The paper analyzes the effects of individual--specific and economy--wide productivity shocks on intertemporal decision--making of risk averse agents. We focus especially on the consequences for long--run growth. By contrasting the most widely used models of modern growth theory, namely the AK-model and the learning by doing-model, it is shown that not only the degree of risk aversion but also the source of income as measured by the factor income distribution is crucial for the impact of the stochastic disturbances. In the presence of a pure capital risk, growth and welfare effects are different from those arising when agents are subject to capital and income risk.

Christiane Clemens

English

Research Papers in Economics

Capital and income risk reconsidered:
The case of non–expected utility
Christiane Clemens
University of Hannover, Department of Economics, Königsworther Platz 1, D-30167 Hannover
e–mail: clemens@vwl.uni-hannover.de
9th May 2000
Abstract
We discuss an economy which is subject to aggregate productivity shocks affecting all factors of produc-
tion. The additional presence of income risk shifts the margin where agents save out of precautionary
motives downwards, such that it does no longer correspond to logarithmic preferences.
Keywords: capital risk, income risk, endogenous growth
JEL classiﬁcation: D8, O4
1 Introduction
The effects of capital and income risk on intertemporal consumption choice of risk averse
agents haveoriginallybeendiscussed by Sandmo(1970) withinan expected utilitysetting.
More recently, this issue has been addressed by Weil (1990, 1993) within the framework
of non–expected utility preferences. But similar to the work of Obstfeld (1994a, b) or
Turnovsky (1999), the author restricts his analysis to a single type of income. The new
feature of the model presented here is that we discuss the effects of capital and income risk
within one consistent framework. Our approach can be viewed as an extension to Obstfeld
(1994b) or Smith (1996). We explicitly take account of non–diversiﬁable labor income
risk and demonstrate that this leads to new insights with respect to the parameter ranges
for the intertemporal elasticity of substitution where phenomena as certainty equivalence
or precautionary saving occur.
2 The model
We assume an economy populated by a continuum
￿
0
￿ 1
￿ of identical, inﬁnitely–lived
agents. Each individual selects the optimal amount of consumption and savings in or-
1der to maximize intertemporal welfare V
￿ t
￿ . Households have iso–elastic, recursive pref-
erences as in Obstfeld (1994b) which allows for a separate analysis of risk aversion and
intertemporal substitution
G
￿
￿ 1
￿ r
￿ V
￿ t
￿
￿
￿
￿
￿
1
￿ r
1
￿ 1
￿ e
C
￿ t
￿ 1
￿ 1
￿ eh
￿ e
￿ bhG
￿
￿ 1
￿ r
￿ EtV
￿ t
￿ h
￿
￿
￿
￿
￿ (1)
Here, C
￿ t
￿ is the intertemporal consumption ﬂow which we assume to be instantaneously
deterministic. The parameter b
￿ 0 is the subjective rate of time preference. Et is the
expectations operator conditional on time–t information. The parameter r
￿ 0 is the coef-
ﬁcient of relative risk aversion and e
￿ 0 denotes the intertemporal elasticity of substitu-
tion. Without loss of generality, we do not discuss the limiting cases where r
￿ 1 and/or
e
￿ 1. The function G
￿
￿
￿
￿
￿ is given by
G
￿
￿
￿
￿
￿
￿
￿
1
￿ r
1
￿ 1
￿ e
￿
￿
￿ 1
￿ r
￿ V
￿ t
￿
￿
￿
1
￿ 1
￿ e
1
￿ r
￿ (2)
Individual production is stochastic. At each increment of time the economy is subject to
an aggregate productivity shock. The representative ﬁrm produces a homogeneous good
according to the following stochastic Cobb–Douglas–technology
dY
￿ t
￿
￿
￿ K
￿ t
￿ aL
￿ t
￿ gA
￿ t
￿ 1
￿ a
￿ dt
￿ sdz
￿ t
￿
￿
￿
￿
￿ (3)
We apply the learning–by–doing setting developed by Romer (1986). The instantaneous
output ﬂow dY
￿ t
￿ is assumed to be generated from physical capital K
￿ t
￿ and labor L
￿ t
￿ .
The factors income shares are given by a and g. We assume a
￿ g
￿
￿
￿ 0
￿ 1
￿ and a
￿ g
￿ 1
to assure the existence of a competitive equilibrium. For analytical convenience, we take
labor to be inelastically supplied, and the labor force is normalized to unity. Nevertheless,
the household receives an income from both factors of production. In terms of Sandmo
(1970), the agent is exposed to capital risk and income risk. In (2), dz
￿ t
￿ is the incre-
ment to a standard Wiener process z
￿ t
￿ with zero mean and the instantaneous variance of
production s2dt.
The production function (2) exhibits human capital externalities. The stock of tech-
nical knowledge A
￿ t
￿ acts as a Harrod–neutral growth parameter and is enhanced by in-
vestment in privately owned capital. In macroeconomic equilibrium A
￿ t
￿ equals K
￿ t
￿ and
aggregate production is linear in capital. Hence, the requirements for ongoing growth
of per capita incomes are met. This assumption together with the assumption stated on
the nature of the random disturbance implies that the economy evolves according to a
stochastic trend.
Individuals save by investing in risky physical capital. The representative agent is
endowed with the initial capital stock K
￿ 0
￿
￿
￿ 0. In each time increment t he receives
capital and labor incomes. His ﬂow budget constraint is given by
dK
￿ t
￿
￿
￿
￿
rK
￿ t
￿
￿
￿ wL
￿ t
￿
 
