Preliminary and Incomplete – Please Do Not Quote

Digestible Microfoundations:
Buffer Stock Saving in a Krusell–Smith World

April 30, 2012
Christopher D. Carroll1
Jiri Slacalek2
Kiichi Tokuoka3


Krusell and Smith (1998) showed that it is possible to construct rational expectations macroeconomic models with serious microfoundations. We argue that three modifications to their framework are required to fulfill its promise. First, we replace their assumption about household income dynamics with a process that matches microeconomic data. Second, our agents have finite lifetimes a la Blanchard (1985), which has both substantive and technical benefits. Finally, we calibrate heterogeneity in time preference rates so that the model matches the observed degree of inequality in the wealth distribution. Our model has substantially different, and considerably more plausible, implications for macroeconomic questions like the aggregate marginal propensity to consume out of an economic ‘stimulus’ program.


Microfoundations, Wealth Inequality, Marginal Propensity to Consume

            JEL codes 

D12, D31, D91, E21




1Carroll: Department of Economics, Johns Hopkins University, Baltimore, MD,,     2Slacalek: European Central Bank, Frankfurt am Main, Germany,,     3Tokuoka: International Monetary Fund, Washington, DC,    

1 Introduction

Macroeconomists have sought credible microfoundations since the dawn of our discipline. Keynes, his critics, and subsequent generations through Lucas (1976) and beyond have agreed on this, if little else.

Since Keynes’s time, consumption modeling has been a battleground between two microfoundational camps. ‘Bottom up’ modelers (e.g. Modigliani and Brumberg (1954); Friedman (1957)) drew wisdom from microeconomic data and argued that macro models should be constructed by aggregation from microeconomic models that matched robust micro facts. ‘Top down’ modelers (e.g., Samuelson (1958); Diamond (1965); Hall (1978)) treated aggregate consumption as reflecting the optimizing decisions of representative agents; with only one such agent (or, at most, one per generation), these models had ‘microfoundations’ under a generous interpretation of the word.

The tractability of representative agent models has made them appealing for business cycle analysis. But such models have never been easy to reconcile with either macroeconomic2 or microeconomic3 evidence on consumption dynamics, nor with microeconomic theory which implies that people who differ from each other (in age, preferences, wealth, liquidity constraints, taxes, and other dimensions) should respond differently to any given shock. If any of these differences matter (and it is hard to see how they could not),4 the aggregate size of a shock is not a sufficient statistic to calculate the aggregate response; information about how the shock is distributed is indispensable.

Bottom-up models, however, also have their problems. Even judged by a sympathetic standard that asks how well they can match measured wealth heterogeneity, bottom-up models have not been as successful as their champions might have initially hoped. For example, bottom-up models calibrated to match the wealth holdings of the median household generally fail to match the large size of the aggregate capital stock, because they seriously underpredict the upper parts of the wealth distribution (Carroll (2000b); Cagetti (2003)). Alternatively, models calibrated to match the aggregate level of wealth greatly overpredict wealth at the median (Hubbard, Skinner, and Zeldes (1994); Carroll (2000b)). A further problem is that (at least until Krusell and Smith (1998)) there has been no common answer to the question of how to analyze systematic macroeconomic fluctuations (business cycles) in bottom-up models.

This paper aims to reconcile the camps. We construct a workhorse model that answers the main objections to both kinds of models by making three modifications to the well-known Krusell–Smith (‘KS’) framework.5 First, we replace KS’s highly stylized assumptions about the nature of idiosyncratic income shocks with a microeconomic labor income process that captures the essentials of the empirical consensus from the labor economics literature about actual income dynamics in micro data (with credibly calibrated transitory and permanent shocks).6 Second, agents in our model have finite lifetimes a la Blanchard (1985), permitting a kind of primitive life cycle analysis and also solving some technical problems created by the incorporation of permanent shocks. Finally, we obtain a necessary extra boost to wealth inequality by calibrating a simple measure of heterogeneity in ‘impatience.’7

The resulting framework differs sharply from the benchmark KS model in its implications for important microeconomic and macroeconomic questions. A timely macroeconomic example is the response of aggregate consumption to an ‘economic stimulus payment,’ interpreted here as a one-time lump sum transfer to households. In response to a $1-per-capita payment, the baseline version of the KS model implies that the annual marginal propensity to consume (MPC) is about 0.05  ,8 almost irrespective of how the cash is distributed across households. In contrast, a version of our model that matches the distribution of liquid financial wealth implies that if the entire tax cut were directed at households in the bottom half of the liquid financial-wealth-to-income distribution, the MPC would be 0.83  , which counts as a big improvement in realism, given the vast body of microeconomic evidence that consistently finds MPCs much greater than the 3–5 percent figure that characterizes representative agent models.9 Furthermore, the model’s differences with the representative agent framework are not peculiar to unusual events like a stimulus payment; to the extent that different kinds of macroeconomic shocks tend systematically to be differently distributed across the population (for example, labor income shocks may affect a less wealthy set of households than capital income shocks), this improvement in realism may also matter for general questions of macroeconomic dynamics.

Section 2 of the paper begins building the model’s structure by adding microeconomic modeling elements to a benchmark representative agent model. Using this model (without macroeconomic dynamics), the section closes by estimating the degree of heterogeneity in impatience necessary to match the degree of inequality in the U.S. wealth distribution; we find that relatively small differences in impatience substantially affect the model’s fit to the wealth data. Section 3 builds up the full version of the model by adding aggregate shocks of the KS type, and presents detailed comparisons of our model with theirs. Section 4 further improves the model by introducing an aggregate income process that is analytically simpler than the KS ‘toy’ aggregate process, that we believe is more empirically plausible as well, and that simplifies model solution and simulation considerably. We offer this final, simpler version of the model as our preferred jumping-off point for future macroeconomic research.

2 The Model without Aggregate Uncertainty

2.1 The Perfect Foresight Representative Agent Model

To establish notation and a transparent benchmark, we begin by briefly sketching a standard perfect foresight representative agent model.

The aggregate production function is

     α      1- α
ZtKKK t ( ℓLLLt)   ,

where Z
  t  is aggregate productivity in period t  , KKK
  t  is capital, ℓ  is time worked per employee, and LLLt  is employment. The representative agent’s goal is to maximize discounted utility from consumption

max       βnu (CCC     )

for a CRRA utility function           1- ρ
u (∙) =  ∙    ∕(1 -  ρ)  .10 The representative agent’s state at the time of the consumption decision is defined by two variables: MMM  t  is market resources, and Zt  is aggregate productivity.

The transition process for MMM  t  is broken up, for clarity of analysis and consistency with later notation, into three steps. Assets at the end of the period are market resources minus consumption, equal to

AAAt =  MMM  t - CCCt,

while next period’s capital is determined from this period’s assets via

KKKt+1   = AAAt.

The final step can be conceived as the transition from the beginning of period t +  1  when capital has not yet been used to produce output, to the middle of that period, when output has been produced and incorporated into resources but has not yet been consumed:

MMM       =   ℸKKK      + Z    KKK α  (ℓLLL    )1- α
   t+1           t+1   ◟ t+1---t+1◝◜--t+1----◞
where r
 t+1   is the interest rate,11   Wt+1   is the wage rate,12 and ℸ  =  1 - δ  is the depreciation factor for capital.

After normalizing by the productivity factor Zt =  Z1t∕(1- α )( ℓLLLt)  ,13 the representative agent’s problem is

                                         [ 1- ρ                ]
V (Mt,  Zt )  =    max    u(Ct ) + β Et   Γt+1 V (Mt+1,  Zt+1 )         (2)

         At   =    Mt  -  Ct                                            (3)

      Kt+1    =    At ∕Γ t+1                                             (4)
      Mt+1    =    ℸKt+1   +  K α  ,                                    (5)
where the non-bold variables are the corresponding bold variables divided by Zt  (e.g., At =  AAAt ∕Zt  , Mt  =  MMM  t∕Zt  ); Γ t+1 = Zt+1 ∕Zt  ; and the expectations operator Et  here signifies the perfection of the agent’s foresight (but will have the usual interpretation when uncertainty is introduced below).

Table 1: Parameter Values and Steady State



 Representative    agent  model
 Time   discount  factor                 β           0.99      JEDC    (2010 )

 Coef  of relative risk aversion         ρ           1         JEDC    (2010 )
 Capital  share                          α           0.36      JEDC    (2010 )

 Depreciation   rate                     δ           0.025     JEDC    (2010 )
 Time   worked   per employee            ℓ           1/0.9     JEDC    (2010 )

 Steady   state
 Capital  –output  ratio               KKK ∕YYY        10.26      JEDC    (2010 )

 E ffective  interest rate              r - δ         0.01      JEDC    (2010 )
 Wage   rate                             W           2.37      JEDC    (2010 )
 Heterogenous    agents  models

 Unempl    insurance   payment           μ           0.15      JEDC    (2010 )
 Unemployment      rate                  u           0.07      Mean   in JEDC    (2010  )

 Probability   of death                  D           0.005     Yields  50  year working   life
 Variance   of log θt,i                   σ2θ       0.010  ×  4  Carroll  (1992 )

Notes: The models are calibrated at the quarterly frequency, and the steady state values are calculated on a quarterly basis.

