The Income Fluctuations Model and Precautionary Savings¶
The basic form of the heterogeneous agent model is built to generate ex-post household heterogeneity and to be consistent (as much as possible) with the neoclassical growth model. However, unlike the neoclassical growth model, there are no aggregate shocks.
The economy is populated by a continuum of households who are ex-ante identical. That means that they have the same preferences and the solve the same problem. These households are subject to idiosyncratic income shocks. These shocks are most often interpreted as variation in the household's labor-income. The households have access to a savings technology in the form of assets (or bonds) that pay the same return \(r\) in all states (corresponding to the value of the income shocks of the households). Households are further subject to a borrowing constraint, so that their (net) assets have to be above a given lower bound \(\underline{a}\).
The problem of the household is (by construction) almost identical to the household problem in the Recursive Competitive Equilibrium of the neoclassical growth problem:
In this problem the household is subject to shocks \(\epsilon\) to their earning ability, as before we assume that \(\epsilon\) follows a Markov process with transition function \(Q\). The household can, in principle, choose how much to work but how \(\epsilon\) determines how labor is translated into earnings. Savings take the form of assets \(a\).
We are already setting the problem in its stationary form, as evidenced by the fact that prices \(\left(r,w\right)\) are constant. We will formalize this idea later when we provide the notion of equilibrium.
Precautionary Savings and the Rate of Return
An important aspect of this problem is that, unlike that of the planner in the neoclassical growth model, the returns to assets are constant. That is, there is no curvature in the budget constraint. This is of course a consequence of the household being a price taker. Tied to this fact is the (newly introduced) disconnect between the individual's income and the return on assets. In the representative agent economy we discussed above, wages and returns moved in response to the same (aggregate) conditions. Here, each household's income moves in response to idiosyncratic factors (captured by \(\epsilon\)) with prices held constant (and equal for all households).
These aspects matter because they introduce a new motive for savings that has been termed "precautionary savings." To understand this motive for savings we start by contrasting the heterogeneous agent model we introduce here with the neoclassical growth model and its representative agent to then contrast them with what is happening in the new model.
Savings and returns in a representative agent world
Savings in the neoclassical growth model are determined by the inter-temporal trade-off of spending resources in the present in order to receive returns in the future. These returns corresponded to the marginal product of the capital being accumulated through savings. The representative household's Euler equation is
In steady state with no aggregate fluctuations (the corresponding version of the stationary recursive competitive equilibrium we will study below) we have the standard result that the return on assets is equal to the (inverse of the) inter-temporal rate of discount
This is a no-arbitrage expression. Without any risk (exogenous variation in future values of consumption and leisure), the household is indifferent between saving an extra unit of goods or not. In other words, because \(\left(1+r\right)\beta=1\) the discounted value of the returns is the same as the present cost of the goods being saved. This indifference condition prevents the (representative) household from engaging in arbitrage. For instance, if \(1+r<\frac{1}{\beta}\) the household would want to borrow infinitely. The linearity (lack of curvature) of the household's budget introduces the same forced discussed before in the context of a firm with constant returns to scale technology. In the same way, the equilibrium level of capital (wealth or assets) is not determined by supply (which is perfectly elastic given the no-arbitrage condition above) but demand-determined. The curvature that sets the steady state level of capital comes from the firm's capital demand, that requires that
Together these conditions provide the standard steady state condition for the neoclassical growth model
equating the steady state marginal product of capital with the rate of inter-temporal discount.
Beyond determining the level of aggregates in steady state, the results above have an important economic interpretation. The sole determinant of the level of capital in the economy is the curvature of the production function (that determines the marginal productivity of capital) relative to the household's rate of discount. The household's saving motive is one of future use of capital. That is, capital is accumulated to be used. This logic is transformed in the heterogeneous agent model as we see below.
Idiosyncratic risk and the stationary rate of return
The introduction of idiosyncratic and uninsurable risk to the model (via income fluctuations and incomplete markets) results in a new saving motive for households. They want to have savings "just in case something happens." Under this logic, assets are not accumulated with the objective of using them (say for production) but rather with the object of not using them! Savings, even in non-state-contingent assets provide a form of self-insurance against idiosyncratic income fluctuations. Households with currently high incomes (relative to their long-run average) accumulate assets in anticipation of potentially low incomes to come (they save hoping that those savings won't be needed, but knowing that they will). Households with low income can partially insure themselves by drawing on their savings. Here is where the borrowing constraint bites, as it prevents better insurance by limiting the debt that households can accumulate. This turns out to be a crucial factor determining the equilibrium aggregates in the economy.
The households' Euler equation is
Even though prices and aggregates do not change, the expectation over future idiosyncratic shocks is still relevant. Moreover, because households are heterogeneous (with different asset and income levels), the Euler equation necessarily implies that some households save while other dissave, depending on the value of their current and expected consumption. Unlike the normal savings motives explained above, the household wants to save because their current income is higher than their expected future income, this will happen for some household regardless of the relationship between the rate of return and the rate of discount. Of course, more households are willing to save if there is a higher rate of return or a higher rate of discount, and vice-versa.