￿ C
￿ t
￿
￿
￿ dt
￿ dk
￿ t
￿
!
￿ (4)
2where w is the wage rate and r is the rate of return to physical capital. In equilibrium
r equals the private marginal product of capital and falls short of the social return. The
stochastic process of capital is deﬁned as dk
￿ t
￿
"
￿
￿
rK
￿ t
￿
#
￿ wL
￿ t
￿
￿
￿ sdz
￿ t
￿ with the corre-
sponding variance of physical capital given by s2
K
￿ E
￿ dk
￿ 2
￿ dt.
The consumer’s problem is to select his rate of consumption in order to maximize his
objective function V
￿ t
￿ as speciﬁed by (1) and (2) subject to his budget constraint (4),
taking prices as given. The solution conjecture usually applied for isoelastic preferences
is that the propensity to consume out of capital µ
￿ C
￿ K is constant in macroeconomic
equilibrium and that investment in capital is determined by constant relative risk aversion.
This guess together with the equilibrium values of factor returns r
￿ a
￿
dt
￿ sdz
￿ and
w
￿ gK
￿
dt
￿ sdz
￿ which can be obtained by the usual marginal productivity conditions,
determines the optimal policy
µ
￿ e
￿ b
￿ a
￿
$
￿
%
￿ a
￿ g
￿
&
1
￿ rs2
￿ a
￿ g
￿
(
’
ea
a
￿ g
￿
e
￿ 1
2
)
+
*
￿ (5)
Substitution of the equilibrium value of the propensity to consume (5) into the budget con-
straint (4) leads to a closed–form expression for the expected growth rate of the economy
y
￿ e
￿ a
￿ b
￿
$
￿
,
￿ a
￿ g
￿ 2rs2
’
e
￿ 1
2
￿
ea
￿ a
￿ g
￿
-
)
￿ (6)
Equation (5) and (6) show that the propensity to consume out of capital as well as the
expected growth rate of the economy are the sum of a drift and a diffusion component,
the latter reﬂecting the agent’s optimal response to technological risk.
3 Growth effects of capital and income risk
The optimal values of the propensity to consume (5) and the expected growth rate (6)
show that one important characteristic of this class of recursive preferences is maintained.
The impact of the coefﬁcient of relative risk aversion r on long–run growth is unaffected
by the additional presence of income risk. As well known from Weil (1990), the degree
of risk aversion just has a size effect on macroeconomic growth.
The main difference between our model and the contributions of Obstfeld (1994b),
Smith (1996) and Weil (1990) becomes apparent if we focus our attention on the sign of
diffusion term. Whether or not the expected growth rate of the risky environment exceeds
the deterministic one is not solely determined by the intertemporal elasticity of substi-
tution. In addition it depends on the factor income distribution. It is easy to verify that
the results of Obstfeld (1994b) can be obtained for a
￿ 1
￿ g
￿ 0, and the results of Smith
(1996), who also accounts for externalities in human capital accumulation, correspond to
the case of g
￿ 0. Both authors discuss a framework where agents receive only capital
incomes. In this case the question of dominance of intertemporal income and substitution
3effects depends entirely on whether the intertemporal elasticity of substitution exceeds or
falls below unity, whereas certainty equivalence can be obtained for e
￿ 1. If the elasticity
of substitution is sufﬁciently low, that is e
. 1, the agent tries to self–insure against the
uncertainty of future income ﬂows. Following Leland (1968) and Sandmo (1970), he has
a motive for precautionary savings.
In our model the income shares of both factors of production enter into the diffusion
term. For instance, with a change in the variance of the technological shock, the optimal
