COST THEORY

Cost theory is related to production theory, they are often used together. However, the

question is usually how much to produce, as opposed to which inputs to use. That is, assume

that we use production theory to choose the optimal ratio of inputs (eg. 2 fewer engineers

than technicians), how much should we produce in order to minimize costs and/or maximize

profits? We can also learn a lot about what kinds of costs matter for decisions made by

managers, and what kinds of costs do not.

I What costs matter?

A Opportunity Costs

Remember from Section (IV) of the Introduction, that in addition to accounting profit,

managers must consider the cost of inputs supplied by the owners (owners capital and labor).

Definition 16 Explicit Costs: Accounting Costs, or costs that would appear as costs in

an accounting statement.

Definition 17 Opportunity Costs: The value of all inputs to a firms production in their

most valuable alternative use.

Recall the example from Section (IV), where the decision was whether or not to buy

the kitchen. the opportunity costs were the $24,000 per year the money could be earning

elsewhere, and the owners time cost of $40,000, which exceeded the accounting profit of

$60,000.

B Fixed costs, variable costs, and sunk costs

Some inputs vary with the amount produced and others do not. The firms computer sys-

tem and accountants may be able to handle a large volume of sales without increasing the

number of computers or accountants, for example. Inputs that do not vary with the amount

produced, like accountants and computers, are called fixed inputs.

Most inputs are fixed only for a certain range of production. A medical office may be

able to handle many additional patients without adding and office assistant or extra phones.

The phones and office assistants are fixed inputs. But, if the number of extra patients is

large enough, the firm needs extra office staff.

Reasons for fixed costs:

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1. Salaried workers. Salaried workers are a fixed input if the worker can work overtime

without additional compensation (doctors paid a fee for service are variable inputs,

salaried medical staff are fixed inputs).

2. Fixed hours at work. An hourly worker sometimes cannot be sent home early if not

enough work is available. Therefore, workers may not be busy and be able to handle

extra work without additional hours.

3. Time to adjust. Some inputs, like machines, take time to purchase and install.

Conversely, unskilled labor may be adjusted more quickly through overtime, temps,

etc. Therefore, by necessity the firm may only be able to vary production by increasing

labor in the short run.

Definition 18 Total Variable Cost The total cost of all inputs that change with the

amount produced (all variable inputs).

Definition 19 Total fixed costs The total cost of all inputs that do not vary with the

amount produced (all fixed inputs).

Consider the Thompson machine company. The firm uses 5 machines to make machine

parts. Because of the time to adjust, machines are a fixed cost, while the number of workers

varies with the amount produced. Labor is a variable cost.

Definition 20 Sunk costs Are costs that have been incurred and cannot be reversed.

Any costs incurred in the past, or indeed any fixed cost for which payment must be

made regardless of the decision is irrelevant for any managerial decision. Suppose you hire

an executive with a $100,000 signing bonus, plus $200,000 salary. After hiring, you may

find the executive does not live up to expectations. However, if the executives marginal

revenue product is $200,001, the executive still generates $1 in profits relative to his salary

and therefore should be retained. But if his MRP is $199,999, the firm loses an extra $1 each

year they keep him, so he should be let go. The bonus is a sunk cost and does not affect the

retention decision.

The principle of sunk costs is equivalent to the saying dont throw good money after

bad.

Sometimes a decision can be made to recover part of a fixed cost. Perhaps one could sell

a factory and recover part of the fixed costs. Then only the difference is sunk. For example,

if we can sell a building for which we paid $500,000 for $300,000, then only $200,000 is sunk.

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Sunk costs are perhaps one of the most psychologically difficult things to ignore. Exam-

ples:

1. Finance. Studies show investors let sunk costs enter their decision making. What

price the stock was purchased at is sunk and therefore irrelevant. What matters is

only whether or not this stock offers the best return for the risk. Yet, investors are

reluctant to sell stocks whose price has gone down.

