Consider an economy that consists of two consumers, named consumer i = 1, 2, and lasts two periods, period 0 and period 1. Consumers have no endowment in period 0 and do not value consumption
in that period. There are two possible events in period 1, Event p, a pandemic, and Event n, the normal
state. The pandemic happens with probability 0.1. Both consumers are endowed with one unit of
the consumption good in Event n. Consumer 1 loses his income in Event p but consumer 2’s income
remains the same. Consumer i’s period-0 expected utility from ci
n units of consumption in Event n
and ci
p units of consumption in Event p is
Ui = 0.9 ⇥ ln ci
n + 0.1 ⇥ " ln ci
p
where > 1 represents risk aversion, " < 1 represents lower demand for consumption during the
pandemic.
[25 marks]
(a) The social planner’s objective is to maximize ✓U1+(1✓)U2 by allocating consumption, where
✓ is the Pareto weight on consumer 1 and 1✓ the Pareto weight on consumer 2. 0 < ✓ < 1. We
say that the social planner’s optimal consumption allocation is Pareto optimal. Write down the
constrained optimization problem of the social planner and solve for the optimal consumption
allocation.
[6 marks]
(b) Consider an economy with a complete set of contracts, which includes a contract that pays 1
unit of the consumption good only in Event n and a contract that pays 1 unit of the consumption
good only in Event p. In period 0, there is a competitive market where all financial contracts are
traded. The market price for the contract that pays one unit of the consumption good only in
event e is qe, for e = n, p. Define a competitive equilibrium for this economy.
[4 marks]
(c) Assume that the consumption allocation in the competitive equilibrium is Pareto optimal. By
Walras Law, we can normalize qn to one. Solve for qp and consumption allocations ci
e for i = 1, 2
and e = n, p in equilibrium and the corresponding Pareto weight on consumer 1, ✓ (the Pareto
weight on consumer 2 being 1 ✓).
[9 marks]
(d) Consider the following two assets: Asset A pays 1 in both events, Asset B pays 10/9 in event
n, 0 in event p. Compare the value of these two assets. Can the risky Asset B be more valuable
than the safe Asset A? How does your answer depend on the demand for consumption during
the pandemic represented by parameter "?
Sample Solution