Download P = 120

A monopolist firm faces a demand with constant elasticity of -2.0. It has a constant
marginal cost of $20 per unit and sets a price to maximize profit. If marginal cost
should increase by 25 percent, would the price charged also rise by 25 percent?
Yes. The monopolist’s pricing rule as a function of the elasticity of demand
for its product is:
(P - MC)
= -
P
1
Ed
or alternatively,
P =
MC

1


+
 1 

E d 


In this example Ed = -2.0, so 1/Ed = -1/2; price should then be set so that:
P =
MC
1 
2 
= 2MC
Therefore, if MC rises by 25 percent, then price will also rise by 25 percent.
When
MC = $20, P = $40. When MC rises to $20(1.25) = $25, the price rises to $50,
a 25 percent increase.
A firm faces the following average revenue (demand) curve:
P = 120 - 0.02Q
where Q is weekly production and P is price, measured in cents per unit. The firm’s
cost function is given by C = 60Q + 25,000. Assume that the firm maximizes profits.
a.
What is the level of production, price, and total profit per week?
The profit-maximizing output is found by setting marginal revenue equal to
marginal cost. Given a linear demand curve in inverse form, P = 120 - 0.02Q,
we know that the marginal revenue curve will have twice the slope of the
demand curve. Thus, the marginal revenue curve for the firm is MR = 120 0.04Q. Marginal cost is simply the slope of the total cost curve. The slope of
TC = 60Q + 25,000 is 60, so MC equals 60. Setting MR = MC to determine the
profit-maximizing quantity:
120 - 0.04Q = 60, or
Q = 1,500.
Substituting the profit-maximizing quantity into the inverse demand function
to determine the price:
P = 120 - (0.02)(1,500) = 90 cents.
Profit equals total revenue minus total cost:
 = (90)(1,500) - (25,000 + (60)(1,500)), or
 = $200 per week.
If the government decides to levy a tax of 14 cents per unit on this product, what
will be the new level of production, price, and profit?
Suppose initially that the consumers must pay the tax to the government.
Since the total price (including the tax) consumers would be willing to pay
remains unchanged, we know that the demand function is
P* + T = 120 - 0.02Q, or
P* = 120 - 0.02Q - T,
where P* is the price received by the suppliers. Because the tax increases the
price of each unit, total revenue for the monopolist decreases by TQ, and
marginal revenue, the revenue on each additional unit, decreases by T:
MR = 120 - 0.04Q - T
where T = 14 cents. To determine the profit-maximizing level of output with
the tax, equate marginal revenue with marginal cost:
120 - 0.04Q - 14 = 60, or
Q = 1,150 units.
Substituting Q into the demand function to determine price:
P* = 120 - (0.02)(1,150) - 14 = 83 cents.
Profit is total revenue minus total cost:
  831,150  601,150  25,000 1450 cents, or
$14.50 per week.
Note: The price facing the consumer after the imposition of the tax is 97 cents.
The monopolist receives 83 cents. Therefore, the consumer and the monopolist
each pay 7 cents of the tax.
If the monopolist had to pay the tax instead of the consumer, we would arrive
at the same result. The monopolist’s cost function would then be
TC = 60Q + 25,000 + TQ = (60 + T)Q + 25,000.
The slope of the cost function is (60 + T), so MC = 60 + T. We set this MC to
the marginal revenue function from part (a):
120 - 0.04Q = 60 + 14, or
Q = 1,150.
Thus, it does not matter who sends the tax payment to the government. The
burden of the tax is reflected in the price of the good.
Suppose that an industry is characterized as follows:
a.
C  100  2Q2
Firm total cost function
MC  4Q
Firm marginal cost function
P  90  2Q
Industry demand curve
MR  90  4Q
Industry marginal revenue curve
.
If there is only one firm in the industry, find the monopoly price, quantity,
and level of profit.
If there is only one firm in the industry, then the firm will act like a
monopolist and produce at the point where marginal revenue is equal to
marginal cost:
MC=4Q=90-4Q=MR
Q=11.25.
For a quantity of 11.25, the firm will charge a price P=90-2*11.25=$67.50.
The level of profit is $67.50*11.25-100-2*11.25*11.25=$406.25.
b.
Find the price, quantity, and level of profit if the industry is competitive.
If the industry is competitive then price is equal to marginal cost, so that 902Q=4Q, or Q=15. At a quantity of 15 price is equal to 60. The level of profit
is therefore 60*15-100-2*15*15=$350.
c.
Graphically illustrate the demand curve, marginal revenue curve, marginal
cost curve, and average cost curve. Identify the difference between the
profit level of the monopoly and the profit level of the competitive industry
in two different ways. Verify that the two are numerically equivalent.
The graph below illustrates the demand curve, marginal revenue curve, and
marginal cost curve. The average cost curve hits the marginal cost curve at a
quantity of approximately 7, and is increasing thereafter (this is not shown in
the graph below). The profit that is lost by having the firm produce at the
competitive solution as compared to the monopoly solution is given by the
