A Production Economy with Aggregate Risk¶
There is, of course one critical aspect missing from the previous models: aggregate risk. In all of them, there is only idiosyncratic variation affecting households, but leaving aggregates unchanged. Hence the focus on the stationary recursive competitive equilibria of those economies.
The next step is to introduce aggregate fluctuations into the Aiyagari economy. Krusell and Smith (1998) is the first paper to accomplish this taking (essentially) the same economy presented in the previous question and making \(z\), the productivity of the firm, stochastic. Assume it follows a Markov process with transition function \(Q_{z}\). We will write the (recursive) household problem later, but first we need to discuss the implications of aggregate shocks.
The re-introduction of aggregate productivity means that we are back at the problem of Section 6, but with the added difficulty of having a time-varying distribution of households. The key issue is, as before, determining what is the relevant state of the household problem. The state must (i) contain all the information needed to solve the problem, including prices, and (ii) be updated to compute continuation values, as well as expectations over prices and other variables.
In Section 6 this problem was solved by keeping track of the aggregate capital stock \(K\) and imposing consistency between the transition function of the aggregate capital stock and the savings function of the representative household \(\left(G_{K}=g_{a}\right)\). The key was that the aggregate capital stock was the only object needed to compute prices and solve the household problem.
Unfortunately, there is no longer an immediate correspondence between aggregate capital and individual assets. Aggregate capital is still enough to compute prices (as discussed at the end of the previous section), but to know its value we have no option but to integrate over the asset holdings of households with respect to the (time-varying) distribution of households
Further, in order to update the aggregate capital, it becomes necessary to update the distribution itself \(\Gamma_{t+1}=T_{t}^{\star}\Gamma_{t}\) using the adjoint Markov operator of the process induced by households' saving choices.
The outcome of this is that aggregate capital is no longer a viable state and that instead the households need to keep track of the whole distribution \(\Gamma\) and its evolution. However, the evolution of the distribution itself is state dependent as it reflects the households' saving choices that respond to the aggregate state of the economy.
Recursive competitive equilibrium
A RCE is a set of a value function \(\left(V\right)\), a policy function \(\left(a^{\prime}\right)\), a state-dependent transition function for the distribution \(\left(T^{\star}\right)\), and price functions \(\left(R,W\right)\) such that:
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Given the functions \(\left(R,W\right)\) and the transition function for the distribution \(\left(T^{\star}\right)\), the value and policy functions solve the households' dynamic programming problem
\[\begin{align*} V\left(\underbrace{\epsilon,a}_{\text{Ind. State}},\underbrace{z,\Gamma}_{\text{Agg. State}}\right) & =\max_{\left\lbrace c,a^{\prime}\right\rbrace }\,u\left(c\right)+\beta E\left[V\left(\epsilon^{\prime},a^{\prime},z^{\prime},\Gamma^{\prime}\right)|\epsilon,z,\Gamma\right]\\ & \text{s.t. }c+a^{\prime}=\left(1+R\left(z,\Gamma\right)\right)a+W\left(z,\Gamma\right)\epsilon\qquad a^{\prime}\geq\underline{a}\qquad\Gamma^{\prime}=T^{\star}\Gamma \end{align*}\]\(T^{\star}\)'s dependence on the aggregate state is omitted from the notation for simplicity. Alternatively, we can write \(\Gamma^{\prime}=H\left(z,\Gamma\right)\) for a transition function \(H\). We can also write prices as functions of productivity and aggregate capital, \(r=R\left(z,K\right)\) and \(w=W\left(z,K\right)\). 1. The transition function for the distribution of agents corresponds to the adjoint Markov operator of the endogenous Markov process defined by the (exogenous) Markov processes for \(\epsilon\) and \(z\) with Transition function \(Q\) and \(Q_{z}\) and the (endogenous) policy function for \(a^{\prime}\).
