= 2 = 1.5. Figure 4.1: WARP violated

Transcription

1 Chapter 4 The Consumer Exercise 4.1 You observe a consumer in two situations: with an income of $100 he buys 5 units of good 1 at a price of $10 per unit and 10 units of good 2 at a price of $5 per unit. With an income of $175 he buys 3 units of good 1 at a price of $15 per unit and 13 units of good 2 at a price of $10 per unit. Do the actions of this consumer conform to the basic axioms of consumer behaviour? x 2 x = (3,13) x = (5,10) p 1 / p 2 = 2 p 1 / p 2 = 1.5 x 1 Figure 4.1: WARP violated Outline Answer At the original price ratio p 1 /p 2 = 2 the choice is x = (5, 10); but at those prices the and with that budget the consumer could have afforded x = (3, 13): x is revealed-preferred to x. But at the new price ratio p 1 /p 2 = 1.5 x is chosen, although x is still affordable: x is revealed-preferred to x. This violates WARP see Figure

2 Microeconomics CHAPTER 4. THE CONSUMER Exercise 4.2 Draw the indiff erence curves for the following four types of preferences: Type A : α log x 1 + [1 α] log x 2 Type B : βx 1 + x 2 Type C : γ [x 1 ] 2 + [x 2 ] 2 Type D : min {δx 1, x 2 } where x 1, x 2 denote respectively consumption of goods 1 and 2 and α, β, γ, δ are strictly positive parameters with α < 1. What is the consumer s cost function in each case? x 2 x 2 A B x 2 x 1 x 2 x 1 C D x 1 x 1 Figure 4.2: Indifference curves: four cases Use the fact that expenditure minimisation for the household and costminimisation for the firm are essentially the same problem. The indifference curves in Figure 4.2 are identical to the isoquants depicted in Exercises 2.4, 2.5. So, substituting the notation in Exercise 2.4 and 2.5we get: Case A: C(p, υ) = C(p, υ) = e υ [ p 1 α ] α [ p2 1 α] 1 α. Case B: C(p, υ) = υ min(p 1 /β, p 2 ) Case C: C(p, υ) = υ min(p 1 / γ, p 2 ) Case D: C(p, υ) = [ p 1 δ + p 2 ] υ. c Frank Cowell

3 Microeconomics Exercise 4.3 Suppose a person has the Cobb-Douglas utility function a i log(x i ) where x i is the quantity consumed of good i, and a 1,..., a n are non-negative parameters such that n j=1 a j = 1. If he has a given income y, and faces prices p 1,..., p n, find the ordinary demand functions. What is special about the expenditure on each commodity under this set of preferences? Outline Answer The relevant Lagrangean is [ α i log x i + ν y ] p i x i (4.1) The first-order conditions yield: x i = y = α i ν, i = 1, 2,..., n. (4.2) p i p i x i (4.3) From the n + 1 equations (4.2,4.3) we get at the optimum: y = n α i/ν = 1/ν. So the demand functions are and expenditure on each commodity i is a constant proportion of income. x i = α iy p i, i = 1, 2,..., n. (4.4) e i := p i x i = α i y, (4.5) c Frank Cowell

4 Microeconomics CHAPTER 4. THE CONSUMER Exercise 4.4 The elasticity of demand for domestic heating oil is 0.5, and for gasoline is 1.5. The price of both sorts of fuel is 60c/ per litre: included in this price is an excise tax of 48c/ per litre. The government wants to reduce energy consumption in the economy and to increase its tax revenue. Can it do this (a) by taxing domestic heating oil? (b) by taxing gasoline? Outline Answer Let p be the untaxed price, and τ the excise tax. Government revenue is T = τx, and the purchase price is p + τ. Clearly an increase in τ would reduce consumption, and τ/[τ + p] = 0.8. The effect on tax revenue is given by T/ τ = x + τ x/ τ = x[ ε]. If (a) ε = 0.5 then this is positive. If (b) ε = 1.5 then it is negative. Exercise 4.5 Define the uncompensated and compensated price elasticities as ε ij := p j x i and the income elasticity D i (p,y), ε ij := p j H i (p,υ) p j x i p j ε iy := y x i D i (p,y). y Show how the Slutsky equation can be expressed in terms of these elasticities and the expenditure share of each commodity in the total budget. Use the fact that each demand function D i is homogeneous of degree zero in all prices and income. Then, using the standard lemma for homogenous functions, we have for each i = 1,..., n : j=1 D i (p, y) p j + y Di (p, y) p j y = 0 D i (p, y) = 0 which implies ε ij + ε iy = 0. j=1 Moreover we can rewrite the Slutsky equation as ε ij = ε ij v j ε iy where v j = p jx j y is the expenditure share of commodity j. c Frank Cowell

