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.76.:ng LIBRARY
Michigan State
University

 

 

 

This is to certify that the

dissertation entitled

Essays in Quantitative Macroeconomics

presented by

Facundo Sepulveda

has been accepted towards fulfillment
. of the requirements for

Ph . D . degree in Economics

 

 

 

Date August 21, 2002

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

PLACE IN RETURN Box to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

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6/01 c:/ClRC/DateDua.p65-p.15

ESSAYS IN QUANTITATIVE MACROECONOMICS
By

Facundo Sepulveda

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Economics

2002

ABSTRACT
ESSAYS IN QUANTITATIVE MACROECONOMICS
By

Facundo Sepulveda

This dissertation contains three essays in quantitative macroeconomics.

The first chapter, “Precautionary savings in general equilibrium”, uses a calibrated
stochastic OLG model to address three questions about US savings and wealth accu-
mulation: first, does an equilibrium display buffer stock savings by agents? Second, is
this equilibrium consistent with savings behavior of US households? And finally, what
level of precautionary savings arises when general equilibrium effects are accounted for?
I find that given observed earnings risk, the rates of time preference that are consistent
with the equilibrium are very close to the interest rate, so no buffer stock behavior is
observed. Moreover, the equilibrium reproduces important facts about savings behavior
of US households. Finally, accounting for general equilibrium effects lowers the size of
precautionary wealth to about 35% of aggregate wealth, or 30 to 50% less than partial
equilibrium estimates.

The second chapter, “Green taxes and double dividends in a dynamic economy”, asks
whether a tax recycling experiment would deliver a double dividend in the US economy.
According to the double dividend hypothesis, environmental taxes may raise revenue
that can be used to lower other (pre—existing) tax distortions apart from decreasing
pollution externalities. This hypothesis is evaluated using a dynamic general equilibrium
model of capital accumulation. i find that, although in the long run pollution may
worsen, the green dividend -higher discounted utility from a cleaner environment- would
be obtained under all tax changes, due to a better environment during most of the
transition. The efficiency dividend however -higher discounted utility from consumption
of traded goods— will obtain only for target levels of the green tax below a critical number.

In the third chapter, “Training and business cycles”, I examine the behavior of skill

acquisition through training at business cycles frequencies. First, a time series of training
is constructed using individual data from the N LSY79 database. After documenting the
cyclical properties of the series, I discuss what features are needed for a RBC model
to successfully reproduce them. I find that training is weakly countercyclical, leads
the cycle, and has a standard deviation of about ten times output. A model where
employment, but not weekly hours, is costly to adjust, is able to account for most of

the documented regularities.

To Carolina, my wife,

and Sebastian, my brother.

ACKNOWLEDGEMENTS

I would like to thank first of all my advisors Rowena Pecchenino and Gerhard Glomm,
both for the time they gave me and for their encouragement. Rowena and Gerhard
taught me how to think about economics and kept alive the academic ambition I brought
to MSU.

I am also in debt with my other committee members, Andrei Shevchenko, Ana Maria
Herrera, and Jeff Biddle, for taking the time to read my dissertation and sharing their
suggestions with me. Carl Davidson taught one of the most challenging courses I took
at MSU, and Jeff Wooldridge made econometrics seem easy. I learnt much from them.

In Argentina, Professor Julio Olivera made me aware for the first time of a sense of
possibility in academic work, and Jorge Katz and Miguel Teubal encouraged and helped
me pursue graduate studies.

A number of friends were responsible for making my life in the States a rich and
worthwhile experience: Bulent, Claudia and Nestor from Rochester: Onur, Rodrigo and
Maite, Noemi, Jenny and Noel, Luz Maria, Eduardo Paulino, Linda Bailey and Zhehui,
Alonso, Luis and Maria Alejandra, at MSU. Two of these friends, Fabio and Daiji, also
shared with me their intellectual enthusiasm, and I owe them in two currencies.

Finally, thanks to my family: Carolina, my parents Sergio and Ana, my grandmother
Irene, Javiera, and Sebastian, my brother. They, with their unconditional love and

support are the main reason why I have completed this work.

Contents

1 Precautionary saving in general equilibrium
1.1 Introduction ..................................

1.2 The model ...................................

1.3 Calibration. ..................................
1.4 Results .....................................
1.5 Conclusions ..................................
2 Green taxes and double dividends in a dynamic economy
2.1 Introduction ..................................
2.2 The model ...................................
2.3 Calibration ..................................
2.4 Results .....................................
2.5 Conclusion ...................................
3 Training and business cycles
3.1 Introduction ..................................
3.2 Data ......................................
3.2.1 Data description ...........................
3.2.2 Stylized facts .............................
3.3 Model .....................................
3.4 Calibration ..................................
3.5 Results .....................................

vi

13

23
23

28
3O
32

40
40
42
42

47

3.6 Conclusion ...................................

A Numerical methods.
A.1 Solution method for the stochastic OLG model ...............

A.2 Solution method for the deterministic OLG model .............
B Standard RBC model with human capital

C First order conditions in the household problem [P2]

vii

70
70
71

73

75

List of Tables

1.1
1.2

1.3
1.4

1.6
1.7

2.1
2.2
2.3

3.1
3.2
3.3
3.4

3.5
3.6
3.7
3.8

Calibration .................................. 14

Carroll-Samwick estimates for sensitivity of wealth holdings with respect

to income risk ................................ 15
General equilibrium estimates of precautionary wealth .......... 16
Partial and general equilibrium estimates of precautionary wealth (‘70) . 17
Reproducing results: ”/0 of wealth that is precautionary .......... 18
Partial vs. general equilibrium effects ................... 19
Selected capital output ratios ........................ 20
Benchmark parameters and data ...................... 34
Euler residuals (er) .............................. 34
Welfare analysis: Compensating variation ("/c) ............... 35
Assignment of training program ....................... 63
Descriptive statistics ............................. 63
Description of variables ........................... 64

Contemporaneous correlation of 1, hours and productivity with GDP-

Contemporaneous correlations of training with GDP, I-Data ....... 65

Cross correlations of I, H and productivity with GDP-Data (correct, gdl)!+ 1)) 66

Cross correlations of training with GDP-Data ((‘0-r'r(;1:t, 51(1thr 1)) ..... 67
Standard deviations-Data .......................... 68

viii

3.9 Baseline parameter values .......................... 68

3.10 Cross correlations with output-Model ((#01‘1‘(:Et,gdpt+j)) .......... 69
3.11 Standard deviations—l\"l(;)(_lel .......................... 69
3.12 Sensitivity analysis .............................. 69
B.1 Variable description .............................. 74

ix

List of Figures

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9

2.1
2.2
2.3
2.4

3.1
3.2
3.3
3.4
3.5

Income profile (Hubbard et a1. [1994], 12-15 years of education) ..... 14
Age-wealth profile: Baseline model ..................... 15
Age-wealth profile: Models 2 to 4 ...................... 16
Age-wealth profile: Models 5 to 7 ...................... 17
Average propensities to save ......................... 18
Wealth/ income profile: Baseline model ................... 19
Wealth/ income profile: Model 2 ....................... 20
Wealth/ income profile: Model 6 ....................... 21
Age-wealth profile: Stochastic and deterministic economies ........ 22
Steady state comparisons: fuel ........................ 35
Steady state comparisons: capital ...................... 36
Transition path: capital ........................... 37
Transition path: fuel ............................. 38
Transition path: GDP and final goods consumption ............ 39
Impulse responses: Time allocations. .................... 58
Impulse responses: GDP, C, I ......................... 59
Training and GDP, period 1. ........................ 60
Tiaining and GDP, period 2 .......................... 61
{6, €HC’,train-ing} pairs consistent with the steady state ........... 62

Chapter 1

Precautionary saving in general
equilibrium

1. 1 Introduction

It is now understood that precautionary motives for accumulating wealth play a key
role in the consumption/savings decisions of households. At least since the work of
Skinner [1988], Hubbard and Judd [1987], and Summers and Carroll [1987], precaution-
ary savings behavior has been extensively studied. primarily as a candidate solution to
problems in the consumption literature, such as the excess sensitivity puzzle (Zeldes
[1989], Caballero [1991]), and the failure of standard finite horizon models to explain
the observed pattern of consumption growth over the life cycle (Skinner [1988]). Alter-
natively, precautionary motives have been advanced to link the decline in the personal
savings rate over the last 20 years to the extension of social insurance programs such as
Medicare and Social Security (Summers and Carroll [1987]).

This paper is concerned with a research agenda fostered by Skinner [1988], Carroll
and Samwick [1998, 1997], Hubbard et al. [1994] (HSZ), Huggett [1996] and others. The
objective is to study whether a model with realistic lifespans, income paths, and risk
exposure can account for the savings/consumption behavior of US households. In this
line of work, Hubbard et al. [1994] showed that in a calibrated model where the interest
rate is close to the rate of time preference, agents would desire to accumulate levels of

wealth similar to those found in the data. Moreover, evidence was reported that other

model statistics such as the age-consumption profile, and the response of consumption
to innovations in income could also reproduce their data counterparts. In a companion
paper (Hubbard et al. [1995]) these authors focus on the importance of asset tested
programs to explain the low accumulation of assets by the lowest. quintile of the wealth
distribution.

The calibration of these models was criticized by Carroll and Samwick (Carroll and
Samwick [1998, 1997]), on the grounds that it produces a level of sensitivity to changes
in income risk so high that it was impossible to reconcile with their empirical findings.
Instead, they propose a calibration where agents have very high levels of impatience, so
that the rate of time preference is well above of the interest rate. In such model, agents
find it optimal to achieve a target level of wealth over (expected) income, which they
keep until late in their life cycle. Carroll and Samwick report that this model displays
a sensitivity to changes in income risk more in line with their empirical results.

One common finding of this literature is that wealth that is held for precautionary
motives accounts for at least 50% of total wealth. However, these estimates are partial
equilibrium in nature, as prices do not respond to changes in aggregate wealth 1. As
shown by Hubbard and Judd [1987] in a model with longevity risk only, and by Aiyagari
[1994] in the context of an infinite horizon model, general equilibrium effects can be
sizable and tend to increase wealth holding, therefore reducing the estimated share of
wealth that is precautionary.

This paper contributes to the literature by taking an alternative path: imposing the
discipline of general equilibrium, we compute the levels of discount rates consistent with
observed levels of interest rates, savings rates, and income/ longevity risk. We show that
the resulting equilibrium produces interest and discount rates that are very close to
each other, so that agents are not buffer stock savers. Moreover, the age specific saving

behavior that emerges is consistent with average asset accumulation by US households,

 

1An exception is Huggett [1996] who carries a general equilibrium analysis and reports a lower
estimate. However, his focus is on wealth distribution, so there is no discussion of this result.

and displays levels of sensitivity to income risk in line with those reported by Carroll
and Samwick. Finally, we compute the level of precautionary savings that arise in this
model, and show that properly accounting for general equilibrium effects considerably
lowers previous estimates.

This chapter has four other sections. In section 1.2 the model is presented. Section
1.3 discusses the calibration procedure. Section 1.4 presents the results, and Section 1.5

concludes.

1.2 The model

We present a large scale OLG model in the tradition of lmrohoroglu et al. [1995], Huggett
[1996] and Rios-Rull [1996]. In this model, a large number of agents of size 1 live for a
maximum of T periods, are endowed with a level of assets (1.1 at the beginning of their life
(t = 1) and face uncertainty regarding labor earnings and lifespan. Each period, agents
take the interest rate and the realization for labor income as given and must allocate
their earnings between consumption and saving, subject to a borrowing constraint.

Agents take prices as given and maximize a utility function of the form:

T t
max E 7 x 3’11. C
cunt“ lt:1[J-:l—Il [J]; ( t)
81.. at“ + C, 2 (1+ r)a.t + nit/alt + q
art-+1 Z 0,

where it, is a random variable with bounded support that represents a shock to labor
endowment, and (15, is a nonstochastic variable that indexes labor productivity for an
agent of age t. Therefore, 112495,, is the unconditional mean of labor earnings at age t, that
can be thought of as the life cycle component of earnings, and it, is labor endowment of

an agent at age t.

An agent of age t survives to t + 1 with probability 7),, With probability 1 — 17,, he
dies and leaves bequests that are evenly distributed among all living agents, each agent
receives q in bequests every period. Survival probabilities {1),},7‘21, in turn define the
cohort shares {ad}; by n, = (1 — mm,“ and 2;, at = 1.

