEC is an effective cartel, the price and
quantity used by the organization should be the same as those of a profit-maximizing

monopolist.

97

Equilibrium implies that quantity demanded equals quantity supplied. Therefore,

0, since a] is the price elasticity of demand for OPEC

from Equation (4.1), 6‘” = 01 x 3
oil, and

a = a. [mm—63> — 11. (4.11)

After obtaining the price elasticity of demand from (4.1) and the collusive behavior
coefficient (63) from (4.9), the value of 6 can be found from Equation (4.11).

In the OPEC setting, the supply function is given by

ln pt = [2’0 + 51 1n Opect + fighiStOTyt + 531‘ + 54brealct + 55dummies¢ + Um, (4.12)

where history, is a dummy variable that controls for wars involving an OPEC country,
and U2) is a normally distributed error term.

As mentioned above, the coefficient 61 is expected to be positive. The coefficient
on the collusive behavior dummy variable, 63, is expected to be positive and significant
if collusive periods have higher prices. The dummy variables, dummies), capture
seasons.

Equations (4.1) and (4.12) constitute a simultaneous equation model (SEM), in
which 1n pt and In opect are the two endogenous variables. If the 1, variable is the
true indicator of collusion for the cartel, the system can be estimated by three stage
least squares. If, instead, the indicator of collusion is assumed to be unknown but
independent of time and following a Bernoulli distribution, then the estimation is
done using the EM algorithm first proposed by Kiefer (1980).19

Specifically, following Porter’s notation, assume that It equals 1 with probability

/\ and 0 with probability 1 -— /\. Rewriting Equations (4.1) and (4.12) in a matrix

 

19While periods of cheating by OPEC members have never been sustained for a prolonged period
of time, we nevertheless hope to further substantiate our findings in future work by adapting the
model for a Markov process on switches instead of using the memoryless Bernoulli distribution of
the current paper.

98

form:

Byt = FXt + ‘1’]; ‘l" Ut, (4.13)

where y, = (1n opechlnpt)’, X, = (1,0ECDt,ln nopeq,historyt,breakhdummiest)’,
Ut = (U11,U2t)’ and dummies; is a 3 x 1 vector consisting of the seasonal dummies.

The error vector U, is identically and independently distributed N (0, E).

B = 1 —al ,\II = 0 ’1‘ = 00 02 a3 0 0 a4 ,2 = of 012
—31 1 53 50 O 0 52 54 P5 012 ‘73
(4.14)

where 04 and fi5 are 1 x 3 vectors.

Because the variable It is unknown, the probability density function is defined as:

f(ytht:It) = [A exp {‘é (Bl/t — 1“)(t‘. \II)2"1(By, — PXt — ‘10}4' (4-15)

(1 — A) exp {—% (By. — rX,)2-1(By, — we)” (271)-1/2|2|-1/2|B|.

With an initial estimate of the regime classification sequence (w‘l’, .., 1129.) where w? =

Pr(It = 1), an initial estimate of /\ is constructed as the mean of the classification
sequence.20 Hence, maximizing the product of the density functions, initial estimates
BO, P0, ‘17", 2° are obtained. Using Bayes’ formula the new classification series is

updated. Thus,

 

0
h =
wtl : PI<I¢ : llyt,Xt,q/0,FO,EO,BO,/\O) : 0 A (ytllt 3‘) ,
A h(ytl[t:1)+(1_ ’\ lh(ytl1t = 0)
(4.16)
where h(yt|1t) is the probability density function of y; given 1;.
Then the switching probability A is updated: A1 = %Z w}. This procedure is

repeated until the correlation between two consecutive estimates of wt exceeds .999.

 

20The previously constructed It is used to construct the initial classification sequence.

