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# Profit Under Monopoly.pdf

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University of Toronto St. George

Mathematics

MAT133Y1

Abe Igelfeld

Fall

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3. Pro▯t Under a Monopoly
Under a monopoly a producer has full control over how many units of a product will reach
market. Since market demand and selling price are related by a demand relation, the monopolist
can also control the price at which the commodity sells. Under a free enterprise system it is
assumed that the monopolist will attempt to maximize his or her pro▯t. How many units should
the monopolist produce?
Let R(x) be the revenue associated with the sale of x units; C(x) the cost to the
monopolist of producing x units. Then pro▯t is de▯ned to be P(x) , with P(x) = R(x)▯C(x) .
In Figure 1 typical revenue and cost curves are shown, and the pro▯t zone is shaded in.
y
C(x)
R(x)
C
0
x x
0
Figure 1
Now P (x) = R (x) ▯ C (x) , and P (x) = 0 if and only if R (x) = C (x) . So as long as
00
P (x) < 0 , the pro▯t will be maximized at the production level for which marginal revenue
equals marginal cost. In Figure 1 this optimal production level is denoted by x ; it 0s the point
where the tangents to the revenue and cost curves are parallel.
Note that the values of price and volume for optimal pro▯t do not depend on the overhead
cost, but if the overhead cost is too high no positive pro▯t may be possible for any realistic
price.
This can be seen in Figure 1 where we have called the overhead cost C . Increasing C
0 0
may eventually place the entire cost curve above the revenue curve so that there would be no
pro▯table region.
Example 1. A publisher estimates the cost of producing and selling x copies of a new science
▯ction book will be, in dollars,
C(x) = 10;000 + 5x :
1 The demand of the book at price p dollars is
x = 4;000 ▯ 200p :
Find the most pro▯table sale price.
Solution. Solving the demand relation for p , we ▯nd
x
p = 20 ▯ dollars :
200
Hence total revenue is
2
x
R(x) = 20x ▯ 200
and pro▯t is
x 2
P(x) = 20x ▯ ▯ 5x ▯ 10;000 :
200
For maximum value,
0 x
P (x) = 15 ▯ 100 = 0
) x = 1500 :
Observe that
00 1
P (x) = ▯ 100 < 0 for all x ;
so that a true maximum value has been found. The price is then p = 20▯1200 = 12:50 (dollars)
and the maximum pro▯t is
2
P(1500) = 15(1500) ▯ (1500) ▯ 10;000
200
= 1;250 (dollars) :
Example 2. A Case Study of Taxation on a Monopoly
In this example we shall construct a simple model to analyse taxation on a monopoly. We
assume that the cost and demand curves for a certain product are both linear, and that a tax
of t dollars per unit is levied on the producer for every unit produced.
2 y
0
y = C + qx
C 0
y = a - bx
x
Figure 2
Let the demand curve be p = a ▯ bx , with a > 0 , b > 0 ; and let the cost curve be
C = C 0 qx , where C 0 denotes the overhead cost and q is postive. With tax t dollars per
unit, the producer’s cost function becomes
C = C + 0x + tx :
The revenue function is xp , so the pro▯t function is
P(x) = xp ▯ C
= x(a ▯ bx) ▯ C 0 (q + t)x
= ▯C +0(a ▯ q ▯ t)x ▯ bx :
P (x) = a ▯ q ▯ t ▯ 2bx
= 0
a ▯ q ▯ t
, x = :
2b
Since P (x) = ▯2b , which is negative, the monopolist’s pro▯t is maximized by producing
a ▯ q ▯ t
2b units. The maximum pro▯t is
▯ ▯
a ▯ q ▯ t (a ▯ q ▯ t)
P 2b = 4b ▯ C 0
This pro▯t is positive provided that
2
(a ▯ q ▯ t)
C 0 ▯ 4b ;
3 which for t = 0 becomes

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