Download Economics 3070 Fall 2014 Problem Set 6 Answers 1. Chapter 9

Economics 3070
Fall 2014
Problem Set 6 Answers
1. Chapter 9, question 1
Last year, the account ledger for an owner of a small drugstore showed the following
information about her annual receipts and expenditures;
Revenues: $1,000000
Wages for hired labor:
$300,000
Utilities: $20,000
Purchases of drugs and other supplies for the store: $500,000
Wages paid to herself: $100,000
She pays a competitive wage to her workers , and the utilities and drugs and other
supplies are all obtained at market prices. She already owns the building, so she
pays no money for its use. If she were to close the business she could avoid all of
her expenses and, of course , would have no revenue. However, she could rent
out her building for $200,000. She could also work elsewhere herself. Her two
employment alternatives include working as a lawyer, earning wages of $100,000,
or working at a local restaurant, earning $20,000. Determine her accounting
profit and her economic profit if she stays working in the drugstore business. If
the two are different explain the difference.
Accounting Profits:
Total Revenues:
Total Accounting Cost:
Profits:
Economic Profits:
Total Revenues:
Total Economic Costs:
Profits
1,000,000
300,000 + 20,000 + 500,000 + 100, 000
80,000
1,000,000
300,000 + 20,000 + 500,000 + 100, 000 + 200,000
-120,000
So, if she continues to work at the grocery store, she earns an accounting profit of 80,000
plus the salary she pays herself 100,000. However, she earn negative economic profits if
she continues to work at the grocery store (-120,000). But if she exits the business, her
salary as a lawyer would be $100,000, and she would receive $200,000 rent for the
building. She would therefore be better off by $120,000 if she worked as a lawyer.
2. 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
STC ( Q ) = 40 + 10 Q + 0.1Q 2 ,
where Q is the number of cars washed per day. The prevailing market price is
$20 per car.
Economics 3070
Fall 2014
a. How many cars should Earl wash to maximize profit?
Earl needs to maximize the profit function:
Max π = P .Q − (40 + 10Q + 0.1Q 2 )
Q
∂π
= P − 10 − .2Q = 0
∂Q
P = 10 + .2Q
MR
SMC
If price equals 20 then:
20 = 10 + 0.2 Q
Q = 50
Therefore, Earl should wash 50 cars to maximize profit.
b. What is Earl’s maximum daily profit?
Earl’s profit is given by π = TR − TC .
π = PQ − (40 + 10 Q + 0.1 Q 2 )
[
= (20 )(50 ) − 40 + (10 )(50 ) + (0.1)(50 )2
]
= 210
Therefore, Earl’s maximum daily profit is $210.
c. Graph SMC , SAC , and the profit-maximizing quantity. On this graph,
indicate the maximum daily profit.
The equations for short-run marginal cost and short-run average cost are
∂STC
= 10 + 0.2Q
∂Q
STC 40
SA C (Q ) =
=
+ 10 + 0.1Q
Q
Q
SMC (Q ) =
The two curves and profit-maximizing quantity ( Q = 50) are depicted in the
diagram below:
Economics 3070
Fall 2014
$
SMC
20
SAC
15.8
14
10
20
50
Q
The shaded area represents the Earl’s profit. It is a rectangle whose height is the
difference between the market price (P = 20) and the average cost of the 50th unit
(SAC (50) = 15.8) and whose width is the 50 units being produced.
d. 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 Pmin . In the short-run, Pmin is the minimum average
nonsunk cost ( ANSC ) . If all fixed costs are sunk, then
ANSC ( Q ) = AVC ( Q )
10 Q + 0.1 Q 2
=
Q
= 10 + 0.1 Q
Since ANSC is increasing in Q , minimum ANSC occurs at Q = 0 and equals 10.
Therefore, Pmin = 10 .
For P ≤ Pmin Earl does not wash any cars, but for P ≥ Pmin Earl operates at a point
where P = SMC :
P = 10 + 0.2Q
Q = 5P − 50
Therefore, Earl’s short-run supply curve is
Economics 3070
Fall 2014
if P ≤ 10
⎧⎪0
s (P ) = ⎨
⎪⎩5P − 50 if P ≥ 10
The graph below depicts this supply curve:
P
s (P )
20
10
50
Q
e. 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
ANSC ( Q ) = SAC ( Q )
=
40
+ 10 + 0.1 Q
Q
Minimum ANSC occurs at the point where ANSC ( Q ) = SMC ( Q ):
40
+ 10 + 0.1 Q = 10 + 0.2 Q
Q
40
= 0 .1 Q
Q
Q 2 = 400
Q = 20
Since ANSC (20) = 14 , Pmin = 14 .
For P ≤ Pmin Earl does not wash any cars, but for P ≥ Pmin Earl operates at a point
where P = SMC . Therefore, Earl’s short-run supply curve is
Economics 3070
Fall 2014
if P ≤ 14
⎧⎪0
s (P ) = ⎨
⎪⎩5P − 50 if P ≥ 14
The graph below depicts this supply curve:
P
s (P )
20
14
20
50
Q
Economics 3070
Fall 2014
3.
Ch 9, Problem 9.12
The coal industry consists of 60 producers, all of whom have an identical shortrun total cost curve,
STC ( Q ) = 64 + 2 Q 2 ,
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
D (P ) = 400 − 5P ,
where D (P ) 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:
ANSC ( Q ) =
32 + 2 Q 2
Q
=
32
+ 2Q
Q
To find the shut-down price (Pmin ), we find the minimum level of ANSC . This occurs
at the quantity at which ANSC = SMC :
32
+ 2Q = 4Q
Q
32
= 2Q
Q
Q =4
Since ANSC (4 ) = 32 / 4 + (2 )(4 ) = 16 , Pmin = 16 .