Except where otherwise noted, our parametric assumptions match those of the papers in the special issue of the Journal of Economic Dynamics and Control (2010, Volume 34, Issue 1, edited by den Haan, Judd, and Julliard) devoted to comparing solution methods for the KS model (the parameters are reproduced for convenience in the top panel of Table 1).14 The model is calibrated at the quarterly frequency. When aggregate shocks are shut down (Zt =  1  and LLLt =  LLL  ), the model has a steady-state solution with a constant ratio of capital to output and constant (gross) interest and wage factors, which we write without time subscript as r  and W  and which are reflected in Table 1.15

Henceforth, we refer to the version of the model solved by the papers in the special JEDC volume as the ‘KS-JEDC’ model, while we call the original KS model solved in Krusell and Smith (1998) ‘KS-Orig’ model. (The only effective difference between the two is the introduction (for realism) of unemployment insurance in the KS-JEDC version, which does not matter much for any substantive results.16 ,  17 )

2.2 The Household Income Process

For our purposes, the principal conclusion of the large literature on microeconomic labor income dynamics is that household income can be reasonably well described as follows. The idiosyncratic permanent component of labor income p  evolves according to

pt+1   =   Gt+1pt ψt+1                             (6)
where G
  t+1   captures the predictable low-frequency (e.g., life-cycle and demographic) components of income growth, and the Greek letter psi mnemonically indicates the permanent shock to income. Actual income is the product of permanent income, a mean-one transitory shock, and the wage rate:
yyy     =   p    ξ   W     .
 t+1        t+1  t+1   t+1

After taking logarithms, this income process is strikingly similar to Friedman (1957)’s characterization of income as having permanent and transitory components. Because this process has been used widely in the literature on buffer stock saving, and though similar to Friedman’s formulation is not identical to it, we henceforth refer to it as the Friedman/Buffer Stock (or ‘FBS’) process.18

Table 2 summarizes the annual variances of log permanent shocks (σ2
 ψ  ) and log transitory shocks ( 2
σξ  ) estimated by a selection of papers from the extensive literature.19 Some authors have used a process of this kind to describe the labor income or wage process for an individual worker (top panel) while others have used it to describe the process for overall household income (bottom panel); it seems to work reasonably well in both cases (though, obviously, with different estimates of the variances). (Recent work by Sabelhaus and Song (2010) using newly available data from Social Security earnings files finds that the variances of both transitory and permanent shocks have declined during the “Great Moderation” period at all ages; they also find distinct life cycle patterns of shocks by age, with young people experiencing higher levels of both kinds of shocks than the middle-aged).

The second-to-last line of the table shows what labor economists would have found, when estimating a process like the one above, if the empirical data were generated by households who experienced an income process like the one assumed by the KS-JEDC model.20 This row of the table makes our point forcefully: The empirical procedures that have actually been applied to empirical micro data, if used to measure the income process households experience in a KS economy, would have produced estimates of σ2ψ  and σ2ξ  that are orders of magnitude different from what the actual empirical literature finds in actual data. This discrepancy naturally makes one wonder whether the KS-JEDC model’s well-known difficulty in matching the degree of wealth inequality is largely explained by its highly unrealistic assumption about the income process.21

Table 2: Estimates of Annual Variances of Log Income, Earning and Wage Shocks

                                                                        Permanent       Transitory
                                                                              2               2
 Authors                                                                     σψ              σξ
 Individual   data
   MaCurdy     (1982 )‡                                                     0.013          0.031

   Topel  (1990 )                                                           0.013          0.017
   Topel  and  Ward   (1992  )                                              0.017          0.013
   Mo  ffitt  and  Gottschalk   (1995 )                                       0.001          0.180
   Meghir   and  Pistaferri (2004  )◇                                       0.031          0.032

   Low,  Meghir,   and  Pistaferri (2005  )                                 0.011            –
   Jensen  and  Shore   (2008 )◇                                            0.054          0.171
   Hryshko   (2010 )                                                        0.038          0.118
   Guvenen    (2009 )                                                       0.015          0.061
 Household    data

   Carroll  (1992 )                                                         0.016          0.027
   Carroll  and  Samwick    (1997 )                                         0.022          0.044

   Storesletten,  Telmer,  and   Yaron  (2004a  )                           0.017          0.063
   Storesletten,  Telmer,  and   Yaron  (2004b  )                      0.008  –0.026       0.316
   Blundell,  Pistaferri, and  Preston   (2008 )                       0.010  –0.030   0.029 –0.055
 Implied   by KS  -JEDC                                                    0.000           0.038
 Implied   by Castaneda,    Diaz -Gimenez,   and  Rios -Rull  (2003 )      0.029           0.005

Notes: ‡ :  MaCurdy (1982) did not explicitly separate ψt  and ξt  , but we have extracted σ2
 ψ  and σ2
 ξ  as implications of statistics that his paper reports. First, we calculate var(logyyyt+d - logyyyt)  and var(log yyyt+d-1 - logyyyt)  using his estimate (we set d = 5  ). Then, following Carroll and Samwick (1997) we obtain the values of σ2
  ψ  and σ2
  ξ  which can match these statistics, assuming that the income process is yyy = p ξ
 t   t t  and p = p   ψ
 t   t-1 t  (i.e., we solve var(logyyy   - log yyy) = dσ2+ 2σ2
       t+d      t     ψ    ξ  and                             2     2
var(log yyyt+d-1 - logyyyt) = (d- 1)σψ + 2σξ  ). ⋆
 :  Moffitt and Gottschalk (1995) estimated the income process with random walk plus ARMA. Using income draws generated by their estimated process and following Carroll and Samwick (1997), we have estimated the variances under the assumption that these income draws were produced by the process yyyt = ptξt  where pt = pt-1ψt  . ◇
 :  Meghir and Pistaferri (2004), Jensen and Shore (2008), Hryshko (2010), and Blundell, Pistaferri, and Preston (2008) assume that the transitory component is serially correlated (an MA process), and report the variance of a subelement of the transitory component. For example, Meghir and Pistaferri (2004) and Blundell, Pistaferri, and Preston (2008) assume an MA(1) process logξt = vt + ϑvt-1  and obtain estimates   2
(σ v,ϑ)  =(0.0300,- 0.2566)  and (0.0286-0.0544,0.1132)  , respectively.  2
σξ  for these four articles reported in this table are calculated by      2  2
(1+ ϑ )σv  using their estimates.

2.3 Finite Lifetimes and the Finite Variance of Permanent Income in the Cross-Section

One might wish to use the FBS income process specified in subsection 2.2 as a complete characterization of household income dynamics, but that idea has a problem: Since each household accumulates a permanent shock in every period, the cross-sectional distribution of idiosyncratic permanent income becomes wider and wider indefinitely as the simulation progresses; that is, there is no ergodic distribution of permanent income in the population.

This problem and several others can be addressed by assuming that the model’s agents have finite lifetimes a la Blanchard (1985). Death follows a Poisson process, so that every agent alive at date t  has an equal probability D  of dying before the beginning of period t +  1  . (The probability of NOT dying is the cancelation of the probability of dying: /
/D =  1 -  D  ). Households engage in a Blanchardian mutual insurance scheme: Survivors share the estates of those who die. Assuming a zero profit condition for the insurance industry, the insurance scheme’s ultimate effect is simply to boost the rate of return (for survivors) by an amount exactly corresponding to the mortality rate.

In order to maintain a constant population (of mass one, uniformly distributed on the unit interval), we assume that dying households are replaced by an equal number of newborns; we write the population-mean operator as           ∫ 1
M  [∙t] =  0 ∙t,ιdι  . Newborns, we assume, begin life with a level of idiosyncratic permanent income equal to the mean level of idiosyncratic permanent income in the population as a whole. Conveniently, our definition of the permanent shock implies that in a large population, mean idiosyncratic permanent income will remain fixed at M [p] =  1  forever, while the mean of p2   is given by22

M [p2]  =   --------------                          (7)
            1 -  //D E [ψ2 ]
and the variance of p  by
  2           2
σ p  =   M  [p  ] - 1.

Of course for all of this to be valid, it is necessary to impose the parametric restriction  /     2
/D E [ψ  ] < 1  (a requirement that does not do violence to the data, as we shall see). Intuitively, the requirement is that, among surviving consumers, income does not spread out so quickly as to overwhelm the compression of the permanent income distribution that arises because of the equalizing force of death and replacement.

Since our goal here is to produce a realistic distribution of permanent income across the members of the (simulated) population, we measure the empirical distribution of permanent income in the cross section using data from the Survey of Consumer Finances (SCF), which conveniently includes a question asking respondents whether their income in the survey year was about ‘normal’ for them, and if not, asks the level of ‘normal’ income.23 This corresponds well with our (and Friedman (1957)’s) definition of permanent income p  (and Kennickell (1995) shows that the answers people give to this question can be reasonably interpreted as reflecting their perceptions of their permanent income), so we calculate the variance of   i    i      i
p  ≡  p ∕M  [p ]  among such households.24

The results from this exercise are reported in Table 3 (with a final row that makes the point that both the KS-Orig and KS-JEDC models assume that permanent shocks did not exist). Substituting these estimates for   2
σ p  into (7) and (8), we obtain estimates of the variance of ψ  . Reassuringly, we can interpret the variances of ψ  thus obtained as being easily in the range of the estimated variances of             2
log (ψ ) =  σψ  in Table 2.25 Such a correspondence, across two quite different methods of measurement, suggests there is considerable robustness to the measurement of the size of permanent shocks. (Below, we will choose a calibration for  2
σψ  that is in the middle range of estimates from either method.)

