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Federal Reserve Bank of Chicago
Insurance in Human Capital Models with
Limited Enforcement
Tom Krebs, Moritz Kuhn, and Mark Wright
April 2016
WP 2016-08
Insurance in Human Capital Models with Limited
Enforcement
∗
Tom Krebs
University of Mannheim and IZA
Moritz Kuhn
University of Bonn and IZA
Mark Wright
FRB Chicago and NBER
April 2016
Abstract
This paper develops a tractable human capital model with limited enforceability of contracts. The model
economy is populated by a large number of long-lived, risk-averse households with homothetic preferences
who can invest in risk-free physical capital and risky human capital. Households have access to a complete set
of credit and insurance contracts, but their ability to use the available financial instruments is limited by the
possibility of default (limited contract enforcement). We provide a convenient equilibrium characterization
that facilitates the computation of recursive equilibria substantially. We use a calibrated version of the model
with stochastically aging households divided into 9 age groups. Younger households have higher expected
human capital returns than older households. According to the baseline calibration, for young households
less than half of human capital risk is insured and the welfare losses due to the lack of insurance range from
3 percent of lifetime consumption (age 40) to 7 percent of lifetime consumption (age 23). Realistic variations
in the model parameters have non-negligible effects on equilibrium insurance and welfare, but the result that
young households are severely underinsured is robust to such variations.
Keywords: Human Capital Risk, Limited Enforcement, Insurance
JEL Codes: E21, E24, D52, J24
∗
We thank our discussant, Andrew Glover, and seminar participants at various institutions and conferences for useful comments. Tom Krebs thanks the German Research Foundation for support under grant
KR3564/2-1. The views expressed herein are those of the authors and not necessarily those of the Federal
Reserve Bank of Chicago or the Federal Reserve System.
I. Introduction
Many households own almost nothing but their human capital. Moreover, there is strong
evidence that human capital investment is risky, while consumption insurance against this
risk is far from complete. In other words, a significant fraction of labor income is the return to
human capital investment, and a voluminous empirical literature has shown that individual
households face large and highly persistent labor income shocks that have strong effects on
individual consumption. In this paper, we argue that one financial friction—limited contract
enforcement—can explain a substantial part of the observed lack of consumption insurance.
Intuitively, in equilibrium households with high human capital returns and little financial
wealth would like to borrow in order to buy insurance and invest in human capital, but they
cannot do so because of borrowing constraints that arise endogenously due to the limited
enforceability of credit contracts.
Our analysis proceeds in two steps. First, we develop a tractable human capital model
with limited contract enforcement and provide a useful equilibrium characterization result.
Second, we show that a calibrated macro model with physical capital, human capital, and
limited contract enforcement can explain the observed lack of consumption insurance for a
large group of households. Moreover, we show that this result is robust to realistic variations in the model parameters describing human capital risk, risk aversion, and contract
enforcement.
The model developed in this paper is a version of the type of human capital model that
has been popular in the endogenous growth literature. More specifically, we consider a
production economy with an aggregate constant-returns-to-scale production function using
physical and human capital as input factors. There are a large number (a continuum) of
individual households with CRRA-preferences who can invest in risk-free physical capital
and risky human capital. Human capital investment is risky due to shocks to the stock
1
of human capital that follow a stationary Markov process with finite support (a Markov
chain). In the main part of the paper, we assume that all shocks are idiosyncratic, but we
also discuss how our theoretical characterization result can be extended to the case in which
idiosyncratic and aggregate shocks co-exists. Households have access to a complete set of
credit and insurance contracts, but their ability to use the available financial instruments is
limited by the possibility of default, which produces endogenous borrowing, or short-sale,
constraints. Defaulting households continue to participate in the labor market, but part of
their labor income might be garnished and they are excluded from financial markets until a
stochastically determined future date.
The tractability of the model derives from two equilibrium characterization results. First,
the consumption-investment choice of households is linear in total wealth (financial wealth
plus human capital) and the portfolio choice of households is independent of wealth. Further,
the solution to the household decision problem can be obtained solving a static maximization
problem. Moreover, the maximization problem of individual households is shown to be
convex so that a simple FOC-approach is applicable. Second, recursive equilibria can be
found by solving a fixed-point problem that is independent of the wealth distribution. Thus,
a rather complex, infinite-dimensional fixed-point problem has been transformed into a much
simpler, finite-dimensional fixed-point problem.
In the quantitative part of the paper, we consider a version of the model with i.i.d. human
capital shocks and stochastically aging households divided into 9 age groups. Household age
affects expected human capital returns and younger households have higher returns than
older households. The model is calibrated to be consistent with the U.S. evidence on labor
market risk and life-cycle earnings. Specifically, we choose the model parameters determining
the life-cycle profile of expected human capital returns so that the implied life-cycle profile
of median earnings growth rates matches the data. Further, in our model, i.i.d. shocks to
2
the stock of human capital translate into a labor income process that follows a logarithmic
random walk; that is, labor income shocks are permanent. The random-walk specification
has often been used in the empirical literature to model the permanent component of labor
income risk, and we use the estimates obtained by this literature to calibrate our model
economy. Finally, for the baseline calibration we use a degree of relative risk aversion of 1
(log-utility) and a level of contract enforcement (exclusion from financial markets in case of
default) in line with the US bankruptcy code. The results of our quantitative analysis can
be summarized as follows.
First, the calibrated model is in line with the observed life-cycle pattern of household
portfolio choices (mix between financial capital and human capital). Second, many young
households are borrowing constrained and substantially under-insured, where we measure
the degree of consumption insurance by one minus the ratio of the volatility of consumption
growth to the volatility of income growth (the insurance coefficient). For example, households
of age group 26 − 30 only insure 40 percent of their human capital risk even though insurance
markets exist and are perfectly competitive. Further, the welfare consequences of the lack of
consumption insurance are severe. For households of age group 26−30, welfare would increase
by 6 percent of lifetime consumption if they had unlimited access to financial markets. Third,
the result that many young households are substantially under-insured is robust to realistic
variations in the model parameters describing human capital risk, risk aversion, and contract
enforcement. However, such parameter variations have non-negligible effects on the extent
of equilibrium insurance, which suggest that the model presented here has the potential to
account for substantial differences in consumption insurance over time and across countries.
In sum, this paper makes a methodological contribution and a substantive contribution.
Theoretically, we develop a general framework and prove a characterization result for recursive equilibria that provides a powerful tool for the quantitative analysis of a wide range
3
of interesting macroeconomic issues. Substantively, we show that, contrary to the results
obtained by most of the previous literature, limited contract enforcement can explain the
observed lack of consumption insurance for a large group of households.
Literature: This paper builds on the large literature on limited commitment/enforcement.
See, for example, Alvarez and Jermann (2000), Kehoe and Levine (1993), Kocherlakota
(1996), and Thomas and Worrall (1988) for seminal theoretical contributions and Krueger
and Perri (2006) and Ligon, Thomas, and Worrall (2002) for highly influential quantitative work. Our theoretical contribution is to develop a tractable model with human capital
accumulation and to show how to avoid the non-convexity problem that often arises in limited enforcement models with production.1 Our substantive contribution is to show that a
calibrated macro model with physical capital and limited contract enforcement generates a
substantial degree of underinsurance. In contrast, previous work on consumption insurance
in limited enforcement models did not consider life-cycle variations in earnings and human
capital investment decisions. As a consequence, in these models there is little reason for
households to borrow, and a common finding of the previous literature has been that consumption insurance is almost perfect in calibrated models with physical capital (Cordoba,
2008, and Krueger and Perri, 2006).2 Finally, we share with Andolfatto and Gervais (2006)
and Lochner and Monge (2011) the focus on human capital accumulation and endogenous
borrowing constraints due to enforcement problems, but we go beyond their work by studying
the interaction between borrowing constraints and insurance.
The current paper is most closely related to Krebs, Kuhn, and Wright (2015), who pro1
Wright (2001) has shown how to circumvent the non-convexity issue in linear production models (AKmodel) with limited enforcement. The model structure we use in this paper is based on the human capital
model with incomplete markets analyzed in Krebs (2003).
2
Krueger and Perri (2006) match the cross-sectional distribution of consumption fairly well, but the
implied volatility of individual consumption is negligible in their model.
4
vide evidence from the life insurance market that human capital returns and insurance are
negatively correlated. Krebs, Kuhn, and Wright (2015) also conduct a quantitative analysis
of under-insurance based on a calibrated macro model similar to the one studied here. The
current analysis goes beyond Krebs, Kuhn, and Wright (2015) in two important dimensions.
First, the theoretical results derived in the current paper cover the case of general CRRApreferences, non-steady state behavior, and aggregate risk. In contrast, Krebs, Kuhn, and
Wright (2015) confine attention to steady state equilibria in economies with log-preferences
and no aggregate risk. Second, in the current paper we provide a comprehensive analysis
of the conditions that generate non-negligible under-insurance in calibrated models with
limited enforcement and risky human capital investment.
Our paper is also related to the voluminous literature on macroeconomic models with
exogenously incomplete markets, and in particular studies of human capital accumulation
(Krebs, 2003, Guvenen, Kuruscu, and Ozkan, 2014, and Huggett, Ventura, and Yaron,
2011). The current paper and Krebs, Kuhn, and Wright (2015) are complementary to the
incomplete-market literature on human capital investment in the sense that they address
similar issues from different angles. Specifically, the incomplete-market approach studies
the effect of human capital risk on investment/saving and consumption behavior when no
insurance beyond self-insurance is available. In contrast, the limited-enforcement approach
analyzes the effect of human capital risk on investment/saving and consumption behavior
when insurance markets are available, but endogenous borrowing constraints due to limited
contract enforcement generate under-insurance.
II. Model
In this section, we develop the model and define the relevant equilibrium concept. The model
is a generalization of Krebs, Kuhn, and Wright (2015), which in turn is based on a combination of the human capital model developed in Krebs (2003) and the limited commitment
5
model with linear technology presented in Wright (2001).3
a) Human Capital Production
Time is discrete, open ended, and indexed by t = 0, 1, . . .. There is a continuum of households
who live for a stochastic amount of time. A household who dies is replaced by a new-born
household so that the mass of all households alive is normalized to one. We denote the cohort
of a household (the period of birth) by n, but will suppress the cohort-index for notational
ease until we discuss the aggregate market clearing conditions. The exogenous state of an
individual household is denoted by st and has several components st = (s1t , . . . , smt ). In
our quantitative application, st has two components, one denoting the age of the household
and a second representing human capital risk. Depending on the application, additional
components can be used to model either ex-ante heterogeneity or ex-post heterogeneity
(risk). For example, Krebs, Kuhn, and Wright (2015) use additional components to model
the family structure of households in detail. For simplicity, we assume that st can only
take on a finite number of values. We assume that for each household of cohort n, the
process {st}∞
t=n is Markov with a stationary transition function and denote the transition
probabilities by π(st+1|st ). Note that household variables should in principle have a cohort
index n in addition to the time index t, but to ease the notation we suppress the cohort
index whenever possible.
There is one good that can be consumed or used as physical capital in production (see
below). Each household can transform one unit of the good into φ(st) units of human capital.
