Professor S. Severinov

Economics 301, UBC

Winter 2012-2013

Answers to Problem Set 4

1. Question for Review 4, Chapter 7 (p.261)

Suppose that labor is the only variable input to the production process. If the marginal cost of

production is diminishing as more units of output are produced, what can you say about the

marginal product of labor.

Answer: The marginal product of labor must be increasing. The marginal cost of production

measures the extra cost of producing one more unit of output. If this cost is diminishing, then it

must be taking fewer units of labor to produce the extra unit of output. If fewer units of labor

are required to produce a unit of output, then the marginal product (extra output produced by

an extra unit of labor) must be increasing.

Mathematically,

the marginal product of labor must be increasing, since MC(q)=wage*dL/dq=wage/MP(L), and since MC

is diminishing then MPL must be increasing for any given w.

2. Exercise 8, Chapter 7 (p.262):

You manage a plant that mass-produces engines by teams of workers using assembly machines. The

technology is summarized by the production function

q = 5 KL

where q is the number of engines per week, K is the number of assembly machines, and L is the

number of labor teams. Each assembly machine rents for r = $10,000 per week, and each team costs

w = $5000 per week. Engine costs are given by the cost of labor teams and machines, plus $2000 per

engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its

design.

a. What is the cost function for your plant — namely, how much would it cost to produce q

engines? What are average and marginal costs for producing q engines? How do average

costs vary with output?

The short-run production function is q = 5(5)L = 25L, because K is fixed at 5. This

implies that for any level of output q, the number of labor teams hired will be

L q

25. The total cost function is thus given by the sum of the costs of capital, labor,

and raw materials:

TC(q) = rK +wL +2000q = (10,000)(5) + (5,000)(q

25) + 2,000 q

TC(q) = 50,000 +2200q.

The average cost function is then given by:

AC(q)TC(q)

q

50,000 2200q

q.

and the marginal cost function is given by:

2200)( dq

dTCqMC .

Marginal costs are constant at $2200 per engine and average costs will decrease as

quantity increases because the average fixed cost of capital decreases.

b. How many teams are required to produce 250 engines? What is the average cost per engine?

To produce q = 250 engines we need

L q

25 or L = 10 labor teams. Average costs are

given by

AC(q 250)50,000 2200(250)

250 2400.

c. You are asked to make recommendations for the design of a new production facility. What

capital/labor (K/L) ratio should the new plant accommodate if it wants to minimize the total

cost of producing at any level of output q?

We no longer assume that K is fixed at 5. We need to find the combination of K and

L that minimizes costs at any level of output q. The cost-minimization rule is given

by

w

MP

r

MP LK .

To find the marginal product of capital, observe that increasing K by 1 unit increases

q by 5L, so MPK = 5L. Similarly, observe that increasing L by 1 unit increases q by

5K, so MPL = 5K. Mathematically,

LK

qMPK 5

and K

L

qMPL 5

.

Using these formulas in the cost-minimization rule, we obtain:

5L

r

5K

w

K

L

w

r

5000

10,000

1

2.

The new plant should accommodate a capital to labor ratio of 1 to 2, and this is the

same regardless of the number of units produced.

3. Exercise 11, Chapter 7 (p. 262):

You manage a plant that mass-produces engines by teams of workers using assembly machines. The

technology is summarized by the production function

q = 5 KL

where q is the number of engines per week, K is the number of assembly machines, and L is the

number of labor teams. Each assembly machine rents for r = $10,000 per week, and each team costs

w = $5000 per week. Engine costs are given by the cost of labor teams and machines, plus $2000 per

engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its

design.

a. What is the cost function for your plant — namely, how much would it cost to produce q

engines? What are average and marginal costs for producing q engines? How do average

costs vary with output?