response does not solely depend on e
/ 1 but moreover on the size of a and g. Let us
now turn towards the question, as to what extent the factor income distribution affects
growth. We discuss a variation of a and g respectively, while holding the other income
share ﬁxed.1 The derivatives can be obtained as follows
¶y
¶a
￿ e
￿ rs2
￿
a
￿ 1
￿ e
￿
￿
￿ g
￿ (7)
and
¶y
¶g
￿ rs2
￿
a
￿ g
￿ e
￿ 1
￿
￿
￿
0
￿ 0
￿ (8)
An increase in the marginal product of capital a raises both, the mean and the volatility
of future income ﬂows. The effect on the drift term is positive, which is not surprising
and well known from the deterministic setting. As expected, the effect on the diffusion
term is of ambiguous sign. From (7) can be seen, that the growth rate increases with a
rise a for every e
. 1. This result corresponds to the ﬁndings of Weil (1990) and Obstfeld
(1994b) for the case of a pure capital risk. The positive intertemporal income effect
outweighs the negative intertemporal substitution effect. But additionally, in the presence
of income risk, the intertemporal effects do not offset for e
￿ 1. Even in this case the
growth enhancing effect from an increase in capital returns prevails. The expected growth
rate in total declines with an increase of a only if the certainty equivalent to capital return
i
￿ a
￿ 1
￿ rs2
￿ is sufﬁciently negative.2
Equation (8) shows that in contrast to the deterministic model, here an increase in the
labor income share in fact has a positive impact on long–run growth. A ceteris paribus
rise in the labor income share leads to an increase in future expected income. Similar to
the deterministic setting, labor income does not enter into the drift term of the expected
growth rate but, what is more important, its riskiness affects the diffusion term. Within the
preference class considered here, the representative agent has a motive for precautionary
saving out of income risk for any size of the elasticity of intertemporal substitution e
￿ 0.
1It is understood that with constant returns to scale in production an increase in the capital income share
is to be accompanied by a proportional decrease in the labor income share and vice versa in order to sustain
a competitive equilibrium. Clemens (1999) demonstrates that the results of this paper hold for the standard
expected utility setting where e
1 1
2 r and for g
1 1
3 a.
2This effect is discussed by Smith (1996). In general it is not compatible with feasible solutions of the
model because the transversality condition is violated in case of a negative certainty equivalent to capital
return (see Clemens and Soretz, 1997).
4Hence, the results match ﬁndings of Weil (1993) for the case of a pure income risk and
CIES–CARA preferences and Sandmo (1970) for the expected utility case.
Combination of the growth effects from a change in the capital and the labor income
share leads to the following conclusion: If the representative consumer is exposed to
capital and income risk the margin of certainty equivalenceis shifted downwards and does
no longer correspond to a parameter value for the intertemporal elasticity of substitution
of e
￿ 1. The higher the labor income share the earlier the household is inclined to save
out of precautionary motives.3
4 Conclusions
This paper discussed the effects of a simultaneous appearance of capital and income risk