The household problem can be expressed in recursive form. Let V,(a,l) be the
maximum value of the objective function for an agent of age t with a level of asset

holdings and labor endowment shock {a, l} . Then, Vt(a, l) is given by:

We. 1) = m,ax{u(c> + max/...(a', n01}
(1.,c
s.t. a' + c = (l + 'r)a + 1096,14» q

(L’ 2 0 (P1),

where a’ is asset holdings for next period. Moreover, since an agent lives at most for T

periods, we have:

VT((I'9Z) = I115}X{U((-')}

s.t. c 2 (1+ T)(L + 10qu + q

The solution to this problem are the optimal policy functions Ct(a, l) and A¢(a,l), for
t = 1, ...,T, that map the state {(1, l} at age t to consumption at age t and assets at the
beginning of age t + 1 respectively.

The representative firm chooses {L, K} to solve:
nlralx F(K, L) — RK — wL (P2).

To complete the description of the economy, we define the capital accumulation tech-
nology in a standard way: K (+1 2 (1 — (5)11} + 1,, where I is aggregate investment and
6 is the depreciation rate.

We are interested in a steady state equilibrium where the aggregate capital stock

is constant, and although there is a large amount of dynamics at the individual level,

4

the distribution of assets and other endogenous variables is time invariant. Since a
meaningful equilibrium concept needs to be expressed in terms of these distributions,
we proceed to define them.

Let (X, B, ‘15) be a probability space. If Z is the support for the stochastic shock
it and asset holdings are restricted to lie in [0, 00), then an individual state :1: = {(L, l}
lies in the state space X = Z x [0.00). Let B be the Borel sets in X. Then, for each
t from 1 to T a distribution \Ilt can be defined such that, for each B E B,\II,(B) is
the probability that an agent of age t will be in a state :1: E B . Together with the
stochastic process for l, the optimal policy function A,(a, 1) defines a transition function
P(B, t) = P'rob(:r,+1 E B |:1:,) that links current and future distributions. The function

‘11, is then derived recursively by:
\II,(B) 2/ P(B,t — 1)(1\II,_1 B e B.
X

Equilibrium Definition: A steady state equilibrium for this economy is a collection
of value functions Vt(.), policy functions C,(.) and At(.), t = 1,...,T; prices for labor
and capital services {21),7‘}; aggregate values for {K, L}; a level of per capita bequests

q; and distributions {\Ilt, 13,} for t = 1, ,T such that,

1. Households maximize utility: given q and prices {u}, r}, the policy functions C¢(.)

and At(.) solve (PI) for all t.

2. Firms maximize profits:
Fl" : R = 7‘ + 6

FL = w.

3. Markets clear:

(0 Zr I": “C: + AIW‘I’t + (I = (1 — (5)1" + F(K, L)

(ii) Zilltébt = L = 1

(Ill) 2! [it i Atdqlt = A,

4. Cross section distributions are consistent with policy functions:

‘I’tH = fptd‘l’t

5. All bequests are distributed:

‘1 = 21(1 - "1!)fAtd‘I’i

Aiyagari [1994] presents a characterization of this problem in the context of an
infinite horizon model: agents will overaccumulate assets, with respect to a complete
markets situation, as a way to partially insure themselves against. the possibility of being
effectively borrowing constrained in the future.

The pattern of wealth accumulation is studied by Carroll [1999] in a life cycle model,
and Deaton [1991] in an infinite horizon economy. An important result is that, when
the growth rate of income is sufficiently high for given levels of risk {/1473}, prudence
{0—3;}, and patience {13}, agents 01:)tirnally choose to achieve a target level of wealth
over earnings, or “buffer stock” of assets 2.

Checking whether this condition is empirically plausible is difficult given the unob-
servable nature of the discount rate '7 = 1&2 The next section presents a calibration

procedure where [3 is determined using the general equilibrium nature of the model.

1.3 Calibration.

The calibration exercise is designed so that the stochastic: model economy displays rel—
evant features of the US economy. In particular, the discount factor [3 is left as a free
parameter that takes on the value needed for the model economy to display target levels

of the interest rate and the savings rate.

 

2The condition in discrete time can be approximated by:
r_——_'I + 0+1)0’2 <
0 2 9*

where g is the growth rate of income, ’7' is the rate of time preference, and 0 is the coefficient of relative
risk aversion.

To calibrate the model we need to define functional forms and parameters. The

functional forms used are as follows:

0 A Cobb-Douglas production function is used for all exercises:

F(K, L) = AKOLI‘“

o Felicity functions are of the CRRA form:

0 The stochastic process for the labor endowment is AR(1):
ln(l,) = pln(lt_1) + 6, 6, ~ N(0,03), Md.

The parameter values are shown in table 1.1. For the earnings process we adopt the
results for households with 12-15 years of education reported by Hubbard et al. [1994] (ta-
ble A.4) using PSID data. Figure 1.1 shows the unconditional means. The stochastic
process associated implies values of .946 for p , and .025 for of. These values are roughly
consistent with findings by MaCurdy [1982], as explained below, but imply a variance
in the change of earnings lower than the values in Abowd and Card [1989], who use the
same dataset.

The baseline economy is also calibrated so as to display the following ratios: an
interest rate of 4% per annum, in line with the calculations reported by Kotlikoff and
Summers [1981], and a savings rate of 19%, chosen to match the rate of investment over
GDP of the US economy in the period 1980-1989. These ratios imply a depreciation
rate of .045 per year.

A comment of the calibration choices is in order. Evidence from longitudinal studies
of earnings and labor supply suggest that the process for (log) earnings can best be

modelled as a near unit root process with autoregressive errors of order 2 (MaCurdy

[1982]. This leads Carroll and Samwick [1997] to calibrate their model using a unit root
process with a variance of innovations equal to 0.01.
As suggested by Skinner [1988], we can summarize the risk to lifetime resources

implied by a AR(l) stochastic process with the following statistic:

T—t -
dam—10’ 2
1.; (1+ 7‘)J]

7i: = 0.2 I
where c5, indexes labor earnings at each age. We compute this statistic (the average
over all ages) for our baseline parameters, and compare it with those for Carroll and
Samwick and MaCurdy, properly accounting for the ARM A specification. We find that
the AR(l) process chosen here implies a similar level of risk to lifetime resources (2.54)
than the ARM A(1, 2) proposed by MaCurdy (2.81) and the specification used by Carroll
and Samwick (3.13). Moreover, increasing the variance of innovations from our baseline
of .025 to .031 would be enough to produce a value of 3.13 for this statistic.

With respect to the interest and savings rates, since the empirical equivalent of the
model interest rate is a risk-free rate, we are tempted to use a number in the order of
1 / 2% per annum, consistent with the return on Treasury Bonds. On the other hand
this rate is also the marginal product of capital, so the historical return on stocks, of
the order of 7% annually but more volatile, may also be appropriate. We therefore
experiment with different values of r.

The problem of interpreting the savings rate lies in the fact that households are not
the only source of savings in the US economy. In fact, from 1980 to 1989 the personal
savings rate averaged 6.7% of GDP3, businesses contributed with 12.6 percentage points
to the average savings rate, and the government dissaved .8% of GDP. Of these aggre-
gates, businesses and government reported as capital consumption allowances 10.2 and
2.3% of GDP respectively. Aggregate gross saving was then in average 18.8% of GDP

in this period, close to our benchmark, but the net saving rate was only 5.9% (ERP

 

3This is the contribution of rivate savin to a re ate savin and is different from the Personal
, 7
Saving Rate in the National Income and Product Accounts, which is calculated as saving as a percentage
of disposable (after tax) income, and does not take into account changes in asset. prices

8

[1999]).

A related issue is that, since we focus on a steady state equilibrium, savings net of
capital consumption (depreciation) allowances is zero in the model. Since introducing
growth considerations is beyond the scope of this paper, we check the robustness of our
results to the choice of savings rate by doing some sensitivity analysis with values of
S/Y from 15% to 24%.

Finally, note that Table1.1 show the levels of K/ Y implied by each choice of {7“, S/ Y }
These levels are roughly consistent with the ratios of Assets/ Income reported by Hub-
bard et al. [1994], but are in general higher than the capital-output ratios calculated for
the US economy. Note that once the interest rate and savings rates are fixed, the depre-
ciation rate and the capital output ratio are defined by the conditions .9 = 6K/ Y and
r = F K — 6. Table 1.7, with the capital output ratios that result from selected {7“, S/ Y}
pairs, show that a lower K / Y is associated with higher levels of saving rates and in-
terest rates. While decreasing K / Y to 3 increases the estimated share of wealth that
is precautionary (see table 1.3), it remains that this variation is small compared to the

differences in precautionary wealth associated with different risk aversion coefficients.

1 .4 Results

In this section, we begin by showing how the calibrated model reproduces important
facts about wealth accumulation by US households. We then examine the implications
of these results for the debate on whether US households are buffer stock savers or not.
Finally, we compute the levels of precautionary wealth that emerge in this model and
decompose it in partial and general equilibrium effects.

We evaluate the ability of the model to mimic US data along three dimensions: the
age/ wealth profile, the age-specific average propensities to save, and the sensitivity of
wealth holdings with respect to income risk.

The paper focuses on age-specific aggregate statistics, rather than distributions,

because we believe that most of the intra cohort heterogeneity cannot be explained
by different histories of shocks that are mean-reverting. It is known that this model
compresses the income distribution, generating too few very rich (see e. g. Carroll [2000])
and too few very poor agents. Realistic models of wealth distribution imply types of
heterogeneity in agents that are absent here: in investment opportunities (for instance
Quadrini [2000]), in time discounting (e.g. Krusell and Smith [1998]). or in productivity
(e.g. Hubbard et al. [1994]. Considering these types of heterogeneity is beyond the
scope of this work.

Figure 1.2 shows the predicted average profile of wealth holdings at each age (in
thousands of 1984 dollars), compared to data reported by Radner [1989] using the 1984
Survey of Income and Program Participation (SIPP) database. The model generated
data is normalized so that average income equals 1984 per capita GDP at current prices.

The fit is extremely good given that the model was calibrated to savings and interest
rates of the period only. The feature that deserves attention is the similarity of the
shapes of the two curves, more than the fact that they overlap. In fact, since SIPP data
comprises only private wealth, while the model’s data is on aggregate wealth, and their
measurement units (households in SIPP versus ‘workers’ in model) differ, there is little
reason why they should overlap. The similarities however suggest that, on average, the
model contains the right elements that shape life cycle savings behavior.

Figures1.3 and 1.4 present the age-wealth profiles for alternative parameterizations
of the model, compared to Radner’s data. Clearly, very high levels of aggregate wealth
can be attained by this model with the appropriate discount rate.

We now turn to examine two direct measures of age-specific saving behavior. F ig-
ure 1.5 compares the life cycle profiles of average propensities to save (APS) generated
by the model with their data counterparts, constructed by Gokhale et al. [1996] using
the Consumer Expenditure Survey (CEX) for various years. The two series correspond

to different definitions of disposable income. Conventional disposable income is the

10

sum of labor income, capital income, and pension income minus net taxes, while in the
alternative definition social security contributions are classified as loans (so that they
are considered savings), and social security benefits are classified as the repayment of
principal (not part of disposable income) plus interest on past social security loans.

It is important to note that, once again, it is the shape of the APS curve and not
the level that matters the most. Data. on household savings is data. on net savings,
since businesses make most of the allowances for consumption of capital. In a growing
economy this measure of savings should be positive in the aggregate. In our model,
since we are focusing on a Steady State equilibrium, net savings are zero.

The model prediction follows more closely the APS observed under the alternative
definition, but it tends to overpredict savings rates at the beginning (until around age
32) of the life cycle. Overall, it displays the characteristic hump shape present in the
data, with a ‘plateau’ from ages 35 to 60, and a drastic decrease after age 60.

Figure 1.6 allows an examination of the sensitivity of wealth holdings with respect
to uncertainty, measured in this case by the conditional variance of earnings of . The
wealth / income profile for the baseline model is shown along with the average age-wealth
profile of an agent facing the same prices as in the baseline model but with half of the
variance (1.2% versus 2.4%). The results for two alternative parameterizations are shown
in figures 1.7 and 1.8.

These simulated changes in the levels of wealth holdings are consistent with those
predicted by the regression coefficients in Carroll and Samwick [1997]. Using differ-
ences in occupation specific income risk, Carroll and Samwick regress various measures
of log net worth on the variance of permanent and transitory income shocks, perma-
nent income, and life-cycle variables (age, married, etc). Using the approximation
[log(Wl)log(W2)]/[vm'1 — var2] suggested by the authors, where W1,W2, and vnrrl,
11012 are wealth holdings and income variances for the baseline and alternative paths

respectively, we can approxin‘iate what the regression coefficients would be in the mod-

11

els. The results, presented in Table 1.2, indicate that reasonable parameterizations of
the model can reproduce these coefficients without difficulty.

