Uninsured Idiosyncratic Risk and Human Capital
Accumulation
*** Preliminary and Incomplete—Comments Welcome***
Michael J. Pries∗
June, 2001
Abstract
In standard models of uninsured idiosyncratic risk—for example, Deaton (1991), or
Aiyagari (1994)—consumers accumulate a small stock of savings to help them self-insure
against shocks to an exogenous income process. Following an adverse shock, consumers
dissave in order to smooth consumption until their income recovers. In reality, agents
do not passively wait for their income to recover. Instead, they often make investments
in human capital that are intended to boost their income. This paper examines a
model in which agents who face uninsured idiosyncratic income risk can both save in a
riskless asset and make human capital investments. To solve the consumer’s problem
computationally, the paper introduces an adaptation of the parameterized expectations
algorithm to the case of a problem with two state variables, two choice variables, and
two constraints.
Keywords: precautionary saving; human capital; self-insurance
JEL Classification: E21; J24
∗
Dept. of Economics, 3105 Tydings Hall, University of Maryland, College Park, MD 20742; email:
[email protected]. I thank Martin Šuster for outstanding assistance with this paper.
Uninsured Idiosyncratic Risk and Human Capital Accumulation
1
1
Introduction
In standard models of incomplete asset markets, consumers who face liquidity constraints
or whose preferences exhibit prudence attempt to self-insure against shocks to their income
by accumulating a stock of a single riskless asset.1 Following an adverse shock to their
exogenous income process, the agents smooth their consumption by drawing down their
stock of saving as they wait for their income to recover.
In reality, agents do not passively wait for their income to recover. Instead, they often
attempt to affect their income by making investments in some form of human capital. For
example, a worker who encounters a dip in the demand for his skills can obtain new skills
through training or can relocate to an area where his skills are still in high demand. The
willingness or capacity of an agent to absorb the costs associated with these investments
may depend on the amount of saving that he has accumulated.
These observations suggest that a richer model of consumption/saving behavior should
account for these important interactions between saving and income. This paper introduces
such a model. In the model, workers transfer resources through time by investing in two
assets: a riskless asset, physical capital, and a risky asset, human capital. Two different
shocks affect the income process. The first type of shock directly affects the size of the stock
of human capital. An adverse realization of this shock depletes the stock of human capital
and provides incentives for new investments. The second type of shock affects the extent
to which the human capital can be fully employed. This shock is more transitory and is
modeled as a two-state Markov process: income is low in the bad state because the human
capital cannot be fully employed.
Numerically solving a model that endogenizes labor income in this way is a challenging
task. Relative to the standard consumption/saving problem, the problem considered here
has an additional continuous state variable (besides the stock of savings, or physical capital,
there is now also the stock of human capital) and an additional choice variable (besides
consumption, the agent must choose the level of investment in human capital). Moreover,
investment in human capital is subject to a non-negativity constraint and the stock of saving
(physical capital) is also subject to a non-negativity constraint. In spite of this complexity,
1
For examples of these models, see Deaton (1991),Aiyagari (1994), or Zeldes (1989).
Uninsured Idiosyncratic Risk and Human Capital Accumulation
2
the paper shows how to solve the model using an extension of the parameterized expectations
algorithm2 that can accommodate a problem with two continuous state variables, two choice
variables, and two constraints.
The degree of persistence of shocks to income is an important question in the consumption/saving literature. If shocks are highly persistent, or even permanent, then market
incompleteness creates greater welfare losses because agents cannot self-insure very effectively. There is some evidence that shocks are indeed extremely persistent. Ruhm (1991)
and Jacobson et al. (1993) show that displaced workers often suffer very persistent drops
in income. Meghir and Pistaferri (2001) provides further evidence from the PSID of the
importance of permanent shocks to income.
Permanent shocks to income are not easily handled in standard models of consumer
behavior under incomplete markets. With permanent shocks, assets typically are a Martingale process and thus the state space of an agent’s maximization problem is unbounded.
This problem is compounded in general equilibrium analysis in which the distribution of
a heterogeneous population of agents must be found because with permanent shocks there
will be no invariant distribution.
Of course, if agents can control their stock of human capital through their decisions,
then permanent shocks do not necessarily imply that their stocks of human capital will
follow a Martingale process. In the model considered here, agents experience diminishing
returns to human capital at the individual level and thus only agents with low stocks will
have incentive to invest. Together with the assumption of a constant rate of deterioration
for human capital, this ensures that an agent’s stock of human capital remains bounded
and possesses an invariant distribution, in spite of the permanent shocks.
An illustrative parameterization of the model is used to explore how well agents can selfinsure by holding a stock of savings (physical capital) that they can utilize to restore their
stock of human capital following an adverse shock. The optimal decision rules dictate that
when human capital is low, agents sell their financial assets in order to take advantage of the
higher expected return to human capital. Essentially, there is a no-arbitrage condition that
determines the agents’ holding of physical capital and human capital, with the expected
return to human capital being higher due to its greater riskiness.
2
See Christiano and Fisher (2000).
Uninsured Idiosyncratic Risk and Human Capital Accumulation
3
For the parameters chosen, self-insurance is not very effective. Following a permanent
negative shock to human capital, both financial assets and human capital (and thus consumption) recover extremely slowly. Accordingly, the invariant distribution reveals that at
any given point in time there are many agents with little savings of physical capital and
with a small stock of human capital. Thus, not only is consumption not smoothed very effectively, but many agents experience productive inefficiencies in the sense that they possess
much less human capital than they would under complete markets.
In order to fully assess the welfare losses associated with market incompleteness, the
general equilibrium version of the model is solved, as well as the complete markets general
equilibrium model. A Cobb-Douglas production technology is assumed and markets for
the two inputs—physical capital and human capital—are perfectly competitive. As with
standard incomplete markets models, the equilibrium rate of return to capital is less than the
subjective rate of time preference due to the fact that agents hold additional precautionary
stocks of financial assets in order to self-insure.3 This result is expected since riskiness of
human capital causes a higher physical capital to human capital ratio under incomplete
markets.