99

4.3 The Data

The data used in this paper spans the period between 197411 and 2004:4, where each
quarterly observation is calculated as the middle month of that quarter. Oil quantities
and prices are obtained from the US. Department of Energy.21 The variable In opeq
is the logarithm of OPEC oil supplied in period t measured in thousands of barrels per
day. The variable In pt is the logarithm of world oil prices measured as the Refiner
Acquisition Cost of imported crude oil in US. dollars per barrel.

The set of demand shifters comprises the logarithm of the GDP for the OECD,
OE C Dt; the logarithm of the quantity of oil sold by the major non-OPEC producers,
1n nopect; a dummy that controls for the US. price decontrols since the beginning of
1981, breakt; and a vector of seasonal dummy variables, dummies}.

As mentioned in the previous section, OECD, is intended as a proxy for world
income. In fact, the real GDP of the OECD amounts to 78.4% of world GDP and its
oil demand accounts for 66% of world’s oil demand. This variable is computed as the
first difference of the logarithm of GDP for OECD countries, measured in millions of
US. dollars. The data source for OECD, is the OECD Economic Outlook.

The variable In nopect is measured in thousands of barrels per day and comprises
the production of Canada, China, Egypt, Mexico, Norway, the USSR. and the
nations that formerly comprised it, the United Kingdom, and the United States. The
variable break, takes the value of 1 for the quarter 1981:1 onwards, and 0 otherwise.

The collusive behavior variable 1,, and the dummy variable history, serve as con-
trols in the supply equation. The variable It takes on the value of 1 when there is
evidence that OPEC was in a cooperative period. This variables is computed using
the Oil Price Chronology of the US. Department of Energy. Specifically, the produc-
tion quotas assigned by OPEC are compared to the actual production levels: If actual

production in period t is at least 5% over the quota established for that period, and

 

2‘ nominal prices were converted to real using CPI from the BLS webpage.

100

there is no evidence that overproduction was a consequence of an increase in world
demand, 1, takes the value of zero. Finally, the dummy variable history, equals 1
when there is a war involving an OPEC country at time t. This variable includes
the Iran—Iraq war, and the invasion of Kuwait in 1990.

Before proceeding to the econometric analysis, consider the historical evolution
and time series properties of the variables of interest. Figure 4.1 plots world, OPEC,
and non-OPEC oil production. The world production has been roughly constant
between 50 and 70 millions barrels per day with a slightly increasing trend. OPEC
decreased its production from 1980 to 1983, but has been increasing it ever since.
N on—OPEC members have had more constant behavior also with an increasing trend
overall, surpassing OPEC’s production in 1979.

In Figure 4.2, real world prices and the collusive behavior indicator 1, are plotted.
The figure also reports major world events that lead to large fluctuations in oil prices.
The Iranian revolution rasulted in a drop of 3.9 million barrels per day of crude oil
production between 1978 and 1981. Even though other OPEC members raised their
production seeking to maintain the same total output, the revolution still caused an
increase in oil prices. This trend was reversed when the US. removed the price and
allocation controls on the oil industry. As a consequence oil prices dropped. This,
together with a decrease in demand, an increase in non—OPEC production, and Saudi
overproduction, resulted in OPEC loosing control of world oil prices by 1982. By
December 1985, and despite several cutbacks in OPEC production, an abundance of
oil in the market was evident. This situation triggered a price decline that ended in
the so-called crash of 1986. Since 1987 oil prices have been more stable, except for
the five-month peak caused by the Kuwait invasion at the end of 1990.

As Figure 4.2 illustrates, the indicator variable 1, seems to be a fair indicator of
collusive behavior. Even though oil prices have been more stable after 1987, the

variable indicates that there has been significant overproduction compared to levels

101

of effective cartel output.

Table 4.1 reports statistics about oil production by OPEC members.22 The OPEC
“ core,” which is formed by Saudi Arabia, the United Arab Emirates (U.A.E.),
Kuwait, Qatar, and Libya, account for over 50% of total OPEC production, with
about 30% of OPEC’s oil being produced by Saudi Arabia alone. Note that minimum
observed values of zero are reported for Iraq and Kuwait. These values correspond
to the following periods of no production: February and March of 1991 for Iraq, and
February to May of 1991 for Kuwait (due to the Persian Gulf War).

