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Lecture

# Profit Under Monopoly.pdf

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School
University of Toronto St. George
Department
Mathematics
Course
MAT133Y1
Professor
Abe Igelfeld
Semester
Fall

Description
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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