However, a somewhat unexpected result is that under incomplete markets not only is
the aggregate stock of physical capital greater than its counterpart, so is the aggregate
stock of human capital and, thus, so is aggregate output. This arises because the high
level of physical capital that comes from precautionary savings drives up the wage rate
paid to human capital services, which apparently more than offsets the riskiness of human
capital. Nevertheless, this result is particularly surprising since the invariant distribution
reveals that many agents have human capital levels far below the complete markets level.
In spite of the fact that aggregate output is greater under incomplete markets, welfare
losses due to market incompleteness are still substantial due to the relative ineffectiveness
of self-insurance.
The existing consumption/saving literature contains few attempts to treat agents’ income processes as endogenous. A few exceptions deserve mention. First, there have been
attempts to endogenize labor supply decisions in the context of a standard incomplete markets consumption/saving problem. Low (1999), for example, finds that when agents face a
3
See Ljungqvist and Sargent (2000), Ch. 14.
Uninsured Idiosyncratic Risk and Human Capital Accumulation
4
labor/leisure choice, precautionary saving is higher than when labor supply is exogenously
determined. On the one hand, when an agent’s wage falls, the cost of leisure declines and the
agent can substitute out of consumption and into leisure, reducing the benefit from holding
a stock of precautionary savings. On the other hand, the cost of accumulating these savings
is lower, since it can be achieved either by reducing consumption or by increasing labor
supply. Low finds that the second effect dominates.
Second, some recent papers in the search and matching literature have begun to explore
the interaction between uninsured idiosyncratic risk and search behavior. While the search
and matching literature has contributed a richer understanding of many features of the endogenous determination of labor income, complete asset markets are a standard assumption
in these models, so that workers’ decisions are entirely unaffected by the income risk that
labor market frictions cause. However, recent papers such as Acemoglu and Shimer (1999,
2000) and Gomes et al. (1998) have begun to explore the interaction between a worker’s
efforts to self-insure and his willingness to invest in search capital. In models like these,
agents with a greater stock of saving can better afford to invest in search capital, and thus
achieve higher wages, because they are in less danger of experiencing extreme consumption
volatility.
Of course, a worker might invest in many different types of human capital following a
negative shock to income opportunities. For example, in addition to investing in search capital, a worker may invest in skills by seeking additional training or may invest in “locational
capital” by incurring the costs of relocating to a region where employment opportunities are
better.4 The model presented here is sufficiently general to allow for these interpretations.
The paper is organized as follows. The next section describes the elements of the model.
Section 3 derives the Euler equations that implicitly describe an individual agent’s optimal decision rules. Section 4 characterizes an equilibrium of the model under incomplete
markets and describes the algorithm used to compute it. Section 5 describes the complete
markets equilibrium of the model. Section 6 uses a numerical solution of an illustrative
parameterization of the model to demonstrate some basic results. Section 7 briefly discusses several possible extensions to consider in future work. The Appendix describes the
numerical methods used to solve for an individual’s optimal decision rules.
4
See Bertola (2000) for an implementation of the latter idea.
Uninsured Idiosyncratic Risk and Human Capital Accumulation
2
5
The Environment
This section lays out the basic setup of the model. Time is discrete and has an infinite
horizon. The model is general equilibrium in that the wage rate, w, and the rate of return
to physical capital, r, are endogenously determined. There is no aggregate uncertainty
in the model. Asset market incompleteness transforms an ex ante identical population
of agents who experience uncertainty at the individual level into a population that is ex
post heterogeneous in their asset holdings and human capital stocks. However, the lack
of aggregate uncertainty implies that all aggregates will be stationary due to the invariant
nature of the distribution of agents across the state space.
There is a continuum of risk-averse agents of measure one whose objective is to maximize
E
∞
β t u(ct )
(1)
t=0
1−β
β
c1−γ
−1
t
1−γ .
where ct is consumption,
> 0 is the rate of time preference, and one-period utility
is isoelastic: u(ct ) =
Agents have access to two means of transferring resources
across time periods. First, they can buy and sell physical capital, kt , which yields a riskless
equilibrium rate of return, r. There is a borrowing limit that constrains these asset holdings
to be positive: kt ≥ 0, ∀t.5 For simplicity, it is assumed that physical capital does not
deteriorate.
Second, agents can invest in human capital. Allow ht to denote the stock of human
capital and it to denote investment in human capital. In the absence of investment in
human capital, the stock follows a geometric random walk with negative drift:
ht+1 = δ(ht + it )et+1
(2)
ln(ht+1 ) = ln(δ) + ln(ht + it ) + t+1 ,
(3)
or
where t+1 is distributed N(0,σ ) and 0 < δ < 1. The shock t generates increases or
5
We could allow for a more general borrowing limit: kt ≥ k, ∀t, with k being any number greater than the
“natural borrowing limit.” Any k above the natural borrowing limit will result in the borrowing constraint
occasionally binding. See Aiyagari (1994).
Uninsured Idiosyncratic Risk and Human Capital Accumulation
6
decreases in the stock of human capital ht that are permanent. Investment in human
capital, it , can be purchased directly with income. In reality human capital may instead
often be acquired as foregone leisure and may also be associated with various forms of
adjustment costs. Nevertheless, the more simple assumption that human capital can be
purchased—made here for the sake of tractability—still captures the general notion that
there are opportunity costs associated with human capital production and is consistent with
more elaborate theories of human capital formation, such as the one found in Ben-Porath
(1967). Finally, human capital is illiquid: it ≥ 0, ∀t.