For P ≤ Pmin the firm does not produce any output, but for P ≥ Pmin the firm operates at
a point where P = SMC :
P = 4Q
Q=
1
P
4
Therefore, each firm’s short-run supply curve is
Economics 3070
Fall 2014
⎧⎪0
s (P ) = ⎨
⎪⎩ 41 P
if P ≤ 16
if P ≥ 16
Since there are 60 firms, the market supply curve is simply
⎧⎪0
S (P ) = ⎨
⎪⎩15P
if P ≤ 16
if P ≥ 16
To find the equilibrium price, we equate market supply and market demand:
S (P ) = D (P )
15P = 400 − 5P
20P = 400
P ∗= 20
Therefore, the equilibrium price is $20.
4. Ch 9, Problem 9.17
Suppose a competitive, profit-maximizing firm operates at a point where its
short-run average cost curve is upward sloping. What does this imply about the
firm’s economic profits? If the profit-maximizing firm operates at a point where
its short-run average cost curve is downward sloping, what does this imply about
the firm’s economic profits?
If the firm operates at a point where its SAC curve is rising, it must mean that the SMC
curve is above the SAC curve. Since the firm sets P = SMC , it must be the case that
P > SAC . Therefore, the firm earns positive economic profit.
If the firm operates at a point where the SAC curve is falling, it must mean that
SMC < SAC . Since the firm sets P = SMC , it must be the case that P < SAC .
Therefore, the firm earns negative economic profit.
However, the fact that the firm is still operating means that SMC ≥ ANSC ; that is, the
firm’s revenue is covering all of its nonsunk costs but only a portion of its sunk costs.
This is why the firm is better off operating – albeit at a loss – than shutting down.
Economics 3070
Fall 2014
5. Ch 9, Problem 9.20
A firm’s short-run supply curve is given by
if P < 10
⎧⎪0
s (P ) = ⎨
⎪⎩3P − 30 if P ≥ 10
What is the equation of the firm’s marginal cost curve SMC ( Q )?
We know that if the firm produces positive output, it produces where P = SMC . In this
case, when the firm produces positive output,
Q = 3P − 30
P = 31 Q + 10
This means that the equation of the firm’s short-run marginal cost curve is
SMC ( Q ) = 31 Q + 10 .
6. Based off Question 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
From a:
P* = -5 + 2Q*
Economics 3070
Fall 2014
From b.
P* = 400/Q* – 5 + Q*
So a. and b. we have two equations and 2 unknons (P* and Q*).
First, lets slove for Q*
So each firm will produce 20.
-5 +2Q* = 400/Q *– 5 + Q*
( Q*)2= 400
Q* = 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: 245 / 20 = 12.25
7. Based on Question Ch 9, Problem 9.24
The Brussels sprouts industry is perfectly competitive, and each producer has the
long-run total cost function
TC ( Q ) = 40 Q − 6 Q 2 +
1 3
Q .
3
The market demand curve for Brussels sprouts is
D(P ) = 2200 − 100P .
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:
( )
P = AC (Q ) (zero economic profit)
D (P ) = n Q (market clearing)
1. P ∗ = MC Q ∗ (profit maximization)
2.
3.
∗
∗
∗
∗
∗
Economics 3070
Fall 2014
( )
( )
The first step is to apply conditions 1 and 2 by setting MC Q ∗ = AC Q ∗ :
40 − 12 Q + Q 2 = 40 − 6 Q + 31 Q 2
2
3
Q 2 = 6Q
2 Q 2 = 18 Q
Q∗ = 9
Therefore, each individual firm produces Q ∗ = 9 units of Brussels sprouts.
( )
To find the equilibrium price, we simply plug Q ∗ = 9 into MC Q ∗ :
MC (9) = 40 − 12(9) + (9)2
= 40 − 108 + 81
= 13
Therefore, the long-run equilibrium price is P ∗ = 13 .
To find the equilibrium number of firms, we apply condition 2 by plugging P ∗ = 13 into
the market demand curve:
D (P ) = 2200 − 100P
D (13) = 2200 − (100)(13)
n ∗ Q ∗ = 2200 − 1300
n ∗ (9) = 900
n ∗ = 100
Therefore, the equilibrium number of firms is n ∗ = 100 .
8. Based on Question Ch 9, Problem 9.32
The long-run total cost for production of exam booklets is given by
TC = wr (120Q − 20Q 2 + Q 3 ) ,
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
Economics 3070
Fall 2014
L(Q, w, r ) =
r (120Q − 20Q 2 + Q3 )
2 w
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*)
wr (120 − 20Q * +(Q *)2 ) = wr (120 − 40Q * +3(Q *)2 )
20Q * 2(Q*)2
=
Q*
Q*
Q* = 10
b.
wr (120 − 40(10) + 3(10)2
P* = MC(Q*) =
w (120 − 400 + 300)
w (20)
c.
Economics 3070
Fall 2014
120*10 − 20*102 + 103
2 w
200 100
=
=
2 w
w
L(Q, w, r ) =
Economics 3070
Fall 2014
d.
D( P*) =
10,000 10,000 500
=
=
P*
20 w
w
e.
D( P*)
n* =
=
Q*
500
w = 50
10
w
f.
Demand for skilled labor: Demand for Lobor by each firm * number of firms.
100 50 5000
*
=
w
w
w
g.
DL = S L
5000
= 50 w
w
w2 = 100
w = 10