Part 3: Heterogeneous Agent Models Section 8

An Endowment Economy¶

Consider an economy like the one described in Huggett (1993). The economy is populated by a continuum of households who receive a stochastic endowment of consumption goods (say, jelly beans) every period. Denote by \(\epsilon\) the amount of jelly beans received by a household. The amount of jelly beans that a household receives follows a Markov process with transition function \(Q\). The jelly beans follow independent but identical processes across households.

Sadly, jelly beans cannot be stored and thus have to be consumed the same period they are received. This is a problem for the households because they would much rather not have their income fluctuate every period. Nevertheless, the households can smooth consumption by contracting with one another, that is by exchanging jelly beans among them. This is, of course, an exchange economy (as opposed to the production economy we study in the next section).

The contracts that households can sign are limited in two ways. First, they cannot depend on the individual realizations of the present or future endowment of jelly beans. That is, the contracts can only specify a fixed amount of jelly beans being transferred today in exchange for a fixed amount in the future, regardless of how many jelly beans the household actually has. This form of market incompleteness can reflect information frictions, as the amount of jelly beans a household receives is their private information. Second, credit balances (receiving jelly beans today in exchange for payment of jelly beans in the future) are restricted, so that there is a limit to how many jelly beans can be obtained by a household in the market.

Therefore, the contracts take the form of saving/debt contracts, where the household exchanges an amount \(a^{\prime}\) of jelly beans today in exchange for \(\left(1+r\right)a^{\prime}\) jelly beans in the future. When \(a^{\prime}>0\) the household is saving and will receive payment in the future. When \(a^{\prime}<0\) the household is in debt and receives jelly beans today in exchange for a payment in the future. This problem can be equivalently formulated in terms of (zero-coupon) bond contracts with a unit bond having a price of \(q\). Letting \(q=\frac{1}{1+r}\) the problem would be buying a bond that pays one unit of future jelly beans paying \(q\) jelly beans today. This latter formulation is the one used in Huggett (1993).

As explained above, it will be the case that households use financial contracts (bonds) to smooth consumption. They save if \(\epsilon\) is high and borrow if \(\epsilon\) is low.

The problem of the households is then

\[\begin{align*} V\left(\epsilon,a\right) & =\max_{\left\lbrace c,a^{\prime}\right\rbrace }\,u\left(c\right)+\beta E\left[V\left(\epsilon^{\prime},a^{\prime}\right)|\epsilon\right]\\ & \text{s.t. }c+a^{\prime}=\left(1+r\right)a+\epsilon\qquad a^{\prime}\geq\underline{a} \end{align*}\]

With respect to the workhorse model above, there is no labor choice and the "wage" is equal to 1, making income equal to \(\epsilon\). As before, households live forever and there is no aggregate risk. The Euler equation of a household is

\[ u^{\prime}\left(c\right)=\beta E_{\epsilon^{\prime}}\left[V_{a}\left(\epsilon^{\prime},a^{\prime}\right)|\epsilon\right]=\beta\left(1+r\right)E\left[u^{\prime}\left(\left(1+r\right)a^{\prime}+\epsilon^{\prime}-g_{a}\left(\epsilon^{\prime},a^{\prime}\right)\right)\right], \]

where \(g_{a}\) is the policy function for savings.

The borrowing constraint must satisfy:

\[ \underline{a}\geq\frac{-\epsilon_{min}}{r}. \]

This is called the natural borrowing limit. It is the highest amount of debt that can be rolled-over forever if the household faces an infinite sequence of the lowest possible endowment (or income shock).

Crucially, the financial contracts (bonds) are in zero net supply in this economy. This is an immediate consequence of the jelly beans not being storable. That is, there are no actual goods being transferred across time, only promises for future payments. The implication of this is that one household's savings are another household's debt. Even though the households trade against the market and not in bilateral meetings, the total amount of saving and debt in this economy most coincide. If this was a representative agent economy, there would be no trade!

Distribution of households

The fact that financial contracts are in zero net supply means that in equilibrium the aggregate level of assets (bonds) must be zero. To calculate this we need to define and obtain the distribution of households over states, \(\Gamma\). Then we can calculate aggregate assets as the integral over the level of assets (bonds) with respect to the distribution of households \(\left(\Gamma\right)\). This is the key equilibrium condition for this economy.

Formally, we define \(\overline{S}\) as the set of exogenous states and \(\left[\underline{a},\overline{a}\right]\) be the domain of assets (Exercise: Show that there is an endogenous upper bound for the distribution of assets. See Huggett 1993, Theorem 2, Aiyagari 1994, footnote 18, Achdou et al. 2022, Proposition 2). Let \(\mathcal{S}\) and \(\mathcal{A}\) be \(\sigma\)-algebras over \(\overline{S}\) and \(\left[\underline{a},\overline{a}\right]\) respectively. The distribution of agents is a measurable function \(\Gamma:\mathcal{S}\times A\to\left[0,1\right]\) that integrates to 1.

We can update the distribution of households by following their actions. Let \(S\times A\in\mathcal{S}\times\mathcal{A}\) be a set in the \(\sigma\)-algebras. We want to know if there are households coming into the set \(S\times A\). From the Markov kernel (or transition function) of \(\epsilon\) we have the probability that \(\epsilon^{\prime}\in S\):

\[ \text{Pr}\left(\epsilon^{\prime}\in S|\epsilon\right)=\int_{\epsilon^{\prime}\in S}Q\left(\epsilon^{\prime}|\epsilon\right)d\epsilon. \]

Then, we define an indicator function to know if \(a^{\prime}\in A\)

\[ g\left(\epsilon,a,A\right)=\begin{cases} 1 & \text{if }a^{\prime}=g_{a}\left(\epsilon,a\right)\in A\\ 0 & \text{otw} \end{cases}. \]

With this we can update the distribution for all \(\left(S,A\right)\in{\mathcal{S}\times A}\):

\[ \Gamma^{\prime}\left(S,A\right)=\int_{\overline{S}}\int_{\overline{A}}\underbrace{g\left(\epsilon,a,A\right)\cdot\text{Pr}\left(\epsilon^{\prime}\in S|\epsilon\right)}_{\text{Markov Kernel}:P\left(\epsilon^{\prime}a^{\prime}|\epsilon,a\right)}\cdot d\Gamma\left(\epsilon,a\right). \]

This is the adjoint Markov operator of the Markov kernel (or transition function) \(Q\) of the joint process for \(\left(\epsilon,a\right)\). The stationary distribution of states across households is the fixed point of this operator (\(\Gamma\) such that \(\Gamma^{\prime}=\Gamma\)).

When the exogenous state is discrete (as is often the case) updating is simpler. We can work with: \(\mathcal{S}=\left\lbrace \left\lbrace \epsilon_{1}\right\rbrace ,\ldots,\left\lbrace \epsilon_{n}\right\rbrace ,\ldots,\left\lbrace \epsilon_{N}\right\rbrace \right\rbrace\) instead of the \(\sigma\)-algebra and update according to

\[ \Gamma^{\prime}\left(\epsilon^{\prime},A\right)=\sum_{\overline{S}}\int_{\overline{A}}g\left(\epsilon,a,A\right)\cdot\underbrace{Q\left(\epsilon^{\prime}|\epsilon\right)}_{\text{Markov Transition Matrix}}\cdot d\Gamma\left(\epsilon,a\right) \]

Stationary recursive competitive equilibrium

A S-RCE is a set of a value function \(\left(V\right)\), a policy function \(\left(a^{\prime}\right)\), a distribution \(\left(\Gamma\right)\), and a price \(\left(r\right)\) such that:

Given \(r\) the value and policy functions solve the agent's problem (the dynamic programming problem above).
Given the policy function, \(\Gamma\) is a fixed point of the adjoint Markov operator of the endogenous Markov process defined above.
Given the distribution and policy functions the asset (or bond) market clears:

\[ \underbrace{\int\int a^{\prime}\left(\epsilon,a\right)\cdot d\Gamma\left(\epsilon,a\right)}_{\text{Net Supply of Assets}}=0\longleftrightarrow\underbrace{\int\int c\left(\epsilon,a\right)\cdot d\Gamma\left(\epsilon,a\right)}_{\text{Demand for Goods}}=\underbrace{\int\int\epsilon\cdot d\Gamma\left(\epsilon,a\right)}_{\text{Supply of Goods }} \]

The definition of equilibrium also gives us an algorithm to find it.

Algorithm: S-RCE Algorithm

Input: Guess for price \((r)\)

Output: \(V,a^{\prime},\Gamma,r\)

Solve the DP problem of the agent given \(r\):
\((V,a^{\prime})=T(V;r)\) (a fixed point problem)
Find stationary distribution by iterating over the adjoint Markov operator
Check market clearing: \(\sum_{i}\sum_j a^{\prime}(\vec{\epsilon}_i,\vec{a}_j)\cdot\Gamma(i,j)\)
Update prices to clear market
Manually by tatonnement or with a Root finder
Repeat (1)-(4) until market clears

However, it is not a given that such an equilibrium exists. A couple of results are needed for this presented in Theorems 1 and 2 of Huggett (1993). If you read the explanation in the paper you will see how much of a group effort the development of these models was. The proofs of each result build on a series of concurrent papers. Interestingly, Huggett does not fully prove that an equilibrium exists. He conjectures that the excess supply of assets is increasing in the interest rate \(r\) (or decreasing in the price of bonds \(q\)) and uses tatonnment to search for an equilibrium. The conjecture is verified computationally but not proven.