What happens then with the relationship between the rate of return and the rate of discount in equilibrium? Aiyagari (1994) shows that it must be that case that
The reason is a combination of the fact that households are heterogeneous (facing idiosyncratic labor income risk) and that the presence of the borrowing constraint that introduces a key asymmetry to the problem. It cannot be that \(1+r>\frac{1}{\beta}\) because then the Euler equation would imply that consumption must have an upward drift (that is, that it must be expected to increase). This means that households want to save always. Because households are infinitely lived, this implies that assets (and consumption) tend to infinity. This cannot be a solution. See Aiyagari (1994, footnote 20) for a formal argument. The same thing occurs if \(1+r=\frac{1}{\beta}\). The reason lies in the desire for self-insurance as the household wants to have constant (expected) marginal utility over time. Once again, because the household is infinitely lived, they face the risk of an arbitrarily long sequence of low-income shocks. In order to insure against that, the household needs an arbitrarily large stock of savings. This is feasible because, with \(1+r=\frac{1}{\beta}\), it is costless for the household to transform current consumption into savings.
Therefore, it must be that \(1+r<\frac{1}{\beta}\) in equilibrium. Under complete markets this would lead the household to accumulate infinite debt. The borrowing constraint prevents this from happening. Instead, there is a positive mass of households who de-accumulate assets and hit the constraint (following a series of low income shocks), while other households accumulate assets when they have higher levels of income. The properties of the optimal saving function in this income-fluctuation problem are explored in Huggett (1993, Theorem I), Aiyagari (1994, Sec. III, The Individual's Problem), and more explicitly in Achdou et al. (2022).
What does this imply for the aggregate level of assets (or capital) in the economy? Huggett (1993) and Aiyagari (1994) show that the level of assets in the incomplete-markets/heterogeneous-agent economy must be larger than in its complete-markets/representative agent counterpart. The additional assets (relative to complete markets) capture the role of precautionary savings. Angeletos (2007) later showed that, even though aggregate assets are always higher under labor-income risk, they can be increased or decreased under capital-income risk (also called investment or production risk), even though \(1+r<\frac{1}{\beta}\) still holds.
The relationship between the rate of return and the rate of discount also has implications for a long-standing debate on the return to equity and the return to safe assets. While the observed risk-free interest rates (i.e. those on Treasury bills), were around 1\%, the average real return to equity was around 7\%. This gap could not be matched by calibrating existing representative-agent models. In particular, these models predicted risk-free rates and equity premiums that were too large and too small, respectively. Huggett (1993) provides a partial answer for the relatively low risk-free rates by introducing uninsurable income risk. Because of the borrowing constraint, agents are restricted in the level of their indebtedness. However, agents are not restricted from accumulating saving. A low risk-free rate is then needed to persuade agents not to accumulate large credit balances so that the credit market can clear, as we will see in the next section. Angeletos (2007) revisits these results in a production economy with returns to private equity.
An example: Log utility
Consider a concrete example where utility is \(u\left(c\right)=\log c\). In this case the Euler equation becomes
Recall that agents are risk averse, so that marginal utility is convex (as utility is concave). In this case this has an immediate consequence for the expected marginal utility of future consumption (see Jensen's inequality),
with larger \(\Delta\) capturing more consumption risk for the agent (note that if there is no risk then \(E_{\epsilon_{i}^{\prime}}\left[\frac{1}{c_{i}^{\prime}}|\epsilon_{i}\right]=\frac{1}{E_{\epsilon_{i}^{\prime}}\left[c_{i}^{\prime}|\epsilon_{i}\right]}\) and \(\Delta=0\).
Replacing back we get a condition relating the expected growth rate of consumption, its drift, with the discounted returns and the risk faced by the agent:
So, the risk faced by consumers makes it so that the expected drift of consumption is necessarily higher than \(\left(1+r\right)\beta\). This is what is capturing the precautionary savings motives.
The offshoot of this is that we require \(\left(1+r\right)\beta<1\). If \(\left(1+r\right)\beta\geq1\), the drift would surely be positive (the expected growth rate would be higher than 1), leading to all agents to accumulate infinite assets and have infinite consumption in the limit (the drift is positive for everyone, always, forever). This ends up violating the transversality conditions of the problem, so that it has no well-defined value. It must be that \(\left(1+r\right)\beta<1\). In this case there is heterogeneity in the drift of consumption between agents. Agents facing little risk, like wealthy agents who can self-insure against income fluctuations, have a low \(\Delta\) (the "Jensen" wedge in the Euler equation), that is, low precautionary motives for savings. Their savings are instead dictated by the usual present bias versus return tradeoff. Because \(\beta\left(1+r\right)<1\), this tradeoff lands in favor of de-accumulating assets and having a negative drift for consumption. This is key because it ensures that there is a stationary distribution pulling back wealthy agents who stop accumulating assets. To the contrary, agents who face more risk, like poorer agents cannot self-insure, want to save despite the incentives for dissaving. This prevents them from getting trapped in the bottom end of the distribution. They save, despite being asset-poor, in order to insure against risk, not because of higher returns.
Closing the model: What about production?
To close the model we need to specify where output comes from. We will do this below looking at the two main alternatives: an endowment economy (without firms) as modeled in Huggett (1993) and a production economy as modeled in Aiyagari (1994). The firms are in any case kept as simple as possible, loading all the heterogeneity into the households.