2. Capital investment. I watched the world series of poker. In one instance, the odds

of drawing a flush (and almost for sure winning the hand) was 1 in 5. The pot was

about $200,000. So the player should call any bet less than or equal to $40,000. Yet

the commentator advised that the player should call regardless of the bet, because he

already had so much money in the pot (sunk costs).

3. Cut your losses? Consider the war in Afghanistan. We have sunk billions, but that

should not enter our decision about whether or not to stay.

4. Pricing in high rent districts. Consider restaurants in a high rent district (say

an airport). Should they take the rent into account when setting prices? No. In fact,

prices are high not because of the rent, but typically because of the lack of competitors.

II Short run costs

We use short run costs primarily to compute how much to produce while maximizing profits.

We use long run costs to answer questions like should the firm expand, contract, merge, etc.

Definition 21 Average Costs: Costs divided by output.

Definition 22 Marginal Costs: The cost of one additional unit of an input.

Here is the notation:

Type of Cost Total Cost equals Variable Costs Plus Fixed Costs

Total TC = TV C +TFC

Average ATC = TCQ

= AV C = TV CQ

+AFC = TFCQ

Marginal MC = dTCdQ

= TCQ

Properties of cost functions in the short run:

1. Total costs of increase with Q, the quantity produced.

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2. Average Costs decline withQ, but eventually rise. The fixed costs are spread over many

more units of production at high Q, reducing average costs. All of the extra workers

required for producing additional units when the factory is near capacity starts to

increase average costs eventually.

3. Marginal costs usually decline then increase, but must eventually increase. At first,

producing one additional unit is may cheaper than the last unit, due to specialization.

However, eventually diminishing returns sets in and the workers just get in each others

way. Then a very large number of additional workers might be needed to produce an

additional unit.

Here is a graph of the cost curves.

0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400

450

500

Q

Cost

($)

Total Cost Functions

Total CostsTotal Fixed CostsTotal Variable Costs

Figure 5: Typical short total cost curves.

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0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

Q

Cost

s pe

r uni

t ($)

Short Run Cost Functions

Average Total CostsAverage Fixed CostsAverage Variable CostsMarginal Costs

Figure 6: Typical short run average and marginal cost curves.

III Examples using short run cost curves.

A Profit maximization with perfect competition

Let us suppose that you are a hypothetical manager of a group of cruise ships. Using data

from previous years, you estimate the short run cost function is (we will see how to do this

estimation below):

TC = 60 +Q2

20(71)

Here Q is the number of cruises the ship takes (not the number of passengers). The cost

of the ship is $60 million, which is sunk. Notice in equation (71) that the cost of the ship

does not depend on the number of cruises the ship takes. Suppose further that each cruise

generates revenue of $3 million.

Maximize profits:

max pi = TR TC = 3Q 60Q2

20(72)

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Take the derivative to get the slope and set the slope equal to zero:

3Q

10= 0 Q = 30. (73)

Notice that the fixed costs have dropped out. The math agrees: fixed costs do not matter

for our decision. In general, to maximize profits we set marginal revenue (here $3) equal to

marginal costs (here Q/10).

MR =MC. (74)

In a competitive industry, firms have no ability to influence the price. Examples include

market makers in stock markets, commodities, and large food markets. For example, an

individual farmer may produce as much corn as desired and sell the corn on the commodities

markets with no change in price. Regardless of the amount produced by an individual firm,

the price is unchanged and MR = P . So in competitive industries, we set:

P = MC. (75)

The firm should produce one more unit if we can sell it for more than it costs to produce.

The firm makes profits in this case. In the next set of notes, we will do the case where firms

have pricing power.

Using a table:

Cruises (Q) Total variable costs (TV C) Marginal costs (MC)0 0 20 = (202)/20 = 20 = (20 0)/(20 0) = 125 = (252)/20 = 31.25 = (31.25 20)/(25 20) = 2.2530 45 2.7535 61.25 3.2540 80 3.75

Table 6: Variable costs on a cruise ship.