\[ \Gamma^{\prime}\left(S_{\epsilon},A\right)=T^{\star}\Gamma\left(S_{\epsilon},A\right)=\int_{\overline{S}}\int_{\overline{A}}\underbrace{g\left(\epsilon,a,z,\Gamma\right)\cdot\text{Pr}\left(\epsilon^{\prime}\in S_{\epsilon}|\epsilon\right)}_{\text{Markov Kernel}:P\left(\epsilon^{\prime},a^{\prime}|\epsilon,a;z,\Gamma\right)}\cdot d\Gamma\left(\epsilon,a\right) \]where
\[ g\left(\epsilon,a,z,\Gamma,A\right)=\begin{cases} 1 & \text{if }a^{\prime}=g_{a}\left(\epsilon,a,z,\Gamma\right)\in A\\ 0 & \text{otw} \end{cases}. \] -
The capital and labor markets clear. That is, aggregate capital reflects the aggregate assets of the households and aggregate labor demand equals the supply of "efficiency units of labor" from the household
\[ K=\int\int a\cdot d\Gamma\left(\epsilon,a\right)\qquad L=\int\int\epsilon\cdot d\Gamma\left(\epsilon,a\right)\left(=1\right). \]Note that the aggregate capital is a function of the distribution of households, \(K\left(\Gamma\right)\). 1. Prices are consistent with firm optimization (in this case cost-minimization)
\[ r=f_{k}\left(z,K,L\right)-\delta\qquad w=f_{L}\left(z,K,L\right). \]This implies
\[ R\left(z,K\right)=f_{k}\left(z,K,1\right)-\delta\qquad W\left(z,K\right)=f_{L}\left(z,K,1\right). \]The version with prices depending on the distribution \(\Gamma\) rather than on aggregate capital makes it explicit that the underlying state determining capital is the distribution \(\Gamma\).
The definition of the RCE makes the problems introduced by aggregate fluctuations apparent. It is impossible to keep track of the whole distribution of households and generate a state-dependent transition function that is consistent with household optimization.
The main break-through in Krusell and Smith (1998) is providing a computational method that can approximate the solution to the RCE. The key of the algorithm is already present in the discussion of the RCE in Section 6 and of the S-RCE of the Aiyagari economy above. The distribution of households is only really needed to compute the aggregate capital level, \(K\). Notably, this corresponds to the first moment of the wealth distribution. The reason only aggregate capital is needed is that it is sufficient to compute prices. The challenge is that we also need to know how aggregate capital evolves, that is, we need a transition function for aggregate capital. However, the evolution of capital depends on the whole distribution.
Krusell and Smith (1998) solve the problem by proposing an approximate law of motion for capital that depends on moments of the wealth distribution. In principle, if enough moments are taken into account they can sufficiently summarize the information in the wealth distribution. If the approximation works the problem of keeping track of the entire distribution can be reduced to the (much simpler) problem of keeping track of a finite set of moments.
In practice, the rational expectation assumption is partly lifted. Instead of forming exact expectations over the distribution of wealth with knowledge of the transition function \(T^{\star}\), households use a (log-)linear model to forecast future moments of the distribution using its current moments and the aggregate productivity. For instance
where the coefficients depend on the values of \(z\). The solution method proposed in the paper finds coefficients that (almost exactly) match the evolution of capital. When \(z\) is discrete there are finitely many coefficients to find. The surprising result in the implementation of the solution is that \(\gamma\left(z\right)=0\), implying that only the first moment is required to approximate the evolution of aggregate capital.
Algorithm: Krusell-Smith Algorithm
Input: Guess for coeficients \(\alpha(z),\beta(z)\)
Output: \(V,a^{\prime},\Gamma\)
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Solve the DP problem of the agent given market clearing prices \((R(z,K),W(z,K))\) and the transition function for \(K\) parameterized by \(\alpha(z),\beta(z)\):
\((V,a^{\prime})=T(V;\alpha,\beta)\) (a fixed point problem). Note that markets will clear by construction. -
Use the policy function and the Markov processes for \(\epsilon\) and \(z\) to simulate a long panel of the economy. In doing this update the distribution using the appropriate Markov operator. Use the distribution to compute aggregate capital. Record the time series for aggregate capital and productivity.
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Regress capital on moments to update the coefficients allowing for state-dependent coefficients by the level of productivity.
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Repeat (1)-(3) until prices converge and guarantee that the \(R^2\) of the regression is close to 1.
Approximate aggregation
The main result in Krusell and Smith (1998) is that, as it turns out, it is sufficient to only keep track of the first moment of the wealth distribution in order to accurately approximate the aggregate states of the model. That is, aggregate capital is (approximately) the relevant aggregate state for the households in the economy, just as in the neoclassical growth model of Section 6. However, while there is exact aggregation in the neoclassical growth model (allowing for a representative agent), there is no such aggregation result in the Aiyagari economy. This led Krusell and Smith to term their result "approximate aggregation." They write
"Our main insight is that the macroeconomic model with heterogeneity features approximate aggregation. By approximate aggregation, we mean that, in equilibrium, all aggregate variables---consumption, the capital stock, and relative prices---can be almost perfectly de- scribed as a function of two simple statistics: the mean of the wealth distribution and the aggregate productivity shock."
This result makes the model solvable in practice by reducing the dimensionality of the problem.