5 Microeconomics Exercise 4.6 You are planning a study of consumer demand. You have a data set which gives the expenditure of individual consumers on each of n goods. It is suggested to you that an appropriate model for consumer expenditure is the Linear Expenditure System: e i = ξ i p i + α i y p j ξ j where p i is the price of good i, e i is the consumer s expenditure on good i, y is the consumer s income, and α 1,..., α n, ξ 1,..., ξ n are non-negative parameters such that n j=1 α j = Find the effect on x i, the demand for good i, of a change in the consumer s income and of an (uncompensated) change in any price p j. 2. Find the substitution effect of a change in price p j on the demand for good i. 3. Explain how you could check that this demand system is consistent with utility-maximisation and suggest the type of utility function which would yield the demand functions implied by the above formula for consumer expenditure. [Hint: compare this with Exercise 4.3] j=1 Outline Answer 1. We have x i = ξ i + α i p i y p j ξ j (4.6) j=1 Notice that (ξ 1,..., ξ n ) play the role of subsistence minima of the n commodities, and so y 0 := n j=1 p jξ j can be considered as the subsistence minimum expenditure, and the remaining budget y y 0 as discretionary expenditure ; α i is then the proportion of discretionary expenditure spent on discretionary purchases of commodity i: p i [x i ξ i] / [y y 0 ]. Compare this with (4.5). From (4.6) we have: x i = α i p i (4.7) y x i p j = α iξ j x i p i = α i p i if j i (4.8) p i [ξ i + y n j=1 p ] jξ j (4.9) p i 2. Apply Slutsky equation using (4.7) and (4.8) to establish dx i dp j = α [ ] i x j ξ j, if j i (4.10) υ=conθtant p i c Frank Cowell

6 Microeconomics CHAPTER 4. THE CONSUMER 3. Check that demand function (4.6) is homogeneous of degree 0 in prices and income, and that the sum of the right-hand side of the equation in the question adds up to total income. Check that cross-substitution effects are symmetric, and that own-price substitution effects are negative. Using the analogy with part (b) we can see that the demand system is similar, but with the commodity origin shifted from 0 to the point (ξ 1,..., ξ n );so we expect the indifference curves from which the demand system was derived will look like Cobb-Douglas contours with the origin shifted to the point (ξ 1,..., ξ n ). The utility function will then be α i log(x i ξ i ). (4.11) c Frank Cowell

7 Microeconomics Exercise 4.7 Suppose a consumer has a two-period utility function of the form labelled type A in Exercise 4.2. where x i is the amount of consumption in period i. The consumer s resources consist just of inherited assets A in period 1, which is partly spent on consumption in period 1 and the remainder invested in an asset paying a rate of interest r. 1. Interpret the parameter α in this case. 2. Obtain the optimal allocation of (x 1, x 2 ) 3. Explain how consumption varies with A, r and α. 4. Comment on your results and examine the income and substitution eff ects of the interest rate on consumption. Outline Answer 1. The parameter α captures the consumer s impatience : the higher is α the more steeply sloped will be the indifference curves in Figure 4.3. Note 1 that 1+r is the price of consumption in period 2 relative to the price of consumption in period 1; so the lifetime budget constraint, expressed in terms of period-1 prices, is: x 1 + x r A (4.12) and so the Lagrangean is: [ α log x 1 + [1 α] log x 2 + λ A x 1 x ] r (4.13) 2. We can be sure an interior maximum will exist (examine the indifference curve in Figure 4.3). First-order conditions are α x 1 1 α x 2 x 1 + x r = λ = λ r = A From these we find λ = 1 A period is: and therefore optimum consumption in each x 1 = αa (4.14) x 2 = [1 + r] [1 α] A (4.15) So we can see that the smaller is α (the lower is the level of impatience), or the larger is r (the rate of interest), the more consumption will be tilted toward period 2. c Frank Cowell