The marginal revenue is the price of the cruise, equal to $3 million. We can see that

marginal revenue equals marginal costs somewhere between 30 and 35 cruises.2

Producing the 30th cruise gives us $3 of revenue, enough to cover the costs of producing

the 30th cruise, which (using table 6) is approximately $2.75. However, the 35th cruise loses

2The table is an approximation, however. In fact, using the true marginal cost of the 30th cruise is

MC = dTC/dQ = Q/10 = 3.

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money. The table estimates that cruise would cost $3.25, and since revenues are $3, the

cruise would lose an estimated $0.25 million.

The fixed costs are irrelevant here. When considering the fixed costs, the firm has negative

profits regardless of how many cruises the firm takes. The maximum profits occurs at 30

cruises:

pi = TR TC = 3 30

(60 +

302

20

)= $15. (76)

We have already paid the fixed costs, so we might as well lose as little as possible.3

B Break Even Analysis

An important consideration when deciding whether to continue operations in a particular

market, expand into a market, or start a new business is a break even analysis. We can do

a break even analysis with a cost function.

In a break even analysis, the question is: how much profit is required to exactly pay off

all fixed costs? Alternatively, how much revenue is required to pay off the average variable

costs and the fixed costs:

pi = 0 = TR TC (77)

0 = P Q TFC TV C (78)

0 = P Q TFC AV C Q (79)

Q =TFC

P AV C(80)

Here I have assumed linear total costs, so that average variable cost is constant. One could

assume (more realistically) that average variable costs depend on Q, and use a table to get

the break even point.

3Note that it is irrelevant how the ship is financed. Interest and principle payments are also sunk costs.

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C Average Costs

Often average cost data is easy to get. It is relatively easy to measure total costs and the

quantity sold to get average costs. However, many managers then incorrectly base decisions

on average costs.

Consider the following data:

Typical Data Calculate this!

Gas Produc-tion (Q)

Total costs(TC)

Average Totalcosts (ATC)

Marginal Costs(MC)

Profits (pi)

20 1271 = 1271/20 = 63.6 8922 1359 61.8 = (1359 1271)/

(22 20) = 44.0137

24 1456 60.7 48.5 17626 1562 60.1 53.0 20628 1675 59.8 56.5 22930 1797 59.9 61.0 24332 1928 60.3 65.5 24834 2067 60.8 69.5 24536 2214 61.5 73.5 23438 2370 62.4 78.0 21440 2534 63.4 82.0 18642 2707 64.5 86.5 14944 2888 65.6 90.5 10446 3077 66.9 94.5 5148 3275 68.2 99.0 -11

Table 7: Average and marginal costs in the gas industry.

The first 3 columns of table 7 give a typical data set one might have in a real business

situation. For example, a gas refinery might produce different quantities monthly depending

on variations in the price of gas and demand. Suppose the price of gas is currently $68.

How much gas should be produced? Setting price equal to average cost (48 units) actually

produces a loss. We could try to minimize average costs, which occurs at 28 units. Still,

this is not the maximum profits. The maximum profits occurs when price equals marginal

costs, about 32-34 units. The key is that given the first 3 columns, one can easily calculate

the marginal costs and calculate the profit maximizing quantity.

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IV Long Run Costs

We use long run costs to decide scale issues, for example mergers, but also the optimal size

of our operations (e.g. optimal plant size). The long run is when all costs are variable.

In the long run, we can build any size factory we wish, based on anticipated demand,

profits, and other considerations. Once the plant is built, we move back to the short run

as described above. Therefore, it is important to forecast the anticipated demand. Too

small a factory and marginal costs will be high as the factory is stretched to over produce.

Conversely too large a factory results in large fixed costs (eg. air conditioning, or taxes).

A Definition

Definition 23 Long Run Average Costs: The minimum cost per unit of producing a

given output level when all costs are variable.

Averagecost ($/unit)

Long Run AverageCosts

Short Run AverageCosts

Q

small plant small plant

Qd

Cost of making

size plant

medium

plant

small

Qd with a

Cost of making

Qd with a

Figure 7: Long run average cost curve.