The accumulation equation for human capital, h, of an individual household is given by
ht+1 = (1 + (st−1, st )) ht + φ(st )xht ,
(1)
3
Angeletos (2007) and Moll (2014) develop tractable models of entrepreneurial activity in which individual
consumption/saving policies are linear in wealth. In all these approaches, tractability is achieved through
the assumption that individual investment returns are independent of household wealth.
6
where xht is human capital investment of the individual household in period t and is an
idiosyncratic human capital shock.
In line with Jones and Manuelli (1990) and Rebelo (1991), the human capital accumulation equation (1) focuses on the goods cost of human capital production. In contrast, Lucas
(1988), Huggett et al. (2011), and Lochner and Monge (2011) assume that the only cost of
human capital production is a time cost. As suggested by Ben-Porath (1967) and Trostel
(1993), in many applications both goods cost and time cost are important components of
the total cost of human capital production. It is straightforward to extend our model to
the case that allows for both goods cost and time cost of human capital production (see our
discussion in Section III.f below).
The term in (1) captures deterministic and random changes in human capital that
are due to depreciation, learning-by-doing, and various shocks to human capital (skills) of
households. For example, a negative human capital shock could can occur when a household
member loses firm- or sector-specific human capital subsequent to job termination (worker
displacement). A decline in health (disability) or death of a household member provide
further examples of negative human capital shocks. In this case, both general and specific
human capital are lost. Internal promotions and upward movement in the labor market
provide two examples of positive human capital shocks.
We impose the restriction that the stock of human capital must be non-negative, or h ≥ 0.
This creates no technical difficulty and our general characterization of the household decision
rule (proposition 1) holds with this constraint imposed, regardless of whether or not it binds.
In our quantitative analysis, this constraint never binds (does not bind for all households
types and uncertainty states). We do not impose the requirement that gross human capital
investment be non-negative, or xh ≥ 0. This is necessary for tractability which, in turn, is
essential for the theoretical and quantitative analysis conducted in this paper. However, in
7
the calibrated model economy used for our quantitative analysis, a number of alternative
formulations of non-negativity constraints on human capital investment are always satisfied
in equilibrium; that is, they hold for all household types at all ages and all realizations
of uncertainty. See the quantitative Section IV for more details. Thus, imposing these
restrictions would not change the conclusions drawn in the quantitative analysis.
b) Household Budget Constraint
An individual household born in period n of type sn begins life with an initial endowment
of human capital, hn and an initial endowment of financial assets, an . Thus, the initial state
of an individual household is a vector (an , hn , sn ). In each period t ≥ n, households can buy
and sell a (sequentially) complete set of financial contracts (assets) with state-contingent
payoffs, and we assume that for each state s there is one contract or Arrow security. We
denote by at+1(st+1 ) the quantity bought (or sold, if negative) in period t of the contract
that pays off one unit of the good in period t + 1 if st+1 occurs, and denote the price of this
contract by qt(st+1 ). A budget-feasible plan has to satisfy the sequential budget constraint
r̃ht z(st )ht + at(st ) = ct + xht +
X
X
at+1(st+1 )qt (st+1)
st+1
at+1(st+1 )qt(st+1 ) ≥ −D̄ht+1
(2)
st+1
ct ≥ 0 , ht+1 ≥ 0.
The variable z denotes an idiosyncratic shock to the productivity of human capital while r̃ht
denotes the (common) rental rate per efficiency unit of human capital. The term D̄ < 1 is
an explicit debt constraint that requires debt not to exceed the value of human capital. The
explicit debt constraint in (2) is sufficient to rule out Ponzi schemes. Since D̄ can be chosen
arbitrarily close to 1 it amounts to the “natural borrowing constraint” in our setting.
Given the initial state (an , hn , sn ), a household of cohort n chooses a plan {ct , at, ht }∞
t=n ,
where each plan is a sequence of functions mapping histories, sn,t , into actions, ct(sn,t ),
8
at+1 (sn,t, .), and ht+1 (sn,t ), where for given sn,t the variable at+1(sn,t , .) is a vector with
components st+1 . Here sn,t = (sn , . . . , st ) denotes the history of individual states st from
period n up to period t. Note that the household level equations (1) and (2) have to hold in
realizations; that is, they have to hold for all histories, sn,t .
c) Preferences
Households have identical preferences over consumption plans. Households are risk-averse
and their preferences allow for a time-additive expected utility representation:
∞
. X t−n
U ({ct }∞
|s
)
=
β E[νt u(ct)|sn ] ,
n
t=n
(3)
t=n
where νt is the probability that a household born in period n is alive in period t and the
expectations is taken over all individual histories
.
E[νt u(ct)|sn ] =
X
νt (sn,t−1 )u(ct(sn,t ))π(sn,t |sn ) .
sn,t |sn
Here π(sn,t|sn ) stands for the history that sn,t occurs given sn , which is given by π(sn,t |sn ) =
π(sn+1 |sn ) × . . . (st |st−1 ). We assume that νt (sn,t−1 ) =
Qt−1
k=n
ρ(sk ), where ρ(sk ) is the survival
probability in period k + 1 of a household who in period k is in state sk . Note that survival
probabilities depend on age, as encoded in st , but do not depend on cohort. We assume that
the one-period utility function exhibits constant relative risk aversion: u(c) =
c1−γ
1−γ
for γ 6= 1
and u(c) = ln c otherwise. In other words, preferences are homothetic in consumption.
d) Participation/Enforcement Constraint
We confine attention to equilibria in which households have no incentive to default. Thus,
household allocations are required to satisfy the sequential enforcement (or participation)
constraints. That is, for all t ≥ n and all sn,t we have:
∞
X
β m−t E[νm u(cm )|sn,t ] ≥ Vd (ht(sn,t−1 ), st)
m=t
9
(4)
where Vd is the continuation value of a household who decides to default in period t. This
default value is determined as follows.
We assume that upon default all debts of the household are canceled and all financial
assets seized so that at(st ) = 0. While in the default state, households are excluded from
purchasing insurance contracts and borrowing (going short). Further, households in default
retain their human capital, can invest in human capital, and earn a wage rate (1 − τ )r̃h
per efficiency unit of human capital, where 0 ≤ τ ≤ 1 is a parameter that measures the
fraction of labor income that is garnished. Thus, the punishment for default is exclusion
from financial markets and possible garnishment of labor income. We assume that households
remain in the default state until a stochastically determined future date that occurs with
probability (1 − p) in each period; that is, the probability of remaining in default is p.
After moving out of the default state, the household’s expected continuation value is V e ,
which depends on h and s at the time of exiting default (a = 0 at that point in time).
For the individual household the function V e is taken as given, but we close the model and
determine this function endogenously by requiring that V e = V , where V is the equilibrium
value function associated with the maximization problem of a household who participates in
financial markets.4
In sum, Vd is the value function associated with the following household maximization
problem
.
Vd (ht (sn,t−1 ), st ) =
max∞
{cm ,hm }m=t
∞
X
(pβ)
m−t
E[νm u(cm )|sn,t ]
m=t
4
The previous literature has usually assumed p = 1 (permanent autarky). See, however, Krueger and
Uhlig (2006) for a model with p = 0 following a similar approach to ours. Note also that the credit (default)
history of an individual household is not a state variable affecting the expected value function, V e ; we assume
that the credit (default) history of households is information that cannot be used for contracting purposes.
This is in line with the U.S. bankruptcy code, which limits the history of past behavior that can be retained
in credit reports.
10
+
∞
X
β m−t 1 − pm−t E[νm Vme (hm (sn,m−1 ), sm )|sn,t ]
m=t
where the continuation plan {cm , hm }∞
m=t has to satisfy the sequential budget constraint
(1 − τ )r̃h,m z(sm )hm = cm + xh,m
hm+1 = (1 + (sm−1 , sm )) hm + φ(sm )xh,m
(5)
cm ≥ 0 , hm+1 ≥ 0
e) Household Decision Problem
For given initial state (an , hn , sn ), a household of cohort n chooses a plan {ct , at+1, ht+1 }∞
t=n .
The set of budget feasible household plans is defined as
.
∞
B(an , hn , sn ) = { {ct , at+1, ht+1 }∞
t=n | {ct , at+1 , ht+1 }t=n satisfies (1), (2), and (4)}
The decision problem of a household of initial type (an , hn , sn ) is
max
{ct ,at+1 ,ht+1 }∞
t=n
U ({ct }∞
t=n |sn )
(6)
s.t. {ct , at+1 , ht+1}∞
t=n ∈ B(an , hn , sn )
where the lifetime utility function, U , is defined in (3).
f) Goods Production and Physical Capital Accumulation
There is one good that can be consumed or used as physical capital in production. Production
of this good is undertaken by a representative firm that rents capital and labor in competitive
markets and uses these input factors to produce output, Yt , according to the aggregate
production function Yt = F (Kt , Ht ). Here Kt is the aggregate stock of physical capital and
Ht is the aggregate level of efficiency-weighted human capital employed by the firm.
The aggregate production function, F , is a standard neoclassical production function,
that is, it has constant-returns-to-scale, satisfies a Inada condition, and is continuous, con11
cave, and strictly increasing in each argument. Given these assumptions on F , the implied intensive-form production function, f (K̃) = F (K̃, 1), is continuous, strictly increasing,
strictly concave, and satisfies a corresponding Inada condition, where we introduced the
”capital-to-labor ratio” K̃ = K/H. Given the assumption of perfectly competitive labor
and capital markets, profit maximization implies
r̃kt = f 0 (K̃t )
(7)
r̃ht = f (K̃t ) + f 0 (K̃t )K̃t ,
where r̃k is the rental rate of physical capital and r̃h is the rental rate of human capital.
Note that r̃h is simply the wage rate per unit of human capital. Clearly, (7) defines rental
rates as functions of the capital-to-labor ratio: r̃k = r̃k (K̃) and r̃h = r̃h (K̃).
The accumulation equation for the aggregate stock of physical capital is
Kt+1 = (1 − δk )Kt + Xkt ,
(8)
where δk is the depreciation rate of physical capital and Xkt is investment in physical capital.
g) Equilibrium
We confine attention to equilibria in which financial contracts are priced in a risk-neutral
manner,
qt(st+1 ) =
π(st+1|st )
,
1 + rf t
(9)
where rf is the interest rate on financial transactions, which is equal to the return on physical
capital investment, rf t = r̃kt −δk . The pricing equation (9) can be interpreted as a zero-profit
condition. More precisely, consider financial intermediaries that sell insurance contracts to
individual households and invest the proceeds in the risk-free asset that can be created from
the complete set of financial contracts and yields a certain return rf . Given that financial
12
intermediaries face linear investment opportunities and assuming no quantity restrictions
on the trading of financial contracts for financial intermediaries, equilibrium requires that
financial intermediaries make zero profit, namely condition (9).