These levels of risk sensitivity are similar to those reported by Carroll and Samwick
[1997], even though the levels of discount rates and other parameters are very close to
those of Hubbard et al. [1994]. In fact, figures 1.6, 1.7, and 1.8 show that the ratio of
wealth/ income chosen by agents, increases from the beginning of the life cycle, instead
of remaining constant for the first part of it -after a target level is reached- as would be
the case in a buffer stock model.

Using the discount factors consistent with Steady State equilibria in the stochastic
economies, we can predict how large would aggregate wealth be in a similar economy
with no income uncertainty, and no income or lifespan uncertainty. We do so by using
a certainty version of the program, described in the appendix.

The results are shown in Table 1.3. It is worth noting that the levels of precautionary
wealth are significantly lower than those found in similar models. Table 1.4 shows that
the difference can be entirely explained by not accounting for general equilibrium effects.
In what follows, we examine this issue more closely.

Two exercises are carried out. First, we calibrate our model to interest rates, discount
rates, and stochastic paths similar to those in Skinner [1988] and Hubbard et al. [1994].
Rather than attempting a detailed replication, we want to find if given these rough
similarities, our model generates similar levels of precautionary wealth. Table 1.5 shows
the results for two different levels of aggregate earnings, and confirms that in partial
equilibrium this type of model generates high levels of precautionary wealth.

Next, we compare the levels of precautionary wealth generated by these models in
general vs. partial equilibrium (Table 1.6). Given the parameters for the stochastic
process and the interest rate chosen in the original papers, we find the discount factor
consistent with a predetermined savings rate (S / Y=.19 for Skinner and .24 for Hubbard

et al.). Next, we find the level of aggregate wealth in a deterministic economy where

12

agents face the same factor prices (columns labelled PE), and finally we allow prices
to change and compute the general equilibrium effects (columns labelled G.E.).

It is clear that the partial vs. general equilibrium nature of the exercise matters, as
was already noted in Hubbard and Judd [1987] and Aiyagari [1994]. In our examples,
a partial equilibrium estimation of the size of precautionary wealth overstates it by 20
to 50%, consistent with the differences between the findings in this paper and those

reported by Skinner [1988] and Hubbard et al. [1994].

1 .5 Conclusions

An important problem in the study of life cycle savings behavior is whether it can be
characterized by a model of buffer stock versus ‘life cycle’ savings. This paper examines
the issue using the discipline of general equilibrium to sort among alternative models.

We find that a model calibrated to the levels of aggregate savings, interest rates, and
risk exposure found in the data displays life cycle patterns of asset accumulation and
overall sensitivity to risk in line with empirical evidence. This model does not predict
buffer stock behavior. Rather, agents find it optimal to increase their wealth/ income
ratios until shortly before retirement.

At the same time, the equilibrium allocation implied a level of precautionary wealth
around 35% of total wealth, far below comparable estimates in the literature. The dif—

ferences can clearly be traced to the partial/ general equilibrium nature of the exercises.

13

Figure 1.1: Income profile (Hubbard et al. [1994], 12-15 years of education)

 

 

 

 

Table 1.1: Calibration

Income Profile

1.6 . , j , ,1
1.4 f: / -4
1.2 i- .-

1 - ‘
0.8 L .
0.6 i- \ .
0.4 ' . L . - 4 A

20 30 40 50 60 7O 80 90

age '-

 

 

 

 

 

 

 

 

 

 

[ Model ] Calibration Choices Implied Parameters [Stochastic Process ]
S/ Y Interest R. 6 i3 6 K/ Y p 03
Baseline 0.19 0.04 3 0.9815 0.0447 4.25 0.946 2.5%
Model 2 0.15 0.04 3 0.9983 0.0286 5.25 0.946 2.5%
Model 3 0.24 0.04 3 0.9556 0.08 3 0.946 2.5%
Model 4 0.19 0.03 3 1.011 0.0335 5.67 0.946 2.5%
Model 5 0.19 0.05 3 0.959 0.0559 3.4 0.946 2.5%
Model 6 0.19 0.04 5 0.9729 0.0447 4.25 0.946 2.5%
Model 7 0.19 0.04 1 0.9763 0.0447 4.25 0.946 2.5%

 

 

 

 

 

 

 

 

 

 

 

14

Figure 1.2: Age-wealth profile: Baseline model

Age—Wealth Profile:Baseiine Model

 

140 V I 1' r v 1 ‘1 r
120 ~ ,
A a
100 " fl *3
Model i
" 30 r / *SIPP84 -

Assets
\
f

 

 

 

40.. . \ J
20. ,2. .
/
at/
20 I so 40 so so 70 90
Age

Table 1.2: Carroll-Samwick estimates for sensitivity of wealth holdings with respect to
income risk

 

 

 

 

 

 

Model Age<50 Age 1-82
C-S:Per. Var. 12.09 13.27
C-SzTr. Var. 7.11 6.6

Baseline 23.76 18.69

Model 2 21.76 14.88

Model 6 8.61 3.99

 

 

 

 

 

15

Assets

Figure 1.3: Age-wealth profile: Models 2 to 4

180

Age-Wealfli Profile: Models 2 €04

 

160

I

.4
M
O

O!
O

N
O
l

 

 

l

L

 

 

Table 1.3: General equilibrium estimates of precautionary wealth

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model Calibration Precautionary wealth (%)
S/Y Interest R. 0 Lifespan All
Uncertain Certain
Baseline 0.19 0.04 3 27.33 30.09
Model 2 0.15 0.04 3 23.42 26.75
Model 3 0.24 0.04 3 31.94 34.08
Model 4 0.19 0.03 3 22.38 25.63
Model 5 0.19 0.05 3 29.63 32.07
Model 6 0.19 0.04 5 49.63 50.71
Model 7 0.19 0.04 1 6.69 11.81

 

 

16

90

Figure 1.4: Age-wealth profile: Models 5 to 7

Age-Wealm Profile: Models 5 to?

 

Assets

   

140 1 j I I ' ‘I’ 7 I
120 3" be], -
3* A
4a,. +M0dol7 :
100 ”'32; ~Mod~el6 4
4; ~Moda|5 .
\— I.
so ,a, 'SIF‘PB4 -
\A
“‘5‘.
so ,
40 I. ,
20 .

 

 

 

Table 1.4: Partial and general equilibrium estimates of precautionary wealth (%)

 

 

 

 

 

 

 

 

Model Partial Eq. (1) General Eq. (2) 100 x ((1) — (2))/(2)
Baseline 44.92 30.09 49.29
Model 2 33.28 26.75 24.42
Model 3 76.03 34.08 123.11
Model 4 30.33 25.63 18.33
Model 5 60.36 32.07 88.25
Model 6 59.22 50.71 16.79
Model 7 39.34 11.81 233.19

 

 

 

 

 

 

17

Average Propensiliesb Save
0-5 ‘ r V i ' -1 r ' I T 7' r N l
. 0 0 hi
0 - / u all; )1 it it "s. * K ‘
o if as
415 ~ 3., .
\‘x

-1 - -Model a \ 4
*Conventional ) ;
-1.5 - oAlternafive a O\ .l
\ r) i
.2 . .1
_2.5 j I 7 . . I . . a. . f

20 30 40 50 60 70 80

Figure 1.5: Average propensities to save

 

 

 

 

 

Table 1.5: Reproducing results: % of wealth that is precautionary

 

Model Lifespan All
Uncertain Certain
HSZ 1 68.29 71.93
HSZ 2 67.17 70.97
Skinner 1 47.86 50.87
Skinner 2 47.04 50.13

 

 

 

 

 

 

 

 

 

 

18

90

MeanWiY Ratio

Figure 1.6: Wealth/income profile: Baseline model

Wealthllncome profile: Baseline Model

 

 

   
 

 

 

 

 

 

18 ' r I l'—' I r
16,»
14.
12 ~
10 r - BaselineVarianoe
r—1IZBaselineVariance

8

6

4

2

0 , I . . -. . , ..

20 30 40 50 60 70

A96
Table 1.6: Partial vs. general equilibrium effects
Model % prec. % Prec. 100 x % prec. % Prec. 100 x
((1) - ((3) -
(2))/ (2) (4))/(4)
PE. (1) GE. (2) PE. (3) GE. (4)

HSZ 26.63 22.38 19 30.33 25.64 18.32
Skinner 41.62 27.63 50.67 44.6 31.06 43.59

 

 

 

 

 

 

 

 

19

 

 

Figure 1.7: \Nealth/ income profile: Model 2

Wealthllnoome profile: Model 2
25 T , ' ~- . .

 

TI

20

«a
a!

a...
0

.~ ineyananoe
-- : aselmeVananoe

MeanW/Y Ratio

 

    

 

 

 

Age

Table 1.7: Selected capital output ratios

 

 

 

 

 

 

 

 

 

 

 

 

Interest Rate
Saving Rate .03 .04 .05 .06
.16 6.7 5 4 3.3
.18 6 4.5 3.6 3
.20 5.3 4 3.2 2.7
.22 4.7 3.5 2.8 2.3
.24 4 3 2.4 2

 

20

 

MeanWN Ratio

Figure 1.8: Wealth/ income profile: Model 6

 

 

   

 

WIWmeprofileModelfi
18 -. » - . . — . . . . .
16 ~
14i-
i
121»
10$- - BaselineVarianoe
-1IZBaselineVariance
Br-
6 ~ ,
4 __ //.x'
J”
...a/
0 . . M ..
20 30 40 50 60 70
Age

21

 

Figure 1.9: Age-wealth profile: Stochastic and deterministic economies

Age-Wealth Profile: Stochastic and Debuninisfic Economies
140 . T . r . .

 

  
 

120 ~ -All Stochastic
- Lifespan Stoch.

 

100

Assets

“8 8 8, S

 

 

 

r09--.

 

22

90

Chapter 2

Green taxes and double dividends
in a dynamic economy

2. 1 Introduction

The possibility that green tax reform may yield a double dividend has become a ma-
jor issue in the environmental policy arena. The double dividend hypothesis is nicely
exposited in Goulder [1995] and Bovenberg [1999]. Apart from decreasing pollution
externalities, a ‘green’ dividend, environmental taxes raise revenue that can be used to
lower other (pre—existing) tax distortions, resulting in a smaller deadweight loss from
the tax system, or ‘efficiency’ dividend. Because of the appealing nature of such reform
Bovenberg calls environmental taxes a ‘no regret option’.

Most of the work on the double dividend problem addresses the question from a
normative perspective and in a static framework. Examples include Bovenberg and
van der Ploeg [1998], Carraro and Soubeyran [1996], Holmlund and Kolm [1995], and
Koskela et al. [1998]. In an influential paper, Bovenberg and de Mooij [1994] use a
model with two goods (one clean, one ’dirty’), and labor as the only input, to examine
whether increasing the tax rate on the polluting good above its Pigovian level, and
reducing labor taxes in a revenue neutral fashion will deliver a welfare gain. In this
version of the double dividend question, the distortionary effect of increasing green
taxes above the level at which the marginal pollution damage is internalized should be

compared to the efficiency gains from reducing other taxes. Bovenberg and De Mooij

23

find that the efficiency dividend does not materialize, since green taxes turn out to be
more distortionary than the labor tax by virtue of their effect on the composition of the
production bundle.

The previous result relies heavily on the static nature of the model. A few papers
have extended the discussion to a dynamic setting. Bovenberg and de Mooij [1997] study
the impact of environmental tax reform on long run growth. In their model, where
production externalities decrease the productivity of capital, a tax shift from output
taxes to pollution taxes always achieves a green dividend, and an efficiency dividend is
obtained only under certain parametric conditions. Perhaps the closest paper to ours
is Bovenberg and Smulders [1996]. In this paper, the transitional dynamics of a growth
model are examined after a tightening of environmental standards occur. Starting from
a situation where green taxes are below their Pigovian levels, the authors study the
conditions under which the efficiency dividend is obtained.

This paper departs from previous literature on the double dividend hypothesis in
that it examines a policy rather than a normative question. We are interested in the
environmental and efficiency effects of green tax reform in the US. As Bovenberg and
Smulders, we compute the transitional path after a policy change, but unlike their paper,
the policy change in our experiment is a revenue-neutral tax reform, intended to address
the double dividend question, and involves an actual calibration to the US economy.

Unlike previous work, this paper focuses on the effects that the higher levels of capi-
tal accumulation resulting from a more efficient tax system may have on environmental
quality. After Lucas [1990], and Hall and Jorgenson [1967], a large body of evidence
suggests that tax changes have a first order effect on the level of the capital stock.
Moreover, this effect is important enough that we would expect it to overshadow the
effects of pollution on productivity emphasized in previous work. Our focus on capital
taxes is in part dictated by the mechanism we want to examine. By choosing to re-

duce capital taxes, as opposed to labor or consumption taxes, we are maximizing the

24

efficiency dividend and therefore, by promoting capital accumulation, betting against
better environmental quality. If it turns out that the green dividend is also achieved by
decreasing capital taxes, then it will most likely be achieved by shifting the tax burden
from any preexisting tax to green taxes.