Table 4.2 reports summary statistics for the variables used in the empirical analy-
sis. The quantity supplied by OPEC accounts for 41.2% of the world production for
this period which is around 59 millions of barrels per day. According to the variable
1,, there are 39 identified c00perative periods out of 124 observations, that is 33.8%
of the sample. The average price per barrel is 35.2 December 2002 US. Dollars.

As a measure of industry concentration, consider the Herfindahl index. This index
is a more accurate measure of concentration than the concentration-ratio since it gives
more weight to large firms. Schmalensee and Willing (1989) state that the Herfind-
ahl index gives the answer to: “By how much would welfare rise if we could perturb
the industry a small amount in the optimal welfare-improving direction?” Using the
US. Department of Justice classification of concentrated industries, Herfindahl in-
dices between 1000 and 1800 are moderately concentrated, those above 1800 indicate
concentrated market structures. Figure 4.3 plots the Herfindahl index for OPEC
across time. Note that although the period 1981—1983 is indicated as concentrated,
the large magnitude of the index reflects the decline in Iran and Iraq’s supply, which
was compensated by Saudi Arabia. In this manner, total OPEC output remained

essentially at the same level, with the resulting increase in share of Saudi Arabia (see

 

22Gabon and Ecuador are excluded from the analysis, even though they were part of OPEC
for most of the period to be examined (Ecuador 1973:11-1992zll, Gabon 1973:12-1995z01), their
combined production was less than 1% of total OPEC oil production.

102

Figure 4.4). Similar behavior can be observed during the Gulf War where Saudi Ara-
bia made up for Kuwait’s and Iraq’s shortages. Nevertheless, when the production
of Iraq, Iran, Kuwait and Saudi Arabia is taken as a whole, the market share remains
fairly constant over time (see Figure 4.5). Indeed, the pattern of Saudi Arabia’s
share is almost identical to that of the index, given that this country has by far the
largest share.

Table 4.3 shows OPEC quotas, production allocations and the Herfindahl index
for four select periods: (i) the last quarter of 1978, before the Iranian Revolution and
the US. decontrol; (ii) 1983 when OPEC started using quotas instead of royalty rates;
(iii) 1990, before the Gulf War and (iv) November 1997.23 As shown in Figure 4.3, in
1983 the Herfindahl index was at its highest, this is reflected in Saudi Arabia’s share
of over 40% of total OPEC production. After 1984, Saudi Arabia’s share declined
to a level between 25% and 30% of total production, even though the output level
had been constantly increasing. At the same time, the share of some of the smaller
producers, that were already producing close to their maximum capacity, decreased.
Over time, other countries, like Venezuela, Nigeria and the U.A.E. slightly increased
their production share, even though OPEC’s output has also increased over time.
Horn the late 1990’s, the index goes back to its average level during the 1970’s.

Lastly, Table 4.4 reports the results for Elliott and J ansson ( 2003) unit root test for
the two main variables of interest, the logarithm of OPEC production, In opec,, and
the logarithm of oil prices, In 10,, as well as for non-OPEC production, In nopec,. The
motivation for using this test over the commonly used Dickey-Fuller test is twofold.
First, this test has been shown to have excellent properties in small samples such as
the one in this paper. Second, Elliott and J ansson (2003) demonstrate that very
large power gains over the Dickey-Fuller test are obtained when relevant stationary

covariates are modeled with the potentially integrated variable. Indeed, given the

 

23For 1978 production as of the fourth quarter of the year, all other years production is quota
allocated. The Herfindahl index is presented in percentage terms.

103

structure of our model the choice of covariates appears to be straightforward. Ac-
cording with equation (4.1), we use the rate of growth of GDP in the OECD, OECD,,
as the stationary covariate in testing for a unit root in the quantities. We include
a constants, but no time trend in testing for a unit root in In opec,, and constants
and time trends when testing In nopec,. For the estimated R2 of 0.43 for In opec,
and 0.20 for In nopec,, the test statistics are 3.2535 and 3.7062 which are less than
the respective critical values of 4.1057 and 5.9976.“ Thus, in both cases we reject
the null of a unit root for a 5% test. Equation (4.12) and the rejection of a unit
root in In opec, would suggest using this variable in testing for a unit root in prices.
Including constants and time trends in both prices and quantities, for the estimated
R2 0.68, the critical value is 12.0812, so with a test value of 9.8399 we reject the
null of a unit root. We reach the same conclusion if instead -not having any prior
information on the stationarity of In opec,- we are cautious and use the first difference
of 1n opec, as a covariate. In that case, even though the estimated R2 is considerably
smaller (R2 =0.01), the test leads us to reject the null of a unit root. Thus, in the
following section we proceed to estimate the model under the assumption that all the

variables in the system (4.13) are stationary.

4.4 Empirical Results

The estimation results are presented in Table 4.5. The first two columns report the
parameter estimates for Equations (4.1) and (4.12) obtained by Three Stage Least
Squares (35' LS ), under the assumption that the constructed cheating variable is an
accurate classification of the regimes. The third and fourth columns correspond
to the estimates obtained using the maximum likelihood algorithm described in the
previous section, when the structural break is not taken into account. The last two

columns present the maximum likelihood estimates when controlling for the structural

 

24The R2 is the correlation between the covariate and the potentially integrated variable.

104

break.

Focusing first on the 3S LS estimates, note that in the demand equation we obtain
a negative price elasticity that is significantly less than one in absolute value, thus
suggesting an inelastic demand for OPEC’s oil. According to these estimates, OPEC
does not maximize profits as a dominant firm given that it does not produce on
the elastic part of its demand curve. When non-OPEC production increases by
1%, OPEC production is estimated to decrease by 1.4%. The coefficient on the
OECD’s GDP is not significantly different from zero. The estimates of the supply
equation predict 28% higher oil prices in cooperative periods, and higher prices when
an OPEC country is involved in a war. None of the seasonal dummies are statistically
significant.

Regarding OPEC’s behavior, using Equation (4.10), we obtain an estimate of 0 =
0.5419, which is consistent with Cournot behavior without the fringe (see Hypothesis
4.1.3). Notice that the closest market structure would be an efficient cartel. This
result also matches the estimated elasticity of demand.

Consider now the estimates obtained when we allow for the regime classification
to be estimated by maximizing the likelihood function, Equation (4.15). If we do
not take into account the structural break in 1981:1 (Columns 3 and 4 in Table 4.5),
the estimated price elasticity in the demand equation is negative and less than one in
absolute value. Hence, the profit maximizing condition does not hold. As it is the
case for 35 LS, OPEC’s production is estimated to decrease with increases in non-
OPEC production. However, raises in the world’s income, as proxied by OECD,,
are estimated to have a negative and statistically significant effect on the demand
for OPEC’s oil ((12 = —0.03). On the other hand, war periods involving an OPEC
country increase prices by 41%. The main difference with respect to the 3S LS
estimate is the effect of cooperative periods on oil prices: the maximum likelihood

estimates give a 14% increase, about half the magnitude obtained from 33 LS .

105

Regarding the probability of being in a cooperative period, the estimate obtained
from the maximum likelihood procedure equals 0.353 (see Table 4.5). This probability
is slightly higher than the initial value at time t (A = 0.336), which is equivalent to
the mean of the constructed collusive variable I, used in the 3S LS estimation. The
corresponding 0 equals .1059, a result also consistent with Cournot behavior with the
presence of a competitive fringe (Hypothesis 4.12).” Note that given this estimated
value for 9, the closest alternative possible market structure would be competitive
behavior (see Hypothesis 4.1.1).These result corroborates the prior beliefs that non-
OPEC producers having a significantly larger market share, take some of OPEC’s
market power.

Finally, the last two columns of Table 4.5 report the estimates obtained when we
take into account the structural break in 1981:1 in the supply equation. The inclusion
of the break, dummy increases the log likelihood by 108.9; thus, providing statistical
evidence of a structural break in the mid-19808. The demand equation presents a
similar intercept for the pre—1981 period, however, the price elasticity of demand is
smaller in magnitude. Now a 1% increase in non-OPEC production decreases OPEC
production by almost 1% which is expected assuming that OPEC and non-OPEC
countries are the only two oil producers. Also the coefficient of GDP for OECD
countries is still negative but not significant, hence OPEC demand does not depend
on the growth rate of OECD countries. Note that the finding of price inelastic oil
demand is consistent with Alhajji (2000), who finds that OPEC is not a dominant
producer since it operates in the elastic part of its demand function.

Regarding the supply equation, the intercept appears to be overestimated by the
ML with no break. More importantly, the inclusion of the break results in a sig-
nificantly larger value for the coefficient on the collusive dummy which is almost the

same as in OLS. In fact, these results are consistent with a more realistic scenario

 

25A t-statistic was constructed using the delta method for the standard error of 0. The only
hypohesis that we failed to reject was Cournot competition with a competitive fringe.

106

where periods of collusion are estimated to result in a 28% increase in oil prices, and
the coefficient on the history dummy reflects a 64% increase in OPEC prices during
periods of wars. Furthermore, when the break is taken into account, the value of
6 equals .116. This value is smaller than the BSLS estimate but larger than the
ML estimate with no break, however still consistent with Cournot behavior with the
competitive fringe (see Hypothesis 4.1.2). In other words, estimates for the ML
model with the break dummy, suggest that OPEC has not been effective in raising
prices over quantity competition levels.

Note that in this model the probability of collusion (A = 0.3524) is closer to
the average number of collusive periods in the data, and lower than the estimated
probability when the structural break is not taken into account. All in all, accounting
for the structural break renders the fringe insignificant since OPEC and non-OPEC
producer appear to have the same market structure. Yet, the statistical significance
of B3 provides evidence of switches between collusive and non-cooperative behavior
at the interior of OPEC. This result is confirmed by the fact that a likelihood ratio
test comparing this model and one where 63 is restricted to zero equals 217.98.

Summarizing, accounting for the structural break originated by the US. price
decontrol leads one to conclude that OPEC behavior is better described by quantity
competition between OPEC and non-OPEC producers (Hypothesis 4.1.2). Regard-
less of the model, the price elasticity of demand indicates that OPEC is not a dominant

producer.

4.5 Conclusion

The estimations in this paper examine world crude oil prices for the period 1974:2-
200224 in order to gain insight into OPEC behavior. It is demonstrated that changes
in oil prices are a result not only of shocks to demand, but also of switches from

collusive to non-collusive behavior among members of the cartel. That is, we present

107

statistical evidence that switches in the behavior of the organization lead to price
changes with prices being significantly higher in periods of cooperative behavior.
Hence, the hypothesis that only cooperative or noncooperative periods are observed
is rejected in favor of switches between regimes.

Indeed, throughout OPEC’s history, there has been evidence of deliberate cheating
among members and our findings suggest that these deviations are significant enough
to drive the overall behavior of the organization away from the collusive outcome, in
spite of efforts from some of its members like Saudi Arabia. In other words, despite
spells of collusive behavior, OPEC cannot be viewed as an effective cartel during the
time period examined. Therefore, the world crude oil market is best described by the
Cournot model of quantity competition where OPEC acts as an inefficient dominant

producer and non-OPEC producers as a competitive fringe.

108

4.6 Appendix

Derivation of Equation (4.5):
From the profit function of member i, 7r, = p,q,=, — C,(q,,), the first order condition
with respect to the production quantity is
871', GP

8
‘— =1), + ":91: — AIC,’ = 0 <=>p,+ £6121: A’IC,.
8% Q1: (In

. . 0p. 0p, BQW 3620
Usm the chain rule we et — = , , ,
g g 91': 8Q” 8Q0 (In.
8Q”, _ 5QO
BQO Wt

 

where = 1.

Then 0p,/0q,, = (kn/362”", hence, we can rewrite the first order condition as

8P1 QW Pt
8Q”, Pt (1"in

using the definition of the world elasticity of demand,

 

a .
p, + 87%.,” = MC, or p, + = Ma,

6.... = 362‘” p
0p 62“”

 

 

and s,- = q, /Q°, the first order condition becomes

0

SS

 

[9 [1+ 1 = MC. (4.17)

1'
6‘"
Writing the first order conditions of each of the possible market structures we obtain:

1. Bertrand competition implies p = M 0,. Note that Bertrand competition maxi-
mizes with respect to prices instead of quantity, however in the derivation prices
are not replaced by the demand function (e.g. p = 1 — q), hence we can still

compare the first order condition, since it is solved as an implicit derivative.

109

2. Cooperative cartel points to: p [1 + 1/6‘”] = M0,. This is the first order condi-

tion for a regular monopoly and can be derived as follows:

71 = (p-C)q (4-18)
6w 8p [ 3P4]
_ = _ + _c=()4:y 1+——— =c. 4.19
(9g qu p p 3917 ( )

The inverse price elasticity of demand is i = 35%, hence we obtain the specified
condition. From Equation (4.17) it implies that s, and 3" equal 1, where 3"
represents the market share of the monopolist, that in this case equals 1 as
there is no fringe. And 3, is the market share of each producer, that in the case

of a monopoly also equals 1 because there is only one producer.

3. Cooperative cartel in presence of a competitive fringe indicates:

p [1 + so/ew] = M0,. This is the same as a dominant producer in the presence
of a competitive fringe, as in Church and Ware (1999). The result is obtained
defining the price elasticity of demand in terms of the elasticity of the dominant
firm and of the competitive fringe as 6‘” = e°s° + 6! (1 — s°), where 6" is the
price elasticity of demand for the good of the competitive fringe. This market
structure can be interpreted from Equation (4.17), s, also equals 1 since there is
only one dominant producer, but 30 represents the market share of that producer

in world production because the competitive fringe has a market share of 1— 3°.

4. Cournot competition in presence of a competitive fringe implies:
p [1 + s,s°/e‘”] = M0,. This is the case of the first order condition derived above

in Equation (4.17).

5. Cournot competition without the fringe indicates: p [1 + s,/e“’] = M 0,. Follow-
ing the same derivation as the previous market structure, but with no compet-

itive fringe, the oligopoly has all the market share (3" = 1), hence the price

110

elasticity of demand depends only on the producers of the oligopoly.

Therefore including the parameter 0,, in Equation (4.17), the first order condition

becomes:

 

9.- 0
p[1+ 6‘: [ =MC,, (4.20)

where 6,, can be tested for each of the five market structures for each member. As
the purpose of this paper is to test for an overall behavior of the organization rather
than for each member country, instead of directly testing Equation (4.20) we find an

aggregate relationship to test OPEC’s behavior.

111

Table 4.1: OPEC Members Production (thousands of barrels per day).