An agent with stock ht supplies xt hαt units of labor services at the wage rate w per
unit. The parameter α satisfies 0 < α < 1 so that additional investments in human capital
are characterized by diminishing returns at the individual level. This ensures that human
capital remains bounded (since δ < 1). The variable xt is a two-state Markov process that
introduces exogenous, temporary shocks to income, with one state representing full employment and the other representing partial employment (or unemployment). Specifically,
xt ∈ x̄ ≡ {xh , xl },
0 < xl < xh
(4)
The transition probabilities for xt are given by the matrix P .
The factor prices r and w are determined in a completely conventional manner. Perfectly
competitive firms rent capital and hire labor services from individuals in order to produce
output using a constant returns to scale production technology:
Y = AK η H (1−η) .
(5)
Note that even though the aggregate production technology exhibits constant returns to
scale in accumulable factors (in contrast to the standard growth model), there is no growth
in equilibrium due to the fact that at the individual level, agents face a diminishing marginal
product of human capital (α < 1).
Uninsured Idiosyncratic Risk and Human Capital Accumulation
3
7
Individual decisions
This section describes the maximization problem that determines individuals’ consumption
and investment decisions and derives the associated Euler equations. These Euler equations
can be used to solve for the decision rules. The appendix describes the numerical techniques
used to achieve the solution.
An agent’s problem can be described by the following Bellman equation:
V (kt , ht , xt ) = max u(ct ) + βEt V (kt+1 , ht+1 , xt+1 )
ct ,it
s.t. kt+1 = (1 + r)kt − ct − it + wxt hαt
(6)
(6a)
ln(ht+1 ) = ln(δ) + ln(ht + it ) + t+1
(6b)
kt+1 ≥ 0
(6c)
it ≥ 0
(6d)
The Kuhn-Tucker conditions for the maximization problem are:
0 = u (ct ) − βEt V1 (kt+1 , ht+1 , xt+1 ) − θt
(7)
0 = −βEt V1 (kt+1 , ht+1 , xt+1 ) + βEt V2 (kt+1 , ht+1 , xt+1 )δet+1 − θt + ψt
(8)
θt kt+1 = 0
kt+1 ≥ 0
θt ≥ 0
ψt it = 0
it ≥ 0
ψt ≥ 0
(9)
where θt is the Lagrange multiplier associated with constraint (6c) and ψt is the Lagrange
multiplier associated with constraint (6d).
Equation (8) effectively says that, unless constraints bind, the value of investing in
physical capital is equal to the value of investing in human capital. If the credit constraint
is binding, then the expected return on investment in human capital is higher, but the agent
is unable to spend more on this investment. If the constraint associated with the illiquidity
of human capital is binding, then the expected return to savings of physical capital is higher
and the agent would prefer to disinvest human capital and invest in physical capital.
Uninsured Idiosyncratic Risk and Human Capital Accumulation
8
The Envelope Theorem provides two additional useful equations:
V1 (kt , ht , xt ) =(1 + r)[βEt V1 (kt+1 , ht+1 , xt+1 ) + θt ]
(10)
ht+1
V2 (kt , ht , xt ) = βEt V1 (kt+1 , ht+1 , xt+1 ) + θt · αwxt h1−α
+ βEt V2 (kt+1 , ht+1 , xt+1 )
.
t
ht + it
(11)
Note that the above derivation utilized
ht+1
ht +it
= δet+1 . Now the consumption Euler Equation
can be obtained by combining (7) and (10) to get:
u (ct ) = βEt V1 (kt+1 , ht+1 , xt+1 ) + θt
= βEt [(1 + r)(βEt+1 V1 (kt+2 , ht+2 , xt+2 ) + θt+1 )] + θt
= (1 + r)βEt u (ct+1 ) + θt
(12)
This is the standard Euler equation from a consumption/savings problem with a borrowing
constraint. When the constraint binds (θt > 0), current marginal utility is high due to the
inability to increase consumption via borrowing.
Turning now to the human capital investment decision, first combine (7) and (8) to get
βEt V2 (kt+1 , ht+1 )
ht+1
= βEt V1 (kt+1 , ht+1 , xt+1 ) + θt − ψt
ht + it
= u (ct ) − ψt
(13)
Then, substituting (7) and (13) into (11) yields
V2 (kt , ht , xt ) = wxt h1−α
u (ct ) + u (ct ) − ψt
t
(14)
This says that the value of a marginal unit of human capital is equal to the sum of the
marginal utility of the extra output and the marginal utility of extra consumption due to
a lower need for investment. The last term (ψt ) reflects the fact that if the illiquidity of
human capital is binding, the value of the marginal unit of ht is diminished.
Finally, to get the Euler equation for the investment decision, substitute equations (7)
Uninsured Idiosyncratic Risk and Human Capital Accumulation
9
and (14) into equation (8):
0 = −(u (ct ) − θt ) + βEt [(wxt h1−α
t+1 + 1)u (ct+1 ) − ψt+1 ]
u (ct ) − ψt = βEt [(wxt h1−α
t+1 + 1)u (ct+1 ) − ψt+1 ]
ht+1
− θt + ψt
ht + it
ht+1
ht + it
(15)
The benefit of not investing, but instead consuming, resources today is equal to the discounted expected marginal utility from consuming the return on investment, including the
extra consumption from not having to invest so much next period. The ψt+1 term reflects
the fact that the future benefit of current investment is lower if high current investment
causes the illiquidity of human capital to bind next period. Similarly,
The Euler equations in (12) and (15) can only be solved numerically. Computing the
solution is technically demanding because it involves jointly solving the two Euler equations
for two policy functions (consumption and human capital investment, or, alternatively,
human capital investment and next period’s stock of physical capital) and two multiplier
functions (θt and ψt ), all of which are functions of a state space consisting of two continuous
variables (kt and ht ) and a discrete variable (xt ). The appendix describes the solution
method employed.
