Part 3 of 7
Heterogeneous Agent Models
Precautionary savings, incomplete markets, production economies, and aggregate risk.
We now want to consider economies with a continuum of agents where heterogeneity plays a role. In the models of the previous questions preferences and technology were such that we could aggregate the behavior of agents into a representative household and a representative firm. In sum, only the aggregate levels of variables like capital (wealth) mattered, and not their distribution across individuals. This abstraction is appropriate for many questions, especially those about the behavior of aggregates and business cycles, but it comes at big costs because it essentially collapses various dimensions of relevant heterogeneity into the behavior of aggregates.
The alternative to representative-agent macroeconomics is what Benjamin Moll calls \textsl{Distributional Macroeconomics}. I encourage you to checkout his lecture notes on this available here: Distributional Macroeconomics Notes. Moll's notes frame the development of macroeconomics towards a state where inequality at the micro-level matters for the behavior of aggregates, and the behavior of aggregates matters for individual agents in different ways.
We can think of two types of heterogeneity to include in our models:
Ex-ante heterogeneity where agents have different types that determine their possible actions, preferences, or technology. A good example of this is models of workers and entrepreneurs, where some agents are (always) workers while other agents are (always) entrepreneurs. There can also be further differences within each type of agent, for example some entrepreneurs can have permanently high- or low-productivity. See Guvenen et al. (2023) for an application of these ideas.
Ex-post heterogeneity where the differences among agents arise from their endogenous reaction to the realization of idiosyncratic and aggregate shocks they face. The differences among agents are then seen in their states. Some are in debt, some have savings; some have high income or human capital, some have low income or human capital; some own housing, some rent (and some houses are larger than others).
Whether the model economy exhibits ex-ante or ex-post heterogeneity (or both), agents are subject to shocks to their ability to generate income (say unemployment and employment as in the original work of Imrohoroglu 1992), or to their age (some are young, some are old), or to their health, etc. The distribution of these shocks along with the endogenous response of the different agents leads to an endogenous distribution across states. In the absence of aggregate shocks, this distribution converges to a stationary distribution of agents. That will be the main objective of our theory: develop models of the distribution of agents across states.
Why do we care so much about the distribution of agents across states? The distribution is what connects the micro-behavior of agents with the macro-aggregates of the economy. The distribution is what allows us to explicitly aggregate individual behavior in our models. We will devote most of our time to models where the distribution does not change in the long run because there are no aggregate shocks. In these models, the lives of individuals are always changing as they face shocks and respond to them, moving through the distribution, but the distribution itself is always the same because the changes in individuals' lives are all uncoordinated.
In terms of the machinery we have developed so far, the stationary distribution does not describe the behavior of a single random variable across time (as it follows a stationary stochastic process) but the cross-sectional behavior of a population. In this context, aggregates are of course constant (as in the definition of a steady state) because they only depend on the distribution, and not on the actions of any individual.
There is a final ingredient of these models. Critical to their working is the notion of market incompleteness. If markets were complete, agents could perfectly insure the idiosyncratic risk they face and there would be no differences in allocations across them beyond those explained by preferences and technology. If all heterogeneity is ex-post heterogeneity, complete markets would effectively do away with these ex-post differences. The most common form of market incompleteness is the absence of state-contingent bonds. Instead, agents can only trade in bonds (or assets) that pay the same regardless of the realization of idiosyncratic shocks. This prevents agents from fully insuring against income fluctuations, for instance.