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Much of economic analysis is concerned with optimization problems. Economics, after all, is the science of choice, and optimization problems are the form in which economists usually model choice mathematically. A general discussion of such problems must be postponed until we have developed the necessary tools from calculus. Here we show how the simple results from this section on maximizing quadratic functions can be used to illustrate some basic economic ideas.

E X A M P L E 1 The price P per unit obtained by a firm in producing and selling Q units is P = 102−2Q, and the cost of producing and selling Q units is C= 2Q +1

2Q2. Then the profit is3

π(Q)= P Q − C = (102 − 2Q)Q − (2Q + 1₂Q2)= 100Q −5₂Q2 Find the value of Q which maximizes profits, and the corresponding maximal profit.

Solution: Using formula (4) we find that profit is maximized at

Q= Q∗= − 100 2#−5 2 $ = 20 with π∗= π(Q∗)= − 100 2 4#−5 2 $ = 1000 This example is a special case of the monopoly problem studied in the next example.

2 _{The function f is symmetric about x= x}

0if f (x0+ t) = f (x0− t) for all x. See Section 5.2. 3 _{In mathematics π is used to denote the constant ratio 3.1415 . . . between the circumference of a}

circle and its diameter. In economics, this constant is not used very often. Also, p and P usually denote a price, so π has come to denote profit.

S E C T I O N 4 . 6 / Q U A D R A T I C F U N C T I O N S 101 E X A M P L E 2 (A Monopoly Problem) Consider a firm that is the only seller of the commodity it produces, possibly a patented medicine, and so enjoys a monopoly. The total costs of the monopolist are assumed to be given by the quadratic function

C = αQ + βQ2, Q≥ 0

of its output level Q, where α and β are positive constants. For each Q, the price P at which it can sell its output is assumed to be determined from the linear “inverse” demand function

P = a − bQ, Q≥ 0

where a and b are constants with a > 0 and b ≥ 0. So for any nonnegative Q, the total revenue R is given by the quadratic function R = P Q = (a − bQ)Q, and profit by the quadratic function

π(Q)= R − C = (a − bQ)Q − αQ − βQ2= (a − α)Q − (b + β)Q2

Assuming that the monopolist’s objective is to maximize the profit function π = π(Q), find the optimal output level QM_{and the corresponding optimal profit π}M_.

Solution: By using (4), we see that there is a maximum of π at

QM = a− α

2(b+ β) with π

M ₌ (a− α)2

4(b+ β) (∗)

This is valid if a > α; if a ≤ α, the firm will not produce, but will have QM _{= 0 and}

πM _{= 0. The two cases are illustrated in Figs. 2 and 3. In Fig. 3, the part of the parabola to}

the left of Q= 0 is dashed, because it is not really relevant given the natural requirement that Q≥ 0. The price and cost associated with QM_{in (∗) can be found by routine algebra.}

π Q QM _2QM π Q QM

Figure 2 The profit function, a > α Figure 3 The profit function, a_{≤ α}

If we put b = 0 in the price function P = a − bQ, then P = a for all Q. In this case, the firm’s choice of quantity does not influence the price at all and so the firm is said to be

perfectly competitive. By replacing a by P in our previous expressions, we see that profit is maximized for a perfectly competitive firm at

Q∗=P − α

2β with π∗=

(P − α)2

4β (∗∗)

provided that P > α. If P ≤ α, then Q∗_{= 0 and π}∗_{= 0.}

Solving the first equation in (_{∗∗) for P yields P = α + 2βQ}∗_{. Thus, the equation}

P= α + 2βQ (∗∗∗)

represents the supply curve of this perfectly competitive firm for P > α. For P _{≤ α, the profit-} maximizing output Q∗_{is 0. The supply curve relating the price on the market to the firm’s choice of} output quantity is shown in Fig. 4; it includes all the points of the line segment between the origin and (0, α), where the price is too low for the firm to earn any profit by producing a positive output.

