Economics 3070 Prof. Barham

Problem Set – Chapter 9 Solutions

1. Based on Ch 9, Problem 9.9 Earl’s Car Wash is a small business that operates in the perfectly competitive car

washing industry in Louisville, Colorado. The short-run total cost of production is

( ) 21.01040 QQQSTC ++= ,

where Q is the number of cars washed per day. The prevailing market price is 20$ per car.

a. How many cars should Earl wash to maximize profit?

Earl needs to maximize the profit function:

2. (40 10 0.1 )

10 .2 0

10 .2

QMax P Q Q Q

P QQP Q

p

p

= - + +

¶= - - =

¶= +

If price equals 20 then:

50

2.01020

=

+=

Q

Q

Therefore, Earl should wash 50 cars to maximize profit. b. Assume that all of the 40$ per day fixed costs are sunk. Derive an expression

for Earl’s short-run supply curve. Then graph the curve.

The first step is to identify the minimum price needed to induce Earl to wash cars; that is, we need to identify minP . In the short-run, minP is the minimum average nonsunk cost ( )ANSC . If all fixed costs are sunk, then

( ) ( )

Q

QQQ

QAVCQANSC

1.010

1.010 2

+=

+=

=

MR

SMC

Economics 3070 Prof. Barham

Since ANSC is increasing in Q , minimum ANSC occurs at 0=Q and equals 10. Therefore, 10min =P . For minPP £ Earl does not wash any cars, but for minPP ³ Earl operates at a point where SMCP = :

10 0.25 50

P QQ P= += -

Therefore, Earl’s short-run supply curve is

( )ïî

ïíì

³-

£=

10 if505

10 if0

PP

PPs

The graph below depicts this supply curve:

c. Assume, instead, that if Earl produces zero output, he can rent or sell his

fixed assets thereby avoiding all his fixed costs. Derive an expression for Earl’s short-run supply curve. Then graph the curve. If all fixed costs are nonsunk, then

( ) ( )

QQ

QSACQANSC

1.01040++=

=

Minimum ANSC occurs at the point where ( ) ( )QSMCQANSC = :

Q

P

( )Ps

10

20

50

Economics 3070 Prof. Barham

20

400

1.040

2.0101.01040

2

=

=

=

+=++

Q

Q

QQ

QQQ

Since ( ) 1420 =ANSC , 14min =P .

For minPP £ Earl does not wash any cars, but for minPP ³ Earl operates at a point where SMCP = . Therefore, Earl’s short-run supply curve is

( )ïî

ïíì

³-

£=

14 if505

14 if0

PP

PPs

The graph below depicts this supply curve:

Q

P

( )Ps

14

20

50 20

Economics 3070 Prof. Barham

2. Ch 9, Problem 9.12

The coal industry consists of 60 producers, all of whom have an identical short-

run total cost curve,

( ) 2264 QQSTC += ,

where Q is the monthly output of a firm and 64$ is the monthly fixed cost. Assume that 32$ of the firm’s monthly 64$ fixed cost can be avoided if the firm produces zero output in a month. The market demand curve for coal production is

( ) PPD 5400 -= ,

where ( )PD is monthly demand at price P . Derive an expression for the market supply curve in this market and determine the short-run equilibrium price. We will begin by deriving the short-run supply curve for a single firm. The firm’s average nonsunk cost is given by the following equation:

( )

QQ

QQQANSC

232

232 2

+=

+=

To find the shut-down price ( )minP , we find the minimum level of ANSC . This occurs at the quantity at which SMCANSC = :

4

232

4232

=

=

=+

Q

QQ

QQQ

Since ( ) ( )( ) 16424/324 =+=ANSC , 16min =P . For minPP £ the firm does not produce any output, but for minPP ³ the firm operates at a point where SMCP = :

PQ

QP

41

4

=

=

Therefore, each firm’s short-run supply curve is

Economics 3070 Prof. Barham

( )ïî

ïíì

³

£=

16 if

16 if0

41 PP

PPs

Since there are 60 firms, the market supply curve is simply

( )ïî

ïíì

³

£=

16 if15

16 if0

PP

PPS

To find the equilibrium price, we equate market supply and market demand:

( ) ( )

20

40020

540015

=

=

-=

=

*P

P

PP

PDPS

Therefore, the equilibrium price is $20. 5. Based off Ch 9, Problem 9.16 The screw and bolt market contains many identical firms, each with a short-run total cost function

STC(Q) = 400-5Q + Q2 , where Q is the firm’s annual output (and all of the firm’s $400 fixed cost is sunk). The market demand curve for this industry is

D(P) = 262.5 – P/2

Where P is the market price. Each firm in the industry is currently earning zero economic profit. How many firms are in this industry, and what is the market equilibrium price?