8 Microeconomics CHAPTER 4. THE CONSUMER x 2 x* 1 + r A x 1 Figure 4.3: Equilibrium in 2-period case 3. The effect of an increase in assets is: x 1 A = α (4.16) x 2 A = [1 + r] [1 α] (4.17) leaving the proportion spent on consumption in each period unaltered. The effect of an increase in the interest rate is: x 1 r x 2 r = 0 (4.18) = [1 α] A (4.19) 4. To find the substitution effect we need to use the Slutsky equation. In a conventional 2-commodity model this would be given by x 1 = dx 1 p 2 dp 2 x x 1 2 (4.20) υ=constant y Taking 1/[1 + r] as the price p 2 of consumption in period 2, with A =lifetime budget y and price of period-1 consumption defined as 1. Noting that in this case dp 2 = 1/[1 + r] 2 dr we can rewrite (4.20) as r = dx 1 dr + x 2 x 1 υ=constant [1 + r] 2 A x 1 (4.21) Rearranging this, the substitution effect for good 1 of an increase in r may c Frank Cowell

9 Microeconomics then be found (using 4.16 and 4.18) as: dx 1 dr = x 1 υ=constant r x 2 x 1 [1 + r] 2 A = x 2 [1 + r] 2 α < 0 (4.22) c Frank Cowell

10 Microeconomics CHAPTER 4. THE CONSUMER Exercise 4.8 Suppose a consumer is rationed in his consumption of commodity 1, so that his consumption is constrained thus x 1 a. Discuss the properties of the demand functions for commodities 2,..., n of a consumer for whom the rationing constraint is binding. Outline Answer Use the standard analysis on the short-run for the firm (see Chapter 2) to get insight on the economics of the consumer under rationing. In the case of the firm has to cope with the side-constraint z 3 = z 3 in the short run; the consumer has to cope with the rationing constraint x 1 a: if the constraint is slack then it is irrelevant (the consumer does not use all his ration); if it is binding, then the problem is just like that of the firm. The solution is at x in Figure 4.4 x 2 x a x 1 Figure 4.4: Ration c Frank Cowell

11 Microeconomics Exercise 4.9 A person has preferences represented by the utility function U(x) = log x i where x i is the quantity consumed of good i and n > Assuming that the person has a fixed money income y and can buy commodity i at price p i find the ordinary and compensated demand elasticities for good 1 with respect to p j, j = 1,..., n. 2. Suppose the consumer is legally precommitted to buying an amount A n of commodity n where p n A n < y. Assuming that there are no additional constraints on the choices of the other goods find the ordinary and compensated elasticities for good 1 with respect to p j, j = 1,...n. Compare your answer to part Suppose the consumer is now legally precommitted to buying an amount A k of commodity k, k = n r,..., n where 0 < r < n 2 and n k=n r p ka k < y. Use the above argument to explain what will happen to the elasticity of good 1 with respect to p j as r increases. Comment on the result. Outline Answer 1. For the specified utility function it is clear that the indifference curves do not touch the axes for any finite x i, so we cannot have a corner solution. The budget constraint is p i x i y. The problem of maximising utility subject to the budget constraint is equivalent to maximising the Lagrangean [ ] log x i + λ y p i x i. The FOC are 1 x i λp i = 0, i = 1,..., n (4.23) and the (binding) budget constraint. From (4.23) we get n λ p i x i = 0. (4.24) and so, using the budget constraint, we find λ = n/y. Substituting the value of λ into (4.23) we find: (a) The ordinary demand function for good i is c Frank Cowell x i = y np i (4.25)

12 Microeconomics CHAPTER 4. THE CONSUMER The indirect utility function V is given by υ = V (p, y) = U(x ) = n log x i. So, from (4.25) we have: ( y n ) υ = log n n (4.26) p 1 p 2 p 3...p n Inverting the relation (4.26) the cost function C is given by y = C(p, υ) = [n n p 1 p 2 p 3...p n e υ ] 1 n = n [p 1 p 2 p 3...p n e υ ] 1 n (4.27) Differentiating (4.27) the compensated demand for good 1 is x 1 = p 1 n n 1 [p 2 p 3 p 4...p n e υ ] 1 n (4.28) (b) From (4.25) we have the elasticities log x 1 log p 1 = 1, y=const log x 1 log p j = 0, j = 2,..., n. y=const (c) From (4.28) we have the compensated elasticities log x 1 log p 1 = 1 n < 0, υ=const n log x 1 log p j = 1 > 0, j = 2,..., n υ=const n 2. The problem now is equivalent to maximising subject to n 1 log x i + log A n n 1 p i x i y, where y := y p n A n. Reusing the method above, the ordinary and compensated demand functions are, respectively, x 1 = y [n 1] p 1 = y p na n [n 1] p 1 (4.29) x 1 = p 2 n n 1 1 [p 2 p 3 p 4...p n 1 e υ 1 n 1 ] (4.30) (a) So now, from (4.29) we have log x 1 log p 1 = 1, y=const log x 1 log p j = 0, j = 2,..., n 1. y=const c Frank Cowell

