Editing Systematic risk (section)

===Example: Arrow–Debreu equilibrium===

The following example is from [[Mas-Colell, Whinston, and Green (1995)]].<ref>{{cite journal |last1=Mas-Colell |first1=A. |last2=Whinston |first2=M. |last3=Green |first3=J. |year=1995 |title=Microeconomic Theory |url=https://archive.org/details/microeconomicthe00masc_944 |url-access=registration |location=New York |publisher=Oxford University Press |pages=692–693 }}</ref> 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 <math>\pi_1*u_i(x_{1i})+\pi_2*u_i(x_{2i})</math> where <math>\pi_1</math> and <math>\pi_2</math> 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 <math>\omega_i,</math> we have <math>\omega_1=(1,0)</math>, <math>\omega_2=(0,1)</math>. 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, <math>p_1/p_2=\pi_1/\pi_2</math>. 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, <math>\omega_1=(2,0)</math>, <math>\omega_2=(0,1)</math>. 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: <math>p_1/p_2<\pi_1/\pi_2</math>, so <math>p_1/\pi_1<p_2/\pi_2</math>. Thus, for example, if the two states occur with equal probabilities, then <math>p_1<p_2</math>. 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.