1. A firm uses capital (K) and labor (L) to produce output (q) using the following production function:
q = f(K, L) = min{2K, 3L}
The wage rate (w) is $30 per unit of labor, and the rental rate of capital (r) is $40 per unit of capital.
a. What is the cost-minimizing combination of capital and labor (K* and L*) that this firm should
use to produce 30 units of output?
. What is this firm’s total cost of production associated with your answer to part a?
c. Illustrate your answers to parts a and b on an appropriate diagram. Clearly label the axes of your
diagram and any curves you draw, and the numerical values of any important points on your
diagram.
d. Derive this firm’s long run cost function, C(q).
e. Clearly explain the relationship between your answers to parts b and d of this question.
f. What is this firm’s long run marginal cost of production?
g. Illustrate your answer to part f on an appropriate diagram. Clearly label the axes of your diagram
and any curves you draw.
2. Consider a firm that uses 3 factors of production:
High-skilled labor (H), Low-skilled labor (L), and capital (K).
This firm’s production function is q = f(H, L, K) = 2HK + LK.
The wage rate for High-skilled workers (wH) is $30 per unit of High-skilled labor, the wage rate for Low-
skilled workers (wL) is $10 per unit of Low-skilled labor, and the rental rate of capital (r) is $20 per unit of
capital.
a. Find the marginal product of High-skilled labor (MPH), the marginal product of Low-skilled labor
(MPL), and the marginal product of capital (MPK) for this firm’s production function.
. Clearly explain (in words) what your answer for the marginal product of Low-skilled labor (MPL)
from part a means.
c. Compare your answers for the marginal product of High-skilled labor (MPH) and the marginal product
of Low-skilled labor (MPL) from part a. Clearly explain (in words) what this means.
d. If this firm uses H = 20, L = 10, and K = 36, how much output can this firm produce?
e. What is this firm’s total cost of production when H = 20, L = 10, and K = 36?
f. H = 20, L = 10, and K = 36 is NOT the cost-minimizing combination of inputs that this firm could
use to produce the amount of output that you found in part d of this question as cheaply as
possible. Clearly explain why not, and suggest how this firm should adjust its inputs to reduce the
cost of producing the amount of output that you found in part d of this question.
g. What is this firm’s cost-minimizing combination of inputs to produce the amount of output that
you found in part d of this question, and how much does it cost?
3. Consider an economy with two industries: Industry 1 and Industry 2.
The production function in Industry 1 is q1 = f1(K1, L1) = ?1
2
3 ∙ ?1
1
3
The production function in Industry 2 is q2 = f2(K2, L2) = ?2
3
4 ∙ ?2
1
4
In this economy, the wage rate (w) is $10 per unit of labor, and the rental rate of capital (r) is $20 per
unit of capital.
a. Find the marginal product of labor (MPL), the marginal product of capital (MPK), band the marginal
ate of technical substitution (MRTS) in Industry 1.
. Find the marginal product of labor (MPL), the marginal product of capital (MPK), and the marginal
ate of technical substitution (MRTS) in Industry 2.
c. Find the value of
?1
∗
?1
∗ in Industry 1, where ?1
∗ is the cost-minimizing amount of capital required to
produce a given quantity (q1), and ?1
∗ is the cost-minimizing amount of labor required to produce a given
quantity (q1).
Hint 1: Your answer should be a number.
Hint 2: You do NOT have enough information to calculate individual values for ?1
∗ and ?1
∗ , but you DO
have enough information to calculate
?1
∗
?1
∗
d. Find the value of
?2
∗
?2
∗ in Industry 2, where ?2
∗ is the cost-minimizing amount of capital required to
produce a given quantity (q2), and ?2
∗ is the cost-minimizing amount of labor required to produce a given
quantity (q2).
Hint 1: Your answer should be a number.
Hint 2: You do NOT have enough information to calculate individual values for ?1
∗ and ?1
∗ , but you DO
have enough information to calculate
?2
∗
?2
∗
e. The following statements are definitions of the terms “capital-intensive” and “labor-intensive.”
Industry i is capital-intensive relative to industry j if, for given values of w and r,
??
∗
??
∗
??
∗
??
∗ .
Industry i is labor-intensive relative to industry j if, for given values of w and r,
??
∗
??
∗
??
∗
??
∗.
Given these definitions, and your answers to parts c and d of this question, is Industry 1 capital-intensive
or labor-intensive, and is Industry 2 capital-intensive or labor-intensive?
Consider an economy with two industries: Industry 1 and Industry 2.
The production function in Industry 1 is q1 = f1(K1, L1) = ?1
2
3 ∙ ?1
1
3
The production function in Industry 2 is q2 = f2(K2, L2) = ?2
3
4 ∙ ?2
1
4
In this economy, the wage rate (w) is $10 per unit of labor, and the rental rate of capital (r) is $20 per
unit of capital.
f. Assume that this economy has 300 total units of capital available (�̅� = 300), and 250 total units of
labor available (�̅� = 250), so that in an equili
ium,
300 = ?1
∗ + ?2
∗ and 250 = ?1
∗ + ?2
∗
Using this information, and your answers to parts c and d of this question, find equili
ium values of ?1
∗,
?2
∗, ?1
∗ , ?2
∗ .
g. If this economy instead had 360 total units of capital available (�̅� = 360), and 250 total units of labor
available (�̅� = 250), then in an equili
ium,
360 = ?1
∗ + ?2
∗ and 250 = ?1
∗ + ?2
∗
Using this information, and your answers to parts c and d of this question, find equili
ium values of ?1
∗,
?2
∗, ?1
∗ , ?2
∗ .
h. If this economy instead had 300 total units of capital available (�̅� = 300), and 280 total units of labor
available (�̅� = 280), then in an equili
ium,
300 = ?1
∗ + ?2
∗ and 280 = ?1
∗ + ?2
∗
Using this information, and your answers to parts c and d of this question, find equili
ium values of ?1
∗,
?2
∗, ?1
∗ , ?2
∗ .
i. Since both Industry 1 and Industry 2 use both capital and labor, an increase in the total available
amount of either factor of production (K or L) should make it possible for both industries to increase
their equili
ium output. Given your answers to parts f, g, and h, of this question, clearly summarize
what happens to the equili
ium output of each industry when the total available amount of each factor
of production increases. In other words, what happens to q1 and q2 when �̅� increases, and what
happens to q1 and q2 when �̅� increases, and what do your answers to these questions have to do with
your answers to part e of this question? [Note: You do not need to use or calculate any numbers to
answer this part of this question, but a full-credit answer will use the vocabulary terms “capital-
intensive” and “labor-intensive.”]