Capital market clearing requires that the aggregate stock of physical capital employed by
the representative firm is equal to the value of financial wealth held by households. Similarly,
labor market clearing requires that the firm’s demand for labor equals the aggregate amount
of efficiency-weighted human capital supplied by households. More precisely, in equilibrium
we have
Kt+1 =
t
XX
E[νn,t+1 qt(st+1 )an,t+1 (st+1 )|st+1] +
st+1 n=0
Ht+1 =
t
X
E[νn,t+1 z(st+1 )hn,t+1 ] +
n=0
Z
ht+1 ,st+1
Z
at+1
at+1 dµnew,t+1 (at+1 )
(10)
z(st+1 )ht+1 dµnew,t+1 (ht+1 , st+1 ) ,
where µnew,t+1 is the distribution of new-born households in period t + 1 over initial states,
(at+1, ht+1 , st+1 ), which is an exogenous object. Note that the expectations in (10) is taken
over all individual histories and all possible initial states. That is, we define
.
E[νn,t+1qt+1 (st+1 )an,t+1(st+1 )|st+1 ] =
Z
X
an ,hn ,sn
νn,t+1 (sn,t )qt (st+1 ; st)an,t+1 (st+1 ; sn,t , an , hn , sn )π(sn,t |sn )dµnew,n (an , hn , sn )
sn,t+1 |sn
and
.
E[νn,t+1 z(st+1 )hn,t+1 ] =
Z
X
an ,hn ,sn
νn,t+1 (sn,t )z(st+1)hn,t+1 (sn,t, an , hn , sn )π(sn,t+1 |sn )dµnew,n (an , hn , sn )
sn,t+1 |sn
Note that we allow the distributions of new-born households, µnew,n , to depend on the cohort
n in order to be permit an endogenous growth path.
The distribution µnew,n has to satisfy an aggregate resource restriction. Specifically, we
assume that the aggregate stock of physical capital (human capital) of new-born households
13
is proportional to the aggregate stock of physical capital (human capital) of households who
have died:
Z
an0 +1
n Z
X
0
λa
X
n=0 an ,hn ,sn sn,n0
0
Z
hn0 +1
0
λh
(11)
0
0
(1−ρ(sn0 ))νn,n0 (sn,n −1 )an,n0 +1 (sn,n , an , hn , sn )π(sn,n |sn )dµnew,n (an , hn , sn )
and
n Z
X
an0 +1 dµnew,n0 +1 (an0 +1 ) =
X
n=0 an ,hn ,sn sn,n0
hn0 +1 dµnew,n0 +1 (hn0 +1 ) =
0
0
0
ρ(sn0 )νn,n0 (sn,n −1 )hn0 +1 (sn,n , an , hn , sn )π(sn,n |sn )dµnew,n (an , hn , sn )
where λa is a parameter that measures the relationship between physical capital of households
born in period n0 + 1 relative to the physical capital of households who leave the model in
period n0 + 1. These parameters summarize to what extent physical capital is passed on to
the next generation and to what extent a new-born generation starts with additional capital
unrelated to the capital of their parents/grandparents. In most cases, we have λa = 1
(closed economy), but other cases are possible. The parameter λh expresses the size of
human capital of new-born households relative to the aggregate stock of human capital in
the economy. Equation (11) imposes a restriction on the exogenous distributions µnew,n .
The aggregate resource constraint states that total output produced is equal to aggregate
consumption plus aggregate investment
Yt = Ct + Xkt + Xht
(12)
where Xkt is aggregate investment in physical capital and Xht is aggregate investment in human capital. As in (10), we compute aggregate variables from the respective household-level
variables by summing over cohorts and averaging over individual histories and possible initial
states. It is straightforward to show that the capital and labor market clearing conditions
(10) in conjunction with the household budget constraint (2) and the capital accumulation
14
equations (1) and (8) imply the goods market clearing condition (12) using the asset pricing formula (9). In our equilibrium analysis we will use focus on the two market clearing
conditions in (10), which can be subsumed to one market clearing condition because of the
constant-returns-to-scale assumption (see below).
Our definition of a sequential equilibrium is standard:
Definition 1 A sequential equilibrium is a sequence of aggregate stocks of physical capital
and (productivity weighted) human capital, {Kt , Ht }, rental rates, {r̃kt , r̃ht }, and a family of
household plans, {ct , at, ht }∞
t=n , one for each cohort n and initial household type (an , hn , sn ),
so that
i) Utility maximization of households: for each initial state, (an , hn , sn ), the plan {ct , at, ht }∞
t=n
solves the household problem (6).
ii) Profit maximization of firms: (Kt , Ht ) maximizes profit for all t, that is, the aggregate
capital-to-labor ratio, K̃t , and rental rates, r̃kt and r̃ht satisfy the first-order conditions (7)
for all t.
iii) Profit maximization of financial intermediaries: financial contracts are priced according to (9).
iv) Market clearing in capital and labor markets: equation (10) holds.
v) Rational expectations: expected continuation value functions are equal to actual continuation value functions: V e = V .
We next turn to the characterization of equilibria.
15
III. Theoretical Results
In this section, we state the two main theoretical results. First, the solution to the individual
household maximization problem is linear in total individual wealth (financial and human).
This partial equilibrium result is stated in proposition 2 and the proof is based on a monotone
operator argument (proposition 1). Second, the distribution of total wealth (financial plus
human), Ω, over household types, s, is a sufficient aggregate state variable. This general
equilibrium result is stated in proposition 3. We begin this section with a discussion of a
convenient change of variables and a definition of recursive equilibria with aggregate state
Ω.
a) Change of Variables
For the characterization of equilibria, it is convenient to introduce new variables that emphasize the fact that individual households solve a standard inter-temporal portfolio choice
problem (with additional participation constraints). To this end, introduce the following
variables:
X
ht
+
qt−1 (st )at(st )
φ(st )
st
ht
at (st )
, θat(st ) =
θht =
φ(st )wt
wt
(
(1 + rht (st−1 , st ))θht + θat(st ) if no default
1 + r(θt , st−1 , st) =
if default
(1 + rhd,t (st−1 , st ))θht
w̃t =
(13)
.
where rht (st−1 , st ) = z(st )φ(st )r̃ht + (st−1, st ) is the return on human capital investment
.
if the household does not default and rhd,t (st−1 , st) = (1 − τ )z(st )φ(st)r̃ht + (st−1, st ) is
the return on human capital investment in case of default. In (13) the variable w̃t stands
for beginning-of-period wealth consisting of the value of human wealth,
wealth,
P
st
ht
,
φ(st )
and financial
qt−1 (st )at(st ). The variable θt = (θht, θat ) denotes the vector of portfolio shares
and (1 + r) is the total return to investment. Recall that for given history of shocks, θht
16
is a number, but θat is a vector with components θat(st ). Using the new notation and
substituting out the investment variables, xkt and xht , the budget constraint (2) and human
capital accumulation equation (1) read
w̃t+1 = (1 + r(θt, st−1 , st )) w̃t − ct
1 = θh,t+1 +
X
X
qt (st+1)θa,t+1 (st+1)
(14)
st+1
qt (st+1 )θa,t+1(st+1 ) ≥ D̄θh,t+1
st+1
ct ≥ 0 , w̃t+1 ≥ 0 , θh,t+1 ≥ 0 .
Clearly, (14) is the budget constraint corresponding to an inter-temporal portfolio choice
problem with linear investment opportunities and no exogenous source of income.
It is convenient to use as individual state variable wealth including current asset payoffs
.
(“cash at hand”) defined as wt = (1 + rt )w̃t. Using this concept of total wealth, the budget
constraint (14) can be written as
wt+1 = (1 + r(θt+1 , st, st+1 )) (wt − ct )
1 = θh,t+1 +
X
X
qt (st+1)θa,t+1 (st+1)
(15)
st+1
qt (st+1 )θa,t+1(st+1 ) ≥ D̄θh,t+1
st+1
ct ≥ 0 , wt+1 ≥ 0 , θh,t+1 ≥ 0 .
Further, the default value function, Vd , can be written as a function of w, and (w, s) is therefore a sufficient state for the enforcement constraint (4). Thus, the household maximization
problem (6) is equivalent to the household maximization problem
max
{ct ,wt+1 ,θt+1 }∞
t=n
U ({ct }∞
t=n |sn )
s.t. {ct, wt+1 , θt+1}∞
t=n ∈ B(wn , sn )
where the budget set is now defined as
.
∞
B(wn , sn ) = { {ct, wt+1 , θt+1 }∞
t=n | {ct , wt+1 , θt+1 }t=n satisfies (4) and (15)}
17
(16)
b) Recursive Equilibrium: Definition
We next define a recursive equilibrium. To this end, we first note that the market clearing
condition (10) can be reduced to the condition
K̃t+1 =
P
st+1
Pt
n=0
R
Pt
E[νn,t+1qt (st+1 )an,t+1(st+1 )|st+1 ] + at+1 at+1 dµnew,t+1 (at+1 )
R
E[νn,t+1 z(st+1)hn,t+1 ] + ht+1 ,st+1 z(st+1 )ht+1 dµnew,t+1 (ht+1 , st+1 )
n=0
(17)
because of the constant-return assumption. In a sequential equilibrium, the expectations
in (17) is taken over all individual histories and all initial states, and it depends in general
explicitly on time t. In a recursive equilibrium, the expectations is taken over individual
states conditional on the aggregate state, and it is time-independent.
The household maximization problem (16) suggests that we can use (w, s) as the individual state variable. For the aggregate state, in general the distribution, µ, over individual
states, (w, s), is the minimal state variable. However, for the current model, the typedependent wealth distribution, Ω ∈ IRn , defined as
hP
i
t
. E h n=0 νn,t wn,t |sit
Ωt (st ) =
.
Pt
E
ν
w
n,t
n,t
n=0
turns out to be sufficient (see below). Here Ωt (st ) is the share of aggregate total wealth owned
by all households of type st . Note that Ω is a distribution since E[Ωt ] =
P
st
Ωt (st ) = 1. Note
further that the distribution µ is an infinite-dimensional object, whereas the distribution Ω
is finite-dimensional.
Below we construct a recursive equilibrium with aggregate state variable Ω that evolves
according to an endogenous law of motion Ω0 = Φ(Ω), where the prime denotes next period’s
variable. We further show that next period’s optimal portfolio choice is independent of w,
which implies that the market clearing condition (17) becomes a condition that defines a
function K̃ 0 = K̃ 0 (Ω). Together with the first-order conditions (7) this defines rental rate
functions r̃k0 = r̃k0 (Ω) and r̃h0 = r̃h0 (Ω). Given our definition of sequential equilibrium and the
variables defined so far, our definition of recursive equilibrium is standard:
18
Definition 2 A recursive equilibrium is a law of motion, Φ, for the aggregate state variable,
Ω, a function K̃ 0 = K̃ 0(Ω), rental rate functions r̃k0 = r̃k0 (Ω) and r̃h0 = r̃h0 (Ω), an expected
value function, V e , and a household policy function, g,5 such that
i) Utility maximization of households: for all household cohorts, n, and household types,
(wn , sn ), the household policy function, g, in conjunction with the law of motion, Φ, generates a plan, {ct , wt+1, θt+1 }∞
t=n , that solves the household maximization problem (16).