The short-run production function is q = 5(5)L = 25L, because K is fixed at 5. This

implies that for any level of output q, the number of labor teams hired will be

L q

25. The total cost function is thus given by the sum of the costs of capital, labor,

and raw materials:

TC(q) = rK +wL +2000q = (10,000)(5) + (5,000)(q

25) + 2,000 q

TC(q) = 50,000 +2200q.

The average cost function is then given by:

AC(q)TC(q)

q

50,000 2200q

q.

and the marginal cost function is given by:

2200)( dq

dTCqMC .

Marginal costs are constant at $2200 per engine and average costs will decrease as

quantity increases because the average fixed cost of capital decreases.

b. How many teams are required to produce 250 engines? What is the average cost per engine?

To produce q = 250 engines we need

L q

25 or L = 10 labor teams. Average costs are

given by

AC(q 250)50,000 2200(250)

250 2400.

c. You are asked to make recommendations for the design of a new production facility. What

capital/labor (K/L) ratio should the new plant accommodate if it wants to minimize the total

cost of producing at any level of output q?

We no longer assume that K is fixed at 5. We need to find the combination of K and

L that minimizes costs at any level of output q. The cost-minimization rule is given

by

w

MP

r

MP LK .

To find the marginal product of capital, observe that increasing K by 1 unit increases

q by 5L, so MPK = 5L. Similarly, observe that increasing L by 1 unit increases q by

5K, so MPL = 5K. Mathematically,

LK

qMPK 5

and K

L

qMPL 5

.

Using these formulas in the cost-minimization rule, we obtain:

5L

r

5K

w

K

L

w

r

5000

10,000

1

2.

The new plant should accommodate a capital to labor ratio of 1 to 2, and this is the

same regardless of the number of units produced.

4. Exercise 4 in Appendix to Chapter 7, p 269.

Suppose the process of producing lightweight parkas by Polly’s Parkas is described by the function

q = 10K.8(L – 40).2

where q is the number of parkas produced, K the number of computerized stitching-machine hours,

and L the number of person-hours of labor. In addition to capital and labor, $10 worth of raw

materials is used in the production of each parka.

We are given the production function: q = F(K,L) = 10K.8(L – 40).2

We also know that the cost of production, in addition to the cost of capital and labor,

includes $10 of raw material per unit of output. This yields the following total cost

function:

TC(q) = wL + rK + 10q

By minimizing cost subject to the production function, derive the cost-minimizing

demands for K and L as a function of output (q), wage rates (w), and rental rates on

machines (r). Use these results to derive the total cost function: that is, costs as a

function of q, r, w, and the constant $10 per unit materials cost.

We need to find the combinations of K and L that will minimize this cost function for

any given level of output q and factor prices r and w. To do this, we set up the

Lagrangian:

= wL + rK + 10q – [10K.8 (L – 40).2 – q]

Differentiating with respect to K, L, and , and setting the derivatives equal to zero:

(1)

K r 10(.8)K

.2(L 40)

.2 0

(2)

L w 10K

.8(.2)(L 40)

.8 0

(3)

10K.8(L40)

.2q0.

Note that (3) has been multiplied by –1. The first 2 equations imply:

r 10 (.8)K .2 (L 40).2 and w10K.8(.2)(L 40) .8 .

or

r

w

4(L 40)

K.

This further implies:

K4w(L 40)

r and L-40=

rK

4w.

Substituting the above equations for K and L–40 into equation (3) yields solutions for

K and L:

q104w

r

.8

(L40).8(L40)

.2 and q=10K

.8rK

4w

.2

.

or

403.30 8.

8.

w

qrL and 2.

2.

6.7 r

qwK .

We can now obtain the total cost function in terms of only r, w, and q by substituting

these cost-minimizing values for K and L into the total cost function:

TC(q)wLrK10q

.106.7

403.30

)(

106.7

403.30

)(

2.8.8.2.

2.

2.

8.

