In this section we consider some other interesting economic applications of optimization theory when there are two variables. Versions of the first example have already appeared in Example 13.1.5 and Problem 13.2.4.
E X A M P L E 1 (Discriminating Monopolist) Consider a firm that sells a product in two isolated geo- graphical areas. If it wants to, it can then charge different prices in the two different areas because what is sold in one area cannot easily be resold in the other. As an example, it seems that express mail or courier services find it possible to charge much higher prices in Europe than they can in North America. Another example—pharmaceutical firms often charge much more for the same medication in the USA than they do in Europe or Canada.
Suppose that such a firm also has some monopoly power to influence the different prices it faces in the two separate markets by adjusting the quantity it sells in each. Economists generally use the term “discriminating monopolist” to describe a firm having this power.
Faced with two such isolated markets, the discriminating monopolist has two independent demand curves. Suppose that, in inverse form, these are
P1=a1−b1Q1, P2=a2−b2Q2 (∗) for market areas 1 and 2, respectively. Suppose, too, that the total cost is proportional to total production:2
C(Q)=α(Q1+Q2) As a function ofQ1andQ2, total profits are
π(Q1, Q2)=P1Q1+P2Q2−α(Q1+Q2)
=(a1−b1Q1)Q1+(a2−b2Q2)Q2−α(Q1+Q2)
=(a1−α)Q1+(a2−α)Q2−b1Q21−b2Q22
2 It is true that this cost function neglects transport costs. But the point to be made is that, even though supplies to the two areas are perfect substitutes in production, the monopolist will generally be able to earn higher profits by charging different prices, if this is allowed.
We want to find the values ofQ1 ≥ 0 andQ2 ≥ 0 that maximize profits. The first-order conditions are
π1(Q1, Q2)=(a1−α)−2b1Q1=0, π2(Q1, Q2)=(a2−α)−2b2Q2=0 with the solutions
Q∗1 =(a1−α)/2b1, Q∗2 =(a2−α)/2b2
Furthermore,π11(Q1, Q2)= −2b1,π12(Q1, Q2)=0, andπ22(Q1, Q2)= −2b2. Hence, for all(Q1, Q2)
π11 ≤0, π22 ≤0, and π11π22 −(π12)2=4b1b2 ≥0
We conclude from Theorem 13.2.1 that ifQ∗1 and Q∗2 are both positive, implying that (Q∗1, Q∗2)is an interior point in the domain ofπ, then the pair(Q∗1, Q∗2)really does maximize profits.
The corresponding prices can be found by inserting these values in(∗)to get P1∗=a1−b1Q∗1= 12(a1+α), P2∗=a2−b2Q∗2 =12(a2+α) The maximum profit is
π∗= (a1−α)2
4b1 +(a2−α)2 4b2
Both demandsQ∗1andQ∗2are positive provideda1 > αanda2> α. In this case,P1∗andP2∗ are both greater thanα. This implies that there is no “dumping”, with the price in one market less than the costα. Nor is there any “cross-subsidy”, with the losses due to dumping in one market being subsidized out of profits in the other market. It is notable that the optimal prices are independent ofb1andb2. More important, note that the prices arenotthe same in the two markets, except in the special case whena1 =a2. Indeed,P1∗> P2∗if and only ifa1> a2. This says that the price is higher in the market where consumers are willing to pay a higher price for each unit when the quantity is close to zero.
E X A M P L E 2 Suppose that the monopolist in Example 1 has the demand functions P1 =100−Q1, P2=80−Q2 and that the cost function isC=6(Q1+Q2).
(a) How much should be sold in the two markets to maximize profits? What are the cor- responding prices?
(b) How much profit is lost if it becomes illegal to discriminate?
(c) The authorities in market 1 impose a tax oft per unit sold in market 1. Discuss the consequences.
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Solution:
(a) Herea1=100,a2 =80,b1=b2 =1, andα=6. Example 1 gives the answers Q∗1 =(100−6)/2=47, Q∗2 =37, P1∗= 12(100+6)=53, P2∗=43 The corresponding profit isP1∗Q∗1+P2∗Q∗2−6(Q∗1+Q∗2)=3578.