Table 3: Variance of Permanent Income

 Dataset                    var (p)  E [ψ2 ]    σ2
 SCF1992                        2.5   1.015   0.015

 SCF1995                        7.5   1.018   0.018
 SCF1998                        3.1   1.015   0.015

 SCF2001                        3.6   1.016   0.016
 SCF2004                        5.2   1.017   0.017
 KS  -Orig  or  KS -JEDC         0     1       0

2.4 The Wealth Distribution with Transitory and Permanent Shocks

We now examine how wealth would be distributed in the steady-state equilibrium of an economy with wage rates and interest rates fixed at the steady state values calibrated in Table 1 of subsection 2.1, an income process like the one described in subsection 2.2, and finite lifetimes per subsection 2.3.

The process of noncapital income of each household follows

 yyt  =   ptξtWt                                         (8)
 pt  =   pt- 1ψt                                        (9)

Wt   =   (1 -  α )Zt(KKKt  ∕ℓLLLt )α,                      (10)
where yyy
 t  is noncapital income for the household in period t  , equal to the permanent component of noncapital income pt  multiplied by a transitory income shock factor ξ
 t  and wage rate W
  t  ; the permanent component of noncapital income in period t  is equal to its previous value, multiplied by a mean-one iid shock ψ
  t  , E [ψ    ] =  1
  t  t+n  for all n  ≥  1  . KKKt  is capital and LLLt  =  1 - ut  is the employment rate (because ut  is the unemployment rate). Since there is no aggregate shock, Z
  t  , KKK
   t  , LLLt  , and Wt  are constant (Zt  =  Z =  1  , KKKt  =  KKK  , LLLt =  LLL  , and W   =  W  =  (1 -  α )(KKK ∕ ℓLLL )α
   t  ).

Following the assumptions in the JEDC volume, the distribution of ξt  is:

ξt =   μ  with  probability  ut                               (11)

   =   (1 -  τt)ℓθt with  probability   1 - ut,               (12)
where μ  > 0  is the unemployment insurance payment when unemployed and τt =  μut∕ ℓLLLt  is the rate of tax collected to pay unemployment benefits (see Table 1 for parameter values).26 The probability of unemployment is constant (ut =  u  ); later we allow u  to vary over time.

The decision problem for the household in period t  can be written using normalized variables; the consumer’s objective is to choose a series of consumption functions c  between now and the end of the horizon that satisfy:

                                    [  1- ρ         ]
v(mt )   =    max   u (ct) + β//D Et   ψt+1 v(mt+1  )             (13)

    at   =    mt -  ct

    at   ≥    0
  k      =    a ∕(//D ψ    )                                       (14)
   t+1         t      t+1
 mt+1    =    (ℸ +  r)kt+1 +  ξt+1                               (15)
where the non-bold ratio variables are defined as the bold (level) variables divided by the level of permanent income pppt =  ptW  (e.g., mt  = mmmt  ∕(ptW  )  ). The only state variable is (normalized) cash-on-hand mt  . The household’s employment status is not a state variable, unlike in the KS-JEDC model, where tomorrow’s employment status depends on today’s status. This substantially simplifies the analysis (which is useful for computational and analytical purposes), arguably without too much sacrifice of realism (except possibly for detailed studies of the behavior of households during extended unemployment spells).

Since households die with a constant probability D  between periods, the effective discount factor is β//D  (in (13)); the effective interest rate is (ℸ +  r )∕//D  (combining (14) and (15)).27

Aside from heterogeneity in impatience (introduced below), three parameters characterize our modifications to the KS-JEDC model: D  , σ2
  θ  , and σ2
  ψ  . D  =  0.005  implies the average length of working life is 1∕0.005  =  200  quarters = 50  years (dating from entry into the labor force at, say, age 25). The variance of log transitory income shocks   2
σ θ  is the value advocated in Carroll (1992) (based on the Panel Study of Income Dynamics (PSID) data),28 as is σ2  =  0.016
  ψ  (but note that this value also matches the median value in Table 3).29 ,  30 Other parameter values (ρ  , α  , δ  , and ℓ  ) are from the JEDC volume (Table 1).

The one remaining unspecified parameter is the time preference factor. As a preliminary theoretical consideration, note that Carroll (2011) (generalizing Deaton (1991) and Bewley (1977)) has shown that models of this kind do not have a well-defined solution unless the condition holds:

(R β)1∕ρ
--------- <   1                               (16)
        (    - ρ )- 1∕ρ
Γ   =    E [ψ    ]      Γ .

Carroll (2011) dubs this inequality the ‘Growth Impatience Condition’ because it guarantees that consumers are sufficiently impatient to prevent the indefinite increase in the ratio of net worth to permanent income when income is growing (see also Szeidl (2006)). This condition is an amalgam of the pure time preference factor, expected growth, the relative risk aversion coefficient, and the real interest factor. Thus, a consumer can be ‘impatient’ in the required sense even if β =  1  , so long as expected income growth is positive.31

We begin by searching for the time preference factor `
β  such that if all households had an identical β  = β`  the steady-state value of the capital-to-output ratio (KKK ∕YYY  ) would match the value that characterized the steady-state of the perfect foresight model.32  `
β  turns out to be 0.9888  (recall that this is at a quarterly, not an annual, rate).

We now ask whether this model with realistically calibrated income and finite lifetimes (henceforth, the model is referred to as the ‘β  -Point’ model) can reproduce the degree of wealth inequality evident in the micro data. An improvement in the model’s ability to match the data is to be expected, since in buffer stock models agents strive to achieve a target ratio of wealth to permanent income. By assuming no dispersion in the level of permanent income across households, KS’s income process disables a potentially vital explanation for variation in the level of target wealth (and, therefore, on average, actual wealth) across households.

Table 4: Proportion of Net Worth by Percentile (in percent)

                                 Micro   Income   Process
               Friedman/Bu    ffer  Stock     KS  -JEDC         KS  -Orig ◇
               --------------------------------------------  ------------------
                 Point       Uniformly      Our  Solution            Hetero

               Discount     Distributed
                Factor ‡      Discount
                              Factors                                           U.S.
               (β -Point )    (β -Dist )                                        Data  *
 Top  1%          10.3          24.9               3.0        3.0     24.0      29.6

 Top  10%         38.6          65.6             23.0        19.0     73.0      66.1
 Top   20%        54.9          81.0             40.1        35.0     88.0      79.5

 Top   40%        75.7          93.1             66.0                           92.9
 Top   60%        88.9          97.4             84.0                           98.7


Notes: KKKt ∕YYYt = 10.3.  ‡
  :  `
β = 0.9888  . ⋆
 :   `
(β,∇ ) = (0.9869,0.0052)  , which implies {`β - ∇, `β + ∇} = {0.9816,0.9921} . ◇
  :  The results are from Krusell and Smith (1998) who solved the models with aggregate shocks. *
  :  U.S. data is the SCF reported in Castaneda, Diaz-Gimenez, and Rios-Rull (2003). Bold quantiles are targeted.

Table 4 shows that compared to the distribution of net worth implied by our solution of the KS-JEDC model solved without an aggregate shock (or the results of the KS-Orig model from Krusell and Smith (1998)),33 our β  -Point model does indeed yield a substantial improvement (compare the first, third and fourth columns to the last column).34 For example, in our β  -Point model, the fraction of total net worth held by the top 1 percent is about 10 percent, while the corresponding statistic is only 3 percent in our solution of the KS-JEDC model (or the KS-Orig model).

The KS-JEDC model’s failure to match the wealth distribution is not confined to the top. In fact, perhaps a bigger problem is that the model generates a distribution of wealth in which most households’ wealth levels are not very far from the wealth target of a representative agent in the perfect foresight version of the model. For example, in steady state about 50  percent of all households in the KS-JEDC model have net worth between 0.5 times mean net worth and 1.5 times mean net worth; in the SCF data from 1992–2004, the corresponding fraction ranges from only 20 to 25 percent.

But while the β  -Point model fits the data better than the original KS model, it still falls short of matching the empirical degree of wealth inequality. The proportion of net worth held by households in the top 1 percent of the distribution is three times smaller in the model than in the data (compare the first and last columns in the table). This failure reflects the fact that, empirically, the distribution of wealth is considerably more unequal than the distribution of permanent income.

