The Ohio State University
Department of Economics
Econ 805–Extra Problems on Production and Uncertainty: Questions and
Answers
Winter 2003
Prof. Peck
(1) In the following economy, there are two consumers, two …rms, and two
goods (labor/leisure and food). For i = 1,2, consumer i is endowed with zero
units of food and 1 unit of leisure,
!
i
= (0
;
1)
.
Letting
x
i
denote consumer
i’s consumption of food and
`
i
denote consumer i’s consumption of leisure, the
utility function is:
log(
x
i
) + log(
`
i
)
.
Let
y
1
denote …rm 1’s output of food and
L
1
denote …rm 1’s labor input (so
that
L
1
must be nonnegative). Then …rm 1’s production function, the frontier
of its production set, is given by:
y
1
=
AL
1
, where the parameter A is a positive
real number. Firm 1 is owned by consumer 1.
Let
y
2
denote …rm 2’s output of food and
L
2
denote …rm 2’s labor input (so
that
L
2
must be nonnegative). Then …rm 2’s production function, the frontier
of its production set, is given by:
y
2
= (
L
2
)
1
=
2
. Firm 2 is owned by consumer
2.
(a) De…ne a competitive equilibrium for this economy.
(b) Calculate the competitive equilibrium price vector and allocation, as a
function of the parameter, A. Assume that we have an interior solution, where
both …rms produce output.
(c) For what values of the parameter, A, will we have a corner solution,
where one of the …rms produces zero output?
Answer:
(a) Normalizing the price of food to be 1 and denoting the price of labor
as
p
, a Competitive Equilibrium is a price vector,
(1
; p
)
, and an allocation,
(
x
1
; `
1
; x
2
; `
2
; y
1
; L
1
; y
2
; L
2
)
, such that:
(i)
(
x
1
; `
1
)
solves:
max log(
x
1
) + log(
`
1
)
subject to
x
1
+
p`
1
=
p
(
x
1
; `
1
)
¸
0
:
This relies on the fact that utility is monotonic and …rm 1 has CRS and receives
zero pro…ts at the CE.
(ii)
(
x
2
; `
2
)
solves
max log(
x
2
) + log(
`
2
)
subject to
x
2
+
p`
2
=
p
+
¼
2
(
x
2
; `
2
)
¸
0
:
1
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(iii)
(
y
1
; L
1
)
solves
max
y
1
¡
pL
1
subject to
y
1
=
AL
1
L
1
¸
0
:
(iv)
(
y
2
; L
2
)
solves
max
y
2
¡
pL
2
subject to
y
2
=
(
L
2
)
1
=
2
L
2
¸
0
:
(v)
x
1
+
x
2
=
y
1
+
y
2
`
1
+
`
2
+
L
1
+
L
2
=
2
:
(Equalities follow from strict monotonicity of utility.)
(b) Starting with the pro…t maximization conditions, for an interior solution
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 Spring '08
 PECK
 Economics, Utility, Consumer, pareto, rst order conditions

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