(b) If price discrimination is not permitted, thenP1 = P2 = P, and Q1 = 100−P, Q2 =80−P, with total demandQ=Q1+Q2=180−2P. ThenP =90−12Q, so profits are
π=
90−12Q
Q−6Q=84Q−12Q2
This has a maximum atQ=84 whenP =48. The corresponding profit is now 3528, so the loss in profit is 3578−3528=50.
(c) With the introduction of the tax, the new profit function is ˆ
π =(100−Q1)Q1+(80−Q2)Q2−6(Q1+Q2)−t Q1
We easily see that this has a maximum atQˆ1=47−12t,Qˆ2=37, with corresponding pricesPˆ1 = 53+ 12t, Pˆ2 = 43. The tax therefore has no influence on the sales in market 2, while the amount sold in market 1 is lowered and the price in market 1 goes up. The optimal profit is easily worked out:
π∗=(53+12t )(47−12t )+43ã37−6(84−12t )−t (47−12t )=3578−47t+14t2 So introducing the tax makes the profit fall by 47t−14t2. The authorities in market 1 obtain a tax revenue which is
T =tQˆ1 =t (47−12t )=47t−12t2
Thus we see that profits fall by14t2more than the tax revenue. This amount14t2repre- sents the so-called deadweight loss from the tax.
A monopolistic firm faces a downward-sloping demand curve. Adiscriminating monopolist such as in Example 1 faces separate downward-sloping demand curves in two or more isolated markets. Amonopsonistic firm, on the other hand, faces an upward-sloping supply curve for one or more of its factors of production. Then, by definition, adiscriminating monopsonistfaces two or more upward-sloping supply curves for different kinds of the same input—for example, workers of different race or gender. Of course, discrimination by race or gender is illegal in many countries. The following example, however, suggests one possible reason why firms might want to discriminate if they were allowed to.
E X A M P L E 3 (Discriminating Monopsonist) Consider a firm using quantitiesL1 andL2 of two kinds of labour as its only inputs in order to produce outputQaccording to the simple production function
Q=L1+L2
Thus, both output and labour supply are measured so that each unit of labour produces one unit of output. Note especially how the two kinds of labour are essentially indistinguishable, because each unit of each type makes an equal contribution to the firm’s output. Suppose, however, that there are two segmented labour markets, with different inverse supply func- tions specifying the wage that must be paid to attract a given labour supply. Specifically, suppose that
w1=α1+β1L1, w2=α2+β2L2
Assume moreover that the firm is competitive in its output market, taking priceP as fixed.
Then the firm’s profits are
π(L1, L2)=P Q−w1L1−w2L2=P (L1+L2)−(α1+β1L1)L1−(α2+β2L2)L2
=(P−α1)L1−β1L21+(P−α2)L2−β2L22 The firm wants to maximize profits. The first-order conditions are
π1(L1, L2)=(P −α1)−2β1L1=0, π2(L1, L2)=(P −α2)−2β2L2 =0 These have the solutions
L∗1= P −α1 2β1
, L∗2 =P −α2 2β2
It is easy to see that the conditions for maximum in Theorem 13.2.1 are satisfied, so that L∗1,L∗2really do maximize profits ifP > α1andP > α2. The maximum profit is
π∗=(P −α1)2
4β1 +(P −α2)2 4β2
The corresponding wages are
w1∗=α1+β1L∗1 =12(P +α1), w∗2 =α2+β2L∗2= 12(P +α2)
Hence,w∗1 =w∗2 only ifα1=α2. Generally, the wage is higher for the type of labour that demands a higher wage for very low levels of labour supply—perhaps this is the type of labour with better job prospects elsewhere.
E X A M P L E 4 (Econometrics: Linear Regression) Empirical economics is concerned with analys- ing data in order to try to discern some pattern that helps in understanding the past, and possibly in predicting the future. For example, price and quantity data for a particular com- modity such as natural gas may be used in order to try to estimate a demand curve. This might then be used to predict how demand will respond to future price changes. The most commonly used technique for estimating such a curve islinear regression.