4
Equilibrium
The solution to the Euler equations derived in the preceding section yields two decision
rules:
κ : kt+1 = κ(kt , ht , xt )
ι : it = ι(kt , ht , xt )
Let λt (kt , ht , xt ) denote the distribution of agents over the relevant state space. The decision
rules κ and ι, together with the stochastic processes xt and t , induce a law of motion T
Uninsured Idiosyncratic Risk and Human Capital Accumulation
10
that controls the evolution of this distribution:
λt+1 (kt+1 , ht+1 , xt+1 ) = T λt (kt , ht , xt )
(16)
=
P (xt+1 |xt = xj )
I kt+1 = κ(kt , ht , xi ) · λt (kt , ht , xj )
j={l,h}
φ ln(ht+1 ) − ln(δ) − ln(ht + ι(kt , ht , kj )) dht dkt
where φ denotes the N (0, σ ) probability density function and I is an indicator function
whose value is 1 if its argument is true, and zero otherwise.
Given a distribution λt , the aggregate stock of physical capital, Kt , and the aggregate
stock of human capital, Ht , as well as other aggregates, can be calculated by simple integration. In the stationary equilibrium that will be considered here, the distribution λt and
all the aggregates will be time-invariant. This is evident in the following definition.
Definition. A competitive stationary equilibrium is a function V , a pair of decision rules κ
and ι, an invariant distribution λ, a law of motion T , an aggregate stock K, an aggregate
stock H, and a pair of pricing functions r and w, such that:
(i) Given r and w, V , κ and ι solve the problem given in (6).
(ii) T is the law of motion induced by κ and ι and the two stochastic processes, xt and
t , described in (16)
(iii) λ = T λ
(iv) Both factor markets clear. That is, r = Aη(K/H)η−1 and w = A(1 − η)(K/H)η ,
where the aggregate stocks K and H are calculated by integrating with respect to λ.
The iterative algorithm used to compute an equilibrium is straightforward and is outlined
below:
1. Set the index i = 0. Select an initial value for the rate of return to physical capital,
r0 (a value satisfying r0 < 1−β
β is appropriate because higher values will generate
unbounded stocks of physical capital).
2. From the production technology and the marginal conditions for physical and human
capital, the equilibrium wage will be related to the equilibrium rate of return in such
a way that suggests the following value for the wage, given ri :
w = A(1 − η)
i
ri
Aη
η
η−1
Uninsured Idiosyncratic Risk and Human Capital Accumulation
11
3. Given ri and wi , compute agents’ optimal decision rules, using the techniques described in the Appendix.
4. Given the decision rules, compute the invariant distribution, λ, and calculate aggregate physical capital and aggregate human capital. These tasks can actually be
achieved in one step with Monte Carlo integration. In other words, simulate kt and
ht , for N agents and T time periods (with N · T very large). Throw out the first t0
realizations in order to eliminate sensitivity to starting values and then calculate
N
T
kn,t
N (T − t0 )
N T
xn,t hαn,t
H i = n=1 t=t0
N (T − t0 )
i
K =
5. If ri − Aη
Ki
Hi
n=1
t=t0
(17)
(18)
η−1
exceeds some pre-specified convergence criterion, then set ri+1 =
i η−1
(1 − ω)ri + ωAη K
(with 0 < ω < 1), increase i to i + 1, and return to step 2.
Hi
Otherwise stop.
5
Complete Markets
In order to make useful evaluations of the welfare cost of market incompleteness, the complete markets equilibrium must be found. To determine the allocation in a complete markets
equilibrium, it is useful to think in terms of the investment and consumption decisions that
a social planner would make. Consumption risk is fully insured and physical capital and
human capital investment decisions are made so that output is maximized. Often, in similar
models, determining the complete markets equilibrium reduces to a simple static problem.
However, in the environment under consideration here the irreversibility of human capital
complicates things. The irreversibility constraint can potentially bind for agents who receive sufficiently large positive shocks () to their human capital stock. This makes the
investment decision a dynamic one because the amount of investment chosen in one period
will affect the probability that the irreversibility constraint will bind in subsequent periods.
It’s clear that the ratio of physical capital to human capital in a complete markets
equilibrium will be such that the return to physical capital, rcm , satisfies (1 + rcm )β = 1.
Uninsured Idiosyncratic Risk and Human Capital Accumulation
Given this rate of return, the equilibrium
K
H
12
ratio can be determined from the production
technology and the marginal condition that equates rcm to the marginal product of physical
capital:
1−β
= rcm = Aη
β
1
1 − β η−1
K
=
Aηβ
H
K
H
η−1
(19)
Then, from the marginal condition that equates the wage to the marginal product of human
capital, the equilibrium wage under complete markets is given by
w
cm
= A(1 − η)
K
H
η
= A(1 − η)
1−β
Aηβ
η
η−1
(20)
The levels of aggregate physical capital, K, and human capital services, H, must still
be determined. This requires that the dynamic problem associated with the human capital
investment decision be solved. A planner who maximizes the present discounted value
of output for an agent with human capital stock ht and employment state xt , solves the
problem summarized in the following Bellman equation:
V cm (ht , xt ) = max wcm xt hαt − it +
it
1
Et V cm (ht+1 , xt+1 )
1 + rcm
(21)
s.t. ht+1 = δ(ht + it )et+1
it ≥ 0
The associated Euler equation is given by
(1 + rcm )(1 − φt ) = Et (wcm xt+1 αhα−1
t+1 + 1 − φt+1 )
ht+1
ht + it
(22)
where φ is the Lagrange multiplier associated with the irreversibility constraint. This Euler
equation dictates that investment in human capital be made up until the point where the
expected return to human capital, given wcm , equals the riskless rate of return to physical
capital, rcm .