In our model, which is described in section 2.2, there are a large number of identical
infinitely lived households. Households value clean consumption goods, dirty consump-
tion goods, like gasoline, and their stock of health. Final goods and services are produced
using capital, which is clean, and a dirty input, fuel. Fuel is produced with capital as
the only input. Health is a function of the stock of pollution. Pollution is augmented by
current fuel use, both in consumption and in production and depreciates like the other
capital stock. The government collects taxes on fuel and on capital income. In the main
policy exercise considered, increased revenue from green taxes is used to reduce the tax
rate on capital earnings. As noted above, by choosing to reduce the highly distortionary
capital taxes, we all but ensure that the efficiency dividend will obtain, and we focus on
the environmental effects of the policy change.

We find that, because a more efficient tax system encourages capital accumulation,
the environment may worsen in the new steady state. However, in all the cases we
consider, a double dividend is obtained during a very long period of time at the beginning
of the transition. We conclude that a tax reform experiment of the nature explored in
this paper is most likely to be Pareto improving.

This chapter has four other sections. In section 2.2 the model is presented. III
section 2.3 functional forms and parameter values are chosen, then section 2.4 presents

the results, and section 2.5 concludes.

2.2 The model

The economy is populated by a large number of infinitely lived individuals. We abstract

from population growth and normalize population size to unity. Preferences of the

25

representative individual are given by

2 Hum, mar; 12,), (2.1)
t:0

where C, is consumption of the single perishable consumption good at time t, ma is the
amount of fuel consumed at time t and h, is the state of health at time t, B is discount
factor which is a real number between zero and one, and u is felicity. We find it useful
here to disaggregate consumption goods into two types: one good, which is associated
with negative pollution externalities, we call fuel, in“, and the other good, which is not
associated with such externalities we refer to as the consumption good, ct. The elasticity
of substitution between these two (consumption) goods will play an important role in
the following analysis. In the utility function specified in equation (2.1) the state of
health, ht, enters as a separate variable. Health here is a capital stock, which is taken
as given by each individual, but which depends upon the aggregate amount of pollution
in the economy.
The relationship between health and the aggregate amount of pollution, z,, is given
by
ht = h(zt). (2.2)

The consumption good is produced via a constant returns to scale technology using two

inputs, capital km and fuel mpt. The production function is given by

y, = f(kpt, mpt). (2.3)

Fuel is produced using capital km, only. The production function for fuel is given by
m, = 9(kmt). (2.4)

There are two capital stocks in this economy, physical capital (kt) that can be used in
the production of the consumption good (km) or fuel (km), and the stock of pollution

(at). These two types of capital evolve according to

514.1 = (1— (5)6] + ii) (2.5)

26

zt+1 : (1—— 6Z)zt + mt, (2.6)

where it is investment in physical capital at time t. In this economy, fuel m, can be used
as an input in the final goods sector, mm, or consumed, met. The initial endowments in
this economy are kg and 20.

The government in this economy collects taxes on capital income at the uniform rate
77,, and taxes on household fuel consumption and fuel use by firms at the rate Tm. All
tax revenue is rebated in a lump sum fashion to the households.

The representative household in this economy solves the problem

max 2 ,Btu(ct, met; ht), (2.7)

. e 30
((1‘) A’f+117r)’Cf)t:0 ‘20

subject to
00 00
ZPthr +7} + (1 + Tmlwtmct) = Zl’tffl - Tklfhkt + 7rmt + TL)»
t=0 t=0
kt+1:(1_ (ilk: '1‘ it:
given
k0) {7):,(1c,wt,hl}f:o,

where pt is the price of final goods at time t, wt is relative price of fuel compared with
final goods at time t, qt is return to capital at time t. Here mm are profits from producing
fuel and T, are the lump sum transfers from the government. The final goods producing
firm solves the problem
max f(kp¢,mp¢) — qtkpt — (1 + T,,,)-'wtmpt. (2.8)
{kph mpt}
The fuel producing firm solves the problem

III'CIX “Hf/(knit) — thtnta (2.9)

mt
We do not allow the government to run a deficit or surplus, so the government budget

constraint each period is

27

T,,,-u.',,(mct + mm) + qu,kt = T,. (2.10)

An equilibrium for this economy is an allocation for the representative household
{chmch kpt+1,kmt+1}f:0, an allocation for the final goods producing firm {kpt,rr2.p,}j’f’__0,

an allocation for the fuel-producing firm {k,,,,}f_:0 and prices [10,, q”, qmt},°:0 satisfying

1. the household’s allocation solves the maximization problem in (2.7),

2. the final goods producing firm solves the maximization in problem (2.8),
3. the fuel-producing firm solves the problem in (2.9),

4. the fuel and capital markets clear, and

5. the government budget constraint (2.10) is satisfied.

2.3 Calibration

In this section we restrict numerically the model by choosing functional forms and
parameter values. For the utility function, we need to choose a functional form that
allows us to match the observed income elasticities for household fuel demand. In most
standard utility functions, such as Cobb-Douglas or CES, the implied income elasticity
is unity. Schmalensee and Stoker [1999] find an estimate of this elasticity of 0.2. Other
estimates of this elasticity are in a neighborhood of 0.2 (Puller and Greening [1999]).

To allow for varying income elasticities we pick the following utility function:

1

1_ 00"“ng + (1 — 9)772.£t)””)1‘0,

 

~11.(Ctamct; hr) =

§>0,p>0,0<6<1,0<n<1,021. (2.11)

We also choose a CBS production function, that allows for a response of input use

to changes in relative prices in accordance with microevidence.

28

f(k,,,,m,,,) = Am; + (1 — x)m;",_]’/°‘, A > 0, a < 1, 0 < x < 1. (2.12)

The production function for fuel is Cobb-Douglas in one input capital, so that
g(k,m) = 151.33,, 0 < a < 1. (2.13)
Finally, the health production function is of the form,
[1(a) = l/zt. (2.14)

\Ne calibrate our model to the US economy. The benchmark parameters we use are
illustrated in Table 2.1. Calibrating the utility function to long-run data is a bit tricky
since preferences are not homothetic and expenditure shares do depend upon the level
of income. We thus pick preference parameters E and p that match observed income
elasticities at the steady state. Most estimates of this income elasticity are in the
neighborhood of 0.2 (See Schmalensee and Stoker [1999]). But we experiment with
higher and lower values for this elasticity. The elasticity of substitution parameter, Oz,
in the CES production function is set equal to —O.5, making substitution between fuel
and capital slightly more difficult than in the Cobb-Douglas case. The case of a 2 ~10
was also considered with no substantive change in the results.

We choose a value for the preference parameter 77 that places a relatively small weight
on health. Since health is not a choice variable for the household, the choice of 17 has an
effect on the allocation only through the effect of fuel consumption me on future health
levels.

The preference and technology parameters {A,9, x} are chosen so that fuel usage

by household out of total usage is 30% 1 and fuel share of GNP is 7% 2, and the

 

1The Statistical Abstract of the United States 1999, table 955 contains data on fuel use which is
broken down into the following categories: residential and commercial, industrial, anti transportation.
We assign 50 percent of fuel use in the residential and commercial category to fuel use in consumption.
Over the period from 1970 - 1997, households used 30.75 percent of all fuel.

2According to the Statistical Abstract of the United States table 958 and table 727, expenditure on
fuel as a fraction of GDP in the US for 1995 is about. 7%.

29

household’s expenditure share for fuel is about 3.5 percent 3. We know very little about
the technology parameter 0). We execute some sensitivity analysis and find that our
qualitative results are robust to changes in '1/1. We assume that the depreciation rate for
capital is 10%, and we choose 6,, to be consistent with the steady state conditions.
The Statistical Abstract of the United States 1999, table 793 contains data on the
state tax rates for gasoline in 1997. Together with a federal gasoline tax of about 18

cents for a gallon, the average tax rate for gasoline is around 50%

2.4 Results

To solve this model, we first obtain the steady state using a Newton-Raphson proce-
dure, then we linearize the first-order conditions around the steady state and solve the
resulting difference equations. The approximation errors that result are very small, as
evidenced by the Euler residuals:

'II.(.(‘f.)
11,.(1 + 1);3(1 + n+1— 6)

 

—1

shown in Table 2.2.

We now report the results of our main experiment, a revenue neutral tax change. In
this experiment, we raise the fuel tax and adjust the capital tax to keep the government
share of GDP constant at 35%. For ease of exposition, we concentrate first on steady—
state comparisons, and examine the transition path later.

Figure 2.1 shows fuel usage in steady state as the tax on fuel increases for the baseline
parametrization. It is clear from this figure that the steady state level of aggregate fuel
consumption is not monotonic in the tax rate on fuel. In fact, fuel use by household is
monotonically declining in the tax rate, as the substitution effect dominates the income
effect because of the small income elasticity. Fuel use by firms, however, increases in

the tax rate. This is an expected result, since higher tax rates on fuel are accompanied

 

3’This is slightly lower than the average share of household income allocated to fuel estimated by
Chernick and Reschovsky [1997].

30

by lower capital tax rates, and therefore higher steady state levels of the capital stock.
When the fuel tax rate is low (high), the former (latter) effect tends to dominate, giving
a hump shaped relationship between tax rates and aggregate fuel usage.

The steady state levels of the capital stock as Tm changes are depicted in figure 2.2.
While the amount of capital devoted to fuel production stays roughly constant, capital
in the final goods producing sector increases as the tax on capital income is reduced and
the tax system becomes more efficient.

Figures 2.3 to 2.5 show the transition path from period 11 (time 1), when the policy
change occurs. At time 1, the higher tax rate on fuel generates, via a substitution effect,
a sharp decrease in fuel consumption (figure 2.3). The lower tax rate on capital earnings,
however, creates incentives to accumulate capital (figure 2.4). Since capital and fuel are
complements in the production of the capital good, fuel consumption by firms increases
monotonically from time 1 (period 11).

Figure 2.5 shows the evolution of GDP and consumption of the final good. At the
time the policy change takes place and the rate of return to capital jumps, more capital
is devoted to produce final goods kp, and the relative price of fuel must fall to avoid
arbitrage opportunities. As capital is accumulated however, the relative price of fuel
increases reducing the share of capital used in final goods production. This composition
effect operates to reduce final goods consumption after period 11, but eventually the
growth of the capital stock takes over the dynamics of this variable. Note that for a
period of about 25 years (period 15-40 in figure 5) the household has a lower level of
consumption of both fuel and the final good.

We now turn to the welfare effects of this policy experiment. To disentangle welfare
changes from different consumption paths and different health stock paths, we first
compute the level of discounted utility during the transition to the new steady state,
assuming that households enjoy the levels of health of the original steady state. we

then calculate by what percentage should consumption (of both fuel and the final good)

31

increase for both discounted utilities (original steady state and transition) to be equal,
and call this number the efficiency dividend. Next, we do the same exercise but now
holding consumption at the level of the original steady state, and comparing discounted
utilities where only the stock of health changes. We call this second number the green
dividend.

Table 2.3 shows both dividends for the baseline case, where Tm increases from .5 to
.55, as well as for alternative tax changes. Note that the green dividend is obtained
in all cases, but the efficiency dividend obtains only for target levels of the green tax
below a critical level, around .5. Above this level, we conjecture, the green tax becomes
more distortionary than the capital earnings tax, and the results mimic those in most
of the green tax literature. Below this level, however, both dividends are realized. With
the caveat that we know little about how health enters into the utility function, we can
appeal to linearity and add both columns of table 2.3 to get the aggregate welfare effect.
This effect is always positive, since the green dividend is always larger -by as much as
two orders of magnitude in the baseline case than the efficiency dividend.

Summarizing, even though in steady state comparisons the efficiency dividend always
holds, and the green dividend is in doubt, the transition path shows that the a lower level
of pollution will be enjoyed for a very long period during the transition. In order to build
a larger capital stock however, a lower level of consumption must be endured for some
years near the beginning of the transition. The green dividend, or higher discounted
utility from a cleaner environment, will then always obtain, while the efficiency dividend,
or higher utility from consumption of market goods, obtains under most, but not all tax

changes.