Suppose it is thought that variabley—say, the quantity demanded—depends upon vari- ablex—say, price or income. Suppose that we have observations(xt, yt)of both variables at timest =1,2, . . . , T. Then the technique of linear regression seeks to fit a linear function
y =α+βx to the data, as indicated in Fig. 1.
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et⫽ yt⫺ (α⫹βxt)
y ⫽α⫹βx y
x (xt, yt)
Figure 1
Of course, an exact fit is possible only if there exist numbersαandβfor whichyt =α+βxt fort =1,2, . . . , T. This is rarely possible. Generally, howeverαandβ may be chosen, one has instead
yt =α+βxt+et, t =1,2, . . . , T
whereetis anerrorordisturbanceterm. Obviously, one hopes that the errors will be small, on average. So the parametersαandβare chosen to make the errors as “small as possible”, somehow. One idea would be to make the sumT
t=1(yt−α−βxt)equal to zero. However, in this case, large positive discrepancies would cancel large negative discrepancies. Indeed, the sum of errors could be zero even though the line is very far from giving a perfect or even a good fit. We must somehow prevent large positive errors from cancelling large negative errors. Usually, this is done by minimizing the quadratic “loss” function
L(α, β)= 1 T
T t=1
et2= 1 T
T t=1
(yt−α−βxt)2 (loss function) (∗)
which equals the mean (or average) square error. Expanding the square gives3 L(α, β)= 1
T t
(yt2+α2+β2x2t −2αyt−2βxtyt+2αβxt)
This is a quadratic function ofαandβ. We shall show how to derive theordinaryleast- squaresestimates ofαandβ. To do so it helps to introduce some standard notation. Write
μx = x1+ ã ã ã +xT
T = 1
T t
xt, μy =y1+ ã ã ã +yT
T = 1
T t
yt
for thestatistical meansofxt andyt, and σxx= 1
T t
(xt−μx)2, σyy = 1 T t
(yt −μy)2, σxy = 1 T t
(xt−μx)(yt−μy) for theirstatistical variances, as well as theircovariance, respectively. In what follows, we shall assume that thext are not all equal. Then, in particular,σxx>0.
3 From now on, we often use
tto denoteT t=1.
Using the result in Example 3.2.2, we have σxx= 1
T t
xt2−μ2x, σyy = 1 T t
yt2−μ2y, σxy = 1 T t
xtyt−μxμy (You should check the last as an exercise.) Then the expression forL(α, β)becomes
L(α, β)=(σyy+μ2y)+α2+β2(σxx+μ2x)−2αμy−2β(σxy+μxμy)+2αβμx
=α2+μ2y+β2μ2x −2αμy −2βμxμy+2αβμx+β2σxx−2βσxy+σyy The first-order conditions for a minimum ofL(α, β)take the form
L1(α, β)=2α−2μy+2βμx =0
L2(α, β)=2βμ2x−2μxμy+2αμx+2βσxx−2σxy=0
Note thatL2(α, β)=μxL1(α, β)+2βσxx−2σxy. So the values ofαandβthat makeL stationary are given by
ˆ
β =σxy/σxx, αˆ =μy − ˆβμx =μy −(σxy/σxx)μx (∗∗) Furthermore,L11=2,L12=2μx,L22=2μ2x+2σxx. ThusL11≥0,L22≥0, and
L11L22−(L12)2=2(2μ2x+2σxx)−(2μx)2 =4σxx=4T−1
t
(xt−μx)2 ≥0 We conclude that the conditions in Theorem 13.2.1(b) are satisfied, and therefore the pair (α,ˆ β)ˆ given by(∗∗)minimizesL(α, β). The problem is then completely solved:
The straight liney = ˆα+ ˆβx, with αˆ and βˆ given by(∗∗), is the one that best fits the observations(x1, y1), (x2, y2), . . . , (xT, yT), in the sense of minimizing the mean square error in(∗).
