1 permanent income hypothesis with equilibrium interest rates consider

Question

1. Permanent income hypothesis with equilibrium interest rates
Consider an economy in which each agent maximises her lifetime utility:
U = Σ B¹ In (C₂),
t=0
where c corresponds to her consumption at period t and 3 € (0, 1) is her subjective
discount rate. Assume that the agent can borrow or lend at a given sequence of
real interest rates {r} and that the agent receives a deterministic sequence of
perishable endowment {y}io. Let be be the asset position of the agent at the start
of period t and assume that bo = 0. The one-period budget constraint of the agent is
c₂ + b₁+1 = y + (1+r)b₁.
There is no possibility of a Ponzi-scheme.
(a) Set up the maximisation problem of the agent and derive the Euler equation.
Give an economic interpretation of your result.
(b) Solve for consumption at period t = 0. Comment on your result.
Now assume that this economy has a total of N agents and that there are two types
of agents, A and B, which are indicated by a superscript. Half of agents are of type
A and have the following endowment profile:
y = {1,0,1,0,1,0,...).
The other agents are of type B and have the endowment profile
y = {0, 1, 0, 1,0, 1,...}.
Therefore, type-A agents have one unit of the endowment goods in even periods,
while type-B agents have one unit of the endowment goods in odd periods./n(c) Write down the market clearing condition for the goods market in each period
and use it to get an equilibrium condition between cand cf for even periods
and for odd periods.
(d) Use the equilibrium condition for the goods market and the Euler equation to
derive the equilibrium real interest rate in each period. Solve for the levels of
consumption and c in equilibrium. Interpret your results.