2.5 Heterogeneous Impatience

As the simplest method to address this defect, we introduce heterogeneity in impatience: Each household is now assumed to have an idiosyncratic (but fixed) time preference factor.35 We do not think of this assumption as only capturing actual variation in pure rates of time preference across people (though such variation surely exists). Instead, we view discount-factor heterogeneity as a shortcut that captures the essential consequences of many other kinds of heterogeneity (e.g., heterogeneity in age, income growth expectations, investment opportunities, tax schedules) as well. To be more concrete, take the example of age. A robust pattern in most countries is that income grows much faster for young people than for older people. According to (16), young people should therefore tend to act, financially, in a more ‘impatient’ fashion than older people. In particular, we should expect them to have lower target wealth-to-income ratios. Thus, what we are capturing by allowing heterogeneity in time preference factors is probably also some portion of the difference in behavior that (in truth) reflects differences in age instead of in time preference factors, and that would be introduced into the model if we had a more complex specification of the life cycle that allowed for different growth rates for households of different ages.36

One way of gauging a model’s predictions for wealth inequality is to ask how well it is able to match the proportion of total net worth held by the wealthiest 20  , 40  , 60  , and 80  percent of the population. Because these statistics have been targeted by other papers (e.g., Castaneda, Diaz-Gimenez, and Rios-Rull (2003)), we adopt a goal of matching them.37

We replace the assumption that all households have the same time preference factor with an assumption that, for some ∇ , time preference factors are distributed uniformly in the population between β` - ∇ and β` + ∇ (for this reason, the model is referred to as the ‘β  -Dist’ model below). Then, using simulations, we search for the values of `β  and ∇ for which the model best matches the fraction of net worth held by the top 20  , 40  , 60  , and 80  percent of the population, while at the same time matching the aggregate capital-to-output ratio from the perfect foresight model. Specifically, defining w
  i  and ω
  i  as the proportion of total aggregate net worth held by the top i  percent in our model and in the data, respectively, we solve the following minimization problem:

min             (w  -  ω )2
β`,∇                i     i

subject to the constraint that the aggregate wealth (net worth)-to-output ratio in the model matches the aggregate capital-to-output ratio from the perfect foresight model (KKKP  F ∕YYY P F  ):38

KKK ∕YYY    =  KKKP  F∕YYY  PF                           (18)
The solution to this problem is   `
(β, ∇ ) =  (0.9869,  0.0052 )  .

The introduction of even such a relatively modest amount of time preference heterogeneity sharply improves the model’s fit to the targeted proportions of wealth holdings (second column of the table). The ability of the model to match the targeted moments does not, of course, constitute a formal test, except in the loose sense that a model with such strong structure might have been unable to get nearly so close to four target wealth points with only one free parameter.39 But the model also sharply improves the fit to locations in the wealth distribution that were not explicitly targeted; for example, the net worth shares of the top 10 percent and the top 1 percent are also included in the table, and the model performs reasonably well in matching them.

Of course, Krusell and Smith (1998) were well aware that their baseline model match the wealth distribution well. They, too, examined whether inclusion of a form of discount rate heterogeneity could improve the model’s match to the data. Specifically, they assumed that the discount factor takes one of the three values (0.9858  , 0.9894  , and 0.9930  ), and that agents anticipate that their discount factor might change between these values according to a Markov process. As they showed, the model with this simple form of heterogeneity (henceforth ‘KS-Orig Hetero’ model) did improve the model’s ability to match the wealth holdings of the top percentiles (see KS-Orig Hetero column in the table). Indeed, as inspection of the long-dashing locus in Figure 1 shows, their model of heterogeneity went a bit too far: It concentrated almost all of the net worth in the top 20 percent of the population (though rather evenly among that top 20 percent). By comparison, the figure shows that our model does a notably better job matching the data across the entire span of wealth percentiles.

The reader might wonder why we do not simply adopt the KS specification of heterogeneity in time preference factors, rather than introducing our own novel form of heterogeneity. The principal answer is that our purpose here is to define a method of explicitly matching the model to the data via statistical estimation of a parameter of the distribution of heterogeneity, letting the data speak flexibly to the question of the extent of the heterogeneity required to match model to data. A second point is that, having introduced finite horizons in order to yield an ergodic distribution of permanent income, it would be peculiar to layer on top of the stochastic death probability a stochastic probability of changing one’s time preference factor within the lifetime; Krusell and Smith motivated their differing time preference factors as reflecting different preferences of alternative generations of a dynasty, but with our finite horizons assumption we have eliminated the dynastic interpretation of the model. Having said all of this, the common point across the two papers is that a key requirement to make the model fit the data is a form of heterogeneity that leads different households to have different target levels of wealth.


Figure 1: Cumulative Distribution of Net Worth

3 KS Aggregate Shocks

In this section, we examine a model with an FBS household income process that also incorporates KS aggregate shocks, and investigate the model’s performance in replicating aggregate statistics. Krusell and Smith (1998) assumed that the level of aggregate productivity alternates between Z   = 1 +  △Z
  t  if the aggregate state is good and Z  =  1 -  △Z
  t  if it is bad; similarly, LLLt  =  1 - ut  where ut =  ug  if the state is good and ut =  ub  if bad. (For reference, we reproduce their assumed parameter values in Table 5.)

Table 5: Parameter Values for Aggregate Shocks



 KS  shocks
 Shock   to productivity               △Z         0.01       Krusell  and  Smith  (1998  )
 Unemployment      (good   state)       u         0.04       Krusell  and  Smith  (1998  )
 Unemployment      (bad  state )        ub        0.10       Krusell  and  Smith  (1998  )


 FBS   shocks
 Variance   of log Ψt                   σ2Ψ        0.00004    Carroll, Slacalek,  and  Tokuoka    (2008 )

The decision problem for an individual household in period t  can be written using normalized variables and the employment status ι
 t  :

                                                [            1- ρ                         ]
v(mt,  ιt;KKKt, Zt )  =    maxc    u(ct) +  β/D/ Et  (Γ t+1 ψt+1 )   v (mt+1,  ιt;KKKt+1,  Zt+1 )
                a   =    m   -  c
                 t         t     t
                at  ≥    0

             kt+1   =    at∕ (//D Γ t+1 ψt+1 )

            mt+1    =    (ℸ  + rt+1 )kt+1 +  yt+1
             r      =    αZ     (KKK     ∕ℓLLL    )α- 1,                                   (19)
               t+1            t+1    t+1     t+1

There are more state variables in this version of the model than in the model with no aggregate shock: The aggregate variables Zt  and KKKt  , and the household’s employment status ιt  whose transition process depends on the aggregate state. Solving the full version of the model above with both aggregate and idiosyncratic shocks is not straightforward; the basic idea for the solution method is the key insight of Krusell and Smith (1998). See Appendix C for details about our solution method.

We now report the results of simulations, both for the model in which all households have the same time preference factor (β  -Point model) and for the version with a uniform distribution of time preference factors (β  -Dist model). While the β  -Point model uses `β  estimated in Section 2, the β  -Dist model uses parameter values reestimated by solving the minimization problem (17) with the KS aggregate shocks ((β`, ∇ ) =  (0.9851,  0.0074 )  ). Results using our solution of the KS-JEDC model (with the KS aggregate shocks, θ  =  1
  t  , ψ  =  1
 t  for all t  , and no death (D =  0  )) are also reported for comparison.

3.1 Some Macroeconomic Statistics

Table 6 shows some aggregate statistics that we think are useful for macroeconomic analysis: The serial correlation of consumption growth, and correlation between consumption growth, income growth, and interest rates at several frequencies. The results are generally similar across the β  -Point, β  -Dist, and KS-JEDC models. They all produce positive ϱ(Δ  log CCCt, Δ log CCCt - 1)  , and high correlation of consumption growth with (current) income growth or (current) interest rates. The serial correlation of consumption growth in our solution of the KS-JEDC model is similar to that reported by Maliar, Maliar, and Valli (2008) who also solved the KS-JEDC model (fourth column of the table).40 ,  41 We also report results for the representative agent model with the KS aggregate income shock parameters (last column), the results of which are very close to those of our solution of the KS-JEDC model.

Table 6: Aggregate Statistics with KS Aggregate Shocks


                                 Friedman/Bu     ffer Stock              KS -JEDC                  None
                                ---------------------------- ---------------------------------
                                 β -Point       β -Dist       Our  Solution    Maliar  et al.  Rep   Agent

 corr (Δ  log CCCt, Δ  log CCCt- 1)     0.14         0.05             0.23             0.28            0.24

 corr (Δ  log CCCt, Δ  log YYY t)        0.91         0.97             0.86                             0.84
 corr (Δ  log CCCt, Δ  log YYY t- 1)     0.12         0.06             0.15                             0.15

 corr (Δ  log CCCt, Δ  log YYY t- 2)     0.09         0.04             0.13                             0.11
 corr (Δ  log CCCt, rt)               0.78         0.66             0.85                             0.86

 corr (Δ  log CCCt, rt- 1)            0.13       - 0.05             0.26                             0.28
 corr (Δ4  log CCCt, Δ4  log YYY t)     0.80         0.92             0.70                             0.67


Notes: Δ4  and Δ8  are one-year and two-year growth rates, respectively.

The classic reference point for consumption growth measurement is the random walk model of Hall (1978), and the large literature that rejects the random walk proposition in favor of models that either contain some ‘rule-of-thumb’ consumers who set spending equal to income in every period (Campbell and Mankiw (1989)) or, more popular recently, models with habit formation or ‘sticky expectations’ (Carroll, Slacalek, and Tokuoka (2008)) that imply serial correlation in consumption growth (see Carroll, Sommer, and Slacalek (2011) for evidence).