The optimal decision rule icm (ht , xt ) can be solved using numerical techniques analogous
Uninsured Idiosyncratic Risk and Human Capital Accumulation
Parameter
β
γ
α
σ
1−δ
x̄
Description
discount factor
coef. of relative risk aversion
elasticity of labor services w.r.t. human capital
variance of shocks to human capital
depreciation rate of human capital
employment rates
P
transition matrix for xt
A
η
productivity parameter in output technology
elasticity of output w.r.t. capital
13
Value
0.96
2
0.85
0.002
0.04
[1
0.2] 0.965 0.035
0.5
0.5
0.1984
1/3
Table 1: Parameter Values
to the ones described in the Appendix. The nature of the resulting decision rule is very
simple. There is a target level of human capital. If the stock ht ever falls below its target
(which will happen when δet < 1), a new investment is made to bring the stock back to
the target. If the stock exceeds the target, investment is zero.
Because of the resulting heterogeneity in human capital levels, the invariant distribution
of agents over values of the human capital stock must be found in order to calculate the
aggregate stock of human capital services. Given the nature of the decision rule, at the
beginning of a period t, prior to the realizations of t , the distribution is a mass point at
the target level of human capital in addition to a continuous measure of agents with values
greater than the target level. Of course, following the shock t , the mass point becomes a
log normal distribution about the target level. A simple iterative procedure can be used to
find the invariant distribution.
Once the aggregate stock of human capital services is calculated by integrating over
the invariant distribution, the aggregate stock of physical capital can easily be solved using
equation (19). Output, consumption, and utility can all be easily calculated as well.
6
An Illustrative Simulation
This section discusses results for an illustrative parameterization of the model. Table 1
shows the parameter values chosen for the simulation. The chosen values are not carefully
Uninsured Idiosyncratic Risk and Human Capital Accumulation
c (k ,h ,x )
t
t
14
i (k ,h ,x )
t h
t
10
t
k
t h
30
150
20
100
10
50
0
100
0
100
(k ,h ,x )
t+1
t
t h
5
0
00
40
50
k
t
50
20
ht
0 0
k
t
c (k ,h ,x )
t
t
50
20
0 0
k
ht
t
i (k ,h ,x )
t l
t
10
t
20
0 0
k
t l
30
150
20
100
10
50
0
100
0
100
ht
(k ,h ,x )
t+1
t
t l
5
0
00
40
50
k
t
0 0
20
h
t
50
k
t
20
0 0
h
t
50
k
t
20
0 0
h
t
Figure 1: Policy functions
calibrated in any particular way, but are generally in line with values standard in the related
literature.
The choice for β reflects a time period of one year. The choices for γ and η are relatively
standard. The matrix P implies that 94.5% of all agents are in the full employment state
(xh = 1) at any point in time. Given η and β, the value for A implies a K/H ratio of
2 in the complete markets setting. The remaining parameters α, σ , and δ are somewhat
non-standard parameters but the values chosen for them do not seem extreme in any way.
Figure 1 provides plots of consumption, human capital investment, and next period’s
physical capital stock as a function of the state space. For agents in the fully employed
state (xh ), those with sufficient stocks of physical capital are willing to sell as much as is
necessary in order to bring their stock of human capital to a level at which the expected
return to human capital is on par with the return to physical capital. Even agents with less
physical capital are willing to convert most of their savings of physical capital into human
capital and keep only a small buffer stock of physical capital.
Uninsured Idiosyncratic Risk and Human Capital Accumulation
15
−3
x 10
6
5
4
3
2
40
1
0
140
30
120
20
100
80
10
60
40
k
20
0
0
h
t
t
Figure 2: Invariant Distribution
Figure 2 displays the invariant distribution of agents (summing over agents in the two
different employment states, xl and xh ). The large fraction of agents with very little saving
in physical capital is noteworthy—it almost appears that there is a “poverty trap.” This
result is surprising given that an agent has strong incentives to avoid this part of the
state space, where consumption is both low and highly volatile. It’s true that shocks to
human capital are permanent, and thus difficult to insure against. However, in contrast to
the conventional model with exogenous income, it is not impossible to insure against the
permanent shocks. Instead, agents do in fact accumulate physical capital so that when they
suffer a negative shock to their stock of human capital they can exchange their physical
capital for human capital, whose return would be higher due to the fact that there are
diminishing returns to human capital at the individual level. Nevertheless, the recovery
from an adverse negative shock is very slow (at least for the chosen parameterization).
This can be seen in figure 3. The figure shows the mean levels of human capital, physical
capital, and consumption of a large group of agents who begin with only 5 units of physical
capital and 5 units of human capital. The agents initially convert most of their physical
capital into human capital and subsequently both stocks rise very gradually (on average).
Uninsured Idiosyncratic Risk and Human Capital Accumulation
16
Mean of physical capital savings
8
6
4
2
0
0
20
40
60
80
100
120
Mean of human capital stock
140
160
180
200
0
20
40
60
80
100
120
Mean of consumption
140
160
180
200
0
20
40
60
80
140
160
180
200
20
15
10
5
1.5
1
0.5
100
120
Figure 3: Example of response to negative human capital shock
Variable
K
H
r
w
gross output
average consumption
average utility
Complete Markets
29.008
14.504
0.04167
0.1667
3.626
2.698
0.629
Incomplete Markets
32.463
15.459
0.04014
0.1699
3.928
2.929
0.599
Table 2: Equilibrium Values
Uninsured Idiosyncratic Risk and Human Capital Accumulation
17
Table 2 shows equilibrium values for both the complete markets economy and the incomplete markets economy. Several aspects of the results deserve mention. First, as expected,
the K/H ratio is higher under incomplete markets, as agents substitute away from the
risky asset. It follows that r is lower and w is higher in the incomplete markets equilibrium. Second, both K and H are higher under incomplete markets, and thus so is gross
output. This result stems from the fact that agents choose to increase their holdings of
physical capital (relative to complete markets) since it can be used to self-insure against
shocks to labor earnings. Apparently, the resulting increase in the return to human capital
outweighs the fact that human capital is less desirable under incomplete markets due to
its riskiness. Nevertheless, it’s somewhat surprising that aggregate human capital services
are greater under incomplete markets given the large number of agents with low levels of
human capital, as indicated by the invariant distribution in figure 2.
