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Chapter 7: The Costs of Production

CHAPTER 7

THE COST OF PRODUCTION

QUESTIONS FOR REVIEW

1. A firm pays its accountant an annual retainer of $10,000. Is this an economic

cost?

Explicit costs are actual outlays. They include all costs that involve a monetary

transaction. An implicit cost is an economic cost that does not necessarily

involve a monetary transaction, but still involves the use of resources. When a

firm pays an annual retainer of $10,000, there is a monetary transaction. The

accountant trades his or her time in return for money. Therefore, an annual

retainer is an explicit cost.

2. The owner of a small retail store does her own accounting work. How would you

measure the opportunity cost of her work?

Opportunity costs are measured by comparing the use of a resource with its

alternative uses. The opportunity cost of doing accounting work is the time not

spent in other ways, i.e., time such as running a small business or participating

in leisure activity. The economic, or opportunity, cost of doing accounting work

is measured by computing the monetary amount that the owner?s time would

be worth in its next best use.

3. Please explain whether the following statements are true or false.

a.

If the owner of a business pays himself no salary, then the accounting cost

is zero, but the economic cost is positive.

True. Since there is no monetary transaction, there is no accounting, or

explicit, cost. However, since the owner of the business could be employed

elsewhere, there is an economic cost.

The economic cost is positive,

reflecting the opportunity cost of the owner?s time. The economic cost is the

value of the next best alternative, or the amount that the owner would earn if

he took the next best job.

b.

A firm that has positive accounting profit does not necessarily have positive

economic profit.

True. Accounting profit considers only the explicit, monetary costs. Since

there may be some opportunity costs that were not fully realized as explicit

monetary costs, it is possible that when the opportunity costs are added in,

economic profit will become negative. This indicates that the firm?s resources

are not being put to their best use.

c.

If a firm hires a currently unemployed worker, the opportunity cost of

utilizing the worker?s services is zero.

False. The opportunity cost measures the value of the worker?s time, which is

unlikely to be zero. Though the worker was temporarily unemployed, the

worker still possesses skills, which have a value and make the opportunity

cost of hiring the worker greater than zero. In addition, since opportunity cost

is the equivalent of the worker?s next best option, it is possible that the worker

might have been able to get a better job that utilizes his skills more efficiently.

Alternatively, the worker could have been doing unpaid work, such as care of

a child or elderly person at home, which would have had a value to those

receiving the service.

4. 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 ?

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

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Chapter 7: The Costs of Production

this cost is diminishing, then it must be taking fewer units of labor to produce

the extra unit of output, since the extra cost refers to the extra cost of the

labor. 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. Note also, that MC=w/MPL, so that if MC is diminishing then MPL

must be increasing for any given w.

5. Suppose a chair manufacturer finds that the marginal rate of technical

substitution of capital for labor in his production process is substantially greater

than the ratio of the rental rate on machinery to the wage rate for assembly-line

labor. How should he alter his use of capital and labor to minimize the cost of

production?

To minimize cost, the manufacturer should use a combination of capital and

labor so the rate at which he can trade capital for labor in his production

process is the same as the rate at which he can trade capital for labor in

external markets. The manufacturer would be better off if he increased his use

of capital and decreased his use of labor, decreasing the marginal rate of

technical substitution, MRTS. He should continue this substitution until his

MRTS equals the ratio of the rental rate to the wage rate. The MRTS in this

case is equal to MPK/MPL. As the manufacturer uses more K and less L, the

MPK will diminish and the MPL will increase, both of which will decrease the

MRTS until it is equal to the ratio of the input prices (rental rate on capital

divided by wage rate).

6. Why are isocost lines straight lines?

The isocost line represents all possible combinations of labor and capital that

may be purchased for a given total cost. The slope of the isocost line is the

ratio of the input prices of labor and capital. If input prices are fixed, then the

ratio of these prices is clearly fixed and the isocost line is straight. Only when

the ratio or factor prices change as the quantities of inputs change is the

isocost line not straight.

