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# Consider an overlapping generation model where housing is the only asset. Representative

### Consider an overlapping generation model where housing is the only asset. Representative

Consider an overlapping generation model where housing is the only asset. Representative agent born in t solves the following optimization problem subject to his budget
constraint:
max log(ct+1) + ? log(ht)
s.t. ptht = wt
pt+1ht(1 - d) = ct+1
where ht denotes the housing service that agents buy in period t. pt and pt+1 represent
the price for 1 unit of housing service in t and t + 1, respectively. wt
is the labor income
agent receives in t (youth), he does not receive any labor income in t + 1 (old).
(a) Solve representative agent’s optimization problem. Derive housing price pt as a function of pt-1 from he housing market clear condition in t. (Hint: demand of housing
in t is from young agent born in t, and supply of housing in t is from old agent born
in t - 1).
(b) Suppose agents can buy houses for investment purpose or for the purpose of living in.
Only when they live in the house, they obtain utilities from housing services. However, if they do not live in the house, houses do not depreciate. Agent’s optimization
problem becomes:

Abhinav 05-Dec-2019