P

Q α

P ! α $ 2 βQ

Figure 4 The supply curve of a perfectly competitive firm

Let us return to the monopoly firm (which has no supply curve). If it could somehow be made to act like a competitive firm, taking price as given, it would be on the supply curve (_{∗∗∗). Given the} demand curve P _{= a − bQ, equilibrium between supply and demand occurs when (∗∗∗) is also} satisfied, and so P_{= a − bQ = α + 2βQ. Solving the second equation for Q, and then substituting} for P and π in turn, we see that the respective equilibrium levels of output, price, and profit would be

Qe= a− α b+ 2β, P e₌2aβ+ αb b+ 2β , π e₌β(a− α)2 (b+ 2β)2

In order to have the monopolist mimic a competitive firm by choosing to be at (Qe_{, P}e_{), it may be} desirable to tax (or subsidize) the output of the monopolist. Suppose that the monopolist is required to pay a specific tax of t per unit of output. Because the tax payment tQ is added to the firm’s costs, the new total cost function is

C= αQ + βQ2+ tQ = (α + t)Q + βQ2

Carrying out the same calculations as before, but with α replaced by α_{+ t, gives the monopolist’s} choice of output as QM_t = ⎧ ⎨ ⎩ a− α − t 2(b_{+ β)}, if a≥ α + t 0, otherwise So QM

t = Qe when (a− α − t)/2(b + β) = (a − α)/(b + 2β). Solving this equation for t yields t _{= −(a − α)b/(b + 2β). Note that t is actually negative, indicating the desirability of}

S E C T I O N 4 . 6 / Q U A D R A T I C F U N C T I O N S 103

subsidizingthe output of the monopolist in order to encourage additional production. (Of course, subsidizing monopolists is usually felt to be unjust, and many additional complications need to be considered carefully before formulating a desirable policy for dealing with monopolists. Still the previous analysis suggests that if it is desirable to lower a monopolist’s price or its profit, this is much better done directly than by taxing its output.)

P R O B L E M S F O R S E C T I O N 4 . 6

1. (a) Let f (x)_{= x}2_{− 4x. Complete the following table and use it to sketch the graph of f :}

x −1 0 1 2 3 4 5

f (x)

(b) Using (3), determine the minimum point of f . (c) Solve the equation f (x)_{= 0.}

2. (a) Let f (x)_{= −}1

2x2− x +23. Complete the following table and sketch the graph of f :

x −4 −3 −2 −1 0 1 2

f (x)

(b) Using (4), determine the maximum point of f . (c) Solve the equation₋1

2x2− x +32= 0 for x. (d) Show that f (x)_{= −}1

2(x− 1)(x + 3), and use this to study how the sign of f (x) varies with x. Compare the result with the graph.

3. Determine the maximum/minimum points by using (3) or (4):

(a) x2_{+ 4x (b) x}2_{+ 6x + 18 (c) −3x}2_{+ 30x − 30}

(d) 9x2_{− 6x − 44 (e) −x}2_{− 200x + 30 000 (f) x}2_{+ 100x − 20 000}

4. Find all the zeros of each quadratic function in Problem 3, and write each function in the form a(x− x1)(x− x2)(if possible).

5. Find solutions to the following equations, where p and q are parameters.

(a) x2_{− 3px + 2p}2_{= 0 (b) x}2_{− (p + q)x + pq = 0 (c) 2x}2_{+ (4q − p)x = 2pq}

6. A model by A. Sandmo in the theory of efficient loan markets involves the function U (x)= 72 − (4 + x)2− (4 − rx)2

7. (a) A farmer has 1000 metres of fence wire with which to make a rectangular enclosure, as illustrated in the figure below. Find the areas of the three rectangles whose bases are 100, 250, and 350 metres.

