Solutions to Problem Set 4
EC720.01  Math for Economists
Peter Ireland
Boston College, Department of Economics
Fall 2009
Due Thursday, October 8
The two welfare theorems of economics tell us that optimal and equilibrium resource alloca
tions coincide – but only under certain conditions. Sometimes when market failures prevent
the theorems from holding, however, government policy can help improve equilibrium out
comes, as the questions below reveal.
1. Optimal Allocations
Consider an economy in which output is produced with capital
k
and labor (“hours worked”)
h
according to the CobbDouglas specification
k
α
h
1

α
, where 0
< α <
1. In this static model,
the capital stock
k
is taken as given, but hours worked
h
and consumption
c
are chosen by
a benevolent social planner in order to maximize the utility ln(
c
)

h
of a representative
consumer, where ln denotes the natural logarithm. Hence, an optimal resource allocation
solves the problem
max
h,c
ln(
c
)

h
subject to
k
α
h
1

α
≥
c.
Define the Lagrangian for this problem as
L
(
h, c, λ
) = ln(
c
)

h
+
λ
(
k
α
h
1

α

c
)
and note that the firstorder conditions
L
1
(
h
*
, c
*
, λ
*
) =

1 +
λ
*
(1

α
)
k
α
(
h
*
)

α
= 0
and
L
2
(
h
*
, c
*
, λ
*
) = 1
/c
*

λ
*
= 0
can be combined with the binding constraint
L
3
(
h
*
, c
*
, λ
*
) =
k
α
(
h
*
)
1

α

c
*

0
to form a system of three equations in the three unknowns
h
*
,
c
*
and
λ
*
. Combine the two
firstorder conditions to obtain
c
*
= (1

α
)
k
α
(
h
*
)

α
and then combine this last expression with the binding constraint to obtain
(1

α
)
k
α
(
h
*
)

α
=
k
α
(
h
*
)
1

α
which can be used to find
h
*
= 1

α.
1
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Finally, substitute this last expression back into the binding constraint to find
c
*
=
k
α
(1

α
)
α
.
2. Equilibrium Allocations
Now consider the same economy, but where perfectly competitive markets for inputs and
outputs replace the social planner in allocating resources.
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 Fall '09
 IRELAND
 Economics, representative, Externality, representative consumer, Social Planner

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