∞
ii) Profit maximization of firms: for any sequence {K̃}∞
t=0 , the rental rate sequences {r̃kt }t=0
and {r̃ht }∞
t=0 are defined by the first-order conditions (7).
iii) Profit maximization of financial intermediaries: financial contracts are priced according
to (9)
iv) Market clearing: for any initial state Ω, the law of motion Φ in conjunction with the
function K̃ 0 generate a sequence {K̃}∞
t=0 that satisfies the market clearing condition (17)
v) Rational expectations: V e = V and Φ is the law of motion induced by g.
c) Characterization of Household Problem (Partial Equilibrium)
The principle of optimality in conjunction with our discussion in the previous section regarding the appropriate aggregate state suggest that the household maximization problem
(16) is equivalent to the Bellman equation
V (w, s, Ω)
=
(
max u (c) + βρ(s)
c,w0 ,θ 0
X
0
0
0
0
V (w , s , Ω ) π(s |s)
)
s0
s.t. w0 = (1 + r(θ0 , s, s0, Ω))(w − c)
X π(s0|s)θa0 (s0 )
1 = θh0 +
1 + rf (Ω)
s0
X
s0
(18)
π(s0|s)θa0 (s0)
≥ −D̄θh0 , θh0 ≥ 0 , w0 ≥ 0
1 + rf (Ω)
5
The function g defines next period’s endogenous state as a function of this period’s endogenous state
and this period’s exogenous shock:wt+1 = g(wt , st).
19
V (w0 , s0 , Ω0) ≥ Vd (w0 , s0, Ω0 )
Ω0 = Φ(Ω)
where the default value function is given by
(
Vd (w, s, Ω) = max
u (c) + βρ(s)p
0
X
c,w
+βρ(s)(1 − p)
ρ(s0)Vd (w0 , s0 , Ω0) π(s0|s)
s0
X
e
0
0
0
0
V (w , s , Ω ) π(s |s)
)
s0
w0 = (1 + rhd (s, s0, Ω))(w − c)
Ω0 = Φ(Ω)
Let T be the operator associated with the Bellman equation (18). In contrast to the standard case without a participation constraint, the Bellman operator, T , defined by equation
(18) is in general not a contraction. However, it is still a monotone operator. Monotone
operators might have multiple fixed points, but under certain conditions we can construct
a sequence that converges to the maximal element of the set of fixed points. This maximal
solution is also the value function (principle of optimality). More precisely, if the condition
that for all s
∀θ0 : βρ(s)
X
(1 + r(θ0 , s, s0, Ω0 ))
1−γ
π(s0|s) < 1
if 0 < γ < 1
(1 + r(θ0 , s, s0, Ω0 ))
1−γ
π(s0|s) < 1
if γ > 1
(19)
s0
∃θ0 : β ρ(s)
X
s0
holds,6 then we have the following results:
Proposition 1. Suppose that condition (19) is satisfied and that the law of motion, Φ, and
the value function of a household in financial autarky, Vd , are continuous. Let T stand for
the operator associated with the Bellman equation (18). Then
6
Note that for the log-utility case, no condition of the type (19) is required.
20
i) There is a unique continuous solution, V0 , to the Bellman equation (18) without participation constraint.
ii) limk→∞ T k V0 = V∞ exists and is the maximal solution to the Bellman equation (18)
iii) V∞ is the value function, V , of the sequential household maximization problem.
Proof . See Appendix.
Consider the case V e = V . Using proposition 2 and an induction argument, we can then
show that the value function, V , has the functional form
V (w, s, Ω) =
(
if γ 6= 1
Ṽ (s, Ω)w1−γ
Ṽ0 (s, Ω) + Ṽ1 (s) ln w otherwise
(20)
and that the corresponding optimal policy function, g, is
c(w, s) = c̃(s, Ω) w
w0(w, s, s0 , Ω) = (1 + r(θ0 , s, s0 , Ω))(1 − c̃(s, Ω)) w
θ0 (w, s, Ω) = θ0 (s, Ω) .
In other words, the value function has the functional form of the underlying utility function,
consumption and next-period wealth are linear functions of current period wealth, and nextperiod portfolio choices are independent of wealth. Moreover, we also show that the intensiveform value function, Ṽ , together with the optimal consumption and portfolio choices, c̃ and
θ, can be found by solving an intensive-form Bellman equation that reads
Ṽ (s, Ω)
=
(
X
c̃1−γ
1−γ
max
Ṽ (s0, Ω0 ) π(s0|s)
+ βρ(s)(1 − c̃)1−γ
(1 + r(θ0 , s, s0, Ω))
0
c̃,θ
1−γ
s0
s.t. 1 = θh0 +
X
s0
X
s0
θa0 (s0)π(s0|s)
1 + rf (Ω)
)
(21)
π(s0|s)θa0 (s0 )
≥ −D̄θh0 , θh0 ≥ 0 , 0 ≤ c̃ ≤ 1
1 + rf (Ω)
21
Ṽ (s0, Ω0 )
Ṽd (s0 , Ω0 )
1
! 1−γ
(1 + r(θ0 , s, s0 , Ω)) ≥ (1 + rhd (s, s0, Ω))θh0
Ω0 = Φ(Ω)
and
(
X
c̃1−γ
1−γ
d
Ṽd (s0, Ω0 )π(s0|s)
Ṽd (s, Ω) = max
+ pβρ(s)(1 − c̃d )1−γ
(1 + rhd (s, s0 , Ω))
c̃d
1−γ
s0
(1 − p)βρ(s)(1 − c̃d )
1−γ
X
0
(1 + rhd (s, s , Ω))
1−γ
0
0
0
Ṽ (s , Ω )π(s |s)
)
s0
for γ 6= 1. In the case of log-utility, the intensive-form Bellman equation reads
Ṽ0 (s, Ω)
=
(
max
ln c̃ + βρ(s)
0
c̃,θ
X
Ṽ0 (s0 )π(s0|s) + βρ(s) ln(1 − c̃)
s0
+ βρ(s)
X
Ṽ1 (s0 ) ln(1 + r(θ0 , s, s0, Ω))π(s0|s)
X
Ṽ1 (s0)π(s0|s)
s0
)
s0
s.t. 1 = θh0 +
X
s0
X
s0
θa0 (s0)π(s0|s)
1 + rf (Ω)
π(s0|s)θa0 (s0)
≥ −D̄θh0 , θh0 ≥ 0
1 + rf (Ω)
e(1−β)(Ṽ0 (s ,Ω )−Ṽd0 (s ,Ω )) (1 + r(θ0 , s, s0, Ω)) ≥ (1 + rhd (s, s0, Ω))θh0
0
0
0
0
Ω0 = Φ(Ω)
and
(
Ṽ0d (s, Ω) = max ln c̃d + β ln(1 − c̃d )
c̃d
+ βρ(s)
X
X
Ṽ1 (s0 )ρ(s0 )π(s0|s)
s0
0
Ṽ1 (s ) ln(1 + rhd (s, s0, Ω))π(s0|s)
s0
+ pβρ(s)
X
0
0
Ṽ0d (s )π(s |s) + (1 − p)βρ(s)
s0
X
s0
22
0
0
Ṽ0 (s )π(s |s)
)
where the coefficients Ṽ1 are the solution to
Ṽ1 (s) = 1 + βρ(s)
X
Ṽ1 (s0)π(s0|s)
s0
Proposition 2. Suppose that condition (19) is satisfied, the law of motion, Φ, is continuous,
and V e = V . Then value function, V , and optimal policy function, g, have the functional
form (20). Moreover, the intensive-form value function, Ṽ , and the corresponding optimal
consumption and portfolio choices, c̃ and θ0 , are the maximal solution to the intensive-form
Bellman equation (21). This maximal solution is obtained by iteratively applying T̃ , the
operator associated with the intensive-form Bellman equation (21), starting from Ṽ0 , the
solution of the intensive-form Bellman equation (22) without participation constraint:
Ṽ = lim T̃ nṼ0 .
n→∞
Proof . See Appendix.
Note that proposition 2 cannot simply be proved by the guess-and-verify method since
multiple solutions to the Bellman equation (21) may exist. Specifically, the operator associated with the Bellman equation is monotone, but not a contraction, and hence multiple
fixed points may exist. However, proposition 2 ensures that we have indeed found the value
function associated with the original utility maximization problem, and also provides us with
a iterative method to compute this solution. Note further that the constraint set in (21) is
linear since the return functions are linear in θ. Thus, the constraint set is convex and we
have transformed the original utility maximization problem into a convex problem. In other
words, the non-convexity problem alluded to in the introduction has been resolved.
d) Characterization of Recursive Equilibria
Proposition 2 shows how to rewrite the maximization problem of individual households
as a recursive problem that is wealth-independent. One implication of the intensive-form
23
representation of the individual maximization problem is that optimal portfolio choices are
independent of wealth, w. This result in turn implies that the market clearing condition
(17) can be re-written as
0
K̃ =
P
s
[ρ(s) + λa (1 − ρ(s))] (1 − θh (s, Ω)) (1 − c̃(s, Ω))Ω(s)
P
z̄ s [ρ(s) + λh ] φ(s)θh (s, Ω)(1 − c̃(s, Ω))Ω(s)
(22)
where we have already incorporated restriction (11) and z̄ stands for the mean of z. Equation
(22) defines a function K̃ 0 = K̃ 0 (Ω), which in turn defines rental rate functions r̃k0 = r̃k0 (Ω)
and r̃h0 = r̃h0 (Ω) using the first-order conditions (7). A second implication of proposition 2 is
that the equilibrium law of motion, Φ, can be explicitly derived:
0
0
Ω (s ) =
P
s
ρ(s)(1 − c̃(s, Ω))(1 + r(θ0 (s, Ω), s0, Ω))π(s0|s)Ω(s) + λΩ0new (s0)
0
0
0
s,s0 ρ(s)(1 − c̃(s, Ω))(1 + r(θ (s, Ω), s , Ω))π(s |s)Ω(s) + λ
P
(23)
where the parameter λ is related to the parameters λa and λh through the restriction (11).
Note that the expression in the denominator of (23) ensures that
P
s0
Ω0 (s0 ) = 1.
In sum, a recursive equilibrium can be found by solving (21) and (22), and using (23) as
the law of motion:
Proposition 3. Suppose that (θ, c̃, Ṽ , K̃ 0 ) is an intensive-form equilibrium, that is, (θ, c̃, Ṽ , K̃ 0)
solves (21) and (22). Then (g, Ṽ , K̃ 0, Φ) is a recursive equilibrium, where g is the individual
policy function associated with (θ, c̃) and Φ the aggregate law of motion defined in (23).
Proof . See Appendix.
Proposition 3 simplifies the computation of recursive equilibria. In our framework, the
infinite-dimensional wealth distribution is not a relevant state variable. Instead, the distribution of wealth shares over household types, Ω, becomes a relevant state variable. Note that Ω
is in many applications a low-dimensional object. For example, suppose that st = (s1t, s2t),
where {s1t} and {s2t} are two independent processes and {s2t} is an i.i.d process. In this
case neither c̃ nor θ depend on s2 and the relevant aggregate state is Ω(s1 ) only.
24
e) Extension: Aggregate Shocks
So far, we have considered economies with only idiosyncratic risk, but it is straightforward
to introduce aggregate risk into the framework. To this end, suppose that there are idiosyncratic shocks, s, and aggregate shocks, S, and that uncertainty is described by a stationary
joint Markov process {st, St } with transition probabilities denoted by π(st+1, St+1 |st , St).
