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International Finance
Fall 2010
assignment: The overlapping generations model
The due date for this assignment is Monday November 22nd.
代写留学生作业Reading assignment: Chapter 3
1. In the overlapping generations model, agents live in 2 periods: they were born in oneperiod (Young) and will die in the next period (Old). The population grows at rate n(for example 2%). Therefore the population dynamic isNt+1 = (1 + n)NtAll agents were born with zero assets and we assume that they leave nothing to theirchildren. The utility of an agent isUt = u
􀀀cY
t


+ u
􀀀
cO
t+1


where u (c) = log c . Also assume that every agent’s labor supply is lY (when Young)
and lO (when Old). The salaries of the agent are then wlY (when Young) and wlO(when Old). The agent can buy government bonds b which gives them an interest r in
the next period. So if in period t the agent buys bt+1 then he will receive (1 + r) bt+1
in period t + 1.
(a) What generation is gonna buy government bonds? Why?
(b) Write down the budget constraint of the agent in each period. What is the
intertemporal budget constraint?
(c) Write down the optimal conditions of the agents. Use them to derive the Euler
equation.
(d) What are the consumption of the agent in each period?
(e) Who is gonna save in the economy? (the Young? the Old?) and how much do
they save?
(f) The production function is Y = AK L1􀀀 . What is output per capita function?
(g) The market clearing condition: what is the supply of labor in the economy? (Hint:
how much labor from the Young? from the Old?) The saving in the economy is
to buy government bonds which are used to build capital stock in the next period.
What is the capital stock in the economy?
(h) From the saving formula that you derived above, write down the formula for the
capital per capita
(i) The depreciation rate of capital is 100% (explain why). Therefore the problem of
the …rm is to maximize
F (K; L) 􀀀 wL 􀀀 rK
Compute the wage and interest rate as functions of the capital per capita.
1
2. In this exercise we will study the 2 social security systems using the overlapping gen-
erations model. The production function is f (k) = k where k is the capital stock per
capita. The utility function is
Ut
􀀀
cY
t ; cO
t+1


= log
􀀀
cY
t


+ log
􀀀
cO
t+1


Individuals are supposed to work when they are young and retired when they are old.
Therefore the individual labor supply is
lY = 1; lO = 0
Population grows at rate n:
(a) The …rst system is called Pay as you go: in this system, each young individuals
pays a lumpt sum tax T, the taxes are used to pay the old generation in that
period.
1. How much does an old individual receive?
2. Write down the budget constraint in each period (Young and Old) of each
person. Use those constraints to write down the intertemporal budget con-
straint.
3. Compute the individual consumptions (how much a person consumes when
Young and then when Old).
4. Compute the aggregate consumption in the …rst period (when the government
starts the Pay as you go system). Does the Pay as you go system increase or
decrease aggregate consumption in the …rst period?
5. Compute the individual saving. Does this system increase or increase the
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