2.5 Conclusion

In this paper we have studied whether a green tax reform actually does deliver a double

dividend in a model calibrated to the US. economy. Our answer is a timid yes. In our

32

model, raising a green tax does indeed allow a pre-existing tax to be decreased, here
a tax on capital income. Cutting capital taxes stimulates investment and growth, so
green taxation does yield one dividend. If capital and fuel are complementary inputs,
however, increasing the capital stock raises the demand for fuel which can, depending
on the degree of complementarity, offset any decline in fuel use due to higher fuel taxes.
While this offsetting effect is important in steady state comparisons, it is dwarfed by
substitution effects that decrease the consumption of fuel and thus deliver a better
environmental quality for a very long period along the transition path. A green dividend
is then always achieved, but the growth dividend is achieved only when the target level

of the environmental tax is below a critical level.

33

Table 2.1: Benchmark parameters and data

 

Ereference parameters ]

 

 

 

 

 

 

 

 

 

 

 

,1” 0.979
a 3
6 0.925
7) 0.9
f 0.83
p 0.17
] Technology parameters

 

[ Final good production 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A .111
a -0.5
x 0.983

fluel production j
E 1
1,6 0.5

[Bepreciation rate T]
6 0.041

1 6,, 0.1

[ Data ,
fig 0.3
mm / ('1’ N P 0.07

 

 

 

 

Table 2.2: Euler residuals (er)

 

 

 

 

 

 

Change of tax rate on fuel max [er]
.5 to .55 4.9 x 10“6
.65 to .7 3 x 10-6
.5 to .6 1.8 x 10‘5
.35 to .4 8 x 10‘
.3 to .35 9.4 x 10—6

 

 

 

 

34

Table 2.3: V‘Velfare analysis: Compensating variation (%)

 

 

 

 

 

 

Change of tax rate on fuel Growth dividend Green dividend
.5 to .55 -.008 .16
.65 to .7 -.026 .15
.5 to .6 -.0084 .33
.35 to .4 .038 .18
.3 to .35 .056 .19

 

 

 

 

 

Figure 2.1: Steady state comparisons: fuel

mc(').mp(-).m(0)

 

 

 

 

 

0.4 T l l l I I T F I
O O O o O O O O O O O O O o O O O C)
o
0.35,; 0 -
03-
E025- a
O.
E.
U
E 0.2— .
045- -
it
if 3!
0.1- ”g "f x a
X * 3‘ )K
3* x
X X M )K 3( X * * 3‘
005 l l l l l 1 l l l
O 01 02 03 04 05 06 07 08 09 1

tax rate on fuel

Figure 2.2: Steady state comparisons: capital

kpf').km(‘).k(0)

 

NV

 

 

 

 

6 I I T T T T l T To
0
o x
003*"
03"!
of?"
55 Ox
0)!
Ox
0*
0*
4- 0“
0"
out
A *
% 0
3K

§3~o
dBK
if

2..

11- -<

0 111111 111

0 OJ 02 03 04 05 06 O] 08 09

mxmmonmm

36

 

Figure 2.3: TTansition path: capital

k(-).km(0).kp(.)

(1995

(199

M)

(k km

(1985
(198

(1975

tltttlttttrlttltn
»:Iltctillttltl

~‘llillllll

(lllltltlllllll

.. “'llllll

.tgnzlurttttlattrltltltl

(197
0 10 20 30 40 50 60 70 80 90 100

penod

37

 

(m.mp.mc)

Figure 2.4: Transition path: fuel

mt).mp(-).m0(')

0.995

0.99

 

0.985

0.98

..........

..........;:;: ...........

.-.......-...;; ...........
...-,,,,._ ------------
-~--......:-:‘ - -----------
..............

...............
............

0.975
0 1O 20 30 40 50 60 70 80 90 100
penod

38

 

Figure 2.5: Transition path: GDP

y(*).c(-)
1.0025 .

and final goods consumption

 

1.002 c

1.0015

I

1.001 r

(v.0)

I

10005

 

0.9995 ‘
0

 

 

40 50 60

70
penod

80 90 100

Chapter 3

Training and business cycles

3. 1 Introduction

This paper studies the behavior of skill acquisition through training at business cycles
frequencies. Beginning with Mincer [1974] and Porath [1967], human capital accumu-
lation has been extensively studied as one of the main determinants of productivity
growth along a worker’s life cycle. Human capital investment also plays an important
role in accounting for cross country differences in growth rates in the empirical litera-
ture spanned by Barro [1991] and Mankiw et al. [1992]. In an influential paper, Lucas
[1988] suggests that human capital investment is the main force driving long run growth.
While the literature on human capital accumulation on both the life cycle and aggregate
growth dimensions is vast, we still have a limited understanding of the mechanics of skill
acquisition over the cycle. This paper contributes to bridging this gap.

Understanding the behavior of skill acquisition during the cycle has potentially im-
portant implications. From a policy perspective, firms implement training programs
-and workers engage in them— after deciding that future benefits in terms of higher pro-
ductivity offset current opportunity costs. If training turns out to have a strong cyclical
component, then the rate of return on the large number of government-sponsored train-
ing programs might be affected by the time at which they are implemented. Moreover,
the cyclical behavior of training may help explain both the phenomenon of procyclical

productivity, and the empirical finding that recessions tend to be followed by periods of

40

 

higher than average productivity growth (see Bean [1990] and Saint-Paul [1996] ). In
both cases, it has been conjectured that training occurs in downturns and the economy
starts a new cycle with higher levels of human capital. Finally, we will argue that the
behavior of skill acquisition is strongly linked with the ease with which firms can adjust
their factors of production, so that the analysis in this paper will shed light on the
cyclical behavior of employment, hours, and labor productivity.

Our empirical knowledge of human capital investments during the cycle is limited
to Dellas and Sakellaris [1996], who study skill acquisition activities through formal
schooling. In that paper, a database of college enrollments is constructed and the
cyclical properties of the series examined. The authors report that college enrollments
are countercyclical, and strongly tied to local labor market conditions.

The theoretical implications of skill acquisition activities are explored in a limited
number of papers. DeJong and Ingram [2001] estimate a real business cycle (RBC)
model with human capital production, investment goods and final goods sectors, and
a rich stochastic specification. In their paper, the authors note that there is a lack of
usable data on skill acquisition at high frequencies, and address this issue by using their
model to infer what the behavior of skill acquisition should be given the realization
of the remaining variables. Using a maximum likelihood procedure, they find that a
countercyclical and highly volatile behavior of skill acquisition time provides the best
fit of their model to existing data.

Einarsson and Marquis [1998] show that adding human capital accumulation can
improve the capacity of a model to match the low observed correlation between hours
and productivity. A second paper, by Perli and Sakellaris [1998], shows that the coun-
tercyclical allocation of resources from a goods producing sector to a sector producing
human capital adds a strong propagation mechanism to the standard model. While these

results are important, we believe that the ability to assess their empirical relevance is

41

hampered by lack of reliable data 1.

This paper is closer in scope to DeJong and Ingram [2001] in that we are interested
in the cyclical behavior of the types of skill acquisition that occur after workers leave
formal schooling. In so doing, our study complements the work by Dellas and Sakellaris
[1996], who focus on skill acquisition through formal schooling. This paper contributes
to the literature in two ways. One, it is the first paper to construct a time series of
training activities, and to document its cyclical properties. Two, it highlights the role
of labor adjustment costs in explaining the cyclical behavior of skill acquisition.

Our results show that training, both on and off-the—job, is weakly countercyclical,
leads the cycle, and has a standard deviation of more than ten times that of output.
We show that a standard RBC model with human capital accumulation is unable to
reproduce this volatility, but a. model with empirically plausible adjustment costs of
employment can.

The rest of this chapter is organized as follows. Section 3.2 describes the data used
and documents the regularities to be explained by a business cycle model. Section 3.3
presents the model. Section 3.4 calibrates the model, and section 3.5 presents the results.

The last section concludes.

3.2 Data

3.2.1 Data description

In this study we use the National Longitudinal Survey of Youth 1979 (NLSY79) as the
source of training data. The N LSY79 is a longitudinal survey of 12686 individuals who

are interviewed every year from 1979 to 1994, and every two years since until 1998. The

 

1The data on human capital used in Einarsson and Marquis [1998] and Perli and Sakellaris [1998]
is a series constructed by Jorgenson et al. [1987]. To construct this series, at every period classes of
workers are aggregated using both their wage levels (which are intended to measure the level of human
capital) and relative weights in the workforce. While wages are at best weakly procyclical, it is well
known that low wage workers drop out of the workforce in higher proportions during recessions, and
return during booms. The resulting index shows a clear countercyclical pattern, but this is influenced
by the effects of the cycle on the composition of the workforce, and it. is unclear to what extent it
measures skill acquisition activities.

42

same respondents are followed every interview year without replacement, so that the
age distribution of the sample ranges from 14 to 22 years in 1979 and from 34 to 44 in
1998.

With this dataset we first construct a quarterly panel from 1978Q1 to 1998Q4, using
questions on the incidence and time spent in training, the type of training provider,
working status (working/not working), industry code, and education level (less than
high school, high school and some college, college graduate). The questions on train-
ing, however, are not consistent across time. From 1979 to 1986, the survey registers
information on up to three training programs in which the respondent enrolled for more
than one month since the date of last interview, and up to two programs in which the
respondent was enrolled at the time of the last interview. In 1987 no training ques-
tions were fielded, and in 1988 no information is recorded about training programs in
which the respondent was enrolled at the time of last interview. From 1988, informa-
tion is recorded on up to four training programs started since the date of last interview,
regardless of the duration of the program 2.

The questions on the type of training provider are used to separate training into
On-The—Job (OJT) and Off-The-Job (OFFJT) training, a distinction that intends to
separate firm-specific skill acquisition (OJT) from investments in general skills (OF -
F JT). The assignment of training programs to either OJT or OFF JT is done according
to whether the program took place at the workplace or not. Table 3.1 details how
the different N LSY training categories are aggregated in On-The—Job and Off-The—Job
training.

Once the panel is constructed, there are 12,686 individuals, with each being observed
for up to 84 quarters, or 1,065,624 observations in total. The NLSY79 oversamples the

military population, and this subsample is dropped (107,520 observations). Further,

 

2To construct this panel, we consider a respondent to be in a. training program in any quarter if he
was enrolled in training for more than one of the three months. For consistency between time periods,
we use a maximum of three training programs per year for each respondent, which has negligible effects
on the resulting series.

43

all observations prior to the respondent last being enrolled in school, and posterior to
the respondent’s last interview are also dropped (414,976 observations) 3. Finally, we
divide the sample into two subperiods to address the problem of data inconsistency, as
explained below.

A total of 543,128 observations remain, and are matched with business cycle indi-
cators: GDP and Investment from the Bureau of Economic Analysis, and industrial
production by 2-digit industry from the Board of Governors of the Federal Reserve
System, for respondents who report working in manufacturing industries (SIC 20-39).4
Table 3.2 shows descriptive statistics for this data, and Table 3.3 describes the nomen-
clature. This panel is then used to produce time series at different levels of aggregation:
by education groups, working status, etc.

The NLSY79 data has the natural advantages of a panel dataset over aggregated
data, and for the purpose of this study it contains extremely detailed information on
training at the individual level. There are also two potential problems associated with it
when used to construct an aggregate time series. The first problem, as explained above,
is that the data collection criteria for training questions previous to 1986 and posterior
to 1988 are not entirely consistent. The second potentially important problem is that
we observe the same individuals at every time period, so we must be careful to filter out
life cycle effects. We discuss these questions in turn.

To address the question of data consistency, we choose to divide our sample in two
subperiods, and report our results separately for both. Additionally, we choose to keep
in our time series only the periods with more than 3500 observations. Since about
3 to 4 percent of the respondents are enrolled in a training program in each period,

this procedure mitigates sampling errors to be amplified in the time series. With these

 

3There is also an oversampling of the economically disadvantaged groups, and this subsample is
kept, but weighted accordingly in the calculations of means, etc

4GDP and Investment downloaded from www.Economagic.com on November 10,2001. Two-digit
industrial production downloaded from the Board of Governors of the Federal Reserve System’s web
site (www.federalreservegov) on October 25,2001.

44

adjustments, we have two periods of valid data, one from 1979Q2 to 1985Q4 (Period 1),
and 1989Q1 to 1997Q1 (Period 2).

For the life cycle problem, we begin by noting that at any period a 9 year window
of the age distribution is observed, and this cross sectional variation can be exploited
in order to separate life cycle effects from business cycle effects. To do this we run
a pooled regression of training variables (incidence and hours) on time dummies, that
capture business cycle effects, and age and age squared, that capture life cycle effects.
While we find, using an F test, significant coefficients for OF FJT that suggest declin-
ing investments in general human capital, we find no such effects for OJT programs.
With these results in hand, we choose to use the time series data without applying any
transformation, other than logging the hours variables, and we will report the results

for both types of training programs.