 

 

Variable Mean Std. Dev. Min Max
Algeria 1168.77 174.18 860.66 1715
Indonesia 1440.76 145.24 1088 1698.66

Iran 3442.39 1236.12 887.33 6604.66
Iraq 1731.6 924.48 85.34 3661
Kuwait 1723.66 590.65 16.89 2889.33
Libya 1412.86 327.94 654.33 2154.66
Nigeria 1891.38 361.67 820 2546.66
Qatar 485.63 153.26 208.33 835
Saudi Arabia 7488.79 1814.88 2686.66 10298
U.A.E. 1909.25 427.78 1053.66 2602
Venezuela 2376.27 479.34 1490 3423.57

 

 

112

Table 4.2: Descriptive Statistics

 

 

Mean Std. Dev. Min Max
OPEC production 25070.93 4557.95 15159.33 33556.67
World Prices 35.19 16.59 11.95 80.45
history .2825 .4519 0 1
I .3387 .4751 O 1
Non-OPEC production 35646.33 4238.63 25110 42443.73
GDP for OECD countries 53.93 34.98 8.4 132.84

 

 

113

Table 4.3: Herfindahl Index for OPEC Members for Regated Years

 

 

 

 

1978 1983 1990 1997
Share Production Share Production Share Production Share Production
Total Production 1 31 1 17.15 1 21.61 1 27.5
Saudi Arabia .32 9.97 .416 7.15 .249 5.38 .318 8.76
Iran .12 3.81 .07 1.2 .145 3.14 .143 3.94
Iraq .09 2.98 .07 1.2 .145 3.14 .047 1.31
Venezuela .07 2.32 .087 1.5 .089 1.94 .093 2.58
Nigeria .07 2.17 .075 1.3 .074 1.61 .074 2.04
Indonesia .05 1.58 .075 1.3 .063 1.37 .052 1.45
Kuwait .07 2.31 .046 .8 .069 1.5 .079 2.19
Libya .06 2.13 .043 .75 .056 1.23 .055 1.52
U.A.E. .05 1.83 .058 1 .05 1.09 .085 2.36
Algeria .04 1.37 .037 .65 .037 .82 .032 .9
Qatar .01 .52 .017 .3 .017 .37 .014 .41
Herfindahl Index .1476 .212 .1342 .1594

 

 

 

114

Table 4.4: Elliott and Jansson Unit Root Test

 

 

Variable Test Value 5% Critical Value R2 Covariate
ln (opec) 3.2535 4.1057 0.4324 OECD
ln(nopec) 3.7062 5.9976 0.2044 OECD

[ii (p) 9.8399 12.0812 0.6825 In (opec)
1n(p) 5.5940 5.7094 0.0104 Aln (opec)

 

 

115

Table 4.5: SEM Estimates

 

 

 

 

 

3SLS* IVILE NILE with breaks
(1) (2) (3)
Variables Demand Supply Demand Supply Demand Supply
constant 26.55911 -3455 1 16.84131 2.3221 21.59461 1.67731
(2.3555) (8.37) (1.1712) (.2658) (2.3028) (.371)
In (opec) 3.7024 1 .0961 .1761
(.8212) (.0238) (.0411)
(71(1)) -.54191 -.80211 -.46961
(.1) (.1883) (.0746)
OECD 2.7172 -.03201 -.0218
(3.8897) (.0309) (.0378)
ln(n0pec) ~1.395 1 -.38931 -.94861
(.2107) (.1029) (.2305)
history 1.6899 1 .41441 .64561
(2941) (.0799) (.0732)
I .2896 1 .14161 .28381
(0841) (.0513) (.0494)
break -.37131
(.1296)
january -.0529 .0773 -.0407 -.0139 -.0439 .0011
(.055) (.0996) (.1125) (.1263) (.0534) (.085)
april -.0565 .139 -.0366 .0076 -.0436 .0257
(..0552) (1020) (.0764) (.0817) (.0596) (.0982)
july -.0253 .0225 -.0101 .007 -.0197 .0013
(.055) (.0982) (.1716) (.1953) (.0736) (.1282)
theta .5419 .1059 .116
(0.0016) (.0286) (.0204)
Loglikelihood 625.9 734.89

 

 

* Standard errors are in parenthesis. 1Significant at 1%.