The relevant aggregate state then becomes (Ωt , St ), where Ωt is defined as before. In a
recursive equilibrium, the evolution of the endogenous aggregate state variable is given by
an endogenous law of motion Ωt+1 = Φ(Ωt , St, St+1 ). Further, the aggregate capital-to-labor
ratio is a function K̃t+1 (Ωt , St) and the rentals rates are function r̃k,t+1 = r̃k (Ωt , St ) and
r̃h,t+1 = r̃h (Ωt , St ). The definition of a recursive equilibrium is, mutatis mutandis, as before.
A straightforward (though lengthy) extension of the subsequent theoretical analysis shows
that a modified version of our general characterization results still hold. In particular,
recursive equilibria can be computed by solving a convex problem that is independent of the
wealth distribution, though clearly the finite-dimensional distribution of relative wealth, Ω,
still enters into the equilibrium conditions.
f) Further Extensions
There a several further extensions of the model that can be incorporated without sacrificing
the tractability of the model. First, we can introduce a time cost in human capital production
if we replace the term φ(st )xht in (1) by φ(st )(ht lht)α x1−α
ht , where lht is the time spent in
human capital production. In the simplest extension, the household allocates time between
working and producing human capital (learning). However, we can also add a labor-leisure
choice as long as preferences remain homothetic in consumption. It is straightforward to
show that the human capital production function φ(st)(ht lht )α x1−α
gives rise to a human
ht
capital accumulation equation (1) that is still linear in xht after the optimal choice of lht has
25
been substituted out.
A second extension is shocks to preferences (taste shocks, health shocks, change in family
structure). These can easily be incorporated by replacing the one-period utility function by
one that depends on the state st . Third, the tractability is preserved in a model with taxes
and transfers as long as these payments are proportional to either financial capital (capital
income) or human capital (labor income). To see this, simply re-define the returns in (13) as
returns after taxes and transfers have been taken into account. Note that taxes and transfers
can be an arbitrary (non-linear) function of the state st .
IV. Quantitative Analysis
In this section, we provide a quantitative analysis based on a special version of the model.
To this end, Section IV.a first presents the model specification for the special case of interest.
Sections IV.b and IV.c then discuss the equilibrium conditions for the special case and our
computational approach. Section IV.d briefly discusses the data and Section IV.e presents
the calibration of the partial equilibrium model. Section IV.f and IV.g discuss the life-cycle
implications of the model with respect to portfolio choice, insurance, and welfare. The next
three sections analyze the model response to changes in contract enforcement, labor market
risk, and risk aversion based on the partial equilibrium version of the model.7 Section IV.k
concludes with a discussion how to calibrate the general equilibrium version of the model.
a) Specification
We set the period length to one year. We assume that the economy is in stationary equilibrium and drop the time index t. We further assume that the exogenous individual state
7
We do not re-calibrate the model when we change the value of one parameter and in this sense we
conduct a comparative statics analysis. Our results barely change if we re-calibrate the model to match all
targets before and after the parameter change.
26
has two components, s = (s1, s2 ). The first component, s1 , denotes the age of a household,
which can take on 9 values, s1 = 1, . . . , 9, corresponding to the following 9 age groups: 25
and younger, 26 - 30, 31 - 36, . . . , 56 - 60, and older than 60. We assume that households
stochastically age with the transitions from one age group to another age group governed
by transition probabilities π(s01|s1). We assume that households cannot move up more than
one age group at a time, and choose π(s1 + 1|s1 ) so that so that households spend on average 5 years in the first 8 age groups. That is, for s1 ≤ 8 we have π(s1|s1 ) = 4/5 and
π(s1 + 1|s1 ) = 1/5. Households in the oldest age group die stochastically and the probability
of death is chosen so that these households live on average a further 25 years. Old households
who leaves the model are replaced by households in the youngest age group.
The second component of the state, s2 , describes human capital risk. Both s1 and s2 affect
human capital accumulation through the -function appearing in the human capital equation
(1) as (s1 , s2) = ϕ(s1 ) − δh + η(s2 ). We interpret ϕ as a learning-by-doing parameter which
depends on age and which, in our calibration below, is stronger for younger households so
that ϕ(s1 ) > ϕ(s1 + 1). The parameter δh is the average depreciation rate of human capital
in the economy. We have set the labor productivity parameter z = 1 so that all labor
income risk is generated through the human capital shock η, which is assumed to be i.i.d.
over time and across households and independent of household age s1 .8 Assuming that the
cost of human capital in terms of consumption goods φ is constant, the return to human
capital is given by rh (s1, s2 ) = φr̃h + ϕ(s1) − δh + η(s2). Normalizing the mean of the human
capital shocks to zero, or
P
s2
η(s2)π(s2) = 0, we find that the expected human capital
returns for a household of age s1 are r̄h (s1 ) =
P
s2
rh (s1, s2 )π(s2) = φr̃h + ϕ(s1 ) − δh . For
the oldest household group, s1 = 9, we assume that human capital returns are low enough
so that they only invest in financial capital yielding a portfolio return equal to the risk-free
rf (retirement).
8
In Krebs, Kuhn, and Wright (2015) we consider the more general version with additional shocks to z.
27
With this specification in hand, we can verify that human capital accumulation decisions
satisfy various non-negativity constraints on human capital investment. For example, in
equilibrium the restriction holds that total human capital investment inclusive of learningby-doing is always non-negative: ϕ(s1)ht + φxht ≥ 0.
b) Equilibrium Conditions
Given the assumption made so far, the intensive-form Bellman equation (21) for households
of age s1 ≤ 8 becomes
Ṽ (s1 )
=
X
c̃1−γ
1−γ
0 0
0 1−γ
0
0
0
Ṽ
(s
+
β(1
−
c̃)
max
(1
+
r(θ
,
s
,
s
))
)π(s
)π(s
|s
)
1
1
2
1
2
1
c̃,θ0 1 − γ
s0 ,s0
1
2
(24)
s.t. 1 = θh0 +
X
θa0 (s01 , s02)π(s02)π(s01 |s1)
1 + rf
s01 ,s02
Ṽ (s01 )
Ṽd (s01 )
1
! 1−γ
, 0 ≤ c̃ ≤ 1 , θh0 ≥ 0
(1 + r(θ0 , s01 , s02)) ≥ (1 + rh (s01 , s02))θh0
∀ (s01 , s02)
with
X
c̃1−γ
1−γ
Ṽd (s1) = max d
Ṽd (s01 )π(s02)π(s01|s1)
+ pβ(1 − c̃d )1−γ
(1 + rhd (s01, s02 ))
c̃d
1−γ
0 0
s1 ,s2
(1 − p)β(1 − c̃d )1−γ
X
(1 + rhd (s01, s02 ))
1−γ
s01 ,s01
π(s02)Ṽ (s1)π(s01|s1 )
for γ 6= 1. In the case of log-utility, the intensive-form Bellman equation (21) becomes
Ṽ (s1 )
=
max
ln(1 − β) +
0
θ
+β
X
s01
β X
β
ln β +
ln(1 + r(θ0 , s01, s02 )π(s02)π(s01|s1)
1−β
1 − β s0 ,s0
Ṽ (s01 )π(s01|s1 )
1
28
2
s.t. 1 = θh0 +
X
s01 ,s02
θa0 (s01, s02 )π(s02)π(s01|s1)
, θh0 ≥ 0
1 + rf
e(1−β)(Ṽ (s1 )−Ṽd (s1 )) (1 + r(θ0 , s01, s02 ) ≥ (1 + rhd (s01 , s02))θh0
0
0
∀ (s01, s02 )
with
β X
β
logβ +
log(1 + rhd (s01 , s02)π(s02)π(s01|s1 )
1−β
1 − β s0 ,s0
Ṽd (s1) = ln(1 − β) +
+pβ
X
1
Ṽd (s01)π(s01|s1 )
+ (1 − p)β
s01
2
X
Ṽ (s01 )π(s01|s1 )
s01
From (24) it immediately follows that the optimal portfolio choice, θ, and the optimal
consumption-saving choice, c̃, only depend on age s1 but not on human capital shocks s2 . In
other words, household consumption and portfolio choices are independent are independent
of i.i.d. shocks. This in turn implies that the relevant aggregate state, Ω, only depends on
age, s1 . The stationary Ω is then determined by the following set of equations, defined first
for s1 = 1:
4X
(1 − c̃(1))(1 + r(θ0 (1), 1, s02 ))π(s02)Ω(1) + λ
Ω(1) = N
5 s0
2
and then ∀s1 with 2 ≤ s1 ≤ 8 :
Ω(s1 ) = N
4X
(1 − c̃(s1))(1 + r(θ0 (s1), s1 , s02))π(s02 )Ω(s1) +
5 s0
2
(25)
1X
(1 − c̃(s1 − 1))(1 + r(θ0 (s1 − 1), s02 ))π(s02)Ω(s1 − 1)
5 s0
2
and then lastly:
1
24
Ω(9) = N
(1 − c̃(9))(1 + rf )Ω(9) + (1 − c̃(8))(1 − θh (8))(1 + rf )Ω(8)
25
5
where N is a normalization constant chosen to ensure
P
s1
Ω(s1 ) = 1. Note that (25) is
the stationary version of (23) for the current model set-up, where we have already used the
assumption that new-born households begin life in age group (state) s1 = 1.
29
Suppose we choose λa = 1. Taking into account that ρ(s1 ) = 1 for s1 = 1, . . . , 8 and
θh (9) = 0, we find that the market clearing condition (22) becomes
K̃ =
P
s1 (1
− c̃(s1))(1 − θh (s1))Ω(s1 )
P
(1 + λh )φ s1 6=9 (1 − c̃(s1 ))θh (s1)Ω(s1 )
(26)
where λ in (25) and λh in (26) are related through
X
1
λ = [1 + r̄(θ(1))] (1 − c̃(9))Ω(9) + λh φ
(1 − c̃(s1 ))θh (s1 )Ω(s1)
25
s1 6=9
(27)
where r̄(θ(1)) is the average investment return for a household of age group s1 = 1. Equations
(24), (25), and (26) determine a stationary recursive equilibrium for this specification of the
model.
c) Computation
For the general equilibrium analysis, one needs to solve the three equations (24), (25), and
(26). The algorithm for doing so works as follows. First, pick an aggregate capital-to-labor
ratio, K̃, which determines the rental rates r̃k and r̃h and therefore also the investment
return function r. Second, given the values for the investment returns, solve the intensiveform household decision problem (24) and recover the stationary state Ω. Third, use the
values for θ, c̃, and Ω, to determine a new value for K̃ using (26). Finally, iterate until
convergence. Note that in the log-utility case with ρ(s1 ) = 1 there is no need to solve for c̃
since we have c̃ = 1 − β.