3.2.2 Stylized facts

We now document the cyclical properties of the variables. The correlations with out-
put (Table 3.4) of hours in production, investment and productivity display well known
regularities: all variables are strongly procyclical, with the exception of labor produc-
tivity in period 2. The correlations between the training series and other business cycle
variables are shown in Table 3.5. The results are consistent for both time periods, and
suggest that both OJT and OF F JT are countercyclical. Table 3.5b, computed using
pooled data, shows that these correlations are statistically significant when estimated
taking advantage of individual variation in the data. 5

A disaggregated analysis (not reported) reveals that this pattern is broadly consistent
across education groups, with hours in training of higher educated workers showing
stronger (larger in absolute value) negative correlations with output and investment.

Only hours in off-the—job training (hofi ) in period 2 shows a weakly procyclical pattern.

 

5Note that the results from tables 3.5a and 3.5b are not comparable: table 3.5a uses log aggregate
hours in training at time t as the training variable, while table 3.5b uses hours in training at time t for
each individual.

These results suggest that skill acquisition activities that would tend to make hours in
on-the-job training (hojt) procyclical, such as those associated with the acquisition of
new capital goods, are unimportant in the aggregate and are small compared to training
activities driven by opportunity cost considerations.

Cross correlations of output with investment, hours and productivity (Table 3.6),
and training variables (Table 3.7) are discussed now. In our sample, hours are not
a leading indicator, but show the strongest correlation with contemporaneous output.
This is also true for investment and productivity, with the noted exception of output
per hour (prodh) in period 2. Table 3.7 indicates that, while training incidence does
not show a clear pattern (it lags output by 2 quarters in period 1, and seems to lead in
period 2), the response of hours in training clearly leads output by as much as 3 quarters
(HOJT, period 1).

Finally, table 3.8 shows the volatility of aggregated variables. The standard deviation
of the (log) hours in training variables is extremely high, as much as 17 times that of
output and hours (see figures 3.3 and 3.4) . Note that this is not an artifact of training
having a growth or life cycle trend. When detrended using a Hodrick-Prescott filter,
the volatility of HA CG decreases to .137 and .174 for periods 1 and 2 respectively, and
a regression of HA CC on year and year squared show insignificant (at 5%) coefficients
for both subperiods.

At any quarter, note that only about 4% of workers are enrolled in training pro-
grams, and in average about 1.58% of aggregate hours are spent in training, so the
proportionally large cyclical variations observed in training hours and incidence need
to be weighted by these scaling factors. But even with this caveat, the high volatility
of training hours makes training a margin with an importance of the same order of

magnitude as employn‘ient in adjusting aggregate hours.

46

3.3 Model

In section 3.2 we presented a description of the high frequency regularities of the training
series. Some of these regularities, such as the high volatility and moderate correlation
with the main cycle indicators, are stark and defy obvious explanations, hence providing
a test of high power that can be used to discriminate among competing models.

A natural place to start is the model by Lucas [1988], where human capital accumu-
lation is the driving force of long-run growth. A close examination of this model, or a
RBC version of it (see Appendix B), shows that it fails along important dimensions. In
particular, it cannot reproduce the high volatility of skill acquisition hours, it tends to
produce too high degrees of countercyclicality, and fails to match the observation that
training leads the cycle.

The standard RBC model with human capital fails to reproduce these facts because,
we believe, it does not incorporate the frictions that make some dimensions of labor
input, such as employment or weeks worked, much costlier to adjust than others, such
as hours per week. In this vein, we will explore the hypothesis that this high volatility
of the training variables is driven by the difficulty in adjusting employment at business
cycles frequencies due, for instance, to the existence of hiring and firing costs. This
difficulty creates a labor hoarding effect, so that the marginal product of labor is adjusted
partially on the extensive margin -employment or weeks worked- but mainly on the
intensive margin of hours devoted to work / skill acquisition.

To examine this hypothesis, we construct a model where it is costly to adjust the
number of weeks worked every year, while marginal adjustments in hours per week can
be done at the prevailing wage. we do this in the spirit of Bils and Cho [1994], who
examine the behavior of employment, hours and effort over the cycle. In this model,

firms produce a single good using a Cobb—Douglas production function

K : AtK?(ltNth)l-O. (3.1)

47

Here Y is output, K is capital, N is weeks worked per quarter, 1 is hours per week
devoted to work, and H is human capital. A, a measure of total factor productivity, is

a random variable that follows an AR(1) process:
At+1 : p/It + ff 6! N 1’V(0, 062). (3.2)

To produce, firms hire capital and efficiency units of labor [N H , and pay the costs of

adjusting weeks. These costs take the quadratic form:

B[N, — NH]2

C(AAQ) = 2

(3.3)

The existence of adjustment costs implies that the firm faces a dynamic problem when
it chooses employment, and it may incur negative profits at time t. While this cost
structure is standard in studies of aggregate employment dynamics (see Hamermesh
and Pfann [1996] for a survey), there is strong evidence that it is a poor approximation
of the structure of labor adjustment costs at the firm level. At least two features of such
a structure are absent in (3.3): (a) a fixed cost of adjusting, that drives a pattern of
lumpy adjustment at the firm level 6, and (b) asymmetric costs of positive (net hirings)
versus negative (net layoffs) changes.

The lumpiness in adjusting employment, however, disappears once data is aggregated
over firms (see Hamermesh [1989]). Since we do not have a panel of firms that would
allow for modelling the firm’s decisions when facing fixed costs of adjustment, and then
aggregating over firms, the approach in this paper is to use “reduced form”-equation 3.3-
to interpret aggregate labor market observations. We believe that this approach is
useful, in that the main mechanism that drive our results, namely the relative difficulty
of adjusting employment versus hours, is explicitly modelled.

The second feature of the firm-level structure of adjustment costs that is missing

in our framework is the asymmetric nature of these costs. The evidence surveyed by

 

6This first point is most clearly made in Hamermesh [1989]. In a study of seven large plants,
employment was found to adjust only after deviations of actual output from expected output reached
60%. Using a flexible parametric specification, the author reports significant fixed and marginal costs
of adjusting employment. The same qualitative results were found in a study of airline technicians
(Hamermesh [1992]).

48

Hamermesh and Pfann [1996] suggests that firing costs are larger than hiring costs, and
that this asymmetry might still be present in aggregate data. We believe that, although
the question is important, it goes beyond the scope of this paper, given that our dataset
does not allow for observing firm-level data.
If we let 7‘, be the interest rate net of depreciation, and using the conventions r0 = 0
and R, = 1', + 6, the firm’s problem can be stated as:
00 t 1

max E0 2( H

{Kf’Ht’ltthhZU i=0 r=ol+Tr

 

) (14t1{t(l(ltNth)I—a — 'IUtltNth [P1]

I Art—N(_1 2
—R,I\,—B 2 ).

Households have preferences defined over consumption {0,},321, weeks worked {Nt},°:l,
and hours per week devoted to on the job activities: work plus skill acquisition activities
{1, +71, Bil. The distinction between weeks and hours per week allow for the study of two
margins of labor input with different cost structures. From the perspective of individual
preferences, casual observation indicates that most workers choose an internal solution
to their problem of allocating time resources on both margins. Moreover, data on the
behavior of weeks and hours per week are available to restrict these preferences. The

utility function is similar to that proposed by Bils and Cho:

 

oo nt+lt)l+cp 1V,l+¢
U=E "'l'gc+mN,—(——
ogflio t t 1+“? 1+¢

). (3.4)
Equation 3.5 shows the law of motion for human capital. Human capital depreciates at
a rate 61;, and is accumulated by devoting time to learning activities. The technology
for producing it is in the spirit of Lucas [1988], although human capital is not used to
produce more human capital, and the specification in 3.5 does not allow for unbounded
growth. Given our focus on business cycles, these simplifications are sensible. Allowing

for the level of human capital to influence the efficiency of training hours (Nmt) would

only tend to increase volatility, via a feedback effect from human capital to training.

("It Aft ) 1+0

H. =H1—6 .
tH l( H)+" 1+6

(3.5)

49

The household problem is then to maximize 3.4 subject to 3.5 and the budget con-

straint 3.6:
, [ I+tp fVl+¢
max 22):” fi’(log C, + ‘Iant-(T—uifi— + f——'—7) [P2]
{CiaNtaltanta1"t+131‘1t+1}f:0 1+<p 1+9)
7’). 1V 1+6
Sf. 0: —Ht+1+Ht(1—6H)+€%
0 : ’lUthNtlt ‘i‘ (1 + Tt)kt — Ct — k¢+1. (3.6)

The description of this economy is completed by making explicit the law of motion

for capital:
[Nt - NH]2

1(t+l=(1— (5)1(1 + it — B 2

(3.7)

Equations 3.1- 3.7 provide a complete description of this economy. We are now ready

to define an equilibrium.

Definition 1 An equilibrium for this economy is a collection of sequences for allocations

{thh Kt,H,,n,,,lt}§’_:1, prices {Rt,rt,u2,}f_:1, and shocks {Atfi’il such that:

1. Given sequences for prices and shocks, the sequence for {N,,ct,K,,,Ht,nt,lt}f_:1

solve the household problem [P2].

2. The allocation for {N¢,K,,Ht,l,}f?_.lsolve the firm problem [P1], given sequences

for prices and shocks.

3. Markets clear, in particular:

0 Goods market: C, + 1¢+ 3W 2 A,K{’(l¢Nth)l‘a.

0 Labor markets:

wt = (1— O)A1Kta(ltNth)—a

'lUt : (1" Q)Athn(Nt)_a(lth)l—O — B(Nt ‘- Nt—l)
0 Capital market: rt = aAth'_l(ltNth)l‘° - 6

50

Since this is an economy without distortions, the second welfare theorem holds. The
above arrangement is then one in a class of alternatives that would yield the same
equilibrium allocation. We could, for instance, have the firm hire 1, + n, and decide
how much to invest in human capital accumulation. The resulting wage. rate would then
reflect a lower marginal product of labor of the aggregated training and working hours.
This motivates the interpretation of the equilibrium allocation as the solution to the

social planner’s problem. This problem is:

 

max 2°: [3'(logc + mN + f 1 ) [P3]
{CtaNtaltanta1"t+17IIt+l}f:0 t o 1 t 1+ 9'9 1+ ('0
. . N 1+0
8.5. (I = —Ht+1 + I1t(1— (SH) + (39%— ((1.)

o = A,K;*(1.N.H,)1—" + (1 — 5m. — 1",... (1.)
__ BiNt- _ Nt—ll2
2

Let /\ and p. be the Langrange multipliers for restrictions (1)) and (a) respectively.

Then the first order conditions are:

 

 

 

 

1
(Ci) 0 = —‘ " /\t (3-8)
Ct
, ”‘9 AK"1NH 1‘“
(N,) 0 2 —fN,¢ — ("XE—9L- + A,(1— a) ’ '(’N‘ 'l (3.9)
’ t
—/\tB[th — Nt_1] + E1A1+113UV¢+1 — [Vt] + [.l.t€Nt0'll.tl+0
A 1’" 1 NH 1‘“
(It) 0 : —‘m.N, (It + 71!)? + A(_(1— 0') t \t ( ll t t) (3.10)
t.
(m) 0 = —mN,(l, + in)? +uteNf+1nf (3.11)
A I'“ l N _ [I 1‘" -
(Km) 0 = —A, + (3Ep\,,1{a ”‘ ‘1.“er 1‘” M) + (1 — (5)} (3.12)
t+
(Ht+l) 0 = ‘lit “f (3(1 - (SillEtlltH (3-13)
At+1Kfi (lt+lN(+1Ht+l)l-a
BE A. 1— . *1
+¢ t t+l( (r) Ht+1
(At) 0 = Atlk'ta(ltftht)l-a + (1 — (S)I\,t — A144 — Ct (3.14)
[Nt — Aft_]]2
—B————
2

(n,N,,)1+o

(Ht) 0: —PIH_1 +Ht(1—(SH)+€ 1+6

(3.15)

Equations 38- 3.15, along with standard transversality conditions for physical and

human capital, define the equilibrium allocation.

3.4 Calibration

In this section we restrict this model quantitatively by choosing values for the four types

of parameters:

0 Preferences {p,cp, [3}.
0 Final good technology {6, a, B}.
0 Human capital technology {6H,6}.

0 Stochastic process for A {p, of}.

These choices are summarized in table 3.9 7.

We start with the preference parameters. In steady state, given a quarterly interest
rate of 1% (see Kotlikoff and Summers [1981]), the discount factor [3 equals .99. In
choosing values for the parameters {(0, (15} that govern the labor supply elasticities, we
follow the discussion in Bils and Cho [1994]. In our model, the responses of labor input
to changes in the wage rate have two components: changes in weeks worked and changes
in hours per week. From the optimality condition on l in the household problem (see
appendix C), the Frisch elasticity of hours per week is 1/(0. Evidence from studies using
microdata, reviewed by Pencavel [1986], indicates that this elasticity is no higher than
.5, which gives a value of 2 for (0. The optimality condition on weeks N in the household
problem provides an expression for the elasticity of weeks with respect to hours per week
worked. Using Canadian data, Reilly [1994], calculates this elasticity at .6, so we set (b

accordingly.