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Think of each short run average cost curve as a factory of different size. We build the

factory, and then operate within the short run average cost curve for the factory that is

built. For each size factory, average total costs are initially decreasing as the fixed costs

are spread over more and more units. Eventually, however, average total costs rise as the

factory becomes over crowded. Suppose we forecast demand at Qd on the graph. Then, from

the graph, we create a small sized factory, as that is the cheapest way to build Qd units.

Therefore, we must have that the long run average cost curve is tangent to the minimum of

the short run average cost curves.

B Deriving a long run average cost curve: Example

Suppose we can build plants of three sizes:

Cost Type Plant SizeAV C Small Medium LargeLabor $3.70 $2.50 $1.10

Materials $1.80 $1.40 $0.90Other $2.00 $2.60 $3.00

TFC $25,000 $75,000 $300,000

Capacity 50,000 100,000 200,000

Table 8: Plant size and LRAC.

What type of plant produces 100,000 units at the lowest long run average costs? We can

build either two small plants, 1 medium sized plant, or one large plant. The average costs

of two small plants are:

ATC = AV C +TFC

Q, (81)

ATC = $3.70 + $1.80 + $2.00 +$25, 000 2

100, 000= $8. (82)

Here, the fixed costs of the small plant is multiplied by two since we need two plants. For a

single medium sized plant:

ATC = $2.50 + $1.40 + $2.60 +$75, 000

100, 000= $7.25. (83)

The medium sized plant is cheaper. Although the factory is more expensive to build than

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two small factories, it can produce units at lower average variable cost. A large plant has:

ATC = $1.10 + $0.90 + $3.00 +$300, 000

100, 000= $8.00. (84)

The large plant can produce at a very low average variable cost of $5 per unit, but is just

too expensive to build.

So for Q = 100, 000 units, we build a single, medium sized factory. The point on the long

run average cost curve for Q = 100, 000 is LRAC = $7.25.

Suppose instead expected demand is Q = 200, 000 units. We can build 1 large, two

medium, or 4 small plants. The cost of these options (per unit) are: $6.50, $7.25, and $8,

respectively. The large plant is the cheapest, and LRAC = $6.50 for Q = 200, 000. We

could also build 1 medium and two small factories, but then the costs would be an average

of $7.25 and $8, which is still worse than a large plant.

Finally long run average costs are minimized when we dont have to build any small or

medium sized plants, at Q = 200,000, 400,000, and so on.

C Factors that affect long run average costs

Long run average costs may be decreasing and then increasing, but also may be strictly

decreasing. Here are some LRAC curves for some industries.

1. Nursing Homes have decreasing LRAC. Nursing homes have many fixed man-

agement costs. Further, larger nursing homes are able to negotiate lower prices for

many raw materials.

2. Cruise Ships have decreasing LRAC. Huge cruise ships have lower average costs

than small cruise ships, economizing on many services provided on the ships.

3. Hospitals have U-shaped LRAC. Cowing and Holtmann (1983) in fact found many

hospitals in New York should in the long run reduce both capital and physicians to

lower average costs.

When the LRAC curve is decreasing, it is in the interest of the industry to consolidate.

A merger with another firm can increase the customer base but reduce the cost per unit,

thus increasing profits.

Reasons for decreasing long run average costs:

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1. Specialization of labor. Producing more units requires more workers. Workers can

then specialize in different parts of the production process, and produce more units.

2. Indivisibilities. A firm may require one accountant, even if there is less than 40 hours

of work to do. But then the firm can increase units produced without increasing the

size of the accounting staff.

3. Pricing Power. Large firms buy in bulk and therefore negotiate lower prices for

inputs.

Notice the reasons are similar to increasing returns to scale.

Reasons for increasing costs:

1. Regulation may exempt smaller firms.

2. Coordination and information problems. In a large firm, many individuals do

not meaningfully affect profits and thus have the wrong incentives. Smaller operations

may know their customers and production processes better.

With decreasing returns, spin-offs and divestments may be optimal.