We solve the partial equilibrium problem (24) by iteration. More precisely, consider the
case γ 6= 1 and define the values Ṽ k (s1 ) and Ṽdk (s1 ), recursively by
Ṽ k+1 (s1) =
1−γ
c̃k (s1 )
1−γ
β 1 − c̃k (s1 )
+
(28)
1−γ X
θhk (s1)(1 + rh (s01 , s02)) + θak (s1 , s01, s02 )
s01 ,s02
30
1−γ
ṽ k (s1)π(s02)π(s01 |s1)
k
c̃ (s1) =
1 − β
X
θhk (s1 )(1
+
rh (s01 , s02 ))
+
θak (s1 , s01, s02 )
1−γ
s01 ,s02
1
γ
π(s02)π(s01|s1 )
and
Ṽdk+1 (s1 )
X
(c̃d (s1 ))1−γ
1−γ k 0
+ β(1 − c̃d (s1 ))1−γ p
=
(1 + rhd (s01 , s02 ))
ṽd (s1)π(s02 )π(s01|s1)
1−γ
s0 ,s0
+β(1 − c̃d (s1))
1−γ
(1 − p)
X
1
2
(1 + rhd (s01, s02 ))
1−γ
π(s02)Ṽ k (s01 )π(s01|s1 )
s01 ,s02
c̃d (s1 ) =
1 − β
1
γ
X
(1 +
rhd (s01 , s02 ))
1−γ
s01 ,s02
π(s02)π(s01|s01 )
where the portfolio choices (θhk (s1 ), θak (s1 )) for each s1 are the solution to
max
θh ,θa
s.t.
X
(θh (1 + rh (s01 , s02)) + θa (s01, s02 ))
1−γ
π(s0)
s01 ,s02
X
θh +
s01 ,s02
θh (1 +
θa (s01, s02 )π(s02)π(s01|s1)
= 1
1 + rf
rh (s01 , s02 ))
+
θa(s01 , s02)
≥ θh (1 +
(29)
rhd (s01, s02 ))
Ṽdk (s01 )
Ṽ k (s01 )
1
! 1−γ
The intensive-from value function and the corresponding optimal portfolio choice are obtained by taking the limit Ṽ = limk→∞ Ṽ k , Ṽd = limk→∞ Ṽ k , and θ = limk→∞ θk . To
solve the portfolio problem (29) for given k and s1, we first fix θhk (s1) = θ̄hk (s1 ) and find
θak (s1 ) solving (29) for given θ̄hk (s1 ). To this end, for each s01 , order the pairs (s01, s02 ) so that
rh (s1 , 1) > rh (s1 , 2) > . . . > rh (s1 , S). Given s1 suppose that the participation constraint is
binding for the first J (s1) states. Then for the first J (s1) states θak (s1 ) is given by
θak (s1 , s01 , s02) = θ̄hk (s1 ) (1 + rhd (s01, s02 ))
Ṽdk (s01)
Ṽ k (s01)
1
! 1−γ
− (1 + rh (s01, s02 )) for (s01 , s02) = 1, . . . , J (s1)
(30)
while for the remaining states, we have
θak (s1, s01 , s02) = r̄k (s1 ) − θ̄hk (s1 )(1 + rh (s01, s02 ))
31
where r̄k (s1) is determined by the portfolio constraint in (29). Using the corresponding
first-order conditions it is easy to see that, for given θ̄hk (s1 ), the solution θak (s1 ) to (29) is
determined by (30), where J (s1) is the smallest number for which the portfolio choice satisfies
the participation constraint. Finally, we find optimal θhk (s1 ) using a standard one-dimensional
optimization routine.
d) Data
For the calibration and the results discussed below, we use data on earnings and financial
wealth drawn from the Survey of Consumer Finances (SCF). The SCF is a triennial survey
of U.S. households and we use data from 1989 to 2013. For most steps, we follow Krebs,
Kuhn, and Wright (2015) with our construction and treatment of the data. Our measure
of earnings (labor income) is wages and salaries plus two-thirds of the farm and business
income (if applicable). Our measure of financial wealth is net worth, defined as the sum of
the consolidated household balance sheet (including net housing wealth). All data has been
deflated using the BLS consumer price index for urban consumers (CPI-U-RS)
We follow Heathcote et al. (2010) for the sample selection and confine attention to
households with household head age 23 years and older. Specifically, we drop the wealthiest
1.47 % of households in each calender year, which makes the sample more comparable to
that of the Panel Study of Income Dynamics (PSID) and the Consumer Expenditure Survey
(CEX). Further, we drop all households that report negative labor income or that report
positive hours worked but have missing labor income or that report positive labor income
but zero or negative hours worked. We also drop in each year households with a wage rate
that is below half the minimum wage of the respective year, where we compute the wage
rate by dividing labor income by total hours worked.
We construct separately for each survey year life-cycle profiles of median log earnings and
32
financial wealth to earnings ratios. We smooth life-cycle profiles separately for each survey
year using linear least squares on a cubic polynomial in age and average the smoothed profiles
across survey years to remove time effects.9 We compute earnings growth rates from age
differences in earnings of the smoothed, cross-sectional earnings profiles.
e) Calibration: Partial Equilibrium
In this section, we calibrate the partial equilibrium version of the economy, that is, we find
values for the expected investment returns rf and r̄h (s1) = φr̃h +ϕ(s1 )−δh without specifying
the production function that generates these returns. We calibrate an annual risk-free rate
of rf = 3%, in line with Kaplan and Violante (2010) and roughly in line with Huggett et al.
(2011) and Krueger and Perri (2006) who use a 4% annual risk-free rate, but also deduct
capital income taxes.
We choose the age-dependent expected human capital returns, r̄h (s1 ), to match life-cycle
profile of earnings growth of the median household in the data for the first 8 age groups.
Specifically, we first construct a life-cycle profile of annual median household earnings and
earnings growth as described in the previous section, and then construct a corresponding
life-cycle profile of earnings growth for the relevant age groups. The result is depicted in
Figure 1 and shows the expected life-cycle pattern. Earnings growth rate are very high for
young households, monotonically decreasing in age, and turn negative for households older
than 50.
We assume that human capital shocks, η, are approximately normally distributed, that
is, we choose the probabilities π(s2) and the realizations η(s2) to approximate a normal
distribution with mean 0 and standard deviation ση = 0.15. The parameter ση measures
human capital risk and our choice of ση = 0.15 is motivated by the following considerations.
9
We only use observations until age 60 for the regression.
33
In the model economy, labor income of an individual household in period t is given by
yht = r̃h ht , so that the growth rate of labor income is equal to the growth rate of human
capital: yh,t+1/yht = hh,t+1 /ht . We can use the equilibrium solution to compute the human
capital growth between year t and year t + 1. Neglecting transitions across age groups s1,
this yields:
ht+1
= β (θh (s1,t−1)(1 + φr̃h + ϕ(s1,t) − δh + η(s2t)) + θa (s1,t−1, s1t , s2t))
ht
(31)
Equation (31) can be written as
ln yh,t+1 = ln yht + d(s1 ) + η̃t ,
(32)
where d(s1 ) is a constant and {η̃t } is a sequence of i.i.d. random variables with
ση̃2(s1 ) = θh2 (s1 )ση2 + var[θa|s1]
(33)
Hence, the logarithm of labor income follows a random walk with drift d and innovation
term η̃t.10 The random walk specification is often used by the empirical literature to model
the permanent component of labor income risk (Carroll and Samwick (1997), Meghir and
Pistaferri (2004), and Storesletten et al. (2004)). Thus, their estimate of the standard
deviation of the error term for the random walk component of annual labor income can be
used to find a value for ση̃2 for given portfolio choices θh and θa. For young households, we
will see below that θh is close to one and insurance payments, θa (s2), are small, so that we
have ση̃2 ≈ ση2. In our baseline calibration, we use ση = .15, which lies on the lower end of the
spectrum of estimates found by the empirical literature. For example, Carroll and Samwick
(1997) find .15, Meghir and Pistaferri (2004) estimate .19, and Storesletten et al. (2004)
have .25 (averaged over age-groups and, if applicable, over business cycle conditions). All
10
We have η̃t instead of η̃t+1 in equation (32), and the latter is the common specification for a random
walk. However, this is not a problem if the econometrician observes the idiosyncratic depreciation shocks
with a one-period lag. In this case, (32) is the correct equation from the household’s point of view, but a
modified version of (31) with η̃t+1 replacing η̃t is the specification estimated by the econometrician.
34
these studies use labor income before transfer payments, which is the relevant variable from
our point of view.
For the baseline calibration, we assume that households who default regain access to
financial markets after 7 years: (1 − p) = 1/7. Finally, we assume a degree of relative risk
aversion of γ = 1 (log-utility) and set the annual discount factor to β = 0.95.11 We choose
the human capital rental rate, φr̃h , to match the average value of the financial wealth to
earnings ratio for households age 23-60 (see Figure 2 below).
f) Portfolio Choice and Human Capital Returns
In this subsection, we examine household’s portfolio allocation between financial assets and
human capital. To this end, we first use current earnings as a proxy for human capital
and construct the life-cycle profile of the ratio of financial wealth to earnings in the model
and in the data.12 Figure 2 depicts the result and shows that the model does an excellent
job of matching this life-cycle profile, though it somewhat over-predicts the financial wealth
holdings of the oldest households. Note that the model matches the life-cycle average of the
financial wealth to earnings ratio by construction since we choose the human capital rental
rate, φr̃h , accordingly. However, we have no additional parameter to match the shape of the
life-cycle profile depicted in Figure 2.
The advantage of using the financial wealth to earnings ratio as a measure of portfolio
choice, as we have done in Figure 2, is the ease with which this variable can constructed
from the data without imposing additional assumptions. The disadvantage is that current
earnings is a very crude proxy of human capital. We therefore construct an alternative
measure of portfolio choice that uses the present value of future lifetime earnings as a proxy
11
An alternative calibration approach is to require the model to match a given expected human capital
return for the young and then use β to match the observed earnings growth rate of the young.
12
In the model, this ratio is computed as
1−θh (s1 )
φr̃h θh (s1 ) .
35
for human capital, where future earnings are discounted at the risk-free rate implied by the
calibrated model. Figure 3 depicts the result and shows that the model lines up reasonably
well with the data. However, the model over-predicts the human capital share, and this
over-prediction becomes more severe with age. The explanation for this is that in the model
older households are almost fully insured against the human capital loss upon retirement,
which means that the model tends to overstate the value of human capital. Clearly, this
type of “model error” becomes more important with age.13
In order to help understand the portfolio allocation decisions of households, Figure 4
presents the excess return to investing in human capital as a function of age. As shown in
the figure, young households face an excess return of almost ten percent, explaining why the
young hold very little financial wealth. Moreover, excess human capital returns in the vicinity
of 10 percent are in line with estimated rates of return to on-the-job-training (Blundell et al.
1999 and Mincer 1994). The excess return available to the oldest working households is less
than one-half of one percent, which explains why they hold so much more financial wealth
than the young.
g) Consumption Insurance and Welfare
The youngest households not only hold little financial wealth, but they are also dramatically
under-insured. Figure 5 plots a measure of consumption insurance, the insurance coefficient,
defined as one minus the ratio of the standard deviation of household consumption growth
to the standard deviation of household income growth. As shown in the figure, households
of the youngest age group are insured against only one third of their income risk, whereas
13
Note that the value of human capital in the model is always equal to the expected present discounted
value of lifetime earnings if future earnings are discounted using the relevant intertemporal marginal rate of
substitution and the model earnings process is used to computed expected lifetime earnings. The difference
between the model implication and the data depicted in Figure 3 arises because i) different discount rates
are used and ii) the model earnings process (in conjunction with the almost full-insurance result for older
households) does not capture the data well after age 60.