7. Assume the marginal cost of production is increasing. Can you determine

whether the average variable cost is increasing or decreasing? Explain.

Marginal cost can be increasing while average variable cost is either increasing

or decreasing. If marginal cost is less (greater) than average variable cost, then

each additional unit is adding less (more) to total cost than previous units

added to the total cost, which implies that the AVC declines (increases).

Therefore, we need to know whether marginal cost is greater than average

variable cost to determine whether the AVC is increasing or decreasing.

8. Assume the marginal cost of production is greater than the average variable

cost. Can you determine whether the average variable cost is increasing or

decreasing? Explain.

If the average variable cost is increasing (decreasing), then the last unit

produced is adding more (less) to total variable cost than the previous units did,

on average. Therefore, marginal cost is above (below) average variable cost.

In fact, the point where marginal cost exceeds average variable cost is also the

point where average variable cost starts to rise.

9. If the firm?s average cost curves are U-shaped, why does its average variable

cost curve achieve its minimum at a lower level of output than the average total

cost curve?

Total cost is equal to fixed plus variable cost. Average total cost is equal to

average fixed plus average variable cost. When graphed, the difference

between the U-shaped total cost and average variable cost curves is the

average fixed cost curve. Thus, falling average variable cost and average fixed

85

Chapter 7: The Costs of Production

cost sum up to a falling average total cost curve. Since average fixed cost

continues to fall as more output is produced, average total cost will continue to

fall even after average variable cost has reached its minimum because the drop

in average fixed cost exceeds the increase in the average variable cost.

Eventually, the fall in average fixed cost becomes small enough so that the rise

in average variable cost causes average total cost to begin to rise.

10. If a firm enjoys economies of scale up to a certain output level, and then cost

increases proportionately with output, what can you say about the shape of the

long-run average cost curve?

When the firm experiences increasing returns to scale, its long-run average cost

curve is downward sloping. When the firm experiences constant returns to

scale, its long-run average cost curve is horizontal. If the firm experiences

increasing returns to scale, then constant returns to scale, its long-run average

cost curve falls, then becomes horizontal.

11. How does a change in the price of one input change the firm?s long-run

expansion path?

The expansion path describes the combination of inputs that the firm chooses

to minimize cost for every output level. This combination depends on the ratio

of input prices: if the price of one input changes, the price ratio also changes.

For example, if the price of an input increases, less of the input can be

purchased for the same total cost, and the intercept of the isocost line on that

input?s axis moves closer to the origin. Also, the slope of the isocost line, the

price ratio, changes. As the price ratio changes, the firm substitutes away from

the now more expensive input toward the cheaper input. Thus, the expansion

path bends toward the axis of the now cheaper input.

12. Distinguish between economies of scale and economies of scope. Why can

one be present without the other?

Economies of scale refer to the production of one good and occur when

proportionate increases in all inputs lead to a more-than-proportionate increase

in output. Economies of scope refer to the production of more than one good

and occur when joint output is less costly than the sum of the costs of

producing each good or service separately. There is no direct relationship

between increasing returns to scale and economies of scope, so production can

exhibit one without the other. See Exercise (14) for a case with constant

product-specific returns to scale and multiproduct economies of scope.

13. Is the firm?s expansion path always a straight line?

No. If the long run expansion path is a straight line this means that the firm

always uses capital and labor in the same proportion. If the capital labor ratio

changes as output is increased then the expansion path is not a straight line.

14. What is the difference between economies of scale and returns to scale?

Economies of scale measures the relationship between cost and output, i.e.,

when output is doubled, does cost double, less then double, or more than

double. Returns to scale measures what happens to output when all inputs

are doubled.

EXERCISES

1. Joe quits his computer-programming job, where he was earning a salary of

$50,000 per year to start his own computer software business in a building that

he owns and was previously renting out for $24,000 per year. In his first year of

business he has the following expenses: salary paid to himself $40,000, rent, $0,

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Chapter 7: The Costs of Production

and other expenses $25,000. Find the accounting cost and the economic cost

associated with Joe?s computer software business.