The two equilibrium conditions that must hold in the short run are: a. Profit Maximization, which implies that P*=MC(Q*) b. Market Supply = Market Demand In addition, we have been told that we must satisfy the zero profit condition. P*=AC(Q*)

c. So, we know that the profit max and zero profit condition together mean that: P=SACmin

Economics 3070 Prof. Barham

From a: P=MC: P* = -5 + 2Q* From zero-profit condition: P=AC: P* = 400/Q* – 5 + Q* So now we have two equations and 2 unknowns (P* and Q*). First, lets slove for Q*

-5 +2Q* = 400/Q *– 5 + Q* ( Q*)2= 400

Q* = 20 So each firm will produce 20.

We can sub the Q* back into the AC(Q*) or MC(Q*) to get P*

P* = MC(Q*) = -5 + 2(20) = 35 So, market equilibrium price will be P* = 35. To determine the number of firms we need to figure out the market output and divide that by the amount each firm produces. Market Output: D(P) : 262.5 – 35 / 2 = 262.5 – 17.5 = 245 So the number of firms are (total output of market/output of a firm): 245 / 20 = 12.25 6. Based on Ch 9, Problem 9.24 The Brussels sprouts industry is perfectly competitive, and each producer has the long-run total cost function

( ) 32

31640 QQQQTC +-= .

The market demand curve for Brussels sprouts is

( ) PPD 1002200 -= .

What is the long-run equilibrium price in this industry? At this price, how much would an individual firm produce? How many firms are in the Brussels sprouts market in a long-run competitive equilibrium? In a long-run equilibrium, the equilibrium price *P , equilibrium output of each firm *Q , and equilibrium number of firms *n satisfy the following three conditions:

1. ( )** = QMCP (profit maximization)

2. ( )** = QACP (zero economic profit)

Economics 3070 Prof. Barham

3. ( ) *** = QnPD (market clearing)

The first step is to apply conditions 1 and 2 by setting ( ) ( )** = QACQMC :

9

182

6

6401240

2

232

2312

=

=

=

+-=+-

*Q

QQ

QQ

QQQQ

Therefore, each individual firm produces 9=*Q units of Brussels sprouts. To find the equilibrium price, we simply plug 9=*Q into ( )*QMC :

( ) ( ) ( )

13

8110840

9912409 2

=

+-=

+-=MC

Therefore, the long-run equilibrium price is 13=*P . To find the equilibrium number of firms, we apply condition 2 by plugging 13=*P into the market demand curve:

( )

( ) ( )( )

( )

100

9009

13002200

13100220013

1002200

=

=

-=

-=

-=

*

*

**

n

n

Qn

D

PPD

Therefore, the equilibrium number of firms is 100=*n .

Economics 3070 Prof. Barham

7. Based on Ch 9, Problem 9.32 The long-run total cost for production of exam booklets is given by

2 3(120 20 )TC wr Q Q Q= - + ,

Where Q is the annual output of a firm, w is the wage rate for skilled assembly labor, and r is the price of capital services. The demand for labor for an individual firm is

2 3(120 20 )( , , )2

r Q Q QL Q w rw

- +=

a) In a long-run competitive equilibrium, how much output will each firm

produce. b) In a long-run competitive equilibrium, what will be the market price? Note

that your answer will be expressed as a function of w. c) In a long-run competitive equilibrium, how much skilled labor will each firm

demand? Again, your answer will be in terms of w. d) Suppose that the market demand curve is given by D(P) = 10,000/p. What is

the market equilibrium quantity as a function of w? e) What is the long-run equilibrium number of firms as a function of w? f) Using your answers to (c) and (e), determine the overall demand for skilled

labor in this industry as a function of w. g) Suppose the supply curve for the skilled labor used in this industry is G(w) =

50w. At what value of w does the supply of skilled labor equal the demand for skilled labor.

NOTE: These answers assume the price of capita, r, is equal to 1.

a) To determine long-run output we can use the profit maximizing and zero profit condition.

MC(Q*) = AC (Q*) 220 * 2( *)

* *Q QQ Q

=

Q* = 10

b.

P* = MC(Q*) =

2(120 40(10) 3(10)

(120 400 300)

(20)

wr

w

w

- +

- +

Economics 3070 Prof. Barham

c.

2 3120*10 20*10 10( , , )2

200 1002

L Q w rw

w w

- +=

= =

d.

10,000 10,000 500( *)* 20

D PP w w

= = =

e.

500( *) 50** 10

D P wnQ w

= = =

f. Demand for skilled labor: Demand for Lobor by each firm * number of firms.

100w* 50w=5000W

g.

2

5000 50

10010

L LD S

wwww

=

=

==