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Constructing an equilibrium
Households live two periods and have prefernces
U(c1)+βU(c2)
where 0<β<1 and U is the utility function and satisfies our usual assumptions. There are N households in the economy. N1 of these have endowments y1 in the first period and no endowment in the second-these agents are called "Type 1". The remaining N2 have no endowment in the firs period and y2 in the second period- these agents are called "Type 2". Hencethe resources of the economy are
N1y1
in the first period and
N2y2 in the second, where
N=N1+N2
Households have access to a credit market where the can borrow (s<0) or save s<0. The type 1 agent faces budget constraints
y1=c11+s1
rs1=c21
where the consumption for the type i agent in period j is denoted cji. The type 2 agent faces budget constraints
0=c12+s2
y2+rs2=c22
The resource constraints are
N1y1=N1c11+N2c12
N2y2=N11c21+N2c22
a) state the maximization problem solved by each type of agent and derive the fist order and second order conditions. Derive the solution using the implicit function theorem.
b) Determine the equilibrium conditions for the three markets using the resource constraints and the budget constraints. provide statement of the equilibrium
c) Assume logarithmic utility U(c)=In(c) and derive a closed form solution for consumption in both periods and savings for both types of agents
d) Solve the social planning problem. Compare the solution of the social planning problem with the competitive equilibrium. Demonstrate that the decentralized solution solves the social planning problem for a particular set of pareto weights. Explain how this is an example of the first welfare theorem.
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