The accounting cost represents the actual expenses, which are $40,000+$0 +

$25,000=$65,000. The economic cost includes accounting cost, but also

takes into account opportunity cost. Therefore, economic will include, in

addition to accounting cost, an extra $24,000 because Joe gave up $24,000

by not renting the building , and an extra $10,000 because he paid himself a

salary $10,000 below market ($50,000-$40,000). Economic cost is then

$99,000.

2. a. Fill in the blanks in the following table.

Units of

Output

0

1

2

3

4

5

6

7

8

9

10

b.

Fixed Variable Total

Cost

Cost

Cost

100

100

100

100

100

100

100

100

100

100

100

0

25

45

57

77

102

136

170

226

298

390

100

125

145

157

177

202

236

270

326

398

490

Marginal Average Average Average

Cost

Fixed CostVariable Total Cost

Cost

??

??

0

??

25

100

25

125

20

50

22.5

72.5

12

33.3

19

52.3

20

25

19.25

44.25

25

20

20.4

40.4

34

16.67

22.67

39.3

34

14.3

24.3

38.6

56

12.5

28.25

40.75

72

11.1

33.1

44.2

92

10

39

49

Draw a graph that shows marginal cost, average variable cost, and average

total cost, with cost on the vertical axis and quantity on the horizontal axis.

Average total cost is u-shaped and reaches a minimum at an output of 7,

based on the above table. Average variable cost is u-shaped also and reaches

a minimum at an output of 3. Notice from the table that average variable cost

is always below average total cost. The difference between the two costs is

the average fixed cost. Marginal cost is first diminishing, to a quantity of 3

based on the table, and then increases as q increases. Marginal cost should

intersect average variable cost and average total cost at their respective

minimum points, though this is not accurately reflected in the numbers in the

table. If the specific functions had been given in the problem instead of just a

series of numbers, then it would be possible to find the exact point of

intersection between marginal and average total cost and marginal and

average variable cost. The curves are likely to intersect at a quantity that is

not a whole number, and hence are not listed in the above table.

3. A firm has a fixed production cost of $5,000 and a constant marginal cost of

production of $500 per unit produced.

a.

What is the firm?s total cost function? Average cost?

The variable cost of producing an additional unit, marginal cost, is constant at

$500, so

VC $500q , and AVC

VC $500q

$500. Fixed cost is $5,000

q

q

87

Chapter 7: The Costs of Production

and average fixed cost is

$5,000

. The total cost function is fixed cost plus

q

variable cost or TC=$5,000+$500q. Average total cost is the sum of average

variable cost and average fixed cost:

b.

ATC $500

$5,000

.

q

If the firm wanted to minimize the average total cost, would it choose to be

very large or very small? Explain.

The firm should choose a very large output because average total cost will

continue to decrease as q is increased. As q becomes infinitely large, ATC will

equal $500.

4. Suppose a firm must pay an annual tax, which is a fixed sum, independent of

whether it produces any output.

a.

How does this tax affect the firm?s fixed, marginal, and average costs?

Total cost, TC, is equal to fixed cost, FC, plus variable cost, VC. Fixed costs do

not vary with the quantity of output. Because the franchise fee, FF, is a fixed

sum, the firm?s fixed costs increase by this fee. Thus, average cost, equal to

FC VC

FC

, and average fixed cost, equal to

, increase by the average

q

q

FF

franchise fee

. Note that the franchise fee does not affect average

q

variable cost. Also, because marginal cost is the change in total cost with the

production of an additional unit and because the fee is constant, marginal cost

is unchanged.

b.

Now suppose the firm is charged a tax that is proportional to the number of

items it produces. Again, how does this tax affect the firm?s fixed, marginal,

and average costs?

Let t equal the per unit tax. When a tax is imposed on each unit produced,

variable costs increase by tq. Average variable costs increase by t, and

because fixed costs are constant, average (total) costs also increase by t.

