cost curves

Chapter 21 NAME

Cost Curves

Introduction. Here you continue to work on cost functions. Total costcan be divided into fixed cost, the part that doesn’t change as outputchanges, and variable cost. To get the average (total) cost, average fixedcost, and average variable cost, just divide the appropriate cost functionby y, the level of output. The marginal cost function is the derivative ofthe total cost function with respect to output—or the rate of increase incost as output increases, if you don’t know calculus.

Remember that the marginal cost curve intersects both the averagecost curve and the average variable cost curve at their minimum points.So to find the minimum point on the average cost curve, you simply setmarginal cost equal to average cost and similarly for the minimum ofaverage variable cost.

Example: A firm has the total cost function C(y) = 100 + 10y. Let usfind the equations for its various cost curves. Total fixed costs are 100, sothe equation of the average fixed cost curve is 100/y. Total variable costsare 10y, so average variable costs are 10y/y = 10 for all y. Marginal costis 10 for all y. Average total costs are (100 + 10y)/y = 10+ 10/y. Noticethat for this firm, average total cost decreases as y increases. Notice alsothat marginal cost is less than average total cost for all y.

21.1 (0) Mr. Otto Carr, owner of Otto’s Autos, sells cars. Otto buysautos for $c each and has no other costs.

(a) What is his total cost if he sells 10 cars? 10c. What if he sells 20

cars? 20c. Write down the equation for Otto’s total costs assuming

he sells y cars: TC(y) = cy.

(b) What is Otto’s average cost function? AC(y) = c. For every

additional auto Otto sells, by how much do his costs increase? c.

Write down Otto’s marginal cost function: MC(y) = c.

(c) In the graph below draw Otto’s average and marginal cost curves ifc = 20.

266 COST CURVES (Ch. 21)

0 10 20 30 40

10

20

30

40AC, MC

Red line

AC=MC=20

Output

(d) Suppose Otto has to pay $b a year to produce obnoxious television

commercials. Otto’s total cost curve is now TC(y) = cy + b , his

average cost curve is now AC(y) = c + b/y , and his marginal cost

curve is MC(y) = c.

(e) If b = $100, use red ink to draw Otto’s average cost curve on thegraph above.

21.2 (0) Otto’s brother, Dent Carr, is in the auto repair business. Dentrecently had little else to do and decided to calculate his cost conditions.He found that the total cost of repairing s cars is TC(s) = 2s2 +10. ButDent’s attention was diverted to other things . . . and that’s where youcome in. Please complete the following:

Dent’s Total Variable Costs: 2s2.

Total Fixed Costs: 10.

Average Variable Costs: 2s.

Average Fixed Costs: 10/s.

Average Total Costs: 2s+ 10/s.

Marginal Costs: 4s.

NAME 267

21.3 (0) A third brother, Rex Carr, owns a junk yard. Rex can use oneof two methods to destroy cars. The first involves purchasing a hydrauliccar smasher that costs $200 a year to own and then spending $1 for everycar smashed into oblivion; the second method involves purchasing a shovelthat will last one year and costs $10 and paying the last Carr brother,Scoop, to bury the cars at a cost of $5 each.

(a) Write down the total cost functions for the two methods, where y is

output per year: TC1(y) = y + 200 , TC2(y) = 5y + 10.

(b) The first method has an average cost function 1 + 200/y and a

marginal cost function 1. For the second method these costs are

5 + 10/y and 5.

(c) If Rex wrecks 40 cars per year, which method should he use?

Method 2. If Rex wrecks 50 cars per year, which method should

he use? Method 1. What is the smallest number of cars per year

for which it would pay him to buy the hydraulic smasher? 48 cars

per year.

21.4 (0) Mary Magnolia wants to open a flower shop, the Petal Pusher,in a new mall. She has her choice of three different floor sizes, 200 squarefeet, 500 square feet, or 1,000 square feet. The monthly rent will be $1 asquare foot. Mary estimates that if she has F square feet of floor spaceand sells y bouquets a month, her variable costs will be cv(y) = y2/F permonth.

(a) If she has 200 square feet of floor space, write down her marginal cost

function: MC = y100 and her average cost function: AC =

200y

+ y200. At what amount of output is average cost minimized?

200. At this level of output, how much is average cost? $2.

(b) If she has 500 square feet, write down her marginal cost function:

MC = y/250 and her average cost function: AC =

(500/y)+ y/500. At what amount of output is average cost min-


imized? 500. At this level of output, how much is average cost?

$2.

(c) If she has 1,000 square feet of floor space, write down her marginal

cost function: MC = y/500 and her average cost function:

AC = (1, 000/y) + y/1, 000. At what amount of output is

average cost minimized? 1,000. At this level of output, how much

is average cost? $2.

(d) Use red ink to show Mary’s average cost curve and her marginal costcurves if she has 200 square feet. Use blue ink to show her average costcurve and her marginal cost curve if she has 500 square feet. Use blackink to show her average cost curve and her marginal cost curve if she has1,000 square feet. Label the average cost curves AC and the marginalcost curves MC.

0 400 600 800 1000

Bouquents

1

2

3

4

Dollars

200 1200

mc

ac

mc

acmc ac

Redlines Blue

linesBlacklines

LRMC=LRAC (yellow line)

(e) Use yellow marker to show Mary’s long-run average cost curve andher long-run marginal cost curve in your graph. Label them LRAC andLRMC.

21.5 (0) Touchie MacFeelie publishes comic books. The only inputs heneeds are old jokes and cartoonists. His production function is

Q = .1J12L3/4,

NAME 269

where J is the number of old jokes used, L the number of hours of cartoon-ists’ labor used as inputs, and Q is the number of comic books produced.