36
older households are insured against roughly 90 percent of their income risk.
Figure 6 examines the welfare consequences of this underinsurance, depicting the equivalent variation of moving to full insurance measured in units of lifetime consumption. As
shown in the figure, the youngest households would require an increase of almost 7.5 percent
in their annual consumption to be as well off as if they had access to full insurance. Thus, for
young households the welfare loss due to lack of insurance against labor market risk are quite
substantial. Further, even for households age 40 these welfare losses amount to 3 percent of
lifetime consumption. For the older working households, however, this equivalent variations
has fallen to less than one-half of one percent.
h) The Effect of Changing Personal Bankruptcy Regimes
Figures 7 through 10 explore the consequences of changing the details of the personal
bankruptcy regime either by increasing the time for which a household is considered bankrupt,
or by allowing for wage garnishment during bankruptcy. Figures 7 and 8 focus on the effects
of changing the duration of bankruptcy from an average of 7 years to an average of 10 years.
As shown in Figure 7, the human capital portfolio share of the youngest households rises
from almost one, denoting no financial wealth, to a value greater than one, denoting negative
net financial assets. This increase is reflected throughout the age distribution, although the
increases are quite modest in size and decline with age. Figure 8 shows that, although human capital investment increases, insurance against human capital risk is almost unchanged,
with the blue and red lines almost atop one another. The reason is that households prefer,
on the margin, to borrow more in order to invest in human capital and not buy any further
insurance. This is also confirmed by the green line in Figure 8 which shows the effect on
risk sharing if the household is constrained from increasing their human capital portfolio. In
this case, risk sharing is increased across all age groups, with the largest effect on the young,
who are most likely to be constrained, and whose consumption insurance rises from about a
37
third of income risk to roughly 40 percent of income risk.
Figures 9 and 10 repeat the above analysis, this time by introducing garnishment of 20
percent of wages while bankrupt. Figure 9 shows that this results in a qualitatively similar increase in human capital portfolio shares with the portfolio share of the very youngest
working households exceeding one by roughly 7 percent. Borrowing levels are also positive
(the human capital share remains larger then one) for households throughout their 30’s and
into their 40’s. Figure 10 also shows that there is no significant increase in risk sharing. If
households are constrained from investing more in human capital (the green line), consumption insurance increases dramatically with the young now insured against almost 60 percent
of their income risk.
i) The Effect of Changing Human Capital Risk
We now consider an increase in the standard deviation of labor income shocks from ση = 0.15
to ση = 0.20. Figure 11 shows that the effect of this increase on human capital investment is
not monotone: young households increase, while older households decrease, their investments
in human capital. This is the result of two offsetting forces. On the one hand, an increase in
labor income risk makes investments in human capital less attractive for a given mean return.
On the other hand, increases in labor income risk make the prospect of declaring bankruptcy
less attractive as the household must bear the full cost of this risk while bankrupt. For young
households, who desire to hold more human capital, the latter effect dominates, while for
older households the former effect dominates.
Figure 12 shows the implications of these choices for consumption insurance. Whereas the
blue line shows that consumption insurance is increased for all household ages, the green
line depicts what would have happened to consumption insurance if the households had
been unable to adjust their human capital holdings. As shown in the figure, the youngest
38
households would have increased their consumption insurance even further, while the oldest
households would have had less consumption insurance as the costs of declaring bankruptcy
were significantly reduced and thus the extent of available insurance constraints was reduced.
j) The Effect of Changing Risk Aversion
Lastly, Figures 13 and 14 illustrate the effects of changing the coefficient of relative risk
aversion from γ = 1 to γ = 2 keeping all factor returns constant at the levels calibrated
in the baseline. Figure 13 shows that greater risk aversion, everything else equal, results
in greater investments in human capital. This result stems from the fact that declaring
bankruptcy is now more expensive, and hence households are able to borrow more for the
purpose of investing in human capital. This offsets the fact that more risk averse individuals
are otherwise less inclined to invest in risky assets.
Figure 14 shows that consumption insurance is also improved for households with higher
risk aversion reflecting both a greater demand for insurance and greater possibilities for
insurance resulting from the fact that default is less desirable for such households. Also
shown in the figure, as depicted by the green line, if that consumption insurance would have
increased still further if households had been unable to increase their investments in risky
human capital.
k) Closing the Model (General Equilibrium)
We now return to the full general equilibrium model (endogenous investment returns) and
show how to complete the model calibration by introducing a production function and
imposing a number of aggregate targets. We use a Cobb-Douglas production function
f (K̃) = AK̃ α, where 0 < α < 1 is capital’s share in output and A is a productivity parameter. In this case, the rental rates of physical capital and human capital are given by
r̃k = αAK̃ α−1
39
(34)
r̃h = (1 − α)AK̃ α
As in Krebs, Kuhn, and Wright (2015), we target an aggregate share of capital income,
r̃k K/Y , of 0.32 so that α = 0.32. We also follow Krebs, Kuhn, and Wright (2015) and target
an aggregate capital-to-output ratio of 2.94. This target in conjunction with the target
rf = r̃k − δk = 0.03 yields r̃k = 0.1085 and δk = 0.0785.
Recall that in the partial equilibrium calibration we have chosen the value of φr̃h to match
the average financial-wealth-to-earnings ratio from the SCF for households age 23-60. Given
that the values for capital’s share in output/income and r̃k and δk are also pinned down, the
only way to have the general equilibrium model match a particular target for the aggregate
capital-to-output ratio is to vary λ in (25), respectively λh in (26), which is the approach
taken Krebs, Kuhn, and Wright (2015).
The calibration approach discussed so far determines φr̃h , the rental rate of human capital
in consumption units, but does not determine separately φ and r̃h . To resolve this indeterminacy, Krebs, Kuhn, and Wright (2015) impose the (somewhat arbitrary) condition that
K̃ = 0.4, where the value 0.4 for the capital-to-labor ratio is in line with the results obtained
in Krebs (2003) using a model with φ = 1 (one unit of the consumption/capital good can
be transformed into one unit of human capital).
Finally, we note that the life-cycle profile of human capital returns, r̄h (s1 ), which is
pinned down by the partial equilibrium calibration, and the human capital rental rate, φr̃h
(see above), imply a life-cycle profile for the difference ϕ(s1 ) − δh . However, the values
for the learning-by-doing parameters and the human capital depreciation rate are still not
separately identified. Krebs, Kuhn, and Wright (2015) choose δh = 0.04, which then pins
down the life-cycle profile of learning-by-doing parameters.
40
Appendix
Proof of Proposition 1
To simplify the notation, suppress the dependence on the aggregate state, Ω, and consider a
household of cohort n = 0. Further, define an action variable xt = (ct , θt+1) and a feasibility
correspondence, Γ, that for every (wt , st) restricts the choice of (wt+1, xt ) according to (15).
Using this notation, the household maximization problem reads
max E
"
∞
X
β tνt u(xt|w0
#
(A1)
t=0
s.t.
(wt+1, xt ) ∈ Γ(wt , st )
E
"
∞
X
m
t
β νm u(xt+m )|w0, s
#
≥ Vd (wt, st )
m=0
The corresponding Bellman equation reads:
V (w, s)
=
s.t.
(
max0 u(x) + βρ(s)
x,w
X
0
0
0
V (w , s )π(s |s)
)
(A2)
s0
(x, w0 ) ∈ Γ(w, s)
V (w0, s0 ) ≥ Vd (w0, s0 )
Define an operator, T , that maps semi-continuous functions into semi-continuous functions as
T V (w, s)
=
max0 {u(x) + βE[V (w0 , s0 )|s]}
x,w
(A3)
s.t. (x, w0) ∈ Γ(w, s)
V (w0 , s0) ≥ Vd (w0 , s0) .
A standard contraction mapping argument shows that there is a unique continuous solution,
V0 , to the Bellman equation (A2) without participation constraint if i) u is continuous, ii)
Γ is compact-valued and continuous, and (19) holds. Extending the argument of Rustichini
41
(1998),14 it can be shown that V∞ = limk→∞ T k V0 exists, is equal to the maximal solution of
the Bellman equation (A2), and is the value function of the sequential maximization problem
(A1) if the following four conditions hold: i) u is continuous, ii) Γ is compact-valued and
continuous, iii) for all states, (w, s), there exists a feasible plan for the sequential problem
(A1) so that the corresponding expected lifetime utility (payoff) is greater than −∞, and
iv) for any given state, (w, s), the value function of the max-problem without participation
constraints satisfies V0∗ (w, s) < +∞. Thus, to prove proposition 1 it suffices to show that
conditions i)-iv) hold.
The continuity of the payoff function, u, is obvious. The correspondence, Γ, is compactvalued since portfolio-choices, θ0 , are elements of a closed and bounded subset of IRm . Closedness follows from the fact that the set is defined by equalities and weak inequalities. Restricting attention to a bounded set can be shown to be without loss of generality. Continuity
of the correspondence Γ is also straightforward to show. A standard argument shows that
conditions iii) and iv) hold if condition (19) is satisfied. This proves proposition 1.
Proof of Proposition 2
As before, let V0 be the solution of the Bellman equation (A2) without the participation constraint. To save space, we only conduct the prove for the case γ 6= 1. Simple guess-and-verify
shows that V0 has the following functional form:
V0 (w, s) = Ṽ0 (s) w1−γ
(A4)
where Ṽ0 is the solution to the intensive-form Bellman equation (22) without participation
constraint. Let the operator T be defined as in (A3). We show by induction that if Vk = T k V0
14
Rustichini (1998) consider a class of dynamic programming problems with participation constraint (incentive compatability constraint) and possibly unbounded utility. However, he requires bi-convergence,
which is always satisfied if lifetime-utility is bounded for all feasible paths (Streufert, 1990). Unfortunately,
in our problem with γ ≥ 1 the requirement of lower convergence is not satisfied, so that Rustichini (1998) is
not directly applicable.
42
has the functional form, then Vk+1 = T k+1 V0 has the functional form. For k = 0 the claim is
true because V0 has the functional form. Suppose now Vk has the functional form. We then
have
Vk+1 (w, s)
=
=
T Vk (w, s)
(
)
X
c1−γ
0
0
0 1−γ
0 1−γ
0
+ ρ(s)
Ṽk (s )(1 + r(θ , s, s )) (w ) π(s |s)
max
w0 ,c,θ0 1 − γ
s0
s.t. w0 = (1 + r(θ0 , s, s0))(w − c)
X θa0 (s0 )π(s0|s)
1 = θh0 +
1 + rf
s0
X
s0
(A5)
π(s0|s)θa0 (s0 )
≥ −D̄θh0 , θh0 ≥ 0 , w0 ≥ 0
1 + rf
Ṽk (s0 ) (1 + r(θh0 , θa(s0 ), s, s0))
1−γ
(w0 )1−γ
≥ Ṽd (s0)(1 + rhd (s, s0 ))θh0 .