The KS-JEDC model produces a relatively high correlation coefficient ϱ (Δ log CCC  ,Δ  log CCC    )
           t          t- 1  , which is closer to the U.S. data (where the statistic is about one-third) than that produced by standard consumption models stemming from Hall (1978).42 As noted already, our β  -Point and β  -Dist models also imply positive ϱ (Δ log CCCt, Δ  log CCCt - 1)  , although not as high as that predicted by the KS-JEDC model. At first blush, it seems puzzling that the KS-JEDC model, which includes neither habits nor sticky expectations, generates a substantial violation of the random walk proposition. This puzzle does not seem to have been noticed in the previous literature on the KS-JEDC model, but after some investigation we determined that the KS-JEDC model’s sticky consumption growth is produced by the high degree of serial correlation in interest rates in the model, which results from the assumption about the process of aggregate productivity shocks (see Appendix D for details). The interesting questions, in a model with time-varying interest rates, are, first, whether one can reliably estimate an intertemporal elasticity of substitution (IES) from the coefficient in a regression of consumption on the predictable component of interest rates (as Hall (1988) attempts to do), and, second, whether consumption growth is serially correlated after accounting for the predictable component related to interest rates (no random walk).43

3.2 The Aggregate Marginal Propensity to Consume

A macroeconomic question of perhaps even greater interest is whether a model that manages to match the distribution of wealth has similar, or different, implications from the KS-JEDC or representative agent models for the reaction of aggregate consumption to an economic ‘stimulus’ payment.

Specifically, we pose the question as follows. The economy has been in its steady-state equilibrium leading up to date t  . Before the consumption decision is made in that period, the government announces the following plan: Effective immediately, every household in the economy will receive a ‘stimulus check’ worth some modest amount $x  (financed by a tax on unborn future generations).44

In theory, the distribution of wealth across recipients of the stimulus checks has important implications for aggregate MPC out of transitory shocks to income. To see why, the solid line of Figure 2 plots our β  -Point model’s individual consumption function in the good (aggregate) state, with the horizontal axis being cash on hand normalized by the level of (quarterly) permanent income. Because the households with less normalized cash have higher MPC,45 the average MPC is higher when a larger fraction of households has less (normalized) cash on hand.


Figure 2: Consumption Functions of β  -Point and β  -Dist Models

There are many more households with little wealth in our β  -Point model than in the KS-JEDC model, as illustrated by comparison of the short-dashing and the long-dashing lines in Figure 1. The greater concentration of wealth at the bottom in the β  -Point model, which is the case in the data (see the histogram in Figure 2), should produce a higher average MPC, given the concave consumption function.

Indeed, the average MPC out of the transitory income (‘stimulus check’) in our β  -Point model is 0.09  in annual terms (first column of Table 7),46 about double the value in the KS-JEDC model (0.05 )  (the fourth column of the table) or the perfect foresight partial equilibrium model (0.04). Our β  -Dist model (second column of the table) produces an even higher average MPC (0.19 )  , since in the β  -Dist model there are more households who possess less wealth, are more impatient, and have higher MPCs (Figure 1 and dashed lines in Figure 2).47 However, this is still at best only at the lower bound of typical empirical MPC estimates which are typically between 0.2  0.5  or even higher (see Table 13 in the Appendix E).

Table 7: Average Marginal Propensity to Consume in Annual Terms

                                                                      KS                                    Friedman/Bu  ffer Stock
                                                              Aggregate  Process                              Aggregate  Process
 Model                                β-Point  β -Dist       β-Dist         KS -JEDC      KS -Orig Hetero   β-Dist       β -Dist
                                                                          Our  Solution    Our Solution

 Wealth  Measure                        Net      Net    Liquid Financial                                    Net    Liquid Financial
 Overall average                        0.09    0.19          0.68              0.05            0.09         0.18         0.69
 By  wealth/permanent  income  ratio
  Top 1%                                0.06    0.05          0.23              0.04            0.04         0.06         0.24
  Top 10%                               0.06    0.06          0.24              0.04            0.04         0.06         0.24
  Top 20%                               0.06    0.06          0.28              0.04            0.04         0.06         0.27

  Top 40%                               0.06    0.07          0.39              0.04            0.05         0.06         0.41
  Top 60%                               0.07    0.09          0.50              0.04            0.06         0.08         0.52
  Bottom  1/2                           0.12    0.28          0.83              0.05            0.13         0.28         0.84
 By  income
  Top 1%                                0.08    0.13          0.36              0.05            0.04         0.15         0.64

  Top 10%                               0.08    0.14          0.48              0.05            0.04         0.15         0.64
  Top 20%                               0.09    0.14          0.52              0.05            0.04         0.16         0.64
  Top 40%                               0.10    0.16          0.57              0.05            0.05         0.17         0.65
  Top 60%                               0.10    0.16          0.61              0.05            0.06         0.18         0.66
  Bottom  1/2                           0.08    0.21          0.76              0.05            0.13         0.18         0.73

 By  employment  status
  Employed                              0.08    0.16          0.65              0.05            0.09         0.16         0.66
  Unemployed                            0.20    0.44          0.95              0.06            0.18         0.35         0.96
 Time  preference parameters
 `β                                      0.9888  0.9851        0.9037                                        0.9869       0.9111
 ∇                                              0.0074        0.0424                                        0.0052       0.0336

Notes: Annual MPC is calculated by 1 - (1- quarterly MPC 4
)  . ‡ : Discount factors are uniformly distributed over the interval  `     `
(β - ∇, β +∇ )  .

Thus far, we have been using total household net worth as our measure of wealth. Implicitly, this assumes that all of the household’s debt and asset positions are perfectly liquid and that, say, a household with home equity of $50,000 and bank balances of $2,000 (and no other balance sheet items) will behave in every respect similarly to a household with home equity of $10,000 and bank balances of $42,000. This seems implausible. The home equity is more illiquid (tapping it requires, at the very least, obtaining a home equity line of credit, which requires an appraisal of the house and some paperwork).

Otsuka (2003) formally analyzes the optimization problem of a consumer with a FBS income process who can invest in an illiquid but higher-return asset (think housing), or a liquid but lower-return asset (cash), and shows, unsurprisingly, that the marginal propensity to consume out of shocks to liquid assets is higher than the MPC out of shocks to illiquid assets. Her results would presumably be even stronger if she had allowed that households hold so much of their wealth in illiquid forms (housing, pension savings), for example, as a mechanism to overcome self-control problems (see Laibson (1997) and many others).48

These considerations suggest that it may be more plausible, for purposes of extracting a predictions about the MPC out of stimulus checks, to focus on matching the distribution of liquid financial assets across households (that is, assets which are of the same kind as represented by the stimulus check, once it has been deposited into a bank account).

When we ask the model to estimate the time preference factors that allow it to best match the distribution of liquid financial assets (instead of net worth),49 estimated parameter values are (`β, ∇ ) =  (0.9037, 0.0424  )  and the average MPC is 0.68  (third column of the table), which lies in the upper part of the range typically reported in the literature (see Table 13), and is considerably higher than when we match the distribution of net worth. This reflects the fact that matching the more skewed distribution of liquid financial assets than that of net worth (Table 8) requires a wider distribution of the time preference factors, which produces even more households with little wealth. The estimated distribution of discount factors lies below that obtained by matching net worth and is considerably more dispersed because of substantially lower median and more unevenly distributed liquid financial wealth (compared to net worth).

Table 8: Proportion of Wealth Held by Percentile (in percent)


            -------Liquid---Financial--Assets-------- ------Net--Worth--------

 Top  1%       42.2    52.7    47.6    49.6     50.6    29.6    34.4     33.9
 Top  10%      79.4    84.8    83.2    85.2     86.1    66.1    68.9     69.7

 Top  20%      90.2    92.8    92.5    93.4     93.8    79.5    82.1     82.9
 Top  40%      97.4    98.1    98.1    98.3     98.6    92.9    94.3     94.7

 Top  60%      99.4    99.6    99.6    99.6     99.7    98.7    99.1     99.0
 Top  80%    100.0    100.0   100.0   100.0   100.0    100.4   100.4   100.2

Notes: Survey of Consumer Finances, ‡ : From Castaneda, Diaz-Gimenez, and Rios-Rull (2003).

Figure 3 shows the cumulative distribution functions of MPCs for the β  -Dist models estimated to match the empirical distribution of net worth and liquid financial assets. The Figure illustrates the high values of implied MPCs obtained for both models, especially the latter.


Figure 3: Distribution of MPCs across Households

MPCs are generally higher among low wealth/income households and the unemployed in both our β  -Point and β  -Dist models (rest of the rows in Table 7). These results provide the basis for a common piece of conventional wisdom about the effects of economic stimulus mentioned in our introduction: If the purpose of the stimulus payments is to stimulate consumption, it makes much more sense to target those payments to low-wealth households than to distribute them uniformly to the population as a whole.

4 A More Plausible and More Tractable ‘Friedman/Buffer Stock’ Aggregate Process

The KS process for aggregate productivity shocks has little empirical foundation; indeed, it appears to have been intended by the authors as an illustration of how one might incorporate business cycles in principle, rather than a serious candidate for an empirical description of actual aggregate dynamics. In this section, we introduce an aggregate income process that is considerably more tractable than the KS aggregate process and is also a much closer match to the aggregate data. We regard the version of our model with this new income process as the ‘preferred’ version for use as a starting point for future research.