A compromise is franchising. Nationalize just the parts for which increasing returns

works.

D Long run competitive equilibrium

In the long run, firms can enter and exit the market, or grow or shrink in size. As long as

economic profits are positive, new entrants arrive as the industry gives higher profits than

alternatives. New entrants increases competition and pushes down the price until economic

profits are zero. Therefore, in the long run:

pi = TR TC = 0, (85)

P Q = TC, (86)

P =TC

Q= LRAC. (87)

Now at the firm level, we set P = LRMC. But since also P = LRAC, marginal and

average costs are equal in the long run. We can therefore minimize long run average costs

and in the long run, we will have P = LRMC = LRAC.

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V Application of long run average costs: Banking mergers

Consider two banks. The first bank services Q = 15 customers and the second (smaller)

bank services Q = 5 customers. The long run average cost function in the industry is:

LRAC = 700 40Q+Q2 (88)

Price is constant at $300 dollars of revenue per customer.

Should these two firms merge? The size of the customer base (Q) which minimizes long

run average costs is:

minLRAC = 700 40Q+Q2 (89)

40 + 2Q = 0 Q = 20 (90)

Costs per unit fall until Q = 20. Thus these two firms can reduce costs by merging from two

firms of size Q = 15 and Q = 5) into one firm with Q = 20.

Profit per unit in each case are:

pi = 300(700 40Q+Q2

)= 400 + 40QQ2 (91)

pi (Q = 15) = 400 + 40 15 152 = 25 (92)

pi (Q = 5) = 400 + 40 5 + 52 = 225 (93)

pi (Q = 20) = 400 + 40 20 202 = 0 (94)

Individually, the two banks lose money but together they have zero economic profits. At

zero economic profits, no incentive exists for firms to enter or exit the market. The choice is

essentially to merge and get bigger or exit the market.

VI Measuring Cost Functions

We use the same procedure as with production functions: Obtain data on total costs and

quantity produced, and use Excel to fit the data. Both total cost and total quantity produced

may appear to be easier to obtain than input data. However, one must remember that costs

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represent opportunity costs, which are not always straightforward.

Some additional issues:

A Choice of Cost Function

One choice is whether to use a linear, quadratic, or cubic function:

TC = a + bQ (95)

TC = a + bQ+ cQ2 (96)

TC = a + bQ+ cQ2 + dQ3 (97)

Under most circumstances, the linear cost function does a reasonable job over a narrow

range of Q (for example in the short run), but the quadratic and cubic terms must matter

theoretically, especially for a wider range of Q. A good strategy might therefore be to

estimate the cubic or quadratic.

If the t-stats are low for the quadratic and cubic terms, then predictions are likely to be

unreliable for Q that falls outside the data. This indicates using some caution before, for

example, committing to large mergers. The following graph illustrates the problem.

1 2 3 4 5 6 760.5

61

61.5

62

62.5

63

63.5

64

Q

Cost

($)

Possible Problem Estimating Cost Functions: Data is to homogeneous

Estimate is inaccurate here

Estimate is accurate here

dataTrue Cost FunctionEstimated Cost Function

Figure 8: Cost estimation problems.

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B Data issues

Some problems with the data that often need correcting:

1. Definition of cost: as mentioned earlier, we use opportunity costs not accounting costs.

2. Price level changes: Historical data is likely to be inaccurate if the price of some inputs

or outputs have changed dramatically.

3. What costs vary with output: Some costs have a very limited relationship with output.

For example, the number of professionals required may vary in some limited way with

output. A firm with $1 million in sales may have two accountants. The firm can

obviously increase output to some degree without needing more accountants (so the

cost would be fixed). But for larger Q additional accountants are needed (like a variable

cost).

4. The cost data needs to match the output data. Often the cost of producing some

output may be accounted for in some other period.

5. The firms technology may change over time.

When estimating long run costs, it is usually preferable to use a cross section of firms

across an industry. An individual firm is unlikely to have changed size significantly enough

to generate data for a wide range of Q.

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