Where does approximate aggregation come from? Why does it work? The key is that even with a single non-state-contingent asset the households can achieve a great deal of insurance, effectively smoothing out the fluctuations in their marginal utility of consumption, in the same way that a representative household would. The outcome of this is that households in the heterogeneous agent economy behave (approximately) as scaled-versions of the representative household (except for those that are borrowing-constrained). Krusell and Smith (1998) explain it as follows:
"The key insight is related to earlier findings from similar models that utility costs from fluctuations in consumption are quite small and that self-insurance with only one asset is quite effective. Self-insurance in our model is not very effective in terms of smoothing individual relative to aggregate consumption; for example, the unconditional standard deviation of individual consumption is about four times that of aggregate consumption, and the unconditional correlation of the consumption of any two agents is very close to zero. However, in utility terms, agents in our stationary equilibria are insured well enough that the marginal propensity to save out of current wealth is almost completely independent of the levels of wealth and labor income, except at the very lowest levels of wealth. Furthermore, although some very poor agents have substantially different marginal savings propensities at any point in time, the fraction of total wealth held by these agents is always very small (this is particularly true in the model with a realistic wealth distribution). Because it is so small, higher-order moments of the wealth distribution simply do not affect the accumulation pattern of total capital, even though these moments do move significantly over time." {[}emphasis added{]}
This is (at its core) the same reason behind the inability of Aiyagari economies based on labor-income risk to generate realistic distributions of wealth, as explained in Benhabib and Bisin (2018) and Stachurski and Toda (2019); Stachurski and Toda (2020). As households' move away from the collateral constraint, their saving functions tend to be linear. This limit behavior means that households can be (approximately) aggregated and is what explains the similar marginal propensity to consume referenced in the passage above.
The approximate aggregation result has also misleadingly led to thinking that heterogeneity does not matter for aggregate fluctuations. That is not the case as shown in Krusell and Smith (1998) and their concurrent paper Krusell and Smith (1997). Approximate aggregation follows from the limit behavior of saving rates in the model, and not from the irrelevance of heterogeneity. Rather, it is because the baseline Aiyagari economy inability to reproduce the observed levels of wealth inequality that Krusell and Smith obtain their (in)famous irrelevance result. They write
"When the representative-agent model is altered only by adding idiosyncratic, uninsurable risk, the resulting stationary wealth distribution is quite unrealistic: there are too few very poor agents, and much too little concentration of wealth among the very richest. For this reason, we consider a version of the model with preference heterogeneity {[}...{]} We show that this model does succeed quite well in matching the key features of the wealth distribution. {[}...{]} in the aggregate, we observe a significant departure from permanent income behavior, in contrast to standard representative-agent models."
In fact, the extended model in Krusell and Smith (1997) also attempts to better match the observed distribution of wealth (an elusive target for the literature). They also find a relevant role for heterogeneity in shaping the aggregates of the economy. The reason wealth concentration matters is that it generates (endogenously) a mass of households that act in a "hand-to-mouth" fashion. These households are not able to effectively insure against aggregate fluctuations and behave markedly different from the representative agent (that is never against the borrowing constraint and behaves like a permanent-income agent, smoothing consumption).
Similarly, Angeletos (2007) shows that introducing capital-income risk in the form of risky portfolios implies significant departures from the representative agent baseline. He writes
"{[}S{]}ignificant general-equilibrium effects on savings and income are both empirically plausible and consistent with low private-equity premia {[}...{]} the macroeconomic effects of idiosyncratic investment risk can be both qualitatively distinct from those of idiosyncratic labor-income risk and quantitatively significant."
Other papers that highlight the importance of heterogeneity for aggregate fluctuations are: Kaplan et al. (2018), Ahn et al. (2018) Kaplan and Violante (2022). See Kaplan and Violante (2018) in the Journal of Economic Perspectives for a non-technical summary of the HANK literature. One important aspect of these papers is highlight the role of portfolio composition and illiquid assets in generating a group of wealthy-hand-to-mouth households who, despite being at the top of the wealth distribution, behave as borrowing constraint agents because their wealth is not readily accessible to produce insurance (as is the case with housing).
Modern solution methods
In a concurrent article, Den Haan (1997) develops an alternative solution method that parametrizes the distribution of wealth on a standard polynomial basis, this reduces the dimensionality of the problem. Rather than track the infinite-dimensional distribution and its transition function, Den Haan's method searches for vector of parameters that approximate the functions on a given polynomial basis. One important advantage of this method is that it does not rely on imprecise Monte Carlo simulations of the model in order to obtain a solution. This method has been extended, for instance in Winberry (2018).
An alternative that has proven to be effective is using Perturbation methods, as in Reiter (2009). The best application of this is Auclert et al. (2021) which is currently the best method to solve heterogeneous agent models with aggregate fluctuations.