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FE II - Week 9 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 state- of-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.   Let ai,j   denote the number of consumption goods that individual i contracts to deliver in the event that state-of-nature j is realized.  The relative price of contracts that specify delivery of goods in state-of-nature 2 is denoted by p. 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.

The individual’s problem is

max      {⇡1u(ci,1)+ ⇡2u(ci,2)}

subject to the budget constraints ci,1  = yi,1 + ai,1 , ci,2  = yi,2 + ai,2  and ai,1 + pai,2  = 0.

1. Write down the Lagrangean for each individual.

2.  Solve for each individual’s optimality conditions.

3. Define an equilibrium.

4. Provide the equilibrium conditions that characterize the equilibrium allocations in the market for contracts.

5. Let the utility function u(c) = ln(c) so that u0 (c) =  .  Solve for the equilibrium price and allocations.