Rip x pim p 100 r

Cip) = 25(100 - p) = 2.500 - 25, jrip) = Rip) - C(p) = 100, - p! - (2.500 - 25,) = I25p- p2 — 2.500

Then maximizing with respect to /> gives

7t'(p*) = 125 - 2p' = 0 giving p* = $62.50, just as before.

Example 6.2 Monopoly Equilibrium I

A monopolist has inverse demand function p = 50 - 2x. The total-cost function is C = 20 + 2.x + 0.5x2. What are the profit-maximizing price and output?

Solution

Profn is

TT(.x) = 50x - 2x: - [20 + 2x + 0.5x2] = 48.r - 2.5.v2 - 20

and so x' = 9.6 p* = 50 - 2(9.6) = $30.80 The level of profit at the maximum is then rtix*) = 48(9.6) - 2.5(9.6 f - 20 = $210.40 See figure 6.10. ■

Example 6.3 Monopoly Equilibrium II

A monopolist has inverse demand function p = 150 - 2.r and total-cost function C - 0. lx3 - 3x2 -F 50x 4- 100. What are the profit-maximizing price and output?

Solution

A similar sequence of steps to those in exercise 6.1 gives

7r(.t) = 150.x: - 2x2 - [O.lx3 - 3x2 + 50* + I01JJ = lOO.r + x2 - O.lx3 - 100

Figure 6.10 Monopoly equilibrium for example 6.2

Solving this quadratic gives

r ~ mj so that a* = 21.89 or x* = -15.23 Since negative outputs are impossible, at* - 21.89. />* = $106.22. and

7rU*)= 100(21.89) -f- (21.89)2 — 0.1 (21.89)' - 100 = $1,519.26 See figure 6.1 I.

Figure 6.11 Monopoly equilibrium for example 6.3 Competitive Firm with Linear Costs

Suppose thai a firm has the cost function C = 5.v. and sells into a perfectly competitive market. This means that it can take the market price p as a given constant, rather than a function of its output. This assumption shows that a competitive firm is so small relative to the size of the market that its output decision has essentially no effect on the market price. Suppose that the market price is p = S8. Then we set up the firm's profit as follows:

7rU) = (8 - 5).r = 3.v If we now apply equation (6.3) to find profit-maximizing output, we have tt'(X') = 3 = 0

which is nonsense! What went wrong?

Figure 6. i 2 shows the mathematical answer to this question. In figure 6.12(a). the firm's profit function always increases with output—it has no maximum. Thus, this firm would want to increase output indefinitely, since doing so always increases profit. In Figure 6.12(b). this is seen as resulting from pricc (= marginal revenue) being always greater than marginal cost, implying that an extra unit of output always adds more to revenue than it does to cost.

Figure 6.12 Competitive firm with constant cost

Figure 6.12 Competitive firm with constant cost

Mathematically this example serves as a warning. If we want to solve a problem by applying the _f'{x*)= 0 condition, we must be sure that the function really does possess a maximum. In other words, this is an example of a function that does not possess a maximum. (Compare figure 6.12(a) with figure 6.1(a).) But does the example have any economic meaning? In fact, it does. Essentially it says that if at least one firm in a competitive market has constant returns to scale (implying a linear cost function of the kind used here), then we might expect perfect competition to break down and be replaced by monopoly or oligopoly, since it pays the firm to expand output indefinitely, and as this happens the market structure must become less competitive as firms are driven out of the market. Then the assumption that firms face a "horizontal demand curve" becomes untenable: we can no longer assume that firms are "too small" to affect price. Thus we need to turn to other market models such as oligopoly and monopoly. In a sense, figure 6.12 shows a logically inconsistent situation: if a firm has constant costs below the market price, then it cannot regard price as a given parameter tor all possible output levels.

Monopoly with Constant-Elasticity Demand and Constant Costs

We begin with an example of constant unit elasticity of demand. Suppose that a monopoly firm faces the demand function x = I Op'1, or in inverse form p= 10,v 1. and has the cost function, C = 5,v, Note that wc have to restrict the domain of the function to X > 0. The demand function is called "constant elasticity" because, if we evaluate the elasticity of demand, we obtain or even more simply, since log a = log 10 - log p d logx _ ( d log p

Since the elasticity is independent of the particular point on the demand curve, we say elasticity is constant

To find the profit-maximizing output, we set up

7T(.V) = R(x)-C(x)= 10-5a and applying equation (6.3) gives

which again is nonsense. Wba: went wrong this time? Again, a figure will suggesr the mathematical answer (see figure 6.13). In figure 6.13(a). the revenue function litx) is a constant, while costs are increasing, and so profit varies inversely with output—the lower is output the higher is profit.

(a) (b)

In tigure 6.13(h). we see why this is so: as output is lowered, price rises by enough to keep revenue constant (p = lO/.v => px = 10), and so, since reducing output leaves revenue unchanged while reducing total costs, it pays to produce as small an output as possible. However, there is a problem here. If the tirm produced zero output, it makes zero profit, so this cannot be the maximum. But then, since between zero and any x. however small, there is an infinity of x-values (x is a real number), so there is in fact no solution to the problem. If we proposed, say, x — 0.1 as a solution, we could immediately show that x = 0.01 yielded a larger profit. Jt = 0.001 still more, and so on.