difference of the two profit levels as calculated in parts a and b above, or
$406.25-$350=$56.25. On the graph below, this difference is represented by
the lost profit area, which is the triangle below the marginal cost curve and
above the marginal revenue curve, between the quantities of 11.25 and 15.
This is lost profit because for each of these 3.75 units extra revenue earned
was less than extra cost incurred. This area can be calculated as 0.5*(6045)*3.75+0.5*(45-30)*3.75=$56.25.
The second method of graphically
illustrating the difference in the two profit levels is to draw in the average
cost curve and identify the two profit boxes. The profit box is the difference
between the total revenue box (price times quantity) and the total cost box
(average cost times quantity). The monopolist will gain two areas and lose
one area as compared to the competitive firm, and these areas will sum to
$56.25.
. Michelle’s Monopoly Mutant Turtles (MMMT) has the exclusive right to sell Mutant
2
Turtle t-shirts in the United States. The demand for these t-shirts is Q = 10,000/P .
The firm’s short-run cost is SRTC = 2,000 + 5Q, and its long-run cost is LRTC = 6Q.
a.
What price should MMMT charge to maximize profit in the short run? What
quantity does it sell, and how much profit does it make? Would it be better
off shutting down in the short run?
MMMT should offer enough t-shirts such that MR = MC. In the short run,
marginal cost is the change in SRTC as the result of the production of another
t-shirt,
i.e.,
SRMC = 5, the slope of the SRTC curve. Demand is:
10,000
Q
,
P2
or, in inverse form,
-1/2
P = 100Q
.
1/2
Total revenue (PQ) is 100Q . Taking the derivative of TR with respect to Q,
-1/2
MR = 50Q
quantity:
.
Equating MR and MC to determine the profit-maximizing
-1/2
5 = 50Q
, or Q = 100.
Substituting Q = 100 into the demand function to determine price:
-1/2
P = (100)(100
) = 10.
The profit at this price and quantity is equal to total revenue minus total cost:
 = (10)(100) - (2000 + (5)(100)) = -$1,500.
Although profit is negative, price is above the average variable cost of 5 and
therefore, the firm should not shut down in the short run. Since most of the
firm’s costs are fixed, the firm loses $2,000 if nothing is produced. If the profitmaximizing quantity is produced, the firm loses only $1,500.
b.
What price should MMMT charge in the long run? What quantity does it sell
and how much profit does it make? Would it be better off shutting down in
the long run?
In the long run, marginal cost is equal to the slope of the LRTC curve, which is
6.
Equating marginal revenue and long run marginal cost to determine the profitmaximizing quantity:
-1/2
50Q
= 6 or Q = 69.44
Substituting Q = 69.44 into the demand equation to determine price:
2 -1/2
P = (100)[(50/6) ]
= (100)(6/50) = 12
Therefore, total revenue is $833.33 and total cost is $416.67. Profit is $416.67.
The firm should remain in business.
c.
Can we expect MMMT to have lower marginal cost in the short run than in
the long run? Explain why.
In the long run, MMMT must replace all fixed factors. Therefore, we can
expect LRMC to be higher than SRMC.
The Metro Electric Company produces and distributes electricity to residential customers in the
metropolitan area. This monopoly firm is regulated, as are other investor owned electric
companies. The company faces the following demand and marginal revenue functions:
P = 0.04 - 0.01Q
MR = 0.04 - 0.02Q
Its marginal cost function is:
MC = 0.005 + 0.0075Q,
where Q is in millions of kilowatt hours and P is in dollars per kilowatt hour. Find the
deadweight loss that would result if this company were allowed to operate as a profit
maximizing firm, assuming that P = MC under regulation.
Solution:
Find the area between the average revenue curve and the marginal cost curve that is
bounded by the rates of production chosen first by the profit maximizing monopoly and
second by the regulated industry having the same cost structure.
Monopoly output is denoted QM and is found where MC = MR.
0.04 - 0.02Q = 0.005 + 0.0075Q
QM = 1.2727 (in millions of KWH)
The regulated industry output takes place where average revenue equals marginal cost.
R
= 0.04 - 0.01Q
Q
0.04 - 0.01Q = 0.005 + 0.0075Q
AR =
Q = 2 (in millions of KWH)
The area representing deadweight loss is the area under the AR curve minus the area
under the MC curve between Q = 2 and Q = 1.2727.
Area under AR is computed by first finding the heights of AR at the two quantities.
At Q = 2, AR = 0.04 - 0.01(2) = 0.04 - 0.02 = 0.02
At Q = 1.2727, AR = 0.04 - 0.01(1.2727) = 0.04 - 0.012727 = 0.02727.
The average AR =
0.02 + 0.02727
= 0.023636
2
Area = (2 - 1.2727)(0.023636) = 0.01719
The area under MC is
At Q = 2, MC = 0.005 + 0.0075(2) = 0.02
At Q = 1.2727, MC = 0.005 + 0.0075(1.2727) = 0.014545
Average height =
0.02 + 0.014545
2
The area under MC is
Area = (2 - 1.2727)(0.01727) = 0.01256
Deadweight loss = 0.01719 - 0.01256
= 0.00463 in millions of dollars
= $4,630 / time period.