ECON 2508 The Net Expenditure on Purchasing Contracts Question
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FE II – Week 10 Tutorial Questions
Consider the following 2 person, 1 good economy with two possible states of nature.
There are two states of nature j 2 {1, 2} and two individuals, i 2 {A, B}. In stateof-nature j = 1 the individual i receives income yi,1 whereas in state-of-nature j = 2,
individual i receives income yi,2 . Let ci,j denote the amount of the consumption good
enjoyed by individual i if the state-of-nature is j. State-of-nature j occurs with probability
?j and ?1 + ?2 = 1.
Prior to learning the state-of-nature, individuals have the ability to purchase (or sell)
contracts that specify delivery of the consumption good in each state-of-nature. There
are two assets. Each unit of asset 1 pays one unit of the consumption good if the stateof-nature is revealed to be state 1. Each unit of asset 2 pays one unit of the consumption
good in each state-of-nature. Let ai,j denote the number of asset j 2 {1, 2} purchased
by individual i. The relative price of asset 2 is p. In other words, it costs p units of
asset 1 to obtain a single unit of asset 2 so that asset 1 serves as the numeraire (its price
is normalized to one and relative prices are expressed in units of asset 1). Individuals
cannot create wealth by making promises to deliver goods in the future so the total net
expenditure on purchasing contracts must equal zero, that is, ai,1 + pai,2 = 0. Individual
is consumption in state-of-nature j is equal to his/her realized income, yj , plus the
realized return from his/her asset portfolio.
The timing is as follows: individuals trade in the asset market, and once trades are
complete, the state-of-nature is revealed and asset obligations are settled.
The individuals objective function is
max
ci,1 ,ci,2 ,ai,1 ,ai,2
{?1 u(ci,1 ) + ?2 u(ci,2 )} .
1. Write down each individuals optimization problem.
2. Write down the Lagrangean for each individual.
3. Solve for each individuals optimality conditions.
4. Define an equilibrium.
5. Provide the equilibrium conditions that characterize the equilibrium allocations in
the market for contracts.
6. Let the utility function u(c) = ln(c) so that u0 (c) = 1c . Solve for the equilibrium
price and allocations.
1
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Tags:
Financial Economics
asset market
purchasing contracts
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