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)
.L
e
t
t
i
n
g
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
.F
i
rm2i
sown
edbycon
sum
e
r
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
2
2
)
, such that:
(i)
(
x
1
1
)
solves:
maxlog(
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
maxlog(
x
2
`
2
)
subject to
x
2
+
p`
2
=
p
+
¼
2
(
x
2
2
)
¸
0
:
1