Another way of looking at this is to note that marginal revenue, R'(x). in this case is zero: in figure 6.13(b), the marginal-revenue "curve" coincides with the horizontal axis, since a change in output produces no change in revenue. Since marginal cost is constant at $5. there can be no output at which marginal revenue equals marginal cost. Again, we see that care must be taken in applying equation (6.3) in a model, because we agatn have a case in which a maximum solution does not exist. A general discussion of the existence of solutions to optimization problems is presented in chapter 13.

Now consider an example with constant elasticity of demand which is greater than one. Let .v = p'2 be the demand curve faced by the monopoly. In inverse form we express this demand function as p = x 1 The elasticity of demand is

In this case the revenue function is not a constant:

R(x) = px = x'uzx =xu: If the cost function is C(.\) = 2x. we have profit given by 7t(x\ = R(x) - C(x) = xwl - 2.x and so irix') = 0 =>■ —--2 = 0

so that x' = I /16 is the level of output which will maximize proht. Unlike the case of unit elasticity of demand, we see in this example that reducing output reduces total revenue and so a positive value of output exists which gives maximum profit.

The third possibility is for the constant elasticity of demand to be less than one. We treat this possibility below.

Example 6.4 Constant Elasticity of Demand Less Than One

Is there a level of output. ,v > 0. which maximizes profit for a monopolist facing the demand function x - p~i,: and cost function C(.r) = 2.v? Discuss in terms of the elasticity of demand.

Solution

Since x - p~U2 =>x = —rrz pl/2x = 1 => p>/2 — x~[ p\ti we get p = x'2 as the inverse demand function. We also have

Thus elasticity is a constant, less than one. Moreover

or R(x) — \/x. For any positive value of .v. revenue will rise if .r falls. Since costs also fall as v falls, then starting from any positive output level, this firm can increase profit by reducing output, provided that the firm does not reduce output to zero. This result follows because elasticity less than one implies that a reduction in output is accompanied by a larger increase, in percentage terms, of price. Therefore revenue rises when output is reduccd as long as a > 0. As in the case of unit elasticity of demand, there is no positive value of output which leads to maximum profit. ■

A Publisher Will Always Set a Higher Price and Sell Fewer Copies of a Book Than the Author Would Like

Take the monopoly firm illustrated in figure 6.9 and assume the good concerned is a book. The author of the book receives a royalty of 10% of the purchase price for each book sold, and so her income is

The publisher's profit must now take into account the royalty paid to the author:

tt(.ï) = R{x) - C(.y) - Yu') = 75je - - ( 10jt - 0. U2) = 65.V - 0.9.V-

We assume thai the author would like to set price and quantity to maximize her income, and so we have the condition y\xA)= 10 — 0.2.* ^ = 0

giving ,<a - 50 as the author's desired sales, and p,\ = 100 — 50 = S50 as her desired price. The publisher, however, chooses the number of books lo satisfy n'(Xp) = 65 - l.8xp = 0

giving desired sales of 36.1 at a price of $63.90.

Thss "conflict of interest" always arises and is not due to the special example chosen. If we let r, 0 < r < 1, denote Ihe royalty rate, we can write the author's income and die publisher's profit respectively as

7T(x) = R(x) - C(x) - rfl(v) = (1 - r )R(x) - C(.v)

Then maximizing Y{x) gives

Thus the author essentially wishes to maximize sales revenue. Maximizing publisher's profit, on the other hand, gives tt'u,,) = (I - r)R'(Xp) - C'(X„) = 0

Then, as long as marginal cost is greater than zero. C > 0, the publisher's desired output xr must differ from ihe author's desired output . Given that we usually assume marginal revenue decreasing with output, that is. R"(x) < 0. equations (6.7) and (6.8) must imply that xA > xp.

Example 6.5 A publisher pays the author of a book a royalty of 15%. Demand for the book is x = 2(X) - 5p and the production cost is C = 10 + 2.v + x2. Find the optimal sales from both the author's and the publisher's perspective.

Solution

The inverse demand function is p = 40 - 0.2x

The author's income is y(x) = 0.15px = 0.15(40 - 0.2x)x = 6.v - 0.03x"

The optimal level of sales from the author's viewpoint, x ,. leads to a maximum value for >'(x). and so y'(.r/,) = 0=> 6 — 0.06 A,! = 0=>,t„ = 100

The profit for the publisher, on the other hand, is

= px - (10 + 2x + xz) - (6a - 0.03.x2) = (40 - 0.2a.)a - (10 + 2x + x2) - (6a - 0.03a-2) = 32x - 1.32.r2 - 10

The optimal level of sales from the publisher's viewpoint. t>. leads to a maximum value for jr(.r). and so n'(xp) = 32 - 2.46x = 0 =?> x? = = 13

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