Further, because total cost increases by t with each additional unit, marginal

costs increase by t.

5. A recent issue of Business Week reported the following:

During the recent auto sales slump, GM, Ford, and Chrysler decided

it was cheaper to sell cars to rental companies at a loss than to lay off

workers. That?s because closing and reopening plants is expensive,

partly because the auto makers? current union contracts obligate

them to pay many workers even if they?re not working.

When the article discusses selling cars ?at a loss,? is it referring to accounting

profit or economic profit? How will the two differ in this case? Explain briefly.

When the article refers to the car companies selling at a loss, it is referring to

accounting profit. The article is stating that the price obtained for the sale of

the cars to the rental companies was less than their accounting cost.

Economic profit would be measured by the difference of the price with the

opportunity cost of the cars. This opportunity cost represents the market

value of all the inputs used by the companies to produce the cars. The article

mentions that the car companies must pay workers even if they are not

working (and thus producing cars). This implies that the wages paid to these

workers are sunk and are thus not part of the opportunity cost of production.

On the other hand, the wages would still be included in the accounting costs.

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Chapter 7: The Costs of Production

These accounting costs would then be higher than the opportunity costs and

would make the accounting profit lower than the economic profit.

6. Suppose the economy takes a downturn, and that labor costs fall by 50

percent and are expected to stay at that level for a long time. Show graphically

how this change in the relative price of labor and capital affects the firm?s

expansion path.

Figure 7.6 shows a family of isoquants and two isocost curves. Units of capital

are on the vertical axis and units of labor are on the horizontal axis. (Note: In

drawing this figure we have assumed that the production function underlying

the isoquants exhibits constant returns to scale, resulting in linear expansion

paths. However, the results do not depend on this assumption.)

If the price of labor decreases while the price of capital is constant, the isocost

curve pivots outward around its intersection with the capital axis. Because the

expansion path is the set of points where the MRTS is equal to the ratio of

prices, as the isocost curves pivot outward, the expansion path pivots toward

the labor axis. As the price of labor falls relative to capital, the firm uses more

labor as output increases.

Capital

Expansion path

before wage fall

4

Expansion path

after wage fall

3

2

1

1

2

3

4

5

L abor

Figure 7.6

7. The cost of flying a passenger plane from point A to point B is $50,000. The

airline flies this route four times per day at 7am, 10am, 1pm, and 4pm. The first

and last flights are filled to capacity with 240 people. The second and third flights

are only half full. Find the average cost per passenger for each flight. Suppose

the airline hires you as a marketing consultant and wants to know which type of

customer it should try to attract, the off-peak customer (the middle two flights) or

the rush-hour customer (the first and last flights). What advice would you offer?

The average cost per passenger is $50,000/240 for the full flights and

$50,000/120 for the half full flights. The airline should focus on attracting

more off-peak customers in order to reduce the average cost per passenger

on those flights. The average cost per passenger is already minimized for the

two peak time flights.

8. 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

89

Chapter 7: The Costs of Production

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 = $5,000 per week. Engine costs are

given by the cost of labor teams and machines, plus $2,000 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?

K is fixed at 5. The short-run production function then becomes q = 25L. This

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

L

q

. The total cost function is thus given by the sum of the costs of

25

capital, labor, and raw materials:

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

q

) + 2,000 q

25

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

The average cost function is then given by:

AC(q)

TC(q) 50,000 2200q

.

q

q

and the marginal cost function is given by:

MC(q)

TC

2200.

q

Marginal costs are constant and average costs will decrease as quantity

increases (due to the fixed cost of capital).

b.

How many teams are required to produce 250 engines?

average cost per engine?

To produce q = 250 engines we need labor teams

costs are given by

AC(q 250)

c.

L

What is the

q

or L=10. Average

25

50,000 2200(250)

2400.

250

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 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

MPK

r

=

MPL

w

.

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,

MPK

Q

Q

5L and MPL

5K .