 

7The parameters {m, f, e} are purely normalization parameters, and therefore not considered in the
discussion that follows.

For the technology parameters ((1,6), we follow the literature in choosing a = .36,
equal to the share of capital in aggregate income, and 6 = .018, which together with a
quarterly interest rate of 1% is consistent with a yearly capital—output ratio equal to 3
(Prescott [1986]). To determine a value for the parameter B, we use evidence on the
size of labor adjustment costs. Burgess and Dolado [1989] report that, for firms in the
UK, these costs amount to 25% of the quarterly payroll, which in our model translate
to .17% of output. We set the parameter B so that 2W = .0017.

To pick values for the human capital production technology, we can use results from
the growing literature on the productivity effects of training. A number of papers at-
tempt to identify the effect of different measures of training on labor productivity. While
most of these papers use discrete measures of training, such as participation dummies
(Bartel [1994], Black and Lynch [1996]), at least two papers use a quantitative right
hand side variable: hours in training (Schonewille [2001]), and training days (Barrett
and O’Connell [1999]); these results can be more readily interpreted in terms of the
parameters of the model.

In our model, the elasticity of human capital with respect to training time (TIN)
is 611(1 + 0). Since the share of human capital augmented labor (lN H ) in income is
(1 — (1),, a 1% increase in training time translates into a (1 — a)6H(1 + 6) percent
increase in labor productivity (Y/lN). since a has been picked, we need to choose
values for 6H and 6. Using British data (UK Labor Force Survey), Schonewille [2001]
reports significant estimates of .04 for 6H(1 + 9) with a measure of hours in training as
the right hand side variable. Barrett and O’Connell [1999], using data on a nationally
representative sample of 1000 Irish firms, find a point estimate of .014 for the elasticity
of labor productivity with respect to the variable training days/ total employment (table
2 in their paper). With a = .36, this implies that 6H(1+ 6) = .021. These estimates are
clearly an upper bound for the value of 6;;(1 + 6), since most estimates are not different

from zero. We choose a small value of .0005 for 6H(1 + 19), which implies (5H 2 0.0005

and 6 = —0.0095. In the next section we will discuss how sensitive the results are to
this choice.

It is unfortunate that we have no useful results that rely on US data. A survey of the
evidence based on US data highlights the difficulty of estimating precisely the training
parameters. Bartel [1994] finds that implementing a new training program increases
firms productivity by as much as 41% over a 3 year span, but old training programs
seem to have no effect. Bishop [1994] reports that new workers enrolled in formal OJT
programs at their previous jobs were more productive by an amount equivalent to 9.5%
of their wage, and that this effect disappears after six months. Black and Lynch [1996]
are unable to find significant effects of their main training variables on firm sales.

We interpret the evidence as indicating that the elasticity of human capital with
respect to training EHC'Jraining = 6;,(1 + 6) lies between zero and the estimate by
Schonewille [2001], 0.04. Besides ensuring that {6”, 6} satisfy this elasticity, we need to

impose that these parameters are consistent with the steady state condition:

_ H145)
”‘su+m—s‘

we will choose two values for EHCJraming: {0.0002, 0.01} and discuss the importance of

choosing €11C¢r,,,,,,,,g accurately in the next section.

2

(I

Finally, we set p = .95 and 0 = .007, borrowing these values from Bils and Cho

[1994]
3.5 Results

We solve the model by linearizing the optimality conditions 3.8— 3.15 around the de-
terministic steady state, and solving for the recursive law of motion. We begin by
describing the main mechanisms at work in the model, using the beginning of a boom
as the starting point. As'the economy takes off, driven by positive TFP shocks, the
marginal product of labor increases, and firms adjust largely by increasing hours and

only gradually adjusting weeks, a process that entails direct costs to firms. I\-"Ioreover,

54

given that the marginal disutility of working activities (n, + 1,) has increased, house-
holds choose to devote less time to acquiring human capital. The response of n, is
large because the increase in hours of work l, had to be large enough to compensate the
slow adjustment of weeks (we may think employment). Since preferences are convex in
hours-per-week of leisure (1 — l, — nt), the large increase in n, implied a sharp increase
in the marginal utility of this dimension of leisure. Figure 3.1, with impulse responses
to a shock in A, for weeks N,, hours at work l, and in training 71,, illustrates this pro-
cess. Training has a large negative response to a TF P shock, and hours-at-work show
a larger response to shocks than weeks, even though the elasticity of labor supply for
hours-at-work (.5) is smaller than that for weeks (.6).

In the goods market, a positive shock increases output, consumption and investment
(figure 3.2). As the effects of the shock on TF P die out, and capital is accumulated
above its steady state level, the interest rate increases, reducing the incentives to invest
and fostering the consumption of the now large capital stock.

The model cross correlations with output are shown in table 3.10. It is clear that,
although the signs are all correct, this model fails to reproduce both the weak level of
countercyclicality of training, and its leading indicator characteristics.

Table 3.11 shows the volatility of selected variables. With the calibration of ta-
ble 3.9, the model successfully reproduces the extremely high levels of volatility of time
in training, and its relative volatility with respect to output and investment, but not
hours.

We now study the sensitivity of our results to changes in the parameter values.
Table 3.12 shows the effects of changing one by one the parameter values of table 3.9,
and guides us as to which are the important calibration choices in this model. We see
that the model fails both in reproducing the correlation between training and gdp, and
in leading gdp. We thus concentrate on the volatility of training.

Row 2 of table 3.12 show the effects of increasing €IIC,traini-ng from the (arbitrary)

C7!
CI!

baseline level of .00016. Calibrating this elasticity to values reported by Schonewille
[2001] and Barrett and O’Connell [1999] (.04 and .02 respectively) would imply a de-
preciation factor 6” higher than 1, but even with figmmmmg = .01 consistency with the
steady state requires the factor (1 + 6) to drop to .016 from .977 in the baseline. In
this case, the standard deviation of training is lower than that of output, and training
displays a positive correlation with output. Figure 3.5 shows the pairs {6,fycymmmg}
consistent with the steady state in a neighborhood of the baseline; it shows that 6, and
therefore chaining, is extremely sensitive to very small changes in gymmmmg around the
baseline: decreasing Eycymming to .00016 increases othng to 177 (l), while increasing
it to .0003 brings Utraim’ng down to .43.

It is clear that the choice of {6H, 6}, that determines how efficiently training time is
converted into new human capital, is crucial for our results. To understand this point,
it is useful to decompose gycjmmmg into (1 + 6) and 6”. While the first term indicates
that a 1% change in the time spent in training translates into a (1 + 6)% change in
human capital investment, 6;; is a scale factor that indicates how large is human capital
investment with respect to the stock of human capitalg. Even if a high (1 + 6) can drive
human capital investment to display large fluctuations with respect to its level, as it
does in our simulations, this level is so small with respect to the stock of human capital
that the ultimate effect of training on human capital and labor productivity is barely
distinguishable from zero".

Changing the size of adjustment costs has predictable effects on the second moments
of training (rows 3 and 4): increasing these costs to one percent of gdp increases the
standard deviation of training by 22%, while bringing B to zero has a small negative
impact on 03m].

While formal training programs of the type considered in this paper are an important

[71(Ng]1+0

8Investment in human capital is 6 1+0 , and in steady state we have (SHH = 6

 

(nN)l+0

1+6
9A stochastic depreciation factor 6” is a possible avenue to improve on the results of this model, as

noted by Einarsson and Marquis [1998]. However, as we know of no procedure to directly estimate the
series {6”}, we believe that examining this possibility is of limited use at this point

56

source of human capital accumulation, learning by doing and other informal means to
acquire skills are also available but difficult to measure. We examine in row 5 whether
increasing the time devoted to skill acquisition has significant effects on the results,
and find that increasing the share of time devoted to skill acquisition from .0158 to .02
increases the volatility of training time by 12%.

Varying the size of the labor elasticities (rows 6 and 7) has also quantitatively minor
effects on GEN. While increasing £Mwage makes it less costly to adjust weeks than hours
per week, increasing €(n+()‘wage has the opposite effect. As explained in section 3.3, 012w
increases in this model when weeks become costly to adjust with respect to hours per

week.

3.6 Conclusion

In this paper we have first characterized the behavior of training at business cycles
frequencies, and proposed a model to reproduce this behavior. We find that training
has a clear countercyclical behavior, leads the cycle, and is extremely volatile, making it
an important margin in allocating time at the frequencies considered. We also found that
our model shows the potential to reproduce the volatility of training, but does not fare as
well in reproducing the remaining stylized facts. Moreover, our calibration exercise has
identified the elasticity of human capital with respect to training as the key parameter to
reproduce this volatility. We now believe that studying the technology used to convert
training into new human capital is of great importance in understanding human capital

accumulation, and plan to work on this issue in the future.

57

Percent Deviation from Steady State

Figure 3.1: Impulse responses: Time allocations.

Weeks(*). training(o), work hrs.(-)

 

I I Y I I I T

O

OJLXXX¥XXXXXXXXX

-10

-12

-14

-16

_1o('\ 1 i 1 1 1

1

 

p—
'—

 

U

1 2 3 4 5 6 7
Years after shock

58

 

U1
1

b

N

Percent Deviation from Steady State
(JO

Figure 3.2: Impulse responses: GDP, C, I.

 

w
I“

00

 

”Ex

X
0000000000000000566
o

l

I

Y(‘). C(O). |(-)

X
Xx*
*X*
*‘x
X***

I

1

I

I

1

9999999

 

I

 

O

1

2

3

4
Years after shock

5

Figure 3.3: Training and GDP, period 1.

. tr__hours . . gdp

.440215 “

n46629"
1_________ __ ____ .___.___.___ . ____.___ ., . _ r“
6 32
time

Agg. Training hours and gdp (centered), 1 st period

60

Figure 3.4: Training and GDP, period 2.

. tr_hours . gdp

.253015

2475071

‘l . __ __ __ l ..__._,_ _ —

45
tlme

Agg. Training hours and gdp (centered), 2nd period

61

77

theta

0.5

Figure 3.5: {6, €HCJ,.,,,,,,~,,9} pairs consistent with the steady state

 

 

 

 

 

 

1 1 I l 1 1 I I
1 1 I r I I I I
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
elasticity x 10-3

62

Table 3.1:

Assignment of training program

 

N LSY Classification

 

Years 1978-1986

 

On The Job Training

Company Training

Years 1988-1998
Formal Company
Training run by
Employer

Seminars at work not
run by employer

 

Off The Job Training

Business College
Nurses Program

Vocational Technical
Institute
Barber-Beauty

Business School
Vocational Technical
Institute
Correspondence
Course

Seminar or training
program outside of
work

 

 

Flight School Vocational Rehabili-
tation Center
Correspondence Other
Other
Dropped Observations Apprenticeship Apprenticeship Pro-

 

 

gram
Government Training
Program

 

Table 3.2: Descriptive statistics

 

 

 

 

 

 

 

 

 

 

 

Variable Obs. Mean Std. Dev. Min. Max.
age 543128 28.236 5.379 14 41
hoff 542836 3.343 33.210 0 1248
ho jt 542778 2.379 30.167 0 2496
job 543128 0.750 0.433 0 1
toff 543128 0.020 0.140 0 1
tojt 543128 0.016 0.126 0 1

 

 

 

 

63

 

Table 3.3: Description of variables

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variable Description

age age, in years

hoff hours spent in off the job training programs

hojt hours spent in on the job training programs

hagg Hours spent in either off or on the job train-
ing

toff Binary training variable, =1 if enrolled in a
off the job training program

tojt Binary training variable, =1 if enrolled in a
on the job training program

tagg Binary training variable, =1 if enrolled in ei-
ther type of training program

HOJT Natural log of time t aggregated hojt

HOF F Natural log of time t aggregated hoff

HAGG Natural log of time t aggregated hagg

TOJT Time t mean tojt

TOF F Time t mean toff

TAGG Time t mean tagg

11 Index of total hours worked in the business
sector, hp filtered

gdp Log real gdp, hp filtered

1 Log real investment, hp filtered

prodh Log real gdp per hour, hp filtered

prodp Log real gdp per person, hp filtered

 

Table 3.4: Coi‘itemporaneous correlation of I, hours and productivity with GDP-Data

 

 

 

 

 

 

 

 

 

