Aggregation: Complete Markets, Preferences, and Technology¶
This course is mostly devoted to the development of macroeconomic models with heterogeneity, where the distribution of assets, consumption, or investment is relevant for the determination of macroeconomic aggregates. However, there are many questions for which these distributional concerns are not of first order importance. Representative agent models are often used to address those questions. Before going into the material of the course, it is useful to stop and ask: When can we aggregate? Why do we use representative agent models?
Note
This section draws heavily Larry Jones lecture notes found in https://sites.google.com/umn.edu/larryjones/teaching
The answer to this questions takes the form of (mathematical) conditions under which the outcome (the equilibrium) of a model with full heterogeneity looks the same as the outcome of a model with a representative agent. So, we consider first a heterogeneous agents economy.
Firms:
The economy has two kinds of firms: investment firms and consumption firms. Both use capital and labor to produce, respectively, investment specific goods for the formation of capital or general consumption goods. The production technologies are potentially firm- and time-specific:
Households:
There are also households indexed by \(i\). They are infinitely lived and supply labor and consume every period. Household \(i\) owns a share \(\theta_{i,h}^{x}\) of investment firm \(h\) and a share \(\theta_{i,j}^{y}\) of consumption firm \(j\). These shares are constant (but we can allow for a market in which they are traded). The entirety of the firms are owned by households so that
We assume that households make investment decisions. The initial endowment of households (that determines their lifetime wealth) is an amount of initial capital \(k_{i,0}\) and a sequence of potential labor supplies (or labor productivities) \(\overline{n}_{i,t}\). The payoff of a household is the lifetime value of their consumption and labor choices captured by a utility \(U^{i}\left(\cdot\right)\), a function of the sequence \(\left(c_{i,t},n_{i,t}\right)_{t=0}^{\infty}\).
Competitive equilibrium:
An equilibrium is a sequence of prices \(\left\lbrace p_{t}^{c},p_{t}^{x},r_{t},w_{t}\right\rbrace _{t=0}^{\infty}\) a sequence of input demands and output from firms \(\left\lbrace x_{h,t},k_{h,t}^{x},n_{h,t}^{x},y_{j,t},k_{j,t}^{y},n_{j,t}^{y}\right\rbrace _{t=0;h=1;j=1}^{t=\infty;h=H;j=J}\) and a sequence of household actions \(\left\lbrace c_{i,t},n_{i,t},k_{i,t+1},x_{i,t}\right\rbrace _{t=0;i=1}^{t=\infty;i=I}\), such that, given the ownership of firms \(\left\lbrace \theta_{i,h}^{x},\theta_{i,j}^{y}\right\rbrace\) and endowments \(\left\lbrace k_{i,0},\left\lbrace \overline{n}_{i,t}\right\rbrace _{t=0}^{\infty}\right\rbrace _{i=1}^{I}\),
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For all \(i=1,\ldots,I\), the sequence \(\left\lbrace c_{i,t},n_{i,t},k_{i,t+1},x_{i,t}\right\rbrace _{t=0}^{t=\infty}\) solves (given endowments and prices)
\[\begin{align*} \max & U^{i}\left(\cdot\right)\\ & \text{s.t. }\sum_{t=0}^{\infty}p_{t}^{c}c_{i,t}+p_{t}^{x}x_{i,t}\leq\sum_{t=0}^{\infty}w_{t}n_{i,t}+r_{t}k_{it}+\theta_{i,h}^{x}\pi_{h,t}^{x}+\theta_{i,j}^{y}\pi_{j,t}^{y}\\ & \phantom{\text{s.t. }}k_{i,t+1}\leq\left(1-\delta\right)k_{i,t}+x_{i,t}\\ & \phantom{\text{s.t. }}n_{i,t}\leq\overline{n}_{i,t} \end{align*}\] -
For all \(h=1,\ldots,H\), the sequence \(\left\lbrace x_{h,t},k_{h,t}^{x},n_{h,t}^{x}\right\rbrace _{t=0}^{t=\infty}\) solves (given prices)
\[ \pi_{h,t}^{x}\,=\,\max\,p_{t}^{x}F_{h,t}^{x}\left(k_{h,t}^{x},n_{h,t}^{x}\right)-w_{t}n_{h,t}^{x}-r_{t}k_{h,t}^{x}\qquad\text{s.t. }x_{h,t}\leq F_{h,t}^{x}\left(k_{h,t}^{x},n_{h,t}^{x}\right) \] -
For all \(j=1,\ldots,J\), the sequence \(\left\lbrace y_{j,t},k_{j,t}^{y},n_{j,t}^{y}\right\rbrace _{t=0}^{t=\infty}\) solves (given prices)
\[ \pi_{j,t}^{y}\,=\,\max\,p_{t}^{c}y_{j,t}-w_{t}n_{j,t}^{y}-r_{t}k_{j,t}^{y}\qquad\text{s.t. }y_{j,t}\leq F_{j,t}^{y}\left(k_{j,t}^{y},n_{j,t}^{y}\right) \] -
Markets clear, so that for all \(t\),
\[\begin{align*} \sum_{i=1}^{I}c_{i,t} & =\sum_{j=1}^{J}y_{j,t} & \sum_{i=1}^{I}x_{i,t} & =\sum_{h=1}^{H}x_{j,t}\\ \sum_{i=1}^{I}k_{i,t} & =\sum_{h=1}^{H}k_{j,t}^{x}+\sum_{j=1}^{J}k_{j,t}^{y} & \sum_{i=1}^{I}n_{i,t} & =\sum_{h=1}^{H}n_{j,t}^{x}+\sum_{j=1}^{J}n_{j,t}^{y} \end{align*}\]
Aggregation
Now that we have our economy and have defined its equilibrium, we can ask about the conditions under which we could obtain the same aggregate allocations coming from representative firms and a representative household.
One immediate way in which we get aggregation is all firms and households are identical. This implies that the production technologies do not depend on the identity of the firm \(\left(F_{h,t}^{x}\left(\cdot\right)=F_{h^{\prime},t}^{x}\left(\cdot\right),\,F_{j,t}^{y}\left(\cdot\right)=F_{j^{\prime},t}^{y}\left(\cdot\right)\right)\), and the utility function and endowments do not depend on the identity of the household \(\left(U^{i}\left(\cdot\right)=U^{i'}\left(\cdot\right),\:\theta_{i,h}^{x}=\theta_{i,j}^{y}=\frac{1}{I},\:k_{i,0}=k_{i^{\prime},0},\:\overline{n}_{i,t}=\overline{n}_{i^{\prime},t}\right)\), with the common utility function strictly concave so that it admits a unique solution. In this case all individual firms and households make the same choices. In this case the distribution of agents in the economy does not matter because there is no actual heterogeneity.
However, there can be heterogeneity while allowing for aggregation. The conditions are nevertheless strong. The basic theory is developed by Gorman (1953); Gorman (1961) for the household side. I also recommend Jackson and Yariv (2024) for a more complete treatment of cases where a representative agent is implicitly assumed when none actually exists.
Firms
Assume firms have constant returns to scale. Then, firms make no profits in equilibrium. If they did they would want to scale up production without bound, along with input demand. Because firms make no profits, differences in ownership \(\left\lbrace \theta_{i,h}^{x},\theta_{i,j}^{y}\right\rbrace\) are now irrelevant. In this case, input demand is determined by cost minimization. See Profit Maximization: A note for more on this.
If we further assume that the production technology is common across firms then the problem is reduced to that of two firms that produce the entirety of output in each sector. The many firms we have are just scaled-down copies of the aggregated firms. We can relax this condition by assuming instead that technology is described by
with \(F_{t}^{x}\) and \(F_{t}^{y}\) constant returns to scale. In this case firms have a common technology scaled up by firm-specific Hicks-neutral productivity.
Households
When utility is common across households \(\left(U^{i}\left(\cdot\right)=U^{i'}\left(\cdot\right)\right)\) and homothetic aggregation holds even under heterogeneous endowments. In this case the distribution of wealth does not matter for aggregates.
Definition 2.1: Homothetic Preferences
A preference relation \(\succeq\) is homothetic if and only if \(x\succeq y\) if and only if \(\lambda x\succeq\lambda y\) for all \(\lambda\geq0\).
The implication of homotheticity is that if two bundles are in the same indifference curve, then any scaled version of the bundles are also in the same indifference curve. That is, if \(U\left(x\right)=U\left(y\right)\) then \(U\left(\lambda x\right)=U\left(\lambda y\right)\). The most common example of homothetic utility functions is homogeneous functions, so that \(U\left(\lambda x\right)=\lambda^{\alpha}U\left(x\right)\) for some \(\alpha\geq0\), for this to happen \(U\left(\cdot\right)\) must be a homogeneous function. In words, the shape of indifference curves does not depend on the scale of the consumption bundles.
How does homotheticity deliver aggregation? It implies that the Marshallian demands are proportional to total income: \(x\left(\vec{p},Y\right)=\phi\left(\vec{p}\right)\times Y\), where \(\vec{p}\) is the vector of prices of the bundle \(x\) and \(Y\) is total resources (or lifetime income or wealth). That is, the demand of individuals with different income are always scaled versions of one another, as the slope of the budget constraint is given by common market prices and their problems differ only in the parallel shifts of their budget constraints. This implies that the allocation of a household that has resources \(Y_{i}\), corresponding to a fraction \(\omega_{i}=\frac{Y_{i}}{Y}\) of the aggregate resources, is equal to a fraction \(\omega_{i}\) of the aggregate allocation:
The implication is that aggregate allocations correspond exactly to the weighted sum over individual allocations:
In our problem above the differences between households correspond only to differences in lifetime resources coming from differences in the endowment of initial capital and potential labor supply (you can see this by expressing the budget constraint in the Beckerian fashion, buying leisure out of the value of the time endowment, and replacing the investment choices).
Gorman shows that exact aggregation (so that aggregates depend only on the aggregate amount of resources and not on their distribution) requires that Engel curves are linear with common slopes across individuals. For this to happen the indirect utility must be of the Gorman polar form,
Homothetic preferences are a special case of this in which \(\varphi_{i}\left(\vec{p}\right)=0\).