13 Microeconomics (as before) but log x 1 log p n = p na n y=const y < 0 The reason for this last result is that if the person is forced to buy the fixed amount A n then changing p n is equivalent to a simple income effect (think about what happens to y ). (b) From (4.28) we have log x 1 log p 1 = 2 n υ=const n 1 < 0, log x 1 1 log p j =, j = 2,..., n 1. υ=const n 1 log x 1 log p n = 0. υ=const 3. The problem is just a generalisation of part 2. The person is maximising m log x i + log A subject to m p ix i y, where m := n r 1, A := n k=n r A k and y := y n k=n r p ka k. Ordinary and compensated demands are x 1 = y = y n k=n r p ka k (4.31) mp 1 mp 1 x 1 = p 1 m m 1 [p 2 p 3 p 4...p m e υ ] 1 m (4.32) and so we have. log x 1 log p 1 = 1 m < 0, (4.33) υ=const m log x 1 log p j = 1 > 0, j = 2,..., m. (4.34) υ=const m log x 1 log p k = 0, k = n r,..., n. (4.35) υ=const Given that m = n r 1 it is clear that as r increases the elasticity (4.33) decreases in absolute value and (4.34) increases. We also have log x 1 log p k = p ka k y=const y < 0, k = n r,..., n The model can be used to illustrate in part the comparative statics of someone who is subject to a quota ration x i A i where the rationing constraint is assumed to be binding in the case of goods n r to n. However, it is not rich enough to allow us to determine which commodities are consumed at a conventional equilibrium with MRS =price ratio, like (4.29), and which will be constrained by the ration. Parts 2 and 3 show clearly how the compensated demand becomes steeper (less elastic with respect to its own price) the more external constraints are imposed as in the short-run problem of the firm. c Frank Cowell

14 Microeconomics CHAPTER 4. THE CONSUMER Exercise 4.10 Show that if the utility function is homothetic, then I CV = I EV Outline Answer Let x 0 be optimal for p 0 at υ 0 and x 1 be optimal for p 1 at υ 0. x 2 αx 1 x 1 αx 0 x 0 υ 0 υ 1 0 x 1 Figure 4.5: Homothetic preferences Because of homotheticity, αx 0 must be optimal for p 0 at υ 1 and αx 1 be optimal for p 1 at υ 1 : see Figure 4.5. Hence C(p 0, υ 0 ) = p 0 i x 0 i, C(p 0, υ 1 ) = α p 0 i x 0 i, C(p 1, υ 0 ) = p 1 i x 1 i, So in this special case we have C(p 1, υ 1 ) = α p 1 i x 1 i I CV = p 1 i x 1 i p 0 i x 0 i and the result follows. I EV = α p 1 i x1 i α p 0 i x0 i c Frank Cowell

15 Microeconomics Exercise 4.11 Suppose an individual has Cobb-Douglas preferences given by those in Exercise Write down the consumer s cost function and demand functions. 2. The republic of San Serrife is about to join the European Union. As a consequence the price of milk will rise to eight times its pre-entry value. but the price of wine will fall by fifty per cent. Use the compensating variation to evaluate the impact on consumers welfare of these price changes. 3. San Serrife economists have estimated consumer demand in the republic and have concluded that it is closely approximated by the demand system derived in part 1. They further estimate that the people of San Serrife spend more than three times as much on wine as on milk. They conclude that entry to the European Union is in the interests of San Serrife. Are they right? Outline Answer 1. Using the results from previous exercises we immediately get [ ] α1 [ ] α2 [ ] αn p1 p2 pn C(p, υ) =.... α 1 This is suffi cient. However, it may be useful to see the proof from first principles. The relevant Lagrangean is [ ] p i x i + λ υ α i log x i (4.36) The first-order conditions are: α 2 α n x i = α iλ p i, i = 1, 2,..., n. (4.37) υ = α i log x i (4.38) From the n equations (4.37) we get at the optimum: λ = n p ix i n α i = p i x i = y (4.39) where y is the budget, or minimised cost and n α i = 1. From (4.38) we get, using (4.37): υ = α i log α i + log λ n α i α i log p i (4.40) Using (4.39) and writing n α i log α i = log A, equation (4.40) gives: c Frank Cowell y = Ae υ p α1 1 pα2 2...pαn n = C(p, υ). (4.41)

