Question

Consider a pure exchange economy with 2 consumers, 1 and 2, who trade financial assets. There are two states of the world, a and b. State a occurs with probability 2/3 and state b occurs with probability 1/3. Two assets are traded: money in state a and money in state b. Consumer 1 is risk neutral (i.e. has von Neumann-Morgenstern utility u(x) = x) and consumer 2 is risk averse with von Neumann-Morgenstern utility v(x)=vx. Consumer 1's endowment is (6,6) and consumer 2's endowment is (0,4); that is, consumer 1 is endowed with 6 dollars in each state while consumer 2 is endowed with 0 dollars in state a and 4 dollars in state b.

(a) Find all Walrasian equilibria of this economy.
(b) How would your answer change if consumer 2 was more risk averse?

Answer 1

(a) The Walrasian equilibria for this economy involve Consumer 1 receiving 3 units of money in state a and 3 units in state b, while Consumer 2 holds 3 units in state b.

(b) If Consumer 2 were more risk averse, their equilibrium allocation would tend towards holding more money in state b, resulting in Consumer 1 also holding more money in state b and less in state a.

Step-by-step explanation:

In the initial scenario, to reach a Walrasian equilibrium, the allocation must satisfy the conditions where each consumer maximizes their utility given the endowment and prices, while the markets clear. Consumer 1, being risk neutral, divides their endowment equally between states a and b. Consumer 2, being risk averse, will allocate more wealth towards the state they prefer more due to their utility function's concavity.

(a) The equilibrium distribution is found by considering the marginal rate of substitution (MRS) of both consumers in each state, equating them to the relative price ratio. In this case, Consumer 1's MRS is 1 in both states, reflecting their risk neutrality, while Consumer 2's MRS is v in state b. Setting these equal to the price ratio yields the equilibrium allocation.

(b) If Consumer 2 were more risk averse, their preference for state b would increase. This change in risk aversion would lead to Consumer 2 favoring state b even more, potentially allocating more wealth to that state. This adjustment would influence Consumer 1's allocation, as their equilibrium is interdependent with Consumer 2's decisions in this exchange economy.