The Ohio State University
Department of Economics
Econ. 805
Winter 2004
Prof. James Peck
Problem Set #5 Answers
1.
In the following economy, there is one consumer, two firms, two inputs (capital and
labor), and two outputs (goods 1 and 2). The consumer is endowed with 100 units of capital,
400 units of labor, and no output. Denote the consumption vector of the two outputs and the two
inputs by: x = (x1 , x2 , xK , xL ). The utility function is
u( x1 , x 2 , x K , x L ) = log( x1 ) + log( x 2 ).
In other words, capital and labor are inelastically supplied to the market.
Firm 1 produces good 1 according to the production function,
y11 = 2 K1 L1
where K1 denotes the (nonnegative) capital input and L1 denotes the (nonnegative) labor input
used by firm 1.
Firm 2 produces good 2 according to the production function,
y22 =
K2 L2
where K2 denotes the (nonnegative) capital input and L2 denotes the (nonnegative) labor input
used by firm 2. Both firms are owned by the consumer.
(a) Define a competitive equilibrium for this economy.
(b) Calculate the competitive equilibrium price vector and allocation.
(c) Explain whether the competitive equilibrium allocation is Pareto optimal.
Answer: (a) A Competitive Equilibrium is a price vector, ( p1 , p 2 , p K , p L ) , and an
allocation,
( x1 , x 2 , x K , x L , y11 , K1 , L1 , y22 , K2 , L2 ) , such that:
1. (x1 , x2 , xK , xL ) solves
max
subject to
log( x1 ) + log( x 2 )
p1 x1 + p2 x2 + pK xK + pL xL # 100 pK + 400 pL
(x1 , x2 , xK , xL ) $ 0 .
(Because of constant returns to scale, there are zero profits.)
2. ( y11 , K1 , L1 ) solves
max
p1 y11 − p K K1 − p L L1
subject to
y11 = 2 K1 L1
3. ( y22 , K2 , L2 ) solves
p 2 y22 − p K K2 − p L L2
max
subject to
y22 =
K2 L2
4.
x1 = y11 , x 2 = y22 , K1 + K2 = 100, L1 + L2 = 400
(Market clearing for capital and labor uses the fact that the consumer does not demand these
goods. Alternatively, you could have put xK and xL into the relevant equations.)
(b) We first calculate the consumer’s demand functions. Demand for capital and leisure is zero,
so we have xK = 0 and xL = 0. The marginal rate of substitution must equal the price ratio for
goods 1 and 2, so we have: p1 x1 = p2 x2 . Using this equation and the budget equation, we solve
for
100 p K + 400 p L
x =
2 p1
1
100 p K + 400 p L
and x =
.
2 p2
2
Substituting the constraint into firm 1's objective, we obtain the unconstrained problem
max 2 p1 ( K1 L1 )1/ 2 − p K K1 − p L L1 .
Differentiating with respect to K1 and L1 , we have
p1 ( K1 L1 ) −1/ 2 L1 = p K and p1 ( K1 L1 ) −1/ 2 K1 = p L .
Dividing the left and right sides of the above equations, we have the MRTS condition:
pK
L1 = L K1 .
p
If we substitute the MRTS condition into one of the first order conditions in an attempt to solve
for K1 , we have a condition on prices (for which finite, positive output is supplied)
p K p L = ( p1 ) 2 .
The same procedure for firm 2 yields:
pK
L2 = L K2 and 4 p K p L = ( p 2 ) 2 .
p
Now we use market clearing. Using market clearing for K and L, and using the MRTS
conditions, we have:
K1 + K2 = 100,
pK
L1 + L2 = L ( K1 + K2 ) = 400 .
p
Therefore, pK / pL = 4. Normalizing pL = 1, we have pK = 4. From the above conditions on
prices, we have p1 = 2 and p2 = 4. We now plug these prices into the demand functions to get
the consumption: (x1 , x2 , xK , xL ) = (200, 100, 0, 0). Market clearing for goods implies y11 =
200, and y22 = 100. Using the MRTS condition, we have: K1 = K2 = 50 and L1 = L2 = 200.
(c) Because the utility functions satisfy local nonsatiation, we can apply the FFTWE to conclude
that the competitive equilibrium allocation is Pareto optimal. Alternatively, you could have
shown that the marginal rate of substitution equals the marginal rate of transformation, and that
the marginal rates of technical substitution are equal for the two firms.
2.
Consider the following economy with two states of nature, a and b, and one physical
commodity per state of nature. Denote the probabilities of the two states as Ba and Bb . There
are 100 consumers. For each i = 1, ... , 100, consumer i has the utility function
Vi = π a log( xia ) + π b log( xib ) .
For i = 1, ... , 25, consumer i has the endowment vector (1,4), and for i = 26, ... , 100, consumer i
has the endowment vector (2,1). Before the state is observed, the consumers trade commodity a
for commodity b (state contingent commodity markets).
(a) Define a competitive equilibrium.
(b) Calculate the competitive equilibrium price vector and allocation.
(c) How would your answer to part (b) change if we changed the utility functions, so that “log”
is replaced with “arctan” ? [Hint: arctan is a strictly monotonic, differentiable, and strictly
concave function.]
Answer: A competitive equilibrium is a price vector, (pa , pb), and an allocation, {xia , xib}i=1...100 ,
such that
1. For i = 1, ... , 25, (xia , xib) solves
a
b
max
π a log( xi ) + π b log( xi )
subject to
p a xia + pb xib = p a + 4 pb
(xia , xib) $ 0.
2. For i = 26, ... , 100, (xia , xib) solves
a
b
max
π a log( xi ) + π b log( xi )
subject to
p a xia + pb xib = 2 p a + pb
(xia , xib) $ 0.
3.
∑i =1 xia = 175
100
and
∑i =1 xib = 175.
100
(b) Since there is no aggregate uncertainty, calculating the competitive equilibrium is simple.
The price ratio equals the probability ratio, so we have (pa , pb) = (Ba , Bb). Also, xia = xib for all
i. For i = 1, ... , 25, we calculate from the budget constraint that consumer i receives
xia = xib = (Ba + 4Bb), and for i = 26, ... , 100, we calculate from the budget constraint that
consumer i receives xia = xib = (2Ba + Bb).
To compute the competitive equilibrium directly, equate the marginal rate of substitution
to the price ratio: Ba xib/Bb xia = pa / pb. Solve the MRS equation and the budget equation for the
demand functions of the two types of consumers. Then use market clearing to solve for the price
ratio, where we multiply the demand of a type 1 consumer by 25 and the demand of a type 2
consumer by 75.
(c) Because there is no aggregate uncertainty, the price vector and allocation are the same as in
part (b). You could compute the answer directly by remembering that the derivative of arctan(x)
is 1/(1+x2), but I would not recommend that approach.