Third, as expected, agents still enjoy lower expected utility under incomplete markets,
in spite of the higher output. Obviously, this result stems from the tremendous variation
in both physical capital and human capital stocks, and thus in consumption, that is seen
in the invariant distribution. A standard way to gauge the welfare loss due to market
incompleteness is to calculate the amount of consumption that would have to be given to
all agents in the incomplete market setting in order to make them as well off (in an expected
utility sense) as agents in the complete markets setting. For this parameterization, that
amount is 0.310 units of consumption, or 0.106% of the average consumption level. It’s
worth emphasizing that this is 0.310 units of consumption in addition to the additional
0.231 units that agents already consume relative to the complete markets case.
7
Extensions
There are several possible interesting extensions or applications of the model presented
above. This section discusses some of them. Of course, a serious examination of any of
them would require a more careful calibration of the model than the simple illustrative
parameterization chosen above.
Social Insurance Policies. There are several policy experiments that deserve consideration. Unemployment insurance and subsidies to education are natural candidates. The
Uninsured Idiosyncratic Risk and Human Capital Accumulation
18
relative merits of the two, in the context of this model, likely depend on the relative importance of temporary and permanent shocks.
Comparison with Exogenous Income Models. Meghir and Pistaferri (2001) suggest
that more realistic statistical models of the income process (for example, they find that an
ARCH type model is appropriate for income) imply consumption behavior that can differ
significantly from that found in consumption models with more simple exogenous income
processes. It may be of interest to examine whether the model considered here can explain
the joint behavior of consumption and income better than standard models with exogenous
income. More fundamentally, one would like to examine how the behavior of consumption
in the model with endogenous income differs from the behavior of consumption in a model
with an exogenous income process identical to the endogenously generated one.
Wealth Distribution. The invariant distribution in figure 2 suggests that this model may
have potential for helping to understand important facts about the distribution of wealth
in the U.S., such as the large spike of people at the lower end of the distribution and the
long thin tail at the upper end. While papers such as Huggett (1996), and Krusell and
Smith Jr. (1998), among others, have used consumption/savings models with exogenous
income to analyze wealth distributions, to my knowledge models that endogenize income
are much rarer in this literature. Perhaps a finite-horizon variant of the model would be
more appropriate for this type of analysis.
Heterogeneous Agents vs. Representative Agent. Krusell and Smith Jr. (1998)
conclude from their model that the heterogeneity that results from market incompleteness
is of very limited importance for understanding aggregate fluctuations. This result stems
from the lack of heterogeneity in the marginal propensity to save—since all agents are
likely to save a similar fraction of their wealth, who holds the wealth is unimportant. As
Gourinchas (2000) has pointed out, allowing for life-cycle dynamics and for more persistent
shocks to labor income produces significant heterogeneity in marginal propensities to save.
Nevertheless, there is still little heterogeneity among the part of the population that controls
most of the financial wealth, so uninsured risk is still likely to be of secondary importance for
understanding aggregate fluctuations. That is, movements in the rate of return to capital
are not likely to result from heterogeneity in saving behavior.
In the model presented here, however, there is substantial heterogeneity in the propensity
Uninsured Idiosyncratic Risk and Human Capital Accumulation
19
to save in physical capital and to invest in human capital and this heterogeneity is likely
to be important for understanding fluctuations in the wage rate. In contrast to financial
assets, human capital is not concentrated among a small pocket of the population and thus
the distribution of holdings of both financial assets and of human capital is likely to be
important for understanding fluctuations.
Equity Premium. Agents’ attitudes towards investments in risky financial assets is likely
to be different in the context of the present model of human capital accumulation. Papers
such as Heaton and Lucas (1996) argue that uninsured idiosyncratic risk is not quantitatively
important for explaining the equity premium. If shocks to human capital are positively
correlated with returns to risky financial assets—as seems very plausible—then agents may
be less inclined to hold the risky assets than they would be in the conventional model with
exogenous labor income. Risky financial assets are less attractive if they yield low returns
when human capital is destroyed not only because an agents’ marginal utility will be high
at those times, but also because the potential return from investment in new human capital
is higher.
8
...
Conclusion
Uninsured Idiosyncratic Risk and Human Capital Accumulation
20
Appendix
This appendix outlines the computational approach used to solve the Euler equations associated with an agent’s maximization problem.6 The approach is based on the parameterized
expectations algorithm used by Christiano and Fisher (2000) to solve the stochastic growth
model with irreversible investment.
The following notation will be used to express the policy functions and multiplier functions as functions of the state space:
it = ι(kt , ht , xt )
kt+1 = κ(kt , ht , xt )
θt = Θ(kt , ht , xt )
ψt = Ψ(kt , ht , xt )
A.1
Residual equations
It is useful to express the consumption Euler equation as a residual equation:
Rc (kt , ht , xt ; ι, κ, Θ) = 0 =uc (kt , ht , xt ; ι, κ) − Θ(kt , ht , xt )−
βEt m κ(kt , ht , xt ; ι, κ), δexp(t+1 )(ht + ι(kt , ht , xt )), xt+1 ; ι, κ
where
and
m kt+1 , ht+1 , xt+1 ; ι, κ = (1 + r)uc kt+1 , ht+1 , xt+1 ; ι, κ
uc kt , ht , xt ; ι, κ) = u (1 + r)kt + wxt hαt − κ(kt , ht , xt ) − ι(kt , ht , xt )
The expectation in the residual equation is taken with respect to the distributions for t+1
and for xt+1 .
We can define the conditional expectation function, e, such that the residual equation
becomes
Rc (kt , ht , xt ; e, d) =e(kt , ht , xt )−
βEt m κ(kt , ht , xt ), δexp(t+1 )(ht + ι(kt , ht , xt )), xt+1 ; ι, κ
where now ι and κ are derived from e and another function, d, to be described below.