The aggregate production function is the same as equation (1), but following Carroll, Slacalek, and Tokuoka (2008), the aggregate state (good or bad) no longer exists in this model (Z  =  1
  t  ). Aggregate productivity is instead captured by LLLt  . Specifically, LLLt =  Pt Ξt  ; Pt  is aggregate permanent productivity, where P     =  P Ψ
  t+1      t t+1   ; Ψ
  t+1   is the aggregate permanent shock; and Ξ
  t  is the aggregate transitory shock (note that Ψ  is the capitalized version of the Greek letter ψ  used for the idiosyncratic permanent shock; similarly (though less obviously), Ξ  is the capitalized ξ  ). Both Ψt  and Ξt  are assumed to be log normally distributed with mean one, and their log variances are from Carroll, Slacalek, and Tokuoka (2008), who have estimated them using U.S. data (Table 5).

Table 9: Aggregate Statistics in β  -Dist Model under ‘Plausible’ Aggregate Process


                               Friedman/Bu    ffer  Stock        KS
                                      Aggregate             Aggregate

                            -----------Process-------------   Process
                               Net     Liquid  Financial        Net

 ϱ (Δ log CCCt, Δ  log CCCt - 1)    0.10          0.24              0.05

 ϱ (Δ log CCCt, Δ  log YYY t)       0.82          0.83              0.97
 ϱ (Δ log CCCt, Δ  log YYY t- 1)    0.08          0.18              0.06

 ϱ (Δ log CCCt, Δ  log YYY t- 2)    0.05          0.13              0.04
 ϱ (Δ log CCCt, rt)               0.46          0.66              0.66

 ϱ (Δ log CCCt, rt- 1)            0.20          0.33            - 0.05
 ϱ (Δ4 log CCCt, Δ4  log YYY t)     0.92          0.92              0.92


The assumption that the structure of aggregate shocks resembles the structure of idiosyncratic shocks is valuable not only because it matches the data better, but also because it makes the model easier to solve. In particular, the elimination of the ‘good’ and ‘bad’ aggregate states reduces the number of state variables to two (mt  and KKKt  ) after normalizing the model by ppptPt  (as elaborated in Carroll, Slacalek, and Tokuoka (2008)). As in Section 2, employment status is not a state variable (in eliminating the aggregate states, we also shut down unemployment persistence, which depends on the aggregate state in the KS-JEDC or KS-Orig model). As before, the main thing the household needs to know is the law of motion of KKK
   t  , which can be obtained by following essentially the same method as described in the Appendix C.

When matching the distribution of net worth, aggregate statistics produced by the β  -Dist model with our preferred (Friedman/Buffer Stock) aggregate process are relatively similar to those under the KS aggregate process, despite the difference in the aggregate process (first and third columns of Table 9). Given that there is no aggregate state in the economy, we are using `β  and ∇ estimated in Section 2 and assuming that the unemployment rate u
  t  is fixed at 0.07 (same as in Section 2). Our preferred version of the β  -Dist model maintains positive ϱ(Δ  log CCC  ,Δ  log CCC    )
           t          t- 1  , and high correlation of consumption growth with income growth or interest rates. We have obtained similar results by matching the distribution of liquid financial assets (second column of the table).

More importantly, the preferred version of the β  -Dist model can produce high MPCs. For example, in the net worth case, the average MPC is 0.18  , which is very close to the estimate under the KS aggregate process (compare second and last but one columns of Table 7). In the liquid financial assets case, the average MPC is higher at 0.69  (last column).50

5 Conclusion

This paper found that the performance of a KS-type model in replicating wealth distribution can be improved significantly by introducing i) a microfounded income process, ii) finite lifetimes, and iii) heterogeneity in time preference factors. Moreover, such modifications improve macroeconomic characteristics of the model by substantially boosting the MPC out of transitory income.


A Derivation of Variance of Permanent Income

The evolution of the square of p  is given by

p       =   p  ψ     (1 -  d     ) + d
 t+1,i       t,i  t+1,i       t+1,i     t+1,i
p2      =   (pt,iψt+1,i(1 -  dt+1,i))2 + 2pt,iψt+1,i dt+1,i(1 - dt+1,i)+d2     ,
 t+1,i                                             ◟ ------◝ ◜-------◞    t+1,i
where dt+1,i = 1  if household i  dies.

Because                 2
Et [(1 -  dt+1,i) ] = 1 -  D  and      2
Et [d t+1,i] =  D  , we have

Et[p2t+1,i] =   Et [(pt,iψt+1,i(1 -  dt+1,i))2] + D
                 2       2
           =   p t,i//D E [ψ ] +  D,
   [  2  ]           2       2
M   p t+1    =   M  [p t]/D/ E [ψ  ] + D.
Finally, the steady state expected level of M  [p2] ≡ lim      M  [p2]
              t→ ∞      t  can be found from
M  [p2]  =   D +  /D E [ψ2 ]M [p2]
     2       ------D-------
M  [p ]  =        /     2 .
             1 - /D E [ψ ]

B Estimating Moffitt and Gottschalk Income Process

This appendix estimates the annual income process la Moffitt and Gottschalk (1995) using quarterly income draws generated by our income process (Section 2) with parameter values from Table 1. Moffitt and Gottschalk (1995) assume log permanent income log (pt)  follows a random walk and log transitory income log(ξt)  ) an ARMA process:

      yyyt  =   ptξt,
log (p )  =   log (p    ) + log (ψ ),
      t             t- 1          t
log (ξt)  =   a1 log (ξt- 1 ) + vt + m1vt - 1
Like Moffitt and Gottschalk (1995), we match the covariance matrix of the annual income draws, and obtain estimate with the same signs as theirs obtained using the PSID data; see Table 10, confirming that our calibration is qualitatively consistent with Moffitt and Gottschalk’s.

Interestingly, even though our true quarterly transitory shock process is just white noise, if we estimate the process on an annual basis we obtain positive AR (a1   ) and negative MA (m1   ) coefficients, reflecting time aggregation. This suggests that the positive a1   and negative m1   reported in Moffitt and Gottschalk (1995) may be (at least) partly due to time aggregation.

Table 10: Estimates of Moffitt and Gottschalk Annual Income Process

                                        σ2       σ2       a        m
 Our  estimates                       0.015    0.025    0.504   - 0.521


C Solution Method to Obtain Law of Motion

C.1 Solution Methods

Broadly speaking, the literature takes one of the following two approaches in solving the KS problem in Section 3:

  1. Relying on simulation to obtain the law of motion of per capita capital
  2. (In principle) not relying on simulation to obtain the law of motion of per capita capital

Table 11 lists some existing articles that solve the KS-JEDC model according to this categorization. All articles in the table except Kim, Kim, and Kollmann (2010) solve the exact KS-JEDC model using various methods.51

The advantage of the first approach is that simulation performed to obtain the law of motion generates micro data, which can be used directly to investigate issues such as wealth distribution. The disadvantage is that this approach is generally subject to cross-sectional sampling variation, because this approach typically performs simulation using a finite number of households. Young (2010) and Den Haan (2010b)’s approaches can also be categorized in the first approach but avoid cross-sectional sampling variation by running nonstochastic simulation that approximates the density of wealth with a histogram.

The advantages of the second approach are: i) there is no cross-sectional sampling variation; ii) it is generally faster than the first approach. Using the second approach, Algan, Allais, and Den Haan (2008) and Reiter (2010) find a wealth distribution function of various moments,52 while Reiter (2010) calculates a matrix for the transition probabilities of individual wealth (see Appendix ?? for details about his technique). Kim, Kim, and Kollmann (2010) use a perturbation method that linearizes the system. Although they are not able to solve the exact same KS-JEDC model and thus modify the form of the utility function, they can solve a related problem very quickly.

We use the first approach because it directly generates various micro data (e.g., individual wealth and MPC), which can be used to examine wealth distribution and the aggregate MPC. Details about our algorithm are in the next subsection.

Table 11: Methods of Solving KS-JEDC Model



 Relying   on simulation
    Young   (2010 )                            Grid -based  method


 (In principle ) not  relying on  simulation
    Algan,  Allais, and  Den   Haan   (2008 )  Parametrization    method

    Reiter  (2010 )                            Parametrization    method
    Kim,  Kim,   and  Kollmann    (2010 )      Perturbation    method


C.2 Our Algorithm

In solving the problem in section 3 we closely follow the stochastic simulation method of Krusell and Smith (1998). Krusell and Smith find that per capita capital today (K
KKt  ) is sufficient to predict per capita capital tomorrow (K
KKt+1   ). Our procedure is as follows:

  1. Solve for the optimal individual decision rules given some ‘beliefs’ π  that determine the (expected) law of motion of per capita capital. The law of motion is takes the log-linear form given by                ′   ′
π =  (π0, π1,π 0,π 1)  :
    log KKKt+1  =  π0 +  π1 log  KKKt

    if the aggregate state in period t  is good (Z  =  1 +  △Z
 t  ), and

                  ′     ′
log KKKt+1  =  π0 +  π1 log  KKKt

    if the aggregate state is bad (Zt =  1 -  △Z  ).