Clearly, the solution to the maximization problem defined by the right-hand-side of (A5) has
the form
0
wk+1
= (1 − c̃k+1 (s))w
(A6)
0
0
= θk+1
(s) ,
θk+1
where the subscript k + 1 indicates step k + 1 in the iteration. Thus, we have Vk+1 (w, s) =
Ṽk+1 (s)w1−γ where Ṽk+1 is defined accordingly.
From proposition 1 we know that V∞ = limk→∞ T k V0 exists and that it is the maximal
solution to the Bellman equation (A2) as well as the value function of the corresponding
sequential maximization problem (A1). Since the set of functions with this functional form
is a closed subset of the set of semi-continuous functions, we know that V∞ has the functional
form. This prove proposition 2.
43
Proof of Proposition 3
From proposition 2 we know that individual households maximize utility subject to the
budget constraint and participation constraint if condition (19) is satisfied. Thus, it remains
to show that the market clearing condition can be written as (22) and that the law of motion
(23) describes the equilibrium evolution of the relative wealth distribution.
For simplicity, we only consider the case of infinitely-live households (one cohort n = 0).
For the aggregate value of financial asset holdings we find:
"
θa,t+1wt+1
E
1 + rt+1
#
= E[θa,t+1(1 − c̃t)wt ]
(A7)
= E[E[θa,t+1(1 − c̃t )wt |st]]
= E[θa,t+1(1 − c̃t)E[wt |st]]
E[θa,t+1(1 − c̃t )E[wt|st ]]
= E [wt]
E [wt]
= E [wt] E [θa,t+1(1 − c̃t )Ω(st )] .
where the first line follows from the budget constraint, the second line from the law of iterated
expectations, the third line from the fact that θa,t+1 and c̃t are independent of wealth and
st−1 , and the last line from the definition of Ω. A similar argument shows that
Ht+1 = E [wt ] z̄E [θh,t+1 (1 − c̃t )Ω(st )] .
Dividing the two expressions shows that K̃ 0 is given by (22).
Finally, the law of motion for Ω can be found as:
E [wt+1 |st+1 ]
(A9)
E [wt+1]
E [(1 + rt+1)(1 − c̃t )wt|st+1 ]
=
E [(1 + rt+1 )(1 − c̃t )wt ]
E [E [(1 + rt+1 )(1 − c̃t)wt |st ] |st+1]
=
E [E [(1 + rt+1 )(1 − c̃t )wt |st]]
E [(1 + rt+1)(1 − c̃t )E [wt |st ] |st+1]
=
E [(1 + rt+1 )(1 − c̃t )E [wt |st]]
Ωt+1 (st+1) =
44
(A8)
E [wt ]
E [(1 + rt+1)(1 − c̃t )E [wt |st ] |st+1]
×
E [(1 + rt+1 )(1 − c̃t )E [wt |st]]
E [wt ]
E [(1 + rt+1)(1 − c̃t )Ωt (st)|st+1 ]
,
=
E [(1 + rt+1 )(1 − c̃t )Ωt (st )]
=
where the second line follows from the budget constraint, the third line from the law of
iterated expectations, the fourth line from the fact that θt+1 and c̃t are independent of
wealth and st−1 , and the last line from the definition of Ω. This completes the proof of
proposition 3.
45
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48
Figure 1: Earnings growth rates
0.08
0.06
0.04
0.02
0
-0.02
25
30
35
40
45
50
55
Notes: Earnings growth rates from the model and the data. The red line with circles shows the model,
the blue line with squares the data. Horizontal axis shows average age within each age group and vertical
axis shows annual earnings growth in percent.
Figure 2: Financial wealth to earnings ratio
5
4
3
2
1
0
25
30
35
40
45
50
55
Notes: Wealth-to-income from the model and the data. The red line with circles shows the model, the
blue line with squares the data. Horizontal axis shows average age within each age group.
i
Figure 3: Human capital portfolio share
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
25
30
35
40
45
50
55
Notes: Human capital portfolio share in total wealth from the model and the data. The red line with
circles shows the model, the blue line with squares the data. Horizontal axis shows average age within
each age group.
Figure 4: Excess human capital returns
0.1
0.08
0.06
0.04
0.02
0
25
30
35
40
45
50
55
Notes: Excess human capital returns over the life-cycle. Horizontal axis shows average age within each
age group.
ii
Figure 5: Consumption insurance
1
0.8
0.6
0.4
0.2
0
25
30
35
40
45
50
55
Notes: Consumption insurance of households over the life-cycle. Consumption insurance is measured by
the insurance coefficient. The insurance coefficient is constructed as one minus the ratio of the variance
of consumption growth over the variance of income growth. Full insurance yields an insurance coefficient
of 1 and no insurance (autarky) an insurance coefficient of 0. Horizontal axis shows average age within
each age group.
Figure 6: Welfare cost of underinsurance
8
7
6
5
4
3
2
1
0
25
30
35
40
45
50
55
Notes: Welfare costs of limited contract enforcement over the life-cycle. Welfare costs are shown as
consumption equivalent variation in percentage points. Horizontal axis shows average age within each
age group.
iii
Figure 7: Human capital portfolio share after change in enforcement
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
25
30
35
40
45
50
55
Notes: Human capital portfolio share in total wealth from the baseline model and from comparative
1
. The red line with circles
statics experiment. The comparative statics experiment changes p from 71 to 10
shows the baseline model, the blue line with squares shows the comparative statics results. Horizontal
axis shows average age within each age group.
Figure 8: Consumption insurance after change in enforcement
1
0.8
0.6
0.4
0.2
0
25
30
35
40
45
50
55
Notes: Consumption insurance of households over the life-cycle for baseline model and from comparative
1
statics experiment. The comparative statics experiment changes p from 17 to 10
. The red line with
circles shows the baseline model, the blue line with squares shows the comparative statics results, the
green line with diamonds shows the comparative statics results with human capital allocation fixed to the
baseline. Consumption insurance is measured by the insurance coefficient. Horizontal axis shows average
age within each age group.
iv
Figure 9: Human capital portfolio share after introduction of wage garnishment
1.2
1.1
1
0.9
0.8
0.7
25
30
35
40
45
50
55
Notes: Human capital portfolio share in total wealth from the baseline model and from comparative statics
experiment. The comparative statics experiment introduces a 20% wage garnishment during default. The
red line with circles shows the baseline model, the blue line with squares shows the comparative statics
results. Horizontal axis shows average age within each age group.
Figure 10: Consumption insurance after introduction of wage garnishment
1
0.8
0.6
0.4
0.2
0
25
30
35
40
45
50
55
Notes: Consumption insurance of households over the life-cycle for baseline model and from comparative statics experiment. The comparative statics experiment introduces a 20% wage garnishment during
default. The red line with circles shows the baseline model, the blue line with squares shows the comparative statics results, the green line with diamonds shows the comparative statics results with human
capital allocation fixed to the baseline. Consumption insurance is measured by the insurance coefficient.
Horizontal axis shows average age within each age group.
v
Figure 11: Human capital portfolio share after change in human capital risk
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
25
30
35
40
45
50
55
Notes: Human capital portfolio share in total wealth from the baseline model and from comparative
statics experiment. The comparative statics experiment changes the standard deviation of human capital
risk from 0.15 to 0.2. The red line with circles shows the baseline model, the blue line with squares shows
the comparative statics results. Horizontal axis shows average age within each age group.
Figure 12: Consumption insurance after change in human capital risk
1
0.8
0.6
0.4
0.2
0
25
30
35
40
45
50
55
Notes: Consumption insurance of households over the life-cycle for baseline model and from comparative
statics experiment. The comparative statics experiment changes the standard deviation of human capital
risk from 0.15 to 0.2. The red line with circles shows the baseline model, the blue line with squares shows
the comparative statics results, the green line with diamonds shows the comparative statics results with
human capital allocation fixed to the baseline. Consumption insurance is measured by the insurance
coefficient. Horizontal axis shows average age within each age group.
vi
Figure 13: Human capital portfolio share after change in risk aversion
1.1
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
25
30
35
40
45
50
55
Notes: Human capital portfolio share in total wealth from the baseline model and from comparative
statics experiment. The comparative statics experiment changes the degree of relative risk aversion γ
from 1 to 2. The red line with circles shows the baseline model, the blue line with squares shows the
comparative statics results. Horizontal axis shows average age within each age group.
Figure 14: Consumption insurance after change in risk aversion
1
0.8
0.6
0.4
0.2
0
25
30
35
40
45
50
55
Notes: Consumption insurance of households over the life-cycle for baseline model and from comparative
statics experiment. The comparative statics experiment changes the degree of relative risk aversion γ
from 1 to 2. The red line with circles shows the baseline model, the blue line with squares shows the
comparative statics results, the green line with diamonds shows the comparative statics results with
human capital allocation fixed to the baseline. Consumption insurance is measured by the insurance
coefficient. Horizontal axis shows average age within each age group.
vii
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
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Working Paper Series (continued)
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A Portfolio-Balance Approach to the Nominal Term Structure
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WP-13-23
WP-13-24
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WP-13-26
Policy Intervention in Debt Renegotiation:
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Working Paper Series (continued)
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Early Public Banks
William Roberds and François R. Velde
WP-14-03
Mandatory Disclosure and Financial Contagion
Fernando Alvarez and Gadi Barlevy
WP-14-04
The Stock of External Sovereign Debt: Can We Take the Data at ‘Face Value’?
Daniel A. Dias, Christine Richmond, and Mark L. J. Wright
WP-14-05
Interpreting the Pari Passu Clause in Sovereign Bond Contracts:
It’s All Hebrew (and Aramaic) to Me
Mark L. J. Wright
WP-14-06
AIG in Hindsight
Robert McDonald and Anna Paulson
WP-14-07
On the Structural Interpretation of the Smets-Wouters “Risk Premium” Shock
Jonas D.M. Fisher
WP-14-08
Human Capital Risk, Contract Enforcement, and the Macroeconomy
Tom Krebs, Moritz Kuhn, and Mark L. J. Wright
WP-14-09
Adverse Selection, Risk Sharing and Business Cycles
Marcelo Veracierto
WP-14-10
Core and ‘Crust’: Consumer Prices and the Term Structure of Interest Rates
Andrea Ajello, Luca Benzoni, and Olena Chyruk
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The Evolution of Comparative Advantage: Measurement and Implications
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Liquidity Traps and Monetary Policy: Managing a Credit Crunch
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3
Working Paper Series (continued)
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Ed Nosal, Yuet-Yee Wong, and Randall Wright
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Preventing Bank Runs
David Andolfatto, Ed Nosal, and Bruno Sultanum
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Credit Supply and the Housing Boom
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti
WP-14-21
The Effect of Vehicle Fuel Economy Standards on Technology Adoption
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Inflation Uncertainty and Disagreement in Bond Risk Premia
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Access to Refinancing and Mortgage Interest Rates:
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Private Takings
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Poor (Wo)man’s Bootstrap
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4
Working Paper Series (continued)
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Simpler Bootstrap Estimation of the Asymptotic Variance of U-statistic Based Estimators
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Bad Investments and Missed Opportunities?
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5
Working Paper Series (continued)
Insurance in Human Capital Models with Limited Enforcement
Tom Krebs, Moritz Kuhn, and Mark Wright
WP-16-08
6
@FedFRASER