16 Microeconomics CHAPTER 4. THE CONSUMER This is the required cost function. The demand functions are known from Exercise 4.2 or are obtained immediately from (4.37) and (4.39): x i = α iy p i, i = 1, 2,..., n. (4.42) 2. Let p denote the original price vector, ˆp the price vector after entry. Observe that ˆp milk = 8p milk ; ˆp wine = 1 2 p wine. So, using (4.41): C(ˆp, υ) = Ae υ ˆp α1 1 ˆpα2 2...ˆpαn n = Ae υ p α1 1 pα2 2...pαn n b = bc(p, υ). (4.43) where b := [8] α milk [ 1 2 ]αwine = 2 3α milk α wine (4.44) Using the definition in the notes the compensating variation is therefore CV(p ˆp) := C(p, υ) C(ˆp, υ) = [1 b] C(p, υ) (4.45) Clearly, the consumer will benefit from p ˆp if CV(p ˆp) > 0: the cost of living interpreted as the cost of hitting the original level of utility goes down. This condition is satisfied if, and only if, b < Notice that, from (4.42), α i = p i x i /y the budget share of commodity i. So, since we are told that α wine > 3α milk it is clear that b < 1. The economists are right! c Frank Cowell

17 Microeconomics Exercise 4.12 In a two-commodity world a consumer s preferences are represented by the utility function U(x 1, x 2 ) = αx x 2 where (x 1, x 2 ) represent the quantities consumed of the two goods and α is a non-negative parameter. 1. If the consumer s income y is fixed in money terms find the demand functions for both goods, the cost (expenditure) function and the indirect utility function. 2. Show that, if both commodities are consumed in positive amounts, the compensating variation for a change in the price of good 1 p 1 p 1 is given by α 2 p 2 [ ]. 4 p 1 p 1 3. In this case, why is the compensating variation equal to the equivalent variation and consumer s surplus? Outline Answer x 2 x 1 Figure 4.6: Possible corner solution First sketch the utility function. Note that the indifference curves touch the axes it is possible that one or other commodity is not consumed at the optimum See Figure 4.6. In this case it is easiest to substitute directly from the budget constraint (binding at the optimum) p 1 x 1 + p 2 x 2 = y into the utility function. The consumer will then choose x 1 to maximise αx y p 1x 1 p 2 c Frank Cowell

18 Microeconomics CHAPTER 4. THE CONSUMER The FOC is which suggests that demands are [ x 1 x 2 ] = α 2 [x 1] 1 2 p 1 p 2 = 0 [ D 1 (p 1, p 2, y) D 2 (p 1, p 2, y) ] = y p 2 [ ] 2 αp2 2p 1 α2 4 p 2 p 1 But this neglects the possibility that we may be at a corner. Note that a strictly positive amount of good 2 requires p 1 > p 1 := α2 4 p 2 2 y So demand functions are given by x 1 = D 1 (p 1, p 2, y) = x 2 = D 2 (p 1, p 2, y) = Also, if p 1 > p 1, maximised utility is [ αp2 2p 1 ] 2 if p 1 > p 1 y p 1 otherwise y p 2 α2 p 2 4 p 1 if p 1 > p 1 0 otherwise (4.46) (4.47) V (p 1, p 2, y) = U(x 1, x 2) = α2 p 2 2p 1 + y p 2 α2 p 2 4p 1 = α2 p 2 4p 1 + y p 2 (4.48) Otherwise V (p 1, p 2, y) = α y p 1. Also note ( for the case p 1 > p 1 ) that (4.48) implies: V 1 (p 1, p 2, y) = α2 p 2 4p 2 1 = x 1 p 2 < 0 V 2 (p 1, p 2, y) = α2 y 4p 1 p 2 = x 2 < 0 2 p 2 V y (p 1, p 2, y) = 1 p 2 > 0 so that we immediately see that Roy s identity holds. To find the cost function write υ = V (p, y) and solve for y in terms of p and υ. This gives c Frank Cowell υ = α2 p 2 C(p, υ) + 4p 1 p 2 C(p, υ) = υp 2 α2 p 2 2 4p 1 (4.49)

