ECON 201, Summer 2016
July 23, 2016
Problem Set 3 Solutions
1. For the following production functions: F (K, L) = 2K 1/3 L2/3 , F (K, L) =
2K + 3L and F (K, L) = min{2K, L}:
(a) Draw any two isoquants.
• F (K, L) = 2 → 2K 1/3 L2/3 = 2 → K 1/3 L2/3 = 1 → K(L) =
1/L2
F (K, L) = 2K 1/3 L2/3 = 4 → K 1/3 L2/3 = 2 → K(L) = 8/L2
• F (K, L) = 2 → 2K + 3L = 6
F (K, L) = 2 → 2K + 3L = 12
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• F (K, L) = 2 → min{2K, L} = 2
F (K, L) = 4 → min{2K, L} = 4
(b) Find whether they have diminishing, constant or increasing returns to labor.
• M PL = 2K 1/3 L1/3
Taking the derivative of the marginal product of labor, we get
∂M PL
< 0. Therefore, there is decreasing marginal returns
∂L
to labor.
• M PL = 3, taking the derivative of the marginal product of
∂M PL
labor we get
= 0. Therefore, there is constant returns
∂L
to labor.
2
•
(
1 if 2K ≥ L
M PL =
0 if 2K < L
and
∂M PL
=
∂L
(
0 if 2K ≥ L
0 if 2K < L
Therefore, this production function exhibits constant returns
to labor.
(c) Find whether they have diminishing, constant or increasing returns to scale.
• For α > 1, we have
F (αK, αL) = 2(αK)1/3 (αL)2/3 = 2αK 1/3 L2/3 = αF (K, L)
Therefore, this production function exhibits constant returns
to scale.
• For α > 0, we have
F (αK, αL) = 2αK + 3αL = α(2K + 3L) = αF (K, L)
Therefore, this production function exhibits constant returns
to scale.
•
F (αK, αL) = min{2αK, αL} = α min{2K, L} = αF (K, L)
Therefore, this production function also exhibits constant returns to scale.
(d) Compute the marginal product of input 1 for x2 = 10 and x1 = 4.
• M PK = (2/3)K −2/3 L1/3 = 2/3(4)−2/3 (10)2/3
• M PK = 2
3
• With (K, L) = (4, 10), we have F (K, L) = 8. When we increase the amount of kapital by one unit, i.e., (K, L) = (5, 10),
we have F (K, L) = 10. Therefore M PK = 2 at this point.
You can also check the marginal production function, i.e:
(
2 if 2K < L
M PK =
0 if 2K ≥ L
and we already know that 2K < L at (4, 10).
(e) Compute the marginal product of input 1 for x2 = 10 and x1 = 6
• M PK = (2/3)K −2/3 L1/3 = 2/3(6)−2/3 (10)2/3
• M PK = 2
• With (K, L) = (6, 10), we have F (K, L) = 10. When we
increase the amount of kapital by one unit, i.e., (K, L) =
(7, 10), we have F (K, L) = 10. Therefore M PK = 0 at this
point. You can also check the marginal production function,
i.e:
(
2 if 2K < L
M PK =
0 if 2K ≥ L
and we already know that 2K ≥ L at (6, 10).
(f) Now assume that the price per capital is r and the price per labour
is w. Calculate the long-run cost functions for each production
function. Find also the average and marginal costs.
• F (K, L) = 2K 1/3 L2/3 , from the optimality condition:
2K
w
M PL
=
=
M PK
L
r
2Kr
L=
w
then let F (K, L) = X,
X =2K 1/3 (
2Kr 2/3
2r
) = 2( )2/3 K
w
w
1 2r
K(X) = ( )−2/3 X
2 w
2rK(x)
1 2r
L(X) =
= ( )1/3 X
w
2 w
4
then we find the total cost:
C(X) = wL(X) + rK(X)
1 2r
1 2r
= w ( )1/3 X + r ( )−2/3 X
2 w
2 w
2/3 1/3 −2/3
= w r (2
+ 2−5/3 )X
Moreover, we know that AC(X) = C(X)/X and M C(X) =
∂C(X)
. In particular, in this case AC(X) = M C(X)
∂X
AC(X) = M C(X) = w2/3 r1/3 (2−2/3 + 2−5/3 )
• F (K, L) = 2K + 3L. From the optimality condition we have:
M PL
3
=
M PK
2
w
3
< , then it is optimal to use only L, i.e.,
r
2
L(X) = X/3 and K(X) = 0.
w
3
If > then it is optimal to use only K, i.e., L(X) = 0 and
r
2
K(X) = X/2.
w
3
If
=
then any combination is optimal, i.e., L(X) =
r
2
X
(X − 2a)/3 and K(X) = 2a where a ∈ [0, ]. Therefore,
2

wX
w
3


if
≤

 3
r
2
C(X) =



 rX if w > 3
2
r
2
w r
in other words, C(X) = min{ , }X and
3 2
w r
AC(X) = min{ , }
3 2