Note in particular that this estimated straight line passes through the mean(μx, μy)of the observed pairs(xt, yt),t =1, . . . , T. Also, with a little bit of tedious algebra we obtain
L(α, β)=(α+βμx−μy)2+σxx(β−σxy/σxx)2+(σxxσyy−σxy2 )/σxx The first two terms on the right are always nonnegative, and withα= ˆαandβ = ˆβ, they are zero, confirming thatαˆ andβˆdo give the minimum value ofL(α, β).
P R O B L E M S F O R S E C T I O N 1 3 . 4
1. (a) Suppose that the monopolist in Example 1 faces the demand functions P1=200−2Q1, P2=180−4Q2
and that the cost function isC = 20(Q1+Q2). How much should be sold in the two markets to maximize total profit? What are the corresponding prices?
(b) How much profit is lost if it becomes illegal to discriminate?
(c) Discuss the consequences of imposing a tax of 5 per unit on the product sold in market 1.
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⊂SM⊃2. A firm produces and sells a product in two separate markets. When the price in marketAisp per ton, and the price in marketBisqper ton, the demand in tons per week in the two markets are, respectively,
QA=a−bp, QB=c−dq
The cost function isC(QA, QB)=α+β(QA+QB), and all constants are positive.
(a) Find the firm’s profitπ as a function of the pricespandq, and then find the pair(p∗, q∗) that maximizes profits.
(b) Suppose it becomes unlawful to discriminate by price, so that the firm must charge the same price in the two markets. What pricepˆwill now maximize profits?
(c) In the caseβ =0, find the firm’s loss of profit if it has to charge the same price in both markets. Comment.
3. In Example 1, discuss the effects of a tax imposed in market 1 oftper unit ofQ1.
⊂SM⊃4. The following table shows the Norwegian gross national product (GNP) and spending on foreign aid (FA) for the period 1970–1973 (in millions of crowns).
Year 1970 1971 1972 1973
GNP 79 835 89 112 97 339 110 156
FA 274 307 436 524
Growth of both GNP and FA was almost exponential during the period. So, approximately:
GNP=Aea(t−t0), t0=1970
Definex=t−t0andb=lnA. Then ln(GNP)=ax+b. On the basis of the table above, one gets the following
Year 1970 1971 1972 1973
y=ln(GNP) 11.29 11.40 11.49 11.61
(a) Using the method of least squares, determine the straight liney=ax+bwhich best fits the data in the last table.
(b) Repeat the method above to estimatecandd, where ln(FA)=cx+d.
(c) The Norwegian government had a stated goal of eventually giving 1% of its GNP as foreign aid. If the time trends of the two variables had continued as they did during the years 1970–1973, when would this goal have been reached?
⊂SM⊃5. (Duopoly) Each of two firmsAandBproduces its own brand of a commodity such as mineral water in amounts denoted byxandy, which are sold at pricespandqper unit, respectively.
Each firm determines its own price and produces exactly as much as is demanded. The demands for the two brands are given by
x=29−5p+4q , y=16+4p−6q
FirmAhas total costs 5+x, whereas firmBhas total costs 3+2y. (Assume that the functions to be maximized have maxima, and at positive prices.)
(a) Initially, the two firms collude in order to maximize their combined profit, as one monopolist would. Find the prices(p, q), the production levels(x, y), and the profits of firmsAandB. (b) Then an anti-trust authority prohibits collusion, so each producer maximizes its own profit,
taking the other’s price as given.
Ifqis fixed, how willAchoosep? (Findpas a functionp=pA(q)ofq.) Ifpis fixed, how willBchooseq? (Findqas a functionq=qB(p)ofp.)
(c) Under the assumptions in part (b), what constant equilibrium prices are possible? What are the production levels and profits in this case?
(d) Draw a diagram withpalong the horizontal axis andqalong the vertical axis, and draw the
“reaction” curvespA(q)andqB(p). Show on the diagram how the two firms’ prices change over time ifAbreaks the cooperation first by maximizing its profit, takingB’s initial price as fixed, thenBanswers by maximizing its profit withA’s price fixed, thenAresponds, and so on.