19 Microeconomics for the case p 1 > p 1 and C(p, υ) = p 1 [ υ α ] 2 otherwise. 2 Using the definition of the compensating variation V (p 1, p 2, y) = α2 p 2 4p 1 + y p 2 V (p 1, p 2, y CV) = α2 p 2 4p 1 CV(p p ) = α2 p 2 [ p 1 + y CV p 2 1 ] p 1 (4.50) As an alternative method use the consumer s cost function (4.49). Clearly we have: C(p, υ) C(p, υ) = α2 p 2 2 4p 1 α2 p 2 2 4p 1 3 In this case the income effect on commodity 1 is zero if p 1 > p 1 see equation (4.46). In the special case of zero income effects CV=EV=CS c Frank Cowell

20 Microeconomics CHAPTER 4. THE CONSUMER Exercise 4.13 Take the model of Exercise Commodity 1 is produced by a monopolist with fixed cost C 0 and constant marginal cost of production c. Assume that the price of commodity 2 is fixed at 1 and that c > α 2 /4y. 1. Is the firm a natural monopoly? 2. If there are N identical consumers in the market find the monopolist s demand curve and hence the monopolist s equilibrium output and price p Use the solution to Exercise 4.12 to show the aggregate loss of welfare L(p 1 ) of all consumers having to accept a price p 1 > c rather than being able to buy good 1 at marginal cost c. Evaluate this loss at the monopolist s equilibrium price. 4. The government decides to regulate the monopoly. Suppose the government pays the monopolist a performance bonus B conditional on the price it charges where B = K L(p 1 ) and K is a constant. Express this bonus in terms of output. Find the monopolist s new optimum output and price p 1. Briefly comment on the solution. Outline Answer 1. It is easy to see that the cost function is subadditive and therefore the firm is a natural monopoly. 2. Because consumers are identical we can just multiply the demand of one consumer by N to get the market aggregates. We use this throughout the answer. If p 2 is normalized to 1 then, given that there are N identical consumers the market demand curve is given by which on rearranging gives [ α q = Nx 1 = N 2p 1 p 1 = α 2 N q ] 2 (4.51) This gives the average revenue curve. So total revenue is α 2 Nq. Given the structure of costs specified in the question the monopolist s profits are p 1 q [C 0 + cq] = α 2 Nq C0 cq (4.52) Differentiating (4.52) we find the FOC characterising the monopolist s optimum as c = α N 4 q (4.53) c Frank Cowell

21 Microeconomics where the expression on the right-hand side of (4.53) is marginal revenue. Using (4.53) we find that the monopolist s equilibrium output is given by [ α 2 q = N 4c] Using (4.51) the price charged will be p 1 = α N 2 N [α/4c] 2 in other words p 1 = 2c (4.54) Maximised profits are p 1q cq C 0 = [ α ] 2 [ α ] 2 2cN cn C0 4c 4c = Nα2 16c C 0 3. Using (4.50) and multiplying by N, the (absolute) loss of welfare of each consumer of having to buy at price p 1 rather than at marginal cost c is given by [ L(p 1 ) = N CV = Nα2 1 4 c 1 ] (4.55) p 1 Using (4.54) to evaluate (4.55) L(p 1) = Nα2 4 = Nα2 8c [ 1 c 1 ] 2c > 0 4. Profits, including the performance bonus, are now p 1 q C 0 cq + B = α 2 Nq C0 cq + B (4.56) where, from the definition in the question and (4.55) B = K L(p 1 ) [ = K Nα2 1 4 c 1 ] p 1 (4.57) Given (4.51) we find that (4.57) can be written as [ B(q) = α2 N 2 q 4 α N 1 ] + K c = α α 2 N Nq + K (4.58) 2 4c and so, substituting from (4.58) into (4.56) we get p 1 q C 0 cq + B = α [ ] Nq cq + K C 0 α2 N 4c c Frank Cowell (4.59)

22 Microeconomics CHAPTER 4. THE CONSUMER The FOC for maximising (4.59) α N 2 q = c and so q = [ α ] 2 N > q 2c p 1 = c < p 1 Of course this is just the solution price equal to marginal cost. The bonus scheme has made the monopolist simulate the outcome of a competitive industry. c Frank Cowell