Economics 1011b Problem Set 6
Professor Aleh Tsyvinski
The problem set is due next Tuesday, April 17, by 5 pm in your TF’s mailbox in Lit
tauer.
Late problem sets will not be accepted.
You can work in groups (discuss the so
lutions, etc).
However, you must write the solutions by yourself.
Please write down all
derivations/explanations. For clarifications (not answers) please contact Leon Berkelmans
([email protected]).
Exercise 1.
Optimal Taxation
Consider an
N
period economy where some government expenditures
g
needs to be
financed in the first period. The government can use taxes on labor income in all periods,
τ
1
...τ
N
and is free to set these taxes with the only constraint that the intertemporal budget
constraint must hold, i.e.
the DPV of tax revenues must equal g, but the government
can choose whether to tax agents more in the first period, more in the second, etc. We also
assume that the government is benevolent, in the sense that it maximizes households’ utility.
Utility is given by:
U
=
N
t
=1
β
t

1
(
c
t

1
2
h
2
t
)
where
c
t
is consumption in period
t
,
h
t
is labor supplied in period
t
, and
β <
1 is the discount
factor. Assume the real interest rate is equal to
r
, with
β
(1 +
r
) = 1. Also assume that the
wage rate is 1.
a. Define a Ramsey equilibrium, defining both the household problem and the government
problem.
Solution:
The household problem is:
max
c
1
...c
N
,h
1
,...,h
N
N
t
=1
β
t

1
(
c
t

1
2
h
2
t
)
subject to:
N
t
=1
c
t
(
1
1 +
r
)
t

1
=
N
t
=1
(1

τ
t
)
h
t
(
1
1 +
r
)
t

1
Substituting the budget constraint into the utility function gives the problem as:
max
h
1
,...,h
N
N
t
=1
β
t

1
((1

τ
t
)
h
t

1
2
h
2
t
)
The solution to this is:
h
t
= (1

τ
t
)
1
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The governments problem is to maximize households welfare, i.e.
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 Spring '07
 Huang
 Economics, Macroeconomics

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