Period 1 Period 2
I 0.91 0.86
H 0.97 0.87
Prodh 0.77 0.01
Prodp 0.79 0.39

 

64

 

Table 3.5:

Contemporaneous correlations of training with GDP, I—Data

Table a. Correlations using time series

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Period 1 Period 2
gdp I gdp I

TOFF 0.0026 -0.2246 -0.1307 -0.0073
(0.9898) (0.2599) (0.4684) (0.9678)
TOJT -0.3919* -0.1924 —0.3564* -0.1957
(0.0432) (0.3364) (0.0417) (0.2750)
TAGG -0.1033 -0.2746 -0.2622 -0.1124
(0.6083) (0.1657) (0.1405) (0.5335)
HOFF -0.2338 -0.3992* 0.2555 0.2577
(0.2405) (0.0391) (0.1512) (0.1477)
HOJT -O.2020 -0.1104 -0.3860* -0.4668*
(0.3124) (0.5837) (0.0265) (0.0062)
HAGG -0.2511 -0.3912* -0.2079 -0.2766
(0.2065) (0.0436) (0.2457) (0.1191)

Table b. Correlations using pooled data

Period 1 Period 2
gdp I gdp I

toff -0.0027 —0.0072* -0.0070* -0.0011
(0.2772) (0.0039) (0.0003) (0.5871)
tojt -0.0036 -0.0022 -0.0151* -0.0074*
(0.1471) (0.3713) (0.0000) (0.0001)
tagg -0.0038 -0.0074* -0.0157* -0.0060*
(0.1232) (0.0031) (0.0000) (0.0019)
hoff -0.0063* -0.0092* 0.0013 0.0039*
(0.0113) (0.0002) (0.5184) (0.0421)
hojt -0.0018 -0.0018 -0.0114* -0.0110*
(0.4766) (0.4788) (0.0000) (0.0000)
hagg -0.0064* -0.0089* -0.0077* -0.0055*
(0.0103) (0.0003) (0.0001) (0.0043)

 

 

 

 

 

 

 

Table 3.6: Cross correlations of I, H and productivity with GDP-Data (corr(:1:t, grlpy+j))

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Period 1

La gs I H prodh prodp
-6 -0.40 -0.18 -0.26 -0.39
-5 -0.34 -0.03 -0.38 -0.44
-4 -0.11 0.19 -0.37 -0.38
-3 0.2 0.46 -0.18 -0.18
—2 0.47 0.69 0.08 0.09
—1 0.76 0.89 0.41 0.45
0 0.91 0.97 0.77 0.79
1 0.74 0.80 0.72 0.78
2 0.49 0.53 0.60 0.68
3 0.27 0.25 0.48 0.53
4 0.02 0.00 0.27 0.34
5 -0.10 -0.14 0.11 0.20
6 —0.10 -0.21 0.04 0.12

Period 2

Lags I H prodh prodp
-6 -0.47 0.03 -0.46 -0.70
-5 -0.34 0.24 -0.62 -0.75
-4 -0.15 0.46 —0.67 -0.65
-3 0.06 0.64 -0.67 -0.51
-2 0.38 0.78 —0.52 -0.25
-1 0.68 0.86 -0.30 0.04
0 0.86 0.87 0.01 0.39
1 0.77 0.71 0.06 0.42
2 0.64 0.53 0.09 0.42
3 0.44 0.32 0.08 0.32
4 0.29 0.13 0.09 0.25
5 0.10 -0.05 0.07 0.14
6 -0.05 -0.22 0.14 0.11

 

 

 

 

 

66

 

 

Table 3.7: Cross correlations of training with GDP-Data (corr(:1:., gdp,+ 1))

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Periodl

Lags TOJT TOFF TAGG HOJT HOF F HAGG
-6 -0.04 0.28 0.26 0.24 0.26 0.32
-5 ~0.29 0.27 0.19 0.05 0.18 0.19
-4 -0.41 0.22 0.11 -0.09 0.08 0.07
-3 -0.35 0.16 0.06 -0.18 -0.01 -0.03
-2 -0.37 0.11 0.01 -0.23 -0.07 -0.10
-1 -0.36 0.05 -0.05 -0.23 -0.13 -0.16
0 —0.39 0.00 -0.10 -0.20 ~0.23 -0.25
1 -0.51 -0.05 -0.18 -0.27 -0.27 -0.32
2 -0.50 -0.14 -0.28 -0.38 -0.30 -0.37
3 -0.40 -0.22 -0.32 -0.45 ~0.26 -0.35
4 -0.25 -0.14 -0.21 —0.44 -0.16 -0.25
5 —0.09 -0.06 -0.08 -0.26 -0.04 -0.10
6 0.19 —0.05 0.00 0.00 -0.01 -0.01

Period 2

Lags TOJT TOFF TAGG HOJT HOFF HAGG
—6 -0.23 -0.33 ~0.29 0.30 -0.22 0.16
-5 -0.27 -0.24 -0.27 0.28 -0.07 0.19
-4 -0.30 —0.17 -0.25 0.21 0.09 0.19
-3 -0.37 -0.21 -0.31 0.05 0.15 0.09
-2 —0.41 -0.20 -0.32 -0.18 0.16 -0.09
-1 -0.35 —0.16 ~0.27 -0.30 0.19 -0.18
0 -0.36 -0.13 -0.26 -0.39 0.26 -0.21
1 -0.33 -0.13 -0.25 -0.42 0.23 -0.24
2 -0.23 -0.08 -0.16 -0.36 0.28 -O.17
3 -0.03 0.04 0.00 —0.19 0.36 0.01
4 0.12 0.14 0.13 -0.07 0.44 0.14
5 0.24 0.21 0.23 0.05 0.39 0.21
6 0.31 0.23 0.29 0.10 0.29 0.19

 

 

 

 

 

 

 

 

 

67

 

 

Table 3.8: Standard deviations-Data

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Period 1 Period 2
gdp 0.020 0.010
I 0.096 0.048
H 0.019 0.015
prodh 0.009 0.008
prodp 0.014 0.009
TOFF 0.006 0.005
TOJT 0.002 0.006
TAGG 0.006 0.010
HOFF 0.270 0.168
HOJT 0.274 0.257
HAGG 0.230 0.177

 

Table 3.9: Baseline parameter values

 

 

 

 

 

 

 

 

 

 

 

 

(f) 2.46
(.0 2

1’3 .99

6 .018
a .36
B 13,450
6;; .0002
6 -.0228
p .95
a? .00712

 

 

68

 

Table 3.10: Cross correlations with output-h'lodel (corr(:1:,, gdp, , 1))

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lags I Hours Training hrs. Labor Prod.
—5 205 .01 .11 .1
-4 .05 .11 .01 .19
-3 .26 .32 a2 .38
—2 .43 .49 238 .54
-1 .66 .70 n62 .73
0 .99 .99 a98 .99
1 .72 .73 a72 .68
2 .54 .54 255 .46
3 .39 .38 «41 .29
4 .20 .19 a22 .08
5 .10 .09 a13 a01

 

 

Table 3.11: Standard deviations-Model

 

 

 

 

 

 

 

 

Var. a
Y .012
I .043
Hours .0024
Training hrs. .159
Labor prod. .01

 

 

Table 3.12: Sensitivity analysis

 

 

 

 

 

 

 

 

 

Calibration 0,,N ('orr(gdp,, n N , ) (1 r gmazr )- {(j'orr(nN,, gdp, +J- )}
Baseline .159 -.98 0
{HC,,,,,,,,,,,, = .01 .0016 1 0
Adjustment costs are 1% of gdp .194 -.98 0
No adjustment costs .153 -.99 0
¥ = .02 .179 -.99 0
.5., = .7 .137 -.98 0
g(,,,,,,,,,,,_.,, = .6 .175 -.99 0

 

 

 

 

69

 

Appendix A

Numerical methods.

To simulate the stochastic OLG economy we use a standard dynamic programming
method. To use this method we discretize the state space. In particular, a seven-state
discrete approximation to the labor endowment process is used. This approximation is
done with the method described in Tauchen [1986] to find a markov transition matrix

for continuous stochastic processes.

A.1 Solution method for the stochastic OLG model

The stochastic OLG model is solved using a variation of the Imrohoroglu et al. [1995]
algorithm to compute the policy functions. Then a Monte Carlo simulation is performed
to compute some of the statistics. The algorithm to compute the policy functions can

be summarized as follows,

1. Make an initial guess for the discount factor [3 and the level of bequests go. For 6

define b = [(11,112] (with ()1 < ()2), and let 6 = (bl + ()2)/2.

2. Starting from age J, and given prices consistent with the calibration, compute
the value functions and associated policy functions that solve (P1) by a single

backward recursion.

3. Define the distribution of assets and shocks for the first cohort j = 1), and using

the policy functions and the transition matrix for the shock, compute recursively

70

the distribution of assets and shocks for ages 2, 3...J.

4. Using the distribution of assets for all cohorts, calculate the implied levels of

aggregate capital and bequests K1, and q1.

5. Compare K1 with the level implied by the calibration K>1<. If convergence fails,
adjust bequests with qo = q1,and 13 by letting bl = ,6 if K1 < K*, and ()2 = 13 if

K1 > K*.

For both solution methods the grid size for assets is set to 5 -10% of average asset
holdings, and the convergence criterion is set to .003. Convergence occurs generally in
79 iterations. Using the policy functions, we then simulate paths for 10.000 agents and
compute the statistics. The original version of this algorithm is discussed in Imrohoroglu

et .11. [1999].

A.2 Solution method for the deterministic OLG model

We solve the deterministic version of the OLG model by a method presented in Rios-
Rull [1999]. It is a ’shooting’ method that uses the fact that the Euler equation can be

expressed as:

1. , r —1/o
1+(1+ my >> MHHJ

("mm + (1)030 + 1))‘1/9 — (w, + q)
l + r

,z3’(1+r))‘1/9
1+r

 

 

at = { }(lt+2

+{

 

}

1. Set technology and preference parameters, {(1.1 = a, a, 6, 6, 6} and guess levels of

aggregate capital K0 and per capita bequests qo.

2. Using K0, and given the production function F (K, L) = AKC‘LI‘“, find prices

{13w}.

3. Using the fact that “1+1 = 0, guess a value for (17‘, and compute {(11}le backwards

using the Euler equation.

71

4. Check whether (11 = (1., otherwise go back to 3 and modify guess for (LT.

5. Calculate aggregate capital K1 and per capita bequests q1 using {at};r=1. If con-

vergence fails, set q0 = q and K0 = K1, and go back to 1.

72

Appendix B

Standard RBC model with human
capital

Model

Utility
(1 — l, — 71.,)l"0
1 - 6

 

U = E, Z;3"(log c, — m
t=0

Resource constraint
0 = AtK,“(Htlt)l_a — ct — K,“ + (1 — (5)19

Law of motion for human capital

(711)“7

l

 

0 = —[[(+1 + Ht(1- (5”) + 8
Stochastic process for A

At+1 = pAg '1‘ 6t 6; N N(0,0;2).

73

Table 81: Variable description

H Human Capital
K Capital
[3 Discount factor

( Consumption

1 Hours at work

71. Hours in skill acquisition

A TF P shock

(5 Rate of depreciation of K
(5” Rate of depreciation of H

74

Appendix C

First order conditions in the
household problem [P2]

 

Problem:
. , 1 W NW
max 22:0 fi‘(log c, + 171N,m—t)— + f t ) [P2]
{(it,N(,lt,'nt,l\rt+l,Ht+1}f_:0 1+¢ 1+¢
(”INOIH)
.t. 0: —H +Hl—6 +,——
3 1+1 1( H) 6 1+ 9
0 = TilthNtft + (I + T0117, — Ct — A7144.
Lagrangian:

(Tit +l()1+¢ + Nt1+¢
1+ (0 1+ (15

n N, 1+6
+lltlHt+1— Ht(1— 5H) ' 8(_l+—)6_

+/\t[’lUthiNtlt + (1 + rt)kt — Ct — [17,44]}

.6 = E, 2:0 6‘{ log c, + mN,

First Order Conditions:

(Ct) 0: l-At

Ct

(N,) 0 : _th¢ _ mw

— At‘lllthlt + [116’th671tl+0
(It) 0 : mN,(l, + 71")? + AflUthNt
(71,) 0 = —mN,(l, + n,)"7 +11.,eN,0Hn[9

“((+1) 0 : "(\1 + [fEtAtHu + NH)

75

(Ht+1) 0 = _/l‘t + (13(1 — (SillEilltH — Ez/\t+1'll’t+1N1+llt+1

(At) 0 = ”(llthNtlt + (1+ 7})11} — Cf —' kH-l
(“vtNtlHo

(fit) 0: —f[(+1+Ht(1—(S][)+€ 1+0

76

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