8.

qqwr

wqrw

qTC

qr

qrww

w

qwrqTC

This process requires skilled workers, who earn $32 per hour. The rental rate on the machines used

in the process is $64 per hour. At these factor prices, what are total costs as a function of q? Does

this technology exhibit decreasing, constant, or increasing returns to scale?

Given the values w = 32 and r = 64, the total cost function becomes:

TC(q) = 19.2q + 1280.

The average cost function is then given by

AC(q) = 19.2 + 1280/q.

To find returns to scale, choose an input combination and find the level of output,

and then double all inputs and compare the new and old output levels. Assume K =

50 and L = 60. Then q1 = 10(50)0.8(60 – 40)0.2 = 416.3. When K = 100 and L = 120, q2

= 10(100)0.8(120 – 40)0.2 = 956.4. Since q2/q1 > 2, the production function exhibits

increasing returns to scale.

a. Polly’s Parkas plans to produce 2000 parkas per week. At the factor prices given above, how

many workers should the firm hire (at 40 hours per week) and how many machines should it

rent (at 40 machine-hours per week)? What are the marginal and average costs at this level

of production?

Given q = 2000 per week, we can calculate the required amount of inputs K and L

using the formulas derived in part a:

403.30 8.

8.

w

qrL and 2.

2.

6.7 r

qwK .

Thus L = 154.9 worker hours and K = 229.1 machine hours. Assuming a 40 hour

week, L = 154.9/40 = 3.87 workers per week, and K = 229.1/40 = 5.73 machines per

week. Polly’s Parkas should hire 4 workers and rent 6 machines per week, assuming

she cannot hire fractional workers and machines.

We know that the total cost and average cost functions are given by:

TC(q) = 19.2q + 1280

AC(q) = 19.2 + 1280/q,

so the marginal cost function is

MC(q) = dq

qdTC )( = 19.2.

Marginal costs are constant at $19.20 per parka and average costs are 19.2 +

1280/2000 or $19.84 per parka.

5. Question for review # 6, Chapter 8, p. 306

At the beginning of the twentieth century, there were many small American automobile manufacturers.

At the end of the century, there were only three large ones. Suppose that this situation is not the result

of lax federal enforcement of antimonopoly laws. How do you explain the decrease in the number of

manufacturers? (Hint: What is the inherent cost structure of the automobile industry?)

Automobile plants are highly capital-intensive, and consequently there are substantial

economies of scale in production. So, over time, the automobile companies that

produced larger quantities of cars were able to produce at lower average cost. They then

sold their cars for less and eventually drove smaller (higher cost) companies out of

business, or bought them to become even larger and more efficient. At very large levels

of production, the economies of scale diminish, and diseconomies of scale may even

occur. This would explain why more than one manufacturer remains.

6. Question for review # 9, Chapter 8, p. 306

True or false: A firm should always produce at an output at which long-run average cost is minimized.

Explain.

False. In the long run, under perfect

competition, firms will produce where long-

run average cost is minimized. In the short

run, however, it may be optimal to produce at

a different level. For example, if price is

above the long-run equilibrium price, the firm

will maximize short-run profit by producing a

greater amount of output than the level at

which LAC is minimized as illustrated in the

diagram. PL is the long-run equilibrium

price, and qL is the output level that

minimizes LAC. If price increases to P in the short run, the firm maximizes profit by

producing q, which is greater than qL, because that is the output level at which SMC

(short-run marginal cost) equals price.

7. Question for review # 13, Chapter 8, p. 306

The government passes a law that allows a substantial subsidy for every acre of land used to grow

tobacco. How does this program affect the long-run supply curve for tobacco?

A subsidy on land used to grow tobacco decreases every farmer’s average cost of producing tobacco

and will lead existing tobacco growers to increase output. In addition, tobacco farmers will make

positive economic profits that will encourage other firms to enter tobacco production. The result is

that both the short-run and long-run supply curves for the industry will shift down and to the right

8. Exercise 5, Chapter 8 (p. 307)

Suppose that a competitive firm’s marginal cost of producing output q is given by MC(q) = 3 + 2q.