(b) Let the base have length 250_{+ x. Then the height is 250 − x (see Fig. 5). What choice of} xgives the maximum area?4

250 $ x

250 " x 250 " x

250 $ x

Figure 5

8. (a) If a cocoa shipping firm sells Q tons of cocoa in the UK, the price received is given by PE = α1−1₃Q. On the other hand, if it buys Q tons from its only source in Ghana, the price it has to pay is given by PG= α2+16Q. In addition, it costs γ per ton to ship cocoa from its supplier in Ghana to its customers in the UK (its only market). The numbers α1, α2, and γ are all positive. Express the cocoa shipper’s profit as a function of Q, the number of tons shipped.

(b) Assuming that α1− α2− γ > 0, find the profit-maximizing shipment of cocoa. What happens if α1− α2− γ ≤ 0?

(c) Suppose the government of Ghana imposes an export tax on cocoa of t per ton. Find the new expression for the shipper’s profits and the new quantity shipped.

(d) Calculate the Ghanaian government’s export tax revenue T as a function of t, and compare the graph of this function with the Laffer curve presented in Section 4.1.

(e) Advise the Ghanaian government on how to obtain as much tax revenue as possible.

HARDER PROBLEM

⊂

⊃

nand b1, b2, . . . , bnbe arbitrary real numbers. We claim that the following inequality (called the Cauchy–Schwarz inequality) is always valid:

(a1b1+ a2b2+ · · · + anbn)2≤ (a12+ a22+ · · · + a2n)(b21+ b22+ · · · + b2n) (5) (a) Check the inequality for a1= −3, a2= 2, b1= 5, and b2= −2. (Then n = 2.)

(b) Prove (5) by means of the following trick: first, define f for all x by f (x)= (a1x+ b1)2+ · · · + (anx+ bn)2

It should be obvious that f (x)_{≥ 0 for all x. Write f (x) as Ax}2_{+ Bx + C, where the} expressions for A, B, and C are related to the terms in (5). Because Ax2_{+ Bx + C ≥ 0} for all x, we must have B2_{− 4AC ≤ 0. Why? The conclusion follows.}

4 _{It is reported that certain surveyors in antiquity wrote contracts with farmers to sell them rectangular} pieces of land in which only the perimeter was specified. As a result, the lots were long narrow rectangles.

S E C T I O N 4 . 7 / P O L Y N O M I A L S 105

4.7 Polynomials

After considering linear and quadratic functions, the logical next step is to examine cubic

functions of the form

f (x)= ax3+ bx2+ cx + d (a, b, c, and d are constants; a̸= 0) (1) It is relatively easy to examine the behaviour of linear and quadratic functions. Cubic functions are considerably more complicated, because the shape of their graphs changes drastically as the coefficients a, b, c, and d vary. Two examples are given in Figs. 1 and 2.

5 10 15 "1 "2 1 2 3 4 y x f(x) ! "x3 _{$ 4x}2 _{" x " 6} y Q y ! C(Q)

Figure 1 A cubic function Figure 2 A cubic cost function

Cubic functions do occasionally appear in economic models. Let us look at an example.

E X A M P L E 1 Consider a firm producing a single commodity. The total cost of producing Q units of the commodity is C(Q). Cost functions often have the following properties: First, C(0) is positive, because an initial fixed expenditure is involved. When production increases, costs also increase. In the beginning, costs increase rapidly, but the rate of increase slows down as production equipment is used for a higher proportion of each working week. However, at high levels of production, costs again increase at a fast rate, because of technical bottlenecks and overtime payments to workers, for example. It can be shown that the cubic cost function C(Q)= aQ3+ bQ2+ cQ + d exhibits this type of behaviour provided that a > 0, b < 0, c >0, d > 0, and 3ac > b2. Such a function is sketched in Fig. 2.

Cubic cost functions whose coefficients have a different sign pattern have also been studied. For instance, a study of a particular electric power generating plant revealed that over a certain period, the cost of fuel y as a function of output Q was given by y_{= −Q}3_{+ 214.2Q}2_{− 7900Q + 320700.} (This cost function cannot be valid for all Q, however, because it suggests that fuel costs would be negative for large enough Q.)

The detailed study of cubic functions is made easier by applying the differential calculus, as will be seen later.