K

L

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

90

Chapter 7: The Costs of Production

5L 5K

K w 5000

1

.

r

w

L r 10,000 2

The new plant should accommodate a capital to labor ratio of 1 to 2. Note

that the current firm is presently operating at this capital-labor ratio.

9.

The short-run cost function of a company is given by the equation

TC=200+55q, where TC is the total cost and q is the total quantity of output, both

measured in thousands.

a.

What is the company?s fixed cost?

When q = 0, TC = 200, so fixed cost is equal to 200 (or $200,000).

b.

If the company produced 100,000 units of goods, what is its average variable

cost?

With 100,000 units, q = 100.

Variable cost is 55q = (55)(100) = 5500 (or

$5,500,000). Average variable cost is

c.

TVC $5500

$55, or $55,000.

q

100

What is its marginal cost per unit produced?

With constant average variable cost, marginal cost is equal to average variable

cost, $55 (or $55,000).

d.

What is its average fixed cost?

At q = 100, average fixed cost is

e.

TFC $200

$2 or ($2,000).

q

100

Suppose the company borrows money and expands its factory. Its fixed cost

rises by $50,000, but its variable cost falls to $45,000 per 1,000 units. The

cost of interest (i) also enters into the equation. Each one-point increase in

the interest rate raises costs by $3,000. Write the new cost equation.

Fixed cost changes from 200 to 250, measured in thousands. Variable cost

decreases from 55 to 45, also measured in thousands. Fixed cost also includes

interest charges: 3i. The cost equation is

C = 250 + 45q + 3i.

10. A chair manufacturer hires its assembly-line labor for $30 an hour and

calculates that the rental cost of its machinery is $15 per hour. Suppose that a

chair can be produced using 4 hours of labor or machinery in any combination. If

the firm is currently using 3 hours of labor for each hour of machine time, is it

minimizing its costs of production? If so, why? If not, how can it improve the

situation? Graphically illustrate the isoquant and the two isocost lines, for the

current combination of labor and capital and the optimal combination of labor and

capital.

If the firm can produce one chair with either four hours of labor or four hours of

capital, machinery, or any combination, then the isoquant is a straight line with

a slope of -1 and intercept at K = 4 and L = 4, as depicted in figure 7.10.

30

2 when plotted with

15

TC

TC

capital on the vertical axis and has intercepts at K

and L

. The cost

15

30

The isocost line, TC = 30L + 15K has a slope of

minimizing point is a corner solution, where L = 0 and K = 4. At that point, total

cost is $60. Two isocost lines are illustrated on the graph. The first one is

91

Chapter 7: The Costs of Production

further from the origin and represents the higher cost ($105) of using 3 labor

and 1 capital. The firm will find it optimal to move to the second isocost line

which is closer to the origin, and which represents a lower cost ($60). In

general, the firm wants to be on the lowest isocost line possible, which is the

lowest isocost line that still intersects the given isoquant.

Capital

isocost lines

4

isoquant

Labor

4

Figure 7.10

11. Suppose that a firm?s production function is

1

1

q 10L2 K 2 . The cost of a unit of

labor is $20 and the cost of a unit of capital is $80.

a.

The firm is currently producing 100 units of output, and has determined

that the cost-minimizing quantities of labor and capital are 20 and 5

respectively.

Graphically illustrate this situation on a graph using

isoquants and isocost lines.

The isoquant is convex. The optimal quantities of labor and capital are given

by the point where the isocost line is tangent to the isoquant. The isocost line

has a slope of 1/4, given labor is on the horizontal axis. The total cost is

TC=$20*20+$80*5=$800, so the isocost line has the equation

$800=20L+80K. On the graph, the optimal point is point A.

capital

point A

isoquant

labor

b.

The firm now wants to increase output to 140 units. If capital is fixed in the

short run, how much labor will the firm require? Illustrate this point on

your graph and find the new cost.

The new level of labor is 39.2.

1

2

1

2

To find this, use the production function

q…