(a) Does this production process exhibit increasing, decreasing, or con-

stant returns to scale? Explain your answer. It exhibits

increasing returns to scale since f(tJ, tL) =

t5/4f(J, L) > tf(J, L).

(b) If the number of old jokes used is 100, write an expression for the

marginal product of cartoonists’ labor as a function of L. MP =3

4L1/4 Is the marginal product of labor decreasing or increasing as the

amount of labor increases? Decreasing.

21.6 (0) Touchie MacFeelie’s irascible business manager, Gander Mac-Grope, announces that old jokes can be purchased for $1 each and thatthe wage rate of cartoonists’ labor is $2.

(a) Suppose that in the short run, Touchie is stuck with exactly 100 oldjokes (for which he paid $1 each) but is able to hire as much labor as hewishes. How much labor would he have to hire in order produce Q comic

books? Q4/3.

(b) Write down Touchie’s short-run total cost as a function of his output

2Q4/3 + 100.

(c) His short-run marginal cost function is 8Q1/3/3.

(d) His short-run average cost function is 2Q1/3 + 100/Q.

Calculus 21.7 (1) Touchie asks his brother, Sir Francis MacFeelie, to study thelong-run picture. Sir Francis, who has carefully studied the appendix toChapter 19 in your text, prepared the following report.

(a) If all inputs are variable, and if old jokes cost $1 each and car-toonist labor costs $2 per hour, the cheapest way to produce exactly

one comic book is to use 104/5 (4/3)3/5 ≈ 7.4 jokes and

104/5 (3/4)2/5 ≈ 5.6 hours of labor. (Fractional jokes are cer-tainly allowable.)


(b) This would cost 18.7 dollars.

(c) Given our production function, the cheapest proportions in which touse jokes and labor are the same no matter how many comic books weprint. But when we double the amount of both inputs, the number of

comic books produced is multiplied by 25/4.

21.8 (0) Consider the cost function c(y) = 4y2 + 16.

(a) The average cost function is AC = 4y + 16y .

(b) The marginal cost function is MC = 8y.

(c) The level of output that yields the minimum average cost of production

is y = 2.

(d) The average variable cost function is AV C = 4y.

(e) At what level of output does average variable cost equal marginal

cost? At y = 0.

21.9 (0) A competitive firm has a production function of the formY = 2L + 5K. If w = $2 and r = $3, what will be the minimum cost of

producing 10 units of output? $6.

Chapter 22 NAME

Firm Supply

Introduction. The short-run supply curve of a competitive firm is theportion of its short-run marginal cost curve that is upward sloping andlies above its average variable cost curve. The long-run supply curve of acompetitive firm is the portion of its short-run marginal cost curve thatis upward-sloping and lies above its long-run average cost curve.

Example: A firm has the long-run cost function c(y) = 2y2 + 200 fory > 0 and c(0) = 0. Let us find its long-run supply curve. The firm’smarginal cost when its output is y is MC(y) = 4y. If we graph output onthe horizontal axis and dollars on the vertical axis, then we find that thelong-run marginal cost curve is an upward-sloping straight line throughthe origin with slope 4. The long-run supply curve is the portion of thiscurve that lies above the long-run average cost curve. When output is y,long-run average costs of this firm are AC(y) = 2y + 200/y. This is a U-shaped curve. As y gets close to zero, AC(y) becomes very large because200/y becomes very large. When y is very large, AC(y) becomes verylarge because 2y is very large. When is it true that AC(y) < MC(y)?This happens when 2y+200/y < 4y. Simplify this inequality to find thatAC(y) < MC(y) when y > 10. Therefore the long-run supply curve isthe piece of the long-run marginal cost curve for which y > 10. So thelong-run supply curve has the equation p = 4y for y > 10. If we want tofind quantity supplied as a function of price, we just solve this expressionfor y as a function of p. Then we have y = p/4 whenever p > 40.

Suppose that p < 40. For example, what if p = 20, how much willthe firm supply? At a price of 20, if the firm produces where price equalslong-run marginal cost, it will produce 5 = 20/4 units of output. Whenthe firm produces only 5 units, its average costs are 2 × 5 + 200/5 = 50.Therefore when the price is 20, the best the firm can do if it produces apositive amount is to produce 5 units. But then it will have total costs of5×50 = 250 and total revenue of 5×20 = 100. It will be losing money. Itwould be better off producing nothing at all. In fact, for any price p < 40,the firm will choose to produce zero output.

22.1 (0) Remember Otto’s brother Dent Carr, who is in the auto repairbusiness? Dent found that the total cost of repairing s cars is c(s) =2s2 + 100.

(a) This implies that Dent’s average cost is equal to 2s + 100/s ,

his average variable cost is equal to 2s , and his marginal cost is

equal to 4s. On the graph below, plot the above curves, and also plotDent’s supply curve.

272 FIRM SUPPLY (Ch. 22)

0 5 10 15 20

20

40

60

Output

Dollars

80

��

��

Supply

mc

ac

avc

Revenue

Costs

Profit

(b) If the market price is $20, how many cars will Dent be willing to

repair? 5. If the market price is $40, how many cars will Dent

repair? 10.

(c) Suppose the market price is $40 and Dent maximizes his profits. Onthe above graph, shade in and label the following areas: total costs, totalrevenue, and total profits.

Calculus 22.2 (0) A competitive firm has the following short-run cost function:c(y) = y3 − 8y2 + 30y + 5.

(a) The firm’s marginal cost function isMC(y) = 3y2−16y+30.

(b) The firm’s average variable cost function is AV C(y) = y2− 8y+

30. (Hint: Notice that total variable costs equal c(y)− c(0).)

(c) On the axes below, sketch and label a graph of the marginal costfunction and of the average variable cost function.

(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.

(e) Marginal cost equals average variable cost when output is 4.

cost curves

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