116

Figure 4.1: Oil Production (thousands of barrels per day)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OPEC Non-OPEC — —World
80000
f
70000 ~-/-~e—
.o J"
60000 ..,-VI“ was“; .. ’
«'0 \Aw VA”
50000
40000
30000 r""~"¢‘ '.~."V".A
20000 ~,‘. ...r~+‘ °
10000
0-
u e e o '1. ‘b o '1, e e c '1,
’94 :8” ’94 $5 ’99: ’99? $62 1&9 v99 ’99 [99“ '99 :89 (‘90 (‘90 (90"

 

 

 

*Prices are measured in dollars of December 2002. The I, dummy series is scaled to match

prices in the same graph, when I,equals 10, OPEC is in a cooperative period

117

Figure 4.2: Real Prices and Cooperative Period Indicator

90 . - ._ 1 w . *_ L 1L D .
80 i - - - f 1- __f 111 1-- L 111 11
1 70 .. 11L . _ 1 1fi_ 1,_ _¥ W 17 1L7 if * 1
1 60 f. —. — ___- __ _ _. _ ___ _111 L11 ~1
501111 . - 1L -_ 1111,11 1L 1. _1
40 .1 ~11- - -— -L “-_.- 1* a . _ _ '
30 — 4* — 1111 1 ___—-- ,, 1
2... - _ 11 V . .1

 

1 10 Wfifi-W-—~m*-x -— an art——

0 1.1x...mammmmmmmnmm
bi Q) Q) Q ‘lv ‘1’ ‘b ’1'

11 _1__1_11_____1 _11

g ‘. ___ _ __

118

Figure 4.3: OPEC’s Herfindahl Index

 

 

 

0 ii IT‘l ’ TTT’TT’D'T‘TIT’II—F’I—WTTT'TI 1 11111 H WWTT—TT'ITTTITTTTTTTT‘I 7171?. IIT‘TWT'ITTW' ITITTITTTTTTTTF II'I'IITTTTTWT—TTTT'T ll

,9" .52 (8% 89° .58" .99“ .92" .99" 49° .99" .99“ .99" ,9" ,9“ ,9“ ,9“

___ 1

119

Figure 4.4: Market Shares

 

 

f—‘ - .opecsfidmraniraki‘ — Safiidi_Arabi;:_- ". . Iran 111 lraq1 1
70 4. f —- -4- “_ - — -4 - 4- .4— -4 1
60 4 f ' _ A w , 11 i 1
I V. - 4J- _ - _r ' - I
50 §—\ 4‘ ,_ 4’ 3Y1 ‘4 r— ..4 —~ \4 . a—l—‘y‘v
40 1 4". ~ 1» f i 4* 4* 111 + _ 1
/ \ J "7 ..
30 s «.7 \fi. _ — 1-1-211-4 4. —. ..L‘s
J A ~-.
20 1/ "‘ \ I». A. J _ _ 1

 
 

 

1
0 11"1 T'TT’“. 'WWT'TTW'HWU‘T—rtfiwrmfinr

«“«bfi’eQe'kd‘ebe‘eQe‘l’e‘ebe‘b
991331999193199'3'9

6491:
QB
\ (5°09

‘19

7 7 _ _11 141 v h_ v k 711 1__ 111. D_11__ 71 _ ________ 111k1 111.11

120

Figure 4.5: Combined Share for Iran, Iraq, Kuwait and Saudi Arabia

 

 

 

 

10‘

 

O ’WM‘T"'1”' 1 1 1 1'11 1 .1 1"TT’IIT'TT'TT’W’TTT11‘1"YYITDWWTDW'TTTTV'HiTI'Tn"[3'13"'I'VTT‘W 1
«V4949’e9e’kei‘ebe‘9 ’1' 099999610"
eeeeeeeeefegeqfeeeee1

 

121

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