Assume that the market price of the firm’s product is $9.

a. What level of output will the firm produce?

The firm should set the market price equal to marginal cost to maximize its profits:

9 = 3 + 2q, or q = 3.

Price

Output

LAC

SMC

PL

P

qL q

SAC

b. What is the firm’s producer surplus?

Producer surplus is equal to the area below the market price, i.e., $9.00, and above the

marginal cost curve, i.e., 3 + 2q. Because MC is linear, producer surplus is a triangle

with a base equal to 3 (since q = 3) and a height of $6 (since 9 – 3 = 6). The area of a

triangle is (1/2)(base)(height). Therefore, producer surplus is (0.5)(3)(6) = $9.

Pr ice

Quant ity

1

2

3

4

5

6

7

8

9

10

1 2 3 4

MC(q) = 3 + 2q

Producer ’s

Surplus

P = $9.00

Producer

Surplus

c. Suppose that the average variable cost of the firm is given by AVC(q) = 3 + q. Suppose that the

firm’s fixed costs are known to be $3. Will the firm be earning a positive, negative, or zero profit

in the short run?

Profit is equal to total revenue minus total cost. Total cost is equal to total variable cost

plus fixed cost. Total variable cost is equal to [AVC(q)]q. Therefore, at q = 3, AVC(q) = 3

+ 3 = 6, and therefore

TVC = (6)(3) = $18.

Fixed cost is equal to $3. Therefore, total cost equals TVC plus TFC, or

C = $18 + 3 = $21.

Total revenue is price times quantity:

R = ($9)(3) = $27.

Profit is total revenue minus total cost:

= $27 – 21 = $6.

Therefore, the firm is earning positive economic profits. More easily, you might recall

that profit equals producer surplus minus fixed cost. Since we found that producer

surplus was $9 in part (b), profit equals 9 – 3 or $6.

9. Exercise 7, Chapter 8 (p.307)

Suppose that a competitive firm’s marginal cost of producing output q is given by MC(q) = 3 + 2q.

Assume that the market price of the firm’s product is $9.

a. What level of output will the firm produce?

The firm should set the market price equal to marginal cost to maximize its profits:

9 = 3 + 2q, or q = 3.

b. What is the firm’s producer surplus?

Producer surplus is equal to the area below the market price, i.e., $9.00, and above the

marginal cost curve, i.e., 3 + 2q. Because MC is linear, producer surplus is a triangle

with a base equal to 3 (since q = 3) and a height of $6 (since 9 – 3 = 6). The area of a

triangle is (1/2)(base)(height). Therefore, producer surplus is (0.5)(3)(6) = $9.

Pr ice

Quant ity

1

2

3

4

5

6

7

8

9

10

1 2 3 4

MC(q) = 3 + 2q

Producer ’s

Surplus

P = $9.00

Producer

Surplus

c. Suppose that the average variable cost of the firm is given by AVC(q) = 3 + q. Suppose that the

firm’s fixed costs are known to be $3. Will the firm be earning a positive, negative, or zero profit

in the short run?

Profit is equal to total revenue minus total cost. Total cost is equal to total variable cost

plus fixed cost. Total variable cost is equal to [AVC(q)]q. Therefore, at q = 3, AVC(q) = 3

+ 3 = 6, and therefore

TVC = (6)(3) = $18.

Fixed cost is equal to $3. Therefore, total cost equals TVC plus TFC, or

C = $18 + 3 = $21.

Total revenue is price times quantity:

R = ($9)(3) = $27.

Profit is total revenue minus total cost:

= $27 – 21 = $6.

Therefore, the firm is earning positive economic profits. More easily, you might recall

that profit equals producer surplus minus fixed cost. Since we found that producer

surplus was $9 in part (b), profit equals 9 – 3 or $6.