  2. Simulate the economy populated by 7,000  households (which experiments determined is enough to suppress idiosyncratic noise) for 1, 100  periods (following Maliar, Maliar, and Valli (2010)). When starting a simulation, p   =  1
  t,i  for all i  , the distribution of m
  t,i  is generated assuming kt,i  is equal to its steady state level (38.0  ) for all i  , and Z  =  1 +  △Z
  t  (the aggregate state is good). (The steady state level of kt,i  is k =  (αβ ∕(1 -  β ℸ ))1∕(1- α)   . With kt,i = 38.0  for all i  , kkk   =  KKK   =  41.2
 t,i     t  .) The newborn households start life with p    = 1
  t,i  and kt,i =  0  .
  3. Estimate ˜π  , which determines the law of motion of per capita capital, using the last 1, 000  periods of data generated by the simulation (we discard the first 100  periods).
  4. Compute an improved vector for the next iteration by ˆπ  = (1 -  η )˜π +  ηπ.  η =  1∕2  is used for the β  -Dist model. (Our experiments found that we can reach the solution faster with η =  1∕2  .)

We repeat this process until ˆπ =  π  with a given degree of precision.53

From the second iteration and thereafter, we use the terminal distribution of wealth (and permanent component of income (p  )) in the previous iteration as the initial one. For the case of the β  -Dist model, the number of households is multiplied by 10 in the final two (or three) iterations to reduce cross-sectional simulation error.54

While we can eventually obtain some solution whatever the initial π  is, we use π  obtained using the representative agent model as the starting point. This can significantly reduce the time needed to obtain the solution.

Parameter values to solve the model are from Table 1 (except for the unemployment rate ut  ) and Table 5. The time preference factors are imposed to be those estimated in Section 2.

C.3 Tricks to Reduce Simulation Errors

In obtaining the aggregate law, we introduce the following tricks to reduce simulation errors (or to speed up the solution given a degree of estimate precision):

C.4 Estimated Laws of Motion

The estimated laws of motions for β  -Point, β  -Dist and KS-JEDC models are given in Table 12. The fit measured with R2   in all specifications exceeds 0.9999.56

Table 12: Estimated Laws of Motion

            log KKKt+1  =  π0 +  π1 log KKKt  + ϵt+1
 Model        β-Point          β -Dist        KS  -JEDC
 State     Good     Bad    Good      Bad    Good      Bad
 π         0.140   0.127    0.154   0.144    0.138   0.122
 π1        0.963   0.965    0.959   0.960    0.963   0.966

Notes: The coefficients for the KS-JEDC model are very close to those estimated in Maliar, Maliar, and Valli (2010).

D Experiment to Understand Sticky Consumption Growth in KS-JEDC Model

Although ϱ(Δ  log CCC  ,Δ  log CCC    )
           t          t- 1  reported in subsection 3.1 may not be high enough relative to that observed in the U.S. data, it is still not clear why simulations produce such a high value.

Previous studies on KS type models have not investigated this issue. Using the KS-JEDC model, we performed an experiment to understand the phenomenon better. In this experiment we assume that the aggregate state switches from good to bad (or from bad to good) every eight quarters.57

Figure 4 plots Δ  log CCC
         t  24 quarters of simulated observations (the state is bad for the first eight quarters, good for the next eight quarters, and bad for the final eight quarters). The figure shows that Δ  log CCC
         t  is very persistent (it is negative in the bad state and positive in the good state), resulting in a relatively high ϱ(Δ  log CCC  ,Δ  log CCC    )
           t          t- 1  .

It is easy to understand that Δ  log CCCt  is higher when the state is good (and vice versa) given the following facts:

While in typical simulation one state does not generally last for exactly eight quarters, we observe sticky aggregate consumption growth (and a relatively high ϱ (Δ log CCCt, Δ  log CCCt - 1)  ) because the same mechanisms are at work as in the experiment above.

In sum, a relatively high ϱ (Δ log CCCt, Δ  log CCCt - 1)  in the KS-JEDC model can be interpreted as a consequence of the persistent behavior of the interest rate rt  . Indeed, denoting εt = Δ  log CCCt -  b0 -  b1 Et - 1[rt]  the residual after controlling for the predictable component of consumption growth related to interest rates, we find that ϱ(εt, εt- 1) =  0.02 is much lower than ϱ (Δ log CCCt, Δ  log CCCt - 1)  .59


Figure 4: Dynamics of Δ logCCCt  in KS-JEDC Model


Figure 5: Dynamics of rt  in KS-JEDC Model

E Empirical Estimates of MPCs

Table 13 summarizes the estimates of MPCs obtained using household-level data on various recent fiscal stimulus measures in the U.S.

Table 13: Estimates of MPCs

                                                                    Consumption     Measure
 Authors                                                    Nondurables      Durables    Total  PCE
 Agarwal,   Liu,  and  Souleles (2007  )                                                      0.4

 Coronado,    Lupton,   and  Sheiner  (2005 )                                             0.28 –0.36
 Johnson,   Parker,  and  Souleles  (2006 )                   0.12 –0.30                  0.50 –0.90

 Johnson,   Parker,  and  Souleles  (2009 )                      0.25
 Lusardi   (1996 )‡                                            0.2 –0.5

 Parker  (1999  )                                                 0.2
 Parker,  Souleles,  Johnson,   and  McClelland    (2011 )    0.12 –0.30

 Sahm,   Shapiro,  and  Slemrod    (2009 )                                                   0.33
 Shapiro   and  Slemrod   (2009 )                                                            0.33

 Souleles  (1999 )                                               0.09           0.54         0.64
 Souleles  (2002 )                                             0.6 –0.9

Notes: ‡ : elasticity.


   AGARWAL, SUMIT, CHUNLIN LIU, AND NICHOLAS S. SOULELES (2007): “The Response of Consumer Spending and Debt to Tax Rebates – Evidence from Consumer Credit Data,” Journal of Political Economy, 115(6), 986–1019.

   ALGAN, YANN, OLIVER ALLAIS, AND WOUTER J. DEN HAAN (2008): “Solving Heterogeneous-agent Models with Parameterized Cross-sectional Distributions,” Journal of Economic Dynamics and Control, 32(3), 875–908.

   BEWLEY, TRUMAN (1977): “The Permanent Income Hypothesis: A Theoretical Formulation,” Journal of Economic Theory, 16, 252–292.

   BLANCHARD, OLIVIER J. (1985): “Debt, Deficits, and Finite Horizons,” Journal of Political Economy, 93(2), 223–247.

   BLUNDELL, RICHARD, LUIGI PISTAFERRI, AND IAN PRESTON (2008): “Consumption Inequality and Partial Insurance,” Manuscript.

   BRAUN, RICHARD ANTON, HUIYU LI, AND JOHN STACHURSKI (2009): “Computing Densities and Expectations in Stochastic Recursive Economies: Generalized Look-Ahead Techniques,” Manuscript.

   CAGETTI, MARCO (2003): “Wealth Accumulation Over the Life Cycle and Precautionary Savings,” Journal of Business and Economic Statistics, 21(3), 339–353.

   CAMPBELL, JOHN Y., AND N. GREGORY MANKIW (1989): “Consumption, Income, and Interest Rates: Reinterpreting the Time-Series Evidence,” in NBER Macroeconomics Annual, 1989, ed. by Olivier J. Blanchard, and Stanley Fischer, pp. 185–216. MIT Press, Cambridge, MA,

   __________  (1991): “The Response of Consumption to Income: A Cross-Country Investigation,” European Economic Review, 35, 723–67.

   CARROLL, CHRISTOPHER D. (1992): “The Buffer-Stock Theory of Saving: Some Macroeconomic Evidence,” Brookings Papers on Economic Activity, 1992(2), 61–156,

   __________  (2000a): “Requiem for the Representative Consumer? Aggregate Implications of Microeconomic Consumption Behavior,” American Economic Review, 90(2), 110–115.

   __________  (2000b): “Why Do the Rich Save So Much?,” in Does Atlas Shrug? The Economic Consequences of Taxing the Rich, ed. by Joel B. Slemrod. Harvard University Press,

   __________  (2011): “Theoretical Foundations of Buffer Stock Saving,” Manuscript, Department of Economics, Johns Hopkins University,

   CARROLL, CHRISTOPHER D., AND ANDREW A. SAMWICK (1997): “The Nature of Precautionary Wealth,” Journal of Monetary Economics, 40(1), 41–71.

   CARROLL, CHRISTOPHER D., JIRI SLACALEK, AND KIICHI TOKUOKA (2008): “Sticky Expectations and Consumption Dynamics,” Manuscript.

   CARROLL, CHRISTOPHER D., MARTIN SOMMER, AND JIRI SLACALEK (2011): “International Evidence on Sticky Consumption Growth,” Review of Economics and Statistics, 93(4), 1135–1145,

   CASTANEDA, ANA, JAVIER DIAZ-GIMENEZ, AND JOSE-VICTOR RIOS-RULL (2003): “Accounting for the U.S. Earnings and Wealth Inequality,” Journal of Political Economy, 111(4), 818–857.

   CORONADO, JULIA LYNN, JOSEPH P. LUPTON, AND LOUISE M. SHEINER (2005): “The Household Spending Response to the 2003 Tax Cut: Evidence from Survey Data,” FEDS discussion paper 32, Federal Reserve Board.