w
3


w/3 if
≤


r
2
M C(X) =



r/2 if w > 3
r
2
This implies if
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• F (K, L) = min{2K, L} → 2K = L → X = 2K Therefore we
X
and L(X) = X. Then
have K(X) =
2
X
r
C(X) = wX + r = (w + )X
2
2
r
AC(X) = M C(X) = w +
2
√
2. A firm has a production function f (x1 , x2 ) = x1 x2 . Prices are as
follows: p = 5 for the output, and w1 = 1 and w2 = 2 for the inputs.
The amount of input 2 is fixed at x2 = 4 and can not be changed.
(a) Compute the profit maximizing choice of inputs and output.
π = pf (x1 , x2 ) − c(x1 , x2 )
√
= 5 4x1 − x1 − 8
√
= 10 x1 − x1 − 8
Taking derivative with respect to y, we have
10
=0
√
2 x1 − 1
√
then, when we solve for x1 , we find x∗1 = 25 and y = 25.4 = 10
(b) Compute profits. Substitute the optimal levels in to the profit
function:
π = 5.10 − 25 − 8 = 17
3. Consider a firm with the same production function of question 2. Suppose that the firm wants to produce 40 units of output. If w1 = 40 and
w2 = 10, what is the cost minimizing input levels that will produce
these 40 units?
From the optimality condition, we have
x1
w1
T RS = − = −
x2
w2
40
x1
=
=4
x2
10
x∗1 = 4x∗2
6
then we need to substitute this into the production function to find the
exact values:
√
x1 x 2
p
= 4x∗2 x2 = 2x∗2 = 40
y = f (x1 , x2 ) =
We have x∗1 = 80 and x∗2 = 20.
4. All tobacco producers in Indonesia have the following cost function
(both in the short-run and in the long-run): C(x) = 8 + (1/8)x2 when
x > 0, and C(0) = 0. Assume that there is pure competition and the
demand function for tobacco is D(p) = 72 − 16p, where p denotes the
price of tobacco.
(a) Find the industry’s short-run supply function if there are four
firms.
x
4
8 x
AC(x) = +
x 8
M C(x) =
We know that M C(x) = AC(x), then x∗ = 8 and p∗ = 2. Therefore, we have the following


0 if p < 2
Xi (p) = {0, 8} if p = 2


4p if p > 2


0 if p < 2
X(p) = {0, 8, 16, 24, 32} if p = 2


16p if p > 2
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(b) Find the long-run competitive equilibrium price, per firm output,
industry output, and number of firms.
We already know that Xi∗ = 8 and p2 = 2. Then D(2) = 72 − 16 ∗
2 = 40 and n∗ = 40/8 = 5. In the long run, there will be 5 firms
in the market for tobacco.
5. Find under which conditions, i.e. for what values of a, b and c the following production functions exhibit increasing,constant and decreasing
returns to scale, where a, b and c > 0.
(a) F (K, L) = cK a Lb
F (αK, αL) = c(αK a )(αLb )
= αa+b cK a Lb = αa+b F (K, L)
If a + b < 1, then this function exhibits diminishing returns to
scale.
If a + b = 1, then this function exhibits constant returns to scale.
If a + b > 1, then this function exhibits increasing returns to scale.
(b) F (K, L) = min{aK, bL}
F (αK, αL) = min{αaK, αbL} = α min{aK, bL} = αF (K, L)
For any value of a and b, this production function exhibits constant
returns to scale.
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(c) F (K, L) = c(aK + bL)
F (αK, αL) = c(αaK + αbL) = αc(aK + bL) = αF (K, L)
For any value of a,b and c, this production function exhibits constant returns to scale.
6.
• 22.2 The cost function given as c(y) = y 3 − 8y 2 + 30y + 5
(a) M C(y) = 3y 2 − 16y + 30
(b) AV C(y) = y 2 − 8y + 30
(c) The graph is as follows:
(d) Average variable cost is falling as output rises if output is less
than 4 and rising as output rises if output is greater than 4
(You can find this threshold value by finding the minimum of
average variable cost).
(e) Marginal cost equals to average variable cost when output is
4.
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(f) The firm will supply 0 output if price is less than 14 (Substitute y = 4 into the marginal cost function)
(g) The smallest positive amount that the firm is ever supply at
any price is 4. At what price the firm supply exactly 6 units of
output? p=42 (Substitute 6 into the marginal cost function).
• 23.8 Suppose all firms in a given industry have the same supply
curve given by Si (p) = p/2. Plot and label the 4 industry supply curves generated by these firms if there are 1,2,3 or 4 firms
operating in the industry.
(a) If all the firms had a cost structre such that if the price was
below $3, they would be losing money, what would be the
equilibrium price and output in the industry if the market
demand was equal to D(p) = 3.5?
p
3.5p = n
2
7
=n
p
10
and we know that p > 3, therefore we have 3.5 and n = 2.
The out put in the industry would be 3.5.
(b) What if the identical conditions as above held except that the
market demand was equal to D(p) = 8 − p? Now what would
be the equilibrium price and output? How many firms would
operate in such a market?
8−p=
pn
2
16
>3
2+n
→ 16 > 6 + 3n → 10 > 3n
16 − 2p = pn → p =
Therefore, we conclude that n = 3, p = 3.2 and output is
8 − 3.2 = 4.8
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