Download a. Calculate the price elasticity of demand when the

In 1998, Americans smoked 470 billion cigarettes, or 23.5 billion packs of cigarettes. The
average retail price was $2 per pack. Statistical studies have shown that the price
elasticity of demand is -0.4, and the price elasticity of supply is 0.5. Using this
information, derive linear demand and supply curves for the cigarette market.
EPD =
P DQ
.
Q DP
You are given information about the value of the elasticity, P, and Q, which means
that you can solve for the slope, which is b in the above formula for the demand
curve.
To find the constant a, substitute for Q, P, and b into the above formula so that
23.5=a-4.7*2 and a=32.9. The equation for demand is therefore Q=32.9-4.7P. To
find the supply curve, recall the formula for the elasticity of supply and follow the
same method as above:
To find the constant c, substitute for Q, P, and d into the above formula so that
23.5=c+5.875*2 and c=11.75. The equation for supply is therefore Q=11.75+5.875P.
Consider a competitive market for which the quantities demanded and supplied (per
year) at various prices are given as follows:
a.
Price
Demand
Supply
($)
(millions)
(millions)
60
22
14
80
20
16
100
18
18
120
16
20
Calculate the price elasticity of demand when the price is $80 and when the price
is $100.
We know that the price elasticity of demand may be calculated using equation 2.1
from the text:
ED
DQ D
QD
P DQ D
=
=
.
DP
Q D DP
P
With each price increase of $20, the quantity demanded decreases by 2. Therefore,
æ DQD ö = -2 = -0.1.
è DP ø 20
At P = 80, quantity demanded equals 20 and
80
ED = æè öø ( -0.1) = -0.40.
20
Similarly, at P = 100, quantity demanded equals 18 and
100 ö
ED = æè
( -0.1) = -0.56.
18 ø
b.
Calculate the price elasticity of supply when the price is $80 and when the price
is $100.
The elasticity of supply is given by:
DQ S
QS
P DQ S
ES =
=
.
DP
Q S DP
P
With each price increase of $20, quantity supplied increases by 2. Therefore,
æ DQS ö = 2 = 0.1.
è DP ø 20
At P = 80, quantity supplied equals 16 and
80
ES = æè öø( 0.1) = 0.5.
16
Similarly, at P = 100, quantity supplied equals 18 and
100 ö
ES = æè
(0.1) = 0.56.
18 ø
c.
What are the equilibrium price and quantity?
The equilibrium price and quantity are found where the quantity supplied equals
the quantity demanded at the same price. As we see from the table, the
equilibrium price is $100 and the equilibrium quantity is 18 million.
d.
Suppose the government sets a price ceiling of $80. Will there be a shortage, and
if so, how large will it be?
With a price ceiling of $80, consumers would like to buy 20 million, but producers
will supply only 16 million. This will result in a shortage of 4 million.
Label the curves in the diagram below.
a.
At what range of prices will the firm earn a negative profit? Positive profit or zero
profit?
b.
At what price will the firm shut down?
A competitive firm has the following short run cost function:
C(q) = q - 8q +30q + 5.
3
a.
2
Find MC, AC, and AVC and sketch them on a graph.
The functions can be calculated as follows:
MC =
¶C
= 3q2 - 16q + 30
¶q
AC =
C
5
2
= q - 8q + 30 +
q
q
AVC =
VC
2
= q - 8q + 30
q
Graphically, all three cost functions are u-shaped in that cost declines initially as
q increases, and then cost increases as q increases. Average variable cost is
below average cost. Marginal cost will be initially below AVC and will then
increase to hit AVC at its minimum point. MC will be initially below AC and will
also hit AC at its minimum point.
b.
At what range of prices will the firm supply zero output?
The firm will find it profitable to produce in the short run as long as price is
greater than or equal to average variable cost. If price is less than average
variable cost then the firm will be better off shutting down in the short run, as it
will only lose its fixed cost and not fixed plus some of variable cost. Here we need
to find the minimum average variable cost, which can be done in two different
ways. You can either set marginal cost equal to average variable cost, or you can
differentiate average variable cost with respect to q and set this equal to zero. In
both cases, you can solve for q and then plug into AVC to find the minimum AVC.
Here we will set AVC equal to MC:
AVC = q 2 - 8q + 30 = 3q 2 - 16q + 30 = MC
2q2 = 8q
q= 4
AVC(q = 4) = 4 2 - 8* 4 + 30 = 14.
Hence, the firm supplies zero output if P<14.
c.
Identify the firm’s supply curve on your graph.
The firm supply curve is the MC curve above the point where MC=AVC. The firm
will produce at the point where price equals MC as long as MC is greater than or
equal to AVC.
d.
At what price would the firm supply exactly 6 units of output?
The firm maximizes profit by choosing the level of output such that P=MC. To
find the price where the firm would supply 6 units of output, set q equal to 6 and
solve for MC:
P = MC = 3q -16q + 30 = 3(6 ) -16(6) + 30 = 42.
2
2
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 profitmaximizing 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.
Suppose that an industry is characterized as follows:
C = 100 + 2Q2
F irm total cost function
MC = 4Q
P = 90 - 2Q
MR = 90 - 4Q
F irm marginal cost function
Indus try demand curve
Indus try marginal revenue curve.
a.
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*(60-45)*3.75+0.5*(4530)*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.
P
MC
lost profit
MR
Demand
Q
11.25 15