on long–run expected growth. Within a CIES–CRRA framework, we found that (a) the
higher the labor income share the earlier agents develop a motive to save out of precau-
tionary motives, and that (b) the case of certainty equivalence does not correspond to
logarithmic intertemporal preferences. Our analysis thus extends the results known from
contributions which focus solely on either one of the risky income sources.
References
Clemens, Christiane (1999). Endogenes Wachstum, Einkommensunsicherheit und
Besteuerung. Metropolis, Marburg.
Clemens, Christiane and Soretz, Susanne (1997). Macroeconomic Effects of Income
Taxation in a Model of Stochastic Growth. Finanzarchiv, N. F., 54 (4), 471–493.
Leland, HayneE. (1968). SavingandUncertainty: The PrecautionaryDemandforSaving.
The Quarterly Journal of Economics, 82, 465–473.
Obstfeld, Maurice (1994a). Evaluating Risky Consumption Paths: The Role of Intertem-
poral Substitutability. European Economic Review, 38, 1471–1486.
Obstfeld, Maurice (1994b). Risk–Taking, Global Diversiﬁcation, and Growth. The Amer-
ican Economic Review, 84, 1310–1329.
Romer, Paul M. (1986). Increasing Returns and Long–Run Growth. Journal of Political
Economy, 94, 1002–1037.
Sandmo, Agnar (1970). The Effect of Uncertainty on Savings Decisions. Review of
Economic Studies, 37, 353–360.
Smith, William T. (1996). Taxes, Uncertainty, and Long–term Growth. European Eco-
nomic Review, 40 (8), 1647–1664.
Turnovsky, Stephen J. (1999). On the Role of Government in a Stochastically Growing
Economy. Journal of Economic Dynamics and Control, 23 (5-6), 873–908.
3Of course, we have to restrict our attention to feasible solutions of the model, especially to positive
growth rates.
5Weil, Philippe (1990). Non–Expected Utility in Macroeconomics. The Quarterly Journal
of Economics, 105, 29–42.
Weil, Philippe (1993). Precautionary Savings and the Permanent Income Hypothesis.
Review of Economic Studies, 60, 367–383.
Appendix
We assume the value function J
￿
K
￿ t
￿
￿
￿ t
￿
4
￿ maxV
￿ t
￿ to represent the maximized value of
lifetime utility. We apply Itô’s Lemma to derive the stochastic differential of V
￿ t
￿ to get
the following Lagrangian
max
C
5 K
6
￿
1
￿ r
1
￿ 1
￿ e
C1
￿ 1
￿ e
￿ bG
￿
￿
￿
￿
￿
￿
￿
,
￿ 1
￿ r
￿ G
7
8
￿
￿
￿
￿
￿
&
J
7
9
￿ K
￿
0
￿ rK
￿ w
￿ C
￿
$
￿
1
2
J
7
:
7
9
￿ K
￿ s2
K
*
￿
(A.1)
Intheframeworkconsideredherethecontinuous–timestochasticBellmanequationcannot
be employedbecauseof the increasing returnsto scaletechnology. Soin additionto taking
the derivative of (A.1) with respect to C, we have to differentiate (A.1) with respect to K
which leads to the following ﬁrst–order conditions (for details see Clemens, 1999)
0
￿ C
￿ 1
￿ e
￿ G
7
9
￿
￿
￿
￿
￿ J
7
;
￿ K
￿ (A.2)
0
￿ G
7
￿
￿
￿
￿
￿
&
J
7
￿ K
￿
0
￿ r
￿ b
￿
￿
￿ J
7
7
￿ K
￿
&
rK
￿ w
￿ C
￿
1
2
¶s2
K
¶K
*
￿
1
2
J
7
<
7
:
7
￿ K
￿ s2
K
*
￿
￿
%
￿ 1
￿ r
￿ G
7
7
9
￿
=
￿
￿
￿ J
7
9
￿ K
￿
&
J
7
;
￿ K
￿
0
￿ rK
￿ w
￿ C
￿
￿
￿
1
2
J
7
:
7
9
￿ K
￿ s2
K
*
￿ (A.3)
The solution conjecture stated above in the text implies the following derivatives of the
value function
J
7
9
￿ K
￿
￿
￿ µ
1
￿ re
1
￿ e C
￿ r
￿ 0
J
7
>
7
￿ K
￿
￿
￿
?
￿ rµµ
1
￿ re
1
￿ e C
￿
￿
@ r
A 1
B
. 0 (A.4)
J
7
:
7
>
7
￿ K
￿
￿
￿ r
￿ r
￿ 1
￿ µ2µ
1
￿ re
1
￿ e C
￿
C
@ r
A 2
B
￿ 0
￿
Substitution of (A.2) and (A.4) into (A.3) leads to the expression for the consumption–
capital ratio (5) of the text.
6

THE IMPACT OF CAPITAL AND INCOME RISK ON LONG-RUN GROWTH

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics