A simple example

Consider an investor who purchases stock in many firms from most global industries. This investor is vulnerable to systematic risk but has diversified away the effects of idiosyncratic risks on his portfolio value; further reduction in risk would require him to acquire risk-free assets with lower returns (such as U.S. Treasury securities). On the other hand, an investor who invests all of his money in one industry whose returns are typically uncorrelated with broad market outcomes (beta close to zero) has limited his exposure to systematic risk but, due to lack of diversification, is highly vulnerable to idiosyncratic risk.

Example: Arrow–Debreu equilibrium

The following example is from Mas-Colell, Whinston, and Green (1995).^[6] Consider a simple exchange economy with two identical agents, one (divisible) good, and two potential states of the world (which occur with some probability). Each agent has expected utility in the form ${\displaystyle \pi _{1}*u_{i}(x_{1i})+\pi _{2}*u_{i}(x_{2i})}$ where ${\displaystyle \pi _{1}}$ and ${\displaystyle \pi _{2}}$ are the probabilities of states 1 and 2 occurring, respectively. In state 1, agent 1 is endowed with one unit of the good while agent 2 is endowed with nothing. In state 2, agent 2 is endowed with one unit of the good while agent 1 is endowed with nothing. That is, denoting the vector of endowments in state i as ${\displaystyle \omega _{i},}$ we have ${\displaystyle \omega _{1}=(1,0)}$, ${\displaystyle \omega _{2}=(0,1)}$. Then the aggregate endowment of this economy is one good regardless of which state is realized; that is, the economy has no aggregate risk. It can be shown that, if agents are allowed to make trades, the ratio of the price of a claim on the good in state 1 to the price of a claim on the good in state 2 is equal to the ratios of their respective probabilities of occurrence (and, hence, the marginal rates of substitution of each agent are also equal to this ratio). That is, ${\displaystyle p_{1}/p_{2}=\pi _{1}/\pi _{2}}$. If allowed to do so, agents make trades such that their consumption is equal in either state of the world.

Now consider an example with aggregate risk. The economy is the same as that described above except for endowments: in state 1, agent 1 is endowed two units of the good while agent 2 still receives zero units; and in state 2, agent 2 still receives one unit of the good while agent 1 receives nothing. That is, ${\displaystyle \omega _{1}=(2,0)}$, ${\displaystyle \omega _{2}=(0,1)}$. Now, if state 1 is realized, the aggregate endowment is 2 units; but if state 2 is realized, the aggregate endowment is only 1 unit; this economy is subject to aggregate risk. Agents cannot fully insure and guarantee the same consumption in either state. It can be shown that, in this case, the price ratio will be less than the ratio of probabilities of the two states: ${\displaystyle p_{1}/p_{2}<\pi _{1}/\pi _{2}}$, so ${\displaystyle p_{1}/\pi _{1}<p_{2}/\pi _{2}}$. Thus, for example, if the two states occur with equal probabilities, then ${\displaystyle p_{1}<p_{2}}$. This is the well-known finance result that the contingent claim that delivers more resources in the state of low market returns has a higher price.

In heterogeneous agent models

While the inclusion of aggregate risk is common in macroeconomic models, considerable challenges arise when researchers attempt to incorporate aggregate uncertainty into models with heterogeneous agents. In this case, the entire distribution of allocational outcomes is a state variable which must be carried across periods. This gives rise to the well-known curse of dimensionality. One approach to the dilemma is to let agents ignore attributes of the aggregate distribution, justifying this assumption by referring to bounded rationality. Den Haan (2010) evaluates several algorithms which have been applied to solving the Krusell and Smith (1998) model, showing that solution accuracy can depend heavily on solution method.^[7][8] Researchers should carefully consider the results of accuracy tests while choosing solution methods and pay particular attention to grid selection.