   DEATON, ANGUS S. (1991): “Saving and Liquidity Constraints,” Econometrica, 59, 1221–1248.

   DEN HAAN, WOUTER J. (2010a): “Assessing the Accuracy of the Aggregate Law of Motion in Models with Heterogeneous Agents,” Jouranl of Economic Dynamics and Control, 34(1), 79–99.

   __________  (2010b): “Comparison of Solutions to the Incomplete Markets Model with Aggregate Uncertainty,” Journal of Economic Dynamics and Control, 34(1), 4–27.

   DEN HAAN, WOUTER J., AND PONTUS RENDAHL (2010): “Solving the Incomplete Markets Model with Aggregate Uncertainty Using Explicit Aggregation,” Journal of Economic Dynamics and Control, 34(1), 69–78.

   DIAMOND, PETER A. (1965): “National Debt in a Neoclassical Growth Model,” American Economic Review, 55, 1126–1150.

   FRIEDMAN, MILTON A. (1957): A Theory of the Consumption Function. Princeton University Press.

   FUHRER, JEFFREY C. (2000): “Habit Formation in Consumption and its Implications for Monetary Policy Models,” American Economic Review, 90(3), 367–390,

   GORMAN, WILLIAM M. (1953): “Community Preference Fields,” Econometrica, 21(1), 63–80.

   GUVENEN, FATIH (2007): “Learning Your Earning: Are Labor Income Shocks Really Very Persistent?,” American Economic Review, 97(3), 687–712.

   __________  (2009): “An Empirical Investigation of Labor Income Processes,” Review of Economic Dynamics, 12.

   HALL, ROBERT E. (1978): “Stochastic Implications of the Life-Cycle/Permanent Income Hypothesis: Theory and Evidence,” Journal of Political Economy, 96, 971–87, Available at

   __________  (1988): “Intertemporal Substitution in Consumption,” Journal of Political Economy, XCVI, 339–357, Available at

   HRYSHKO, DMYTRO (2010): “RIP to HIP: The Data Reject Heterogeneous Labor Income Profiles,” Manuscript, University of Alberta.

   HUBBARD, R. GLENN, JONATHAN S. SKINNER, AND STEPHEN P. ZELDES (1994): “The Importance of Precautionary Motives for Explaining Individual and Aggregate Saving,” in The Carnegie-Rochester Conference Series on Public Policy, ed. by Allan H. Meltzer, and Charles I. Plosser, vol. 40, pp. 59–126.

   JENSEN, SHANE T., AND STEPHEN H. SHORE (2008): “Changes in the Distribution of Income Volatility,” Manuscript.

   JOHNSON, DAVID S., JONATHAN A. PARKER, AND NICHOLAS S. SOULELES (2006): “Household Expenditure and the Income Tax Rebates of 2001,” American Economic Review, 96(5), 1589–1610.

   __________  (2009): “The Response of Consumer Spending to Rebates During an Expansion: Evidence from the 2003 Child Tax Credit,” working paper, The Wharton School.

   KAPLAN, GREG, AND GIOVANNI L. VIOLANTE (2011): “A Model of the Consumption Response to Fiscal Stimulus Payments,” NBER Working Paper Number W17338.

   KENNICKELL, ARTHUR B. (1995): “Saving and Permanent Income: Evidence from the 1992 SCF,” Board of Governors of the Federal Reserve System.

   KIM, JINILL, SUNGHYUN HENRY KIM, AND ROBERT KOLLMANN (2010): “Solving the Incomplete Market Model with Aggregate Uncertainty Using a Perturbation Method,” Journal of Economic Dynamics and Control, 34(1), 50–58.

   KIRMAN, ALAN P. (1992): “Whom or What Does the Representative Individual Represent?,” The Journal of Economic Perspectives, 6(2), 117–136.

   KRUSELL, PER, AND ANTHONY A. SMITH (1998): “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy, 106(5), 867–896.

   LAIBSON, DAVID (1997): “Golden Eggs and Hyperbolic Discounting,” Quarterly Journal of Economics, CXII(2), 443–477.

   LOW, HAMISH, COSTAS MEGHIR, AND LUIGI PISTAFERRI (2005): “Wage Risk and Employment Over the Life Cycle,” Manuscript, Stanford University.

   LUCAS, ROBERT E. (1976): “Econometric Policy Evaluation: A Critique,” in The Phillips Curve and Labour Markets, ed. by Karl Brunner, and Allan H. Meltzer, vol. 1, pp. 19–46. Journal of Monetary Economics (Supplement).

   LUSARDI, ANNAMARIA (1996): “Permanent Income, Current Income, and Consumption: Evidence from Two Panel Data Sets,” Journal of Business and Economic Statistics, 14(1), 81–90.

   MACURDY, THOMAS (1982): “The Use of Time Series Processes to Model the Error Structure of Earnings in a Longitudinal Data Analysis,” Journal of Econometrics, 18(1), 83–114.

   MALIAR, LILIA, SERGUEI MALIAR, AND FERNANDO VALLI (2008): “Comparing Numerical Solutions of Models with Heterogeneous Agents (Models B): a Grid-based Euler Equation Algorithm,” Manuscript.

   __________  (2010): “Solving the Incomplete Markets Model with Aggregate Uncertainty Using the Krusell Smith Algorithm,” Journal of Economic Dynamics and Control, 34(1), 42–49.

   MEGHIR, COSTAS, AND LUIGI PISTAFERRI (2004): “Income Variance Dynamics and Heterogeneity,” Journal of Business and Economic Statistics, 72(1), 1–32.

   MODIGLIANI, FRANCO, AND RICHARD BRUMBERG (1954): “Utility Analysis and the Consumption Function: An Interpretation of Cross-Section Data,” in Post-Keynesian Economics, ed. by Kenneth K. Kurihara, pp. 388–436. Rutgers University Press, New Brunswick, N.J.

   MOFFITT, ROBERT A., AND PETER GOTTSCHALK (1995): “Trends in the Covariance Structure of Earnings in the U.S.: 1969-1987,” Manuscript.

   NIELSEN, HELENA SKYT, AND ANNETTE VISSING-JORGENSEN (2006): “The Impact of Labor Income Risk on Educational Choices: Estimates and Implied Risk Aversion,” Manuscript.

   OTSUKA, MISUZU (2003): “Household Portfolio Choice with Illiquid Assets,” manuscript, Johns Hopkins University.

   PARKER, JONATHAN A. (1999): “The Reaction of Household Consumption to Predictable Changes in Social Security Taxes,” American Economic Review, 89(4), 959–973.

   PARKER, JONATHAN A., NICHOLAS S. SOULELES, DAVID S. JOHNSON, AND ROBERT MCCLELLAND (2011): “Consumer Spending and the Economic Stimulus Payments of 2008,” NBER Working Paper Number W16684.

   REITER, MICHAEL (2010): “Solving the Incomplete Markets Model with Aggregate Uncertainty by Backward Induction,” Journal of Economic Dynamics and Control, 34(1), 28–35.

   SABELHAUS, JOHN, AND JAE SONG (2010): “The Great Moderation in Micro Labor Earnings,” Journal of Monetary Economics, 57(4), 391–403,

   SAHM, CLAUDIA R., MATTHEW D. SHAPIRO, AND JOEL B. SLEMROD (2009): “Household Response to the 2008 Tax Rebate: Survey Evidence and Aggregate Implications,” NBER Working Paper Number W15421.

   SAMUELSON, PAUL A. (1958): “An exact consumption loan model of interest with or without the social contrivance of money,” Journal of Political Economy, 66, 467–482.

   SHAPIRO, MATTHEW W., AND JOEL B. SLEMROD (2009): “Did the 2008 Tax Rebates Stimulate Spending?,” American Economic Review, 99(2), 374–379.

   SOLOW, ROBERT M. (2003): “Dumb and Dumber in Macroeconomics,” Manuscript, MIT.

   SOULELES, NICHOLAS S. (1999): “The Response of Household Consumption to Income Tax Refunds,” American Economic Review, 89(4), 947–958.

   __________  (2002): “Consumer Response to the Reagan Tax Cuts,” Journal of Public Economics, 85, 99–120.

   STORESLETTEN, KJETIL, CHRIS I. TELMER, AND AMIR YARON (2004a): “Consumption and Risk Sharing Over the Life Cycle,” Journal of Monetary Economics, 51(3), 609–633.

   __________  (2004b): “Cyclical Dynamics in Idiosyncratic Labor-Market Risk,” Journal of Political Economy, 112(3), 695–717.

   SZEIDL, ADAM (2006): “Invariant Distribution in Buffer-Stock Saving and Stochastic Growth Models,” Manuscript, University of California at Berkeley.

   TOPEL, ROBERT H. (1990): “Specific Capital, Mobility and Wages: Wages Rise with Job Seniority,” NBER Working Paper Number W3294.

   TOPEL, ROBERT H., AND MICHAEL P. WARD (1992): “Job Mobility and the Careers of Young Men,” Quarterly Journal of Economics, 107(2), 439–479.

   YOUNG, ERIC R. (2010): “Solving the Incomplete Markets Model with Aggregate Uncertainty Using the Krusell-Smith Algorithm and Non-Stochastic Simulations,” Journal of Economic Dynamics and Control, 34(1), 36–41.