The investment Euler equation can also be expressed as a residual equation:
Ri (kt , ht , xt ; ι, κ, Ψ) = 0 =[uc kt , ht , xt ; ι, κ − Ψ(kt , ht , xt ))](ht + it )−
βEt n κ(kt , ht , xt ), δexp(t+1 )(ht + ι(kt , ht , xt )), xt+1 ; ι, κ, Ψ
6
Fortran and Matlab programs that implement the solution approach outlined in the Appendix are
available from the author upon request.
Uninsured Idiosyncratic Risk and Human Capital Accumulation
21
where
n kt+1 , ht+1 , xt+1 ; ι, κ, Ψ = (wxt+1 αhα−1
t+1 + 1)uc kt+1 , ht+1 , xt+1 , ι, κ −
Ψ(kt+1 , ht+1 , xt+1 ) ht+1
We can define the conditional expectation function, d, such that the residual equation
becomes
Ri (kt , ht , xt ; e, d) =d(kt , ht , xt )−
βEt n κ(kt , ht , xt ), δ exp(t+1 )(ht + ι(kt , ht , xt )), xt+1 ; ι, κ, Ψ
A.2
Backing out the policy functions
Suppose that d and e were known. Then, if κ were known as well, for every point in the
state space ῑ could be implicitly defined according to
uc kt , κ(kt , ht , xt ), ῑ(kt , ht , xt ), xt (ht + ῑ(kt , ht , xt )) = d(kt , ht , xt )
and then ι and Ψ would be given by
ῑ(kt , ht , xt ) if ῑ(kt , ht , xt ) > 0
ι(kt , ht , xt ) =
0
otherwise
Ψ(kt , ht , xt ) =uc kt , κ(kt , ht , xt ), ι(kt , ht , xt ), xt −
d(kt , ht , xt )
(ht + ι(kt , ht , xt ))
Conversely, if ι were known, the functions κ and θ could be determined in a similar
fashion for every point in the state space. Of course, neither ι nor κ are known, so the
solution approach must find them jointly, along with Θ and Ψ. To this end, define z̄ as the
sum of next period’s physical capital stock and of human capital investment (it + kt ) that
satisfies the consumption Euler equation:
−1 (A.1)
z̄(kt , ht , xt ) = max[0, (1 + r)kt + wxt hαt − u
e(kt , ht , xt )]
Given z̄, define κ̄ as the choice of asset holdings that satisfies the investment Euler equation:
κ̄(kt , ht , xt ) = ht + z̄(kt , ht , xt ) −
u
d
.
(1 + r)kt + wxt hαt − z̄(kt , ht , xt )
(A.2)
It is straightforward to show then that for each point in the state space the optimal decision
Uninsured Idiosyncratic Risk and Human Capital Accumulation
rules κ and ι can be determined from z̄ and κ̄ as follows
ι
κ̄ ≥ z̄ = 0 ⇒
κ
ι
z̄ > κ̄ > 0 ⇒
κ
ι
κ̄ ≥ z̄ > 0 ⇒
κ
ι
z̄ > 0 ≥ κ̄ ⇒
κ
ι
z̄ = 0 ≥ κ̄ ⇒
κ
where ι̃ solves
ι̃ =
A.3
22
=0
=0
= z̄ − κ̄
= κ̄
=0
= z̄
= ι̃
=0
= max[0, ι̃]
=0
d
−h
u (1 + r)a + wxhα − ι̃
Solution algorithm
The conditional expectation functions, e and d, are approximated by a linear combination
of orthogonal basis functions. In particular, a tensor product of degree-N Chebyshev polynomials is used.7 Using Tl (ω) to denote the lth order polynomial,8 evaluated at ω, then the
approximations are given by
ê(a, h, x, Γ) =
N N
e,x
γs,q
Ts (a)Tq (h)
s=0 q=0
ˆ h, x, Γ) =
d(a,
N N
d,x
γs,q
Ts (a)Tq (h)
s=0 q=0
e,x e,x d,x d,x
, γs,q , γs,q , γs,q |x = {xl , xh }; s =
where Γ is a vector containing the 4(N +1)2 coefficients, {γs,q
0, . . . , N ; q = 0, . . . , N }.
The model is solved by an orthogonal collocation approach.9 This entails solving for
the 4(N + 1)2 coefficients that satisfy Rc (ki , hj , xk ; Γ) = 0 and Ri (ki , hj , xk ; Γ) = 0 for
i = 1, . . . , M , j = 1, . . . , M , and k = l, h (where M = N + 1).
The following algorithm achieves this solution method:
7
The degree of the polynomials in the two dimensions need not be the same.
The Chebyshev polynomials can be computed using the recursion formula Tl (ω) = 2zTl−1 (ω) − Tl−2 (ω),
with T1 (ω) = ω and T0 (ω) = 1.
9
For more on this, see Judd (1998).
8
Uninsured Idiosyncratic Risk and Human Capital Accumulation
23
1. Select the approximation nodes. Compute M = N + 1 Chebyshev interpolation nodes
on [−1, 1]:
2k − 1
π)
k = 1, . . . , M
yk = −cos(
2M
Assume minimum and maximum values for a and for h: a ∈ [a, ā] and h ∈ [h, h̄] (it
can be verified later whether the range of values considered is broad enough). Then
adjust the Chebyshev interpolation nodes to these intervals:
ā − a
)+a
2
h̄ − h
)+h
hi = (yi + 1)(
2
ki = (yi + 1)(
i = 1, . . . , M
i = 1, . . . , M
2. Select an initial set of values for the coefficients (Γ).
ˆ t , ht , xt ; Γ) for each node of the state
3. Given coefficients, calculate ê(kt , ht , xt ; Γ) and d(k
space, (ki , hj , xk ). Then calculate approximations of the policy functions, ι̂(kt , ht , xt ; Γ)
and κ̂(kt , ht , xt ; Γ), as well as the multiplier functions, Θ̂(kt , ht , xt ; Γ) and Ψ̂(kt , ht , xt ; Γ),
as follows. First, compute
−1 ê(ki , hj , xk ; Γ) ]
z̄(ki , hj , xk ; Γ) = max[0, (1 + r)ki + wxhαj − u
κ̄(ki , hj , xk ; Γ) = hj + z̄(ki , hj , xk ; Γ) −
Then, κ̂ and ι̂ are determined as follows:
κ̄(ki , hj , xk ) ≥ z̄(ki , hj , xk ; Γ) = 0 ⇒
z̄(ki , hj , xk ; Γ) > κ̄(ki , hj , xk ; Γ) > 0 ⇒
κ̄(ki , hj , xk ; Γ) ≥ z̄(ki , hj , xk ; Γ) > 0 ⇒
z̄(ki , hj , xk ; Γ) > 0 ≥ κ̄(ki , hj , xk ; Γ) ⇒
z̄(ki , hj , xk ; Γ) = 0 ≥ κ̄(ki , hj , xk ; Γ) ⇒
d(ki , hj , xk ; Γ)
.
u (1 + r)ki + wxhαj − z̄(ki , hj , xk ; γ)
ι̂(ki , hj , xk ; Γ) = 0
κ̂(ki , hj , xk ; Γ) = 0
ι̂(ki , hj , xk ; Γ) = z̄(ki , hj , xk ; Γ) − κ̄(ki , hj , xk ; Γ)
κ̂(ki , hj , xk ; Γ) = κ̄(ki , hj , xk ; Γ)
ι̂(ki , hj , xk ; Γ) = 0
κ̂(ki , hj , xk ; Γ) = z̄(ki , hj , xk ; Γ)
ι̂(ki , hj , xk ; Γ) = ι̃(ki , hj , xk ; Γ)
κ̂(ki , hj , xk ; Γ) = 0
ι̂(ki , hj , xk ; Γ) = max[0, ι̃(ki , hj , xk ; Γ)]
κ̂(ki , hj , xk ; Γ) = 0
Uninsured Idiosyncratic Risk and Human Capital Accumulation
24
where ι̃(ki , hj , xk ; Γ) solves
ι̃(ki , hj , xk ; Γ) =
u
ˆ i , hj , xk ; Γ)
d(k
− hj
(1 + r)ki + wxk hαi − ι̃(ki , hj , xk ; Γ)
Finally,
Θ̂(ki , hj , xk ; Γ) = uc ki , κ̂(ki , hj , xk ; Γ), ι̂(ki , hj , xk ; Γ), x − ê(ki , hj , xk ; Γ)
ˆ i , hj , xk ; Γ)
d(k
Ψ̂(ki , hj , xk ; Γ) = uc ki , κ̂(ki , hj , xk ; Γ), ι̂(ki , hj , xk ; Γ), x −
hj + ι̂(ki , hj , xk ; Γ)
4. Use the results from step 3 to calculate “data.” That is, data for the conditional
expectation e can be computed as
y e (ki , hj , xk ; Γ) =βEm(κ̂(ki , hj , xk ; Γ), δ(hj + ι̂(ki , hj , xk ; Γ))e , x )
=βp(x = xh |x) m(κ̂(ki , hj , xk ; Γ), δ(hj + ι̂(ki , hj , xk ; Γ))e , xh )dF ()+
βp(x = xl |x) m(κ̂(ki , hj , xk ; Γ), δ(hj + ι̂(ki , hj , xk ; Γ))e , xl )dF ()
and data for the conditional expectation d can be computed as
y d (ki , hj , xk ; Γ) =βEn(κ̂(ki , hj , xk ; Γ), δ(hj + ι̂(ki , hj , xk ; Γ))e , x )
=βp(x = xh |x) n(κ̂(ki , hj , xk ; Γ), δ(hj + ι̂(ki , hj , xk ; Γ))e , xh )dF ()+
βp(x = xl |x) n(κ̂(ki , hj , xk ; Γ), δ(hj + ι̂(ki , hj , xk ; Γ))e , xl )dF ().
The integral is computed via Gauss-Hermite quadrature. Note that computing the
integrand requires solving for both ι̂(κ̂(ki , hj , xk ; Γ), δ(hj + ι̂(ki , hj , xk ; Γ))e , xk ) and
κ̂(κ̂(ki , hj , xk ; Γ), δ(hj + ι̂(ki , hj , xk ; Γ))e , xk ), at every combination of i = 1, . . . , N ,
j = 1, . . . , N , and k = {l, h}, and at each Gauss-Hermite integration node for .
Because the problem is an infinite horizon problem, the policy functions are timeautonomous and thus these can also be obtained from the current approximations of
e and d.
5. The residual equations can now be expressed as
N N
e,xk
γs,q
Ts (ki )Tq (hj ) − y e (ki , hj , xk ; Γ) = 0
(A.3)
d,xk
γs,q
Ts (ki )Tq (hj ) − y d (ki , hj , xk ; Γ) = 0
(A.4)
s=0 q=0
N N
s=0 q=0
Uninsured Idiosyncratic Risk and Human Capital Accumulation
25
for i = 1, . . . , N , j = 1, . . . , N , and k = {l, h}. Due to the orthogonality property of
the polynomials, these equations can be expressed as
M M
cs,q Ts (ki )Tq (hj )y e (ki , hj , xk ; Γ)
M
(A.5)
M M
cs,q =
Ts (ki )Tq (hj )y d (ki , hj , xk ; Γ)
M
(A.6)
e,xk
=
γs,q
i=1 j=1
d,xk
γs,q
i=1 j=1
where
cs,q


1
= 2


4
if q = s = 1
if q = 1 = s or s = 1 = q
if q = 1 and s = 1
(A.7)
6. Use a non-linear equation solver to determine the coefficients Γ that solve the residual
equations.
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