The Firm and the Market

Transcription

1 Chapter 3 The Firm ad the Market Exercise 3.1 (The pheomeo of atural moopoly ) Cosider a idustry i which all the potetial member rms have the same cost fuctio C. Suppose it is true that for some level of output q ad for ay oegative outputs q; q 0 of two such rms such that q + q 0 q the cost fuctio satis es the subadditivity property C (w; q + q 0 ) < C (w; q) + C (w; q 0 ). 1. Show that this implies that for all itegers N > 1 C (w; q) < NC w; q, for 0 q q N 2. What are the implicatios for the shape of average ad margial cost curves? 3. May oe coclude that a moopoly must be more e ciet i producig this good? Outlie swer 1. If q 0 = q the Hece ad so, puttig q 0 = 2q we have C (w; 2q) < 2C (w; q). C (w; q) + C (w; 2q) < 3C (w; q). C (w; 3q) < C (w; q) + C (w; 2q) < 3C (w; q). The result the follows by iteratio. 2. If there are ecoomies of scale the average cost of productio is decreasig ad margial cost will always be below it. Nevertheless subadditivity does ot imply ecoomies of scale ad therefore we ca also observe a stadard U-shaped average cost curve. 3. It is cheaper to produce i a sigle plat rather tha usig two idetical plats. 31

2 Microecoomics CHPTER 3. THE FIRM ND THE MRKET Exercise 3.2 I a particular idustry there are pro t-maximisig rms each producig a sigle good. The costs for rm i are C 0 + cq i where C 0 ad c are parameters ad q i is the output of rm i. The goods are ot regarded as beig exactly idetical by the cosumers ad the iverse demad fuctio for rm i is give by p i = q 1 P i j=1 q j where measures the degree of substitutability of the rms products, 0 < ssumig that each rm takes the output of all the other rms as give, write dow the rst-order coditios yieldig rm 1 s output coditioal o the outputs q 2 ; :::; q. Hece, usig the symmetry of the equilibrium, show that i equilibrium the optimal output for ay rm is q i = [ 1] 2 c ad that the elasticity of demad for rm i is + 2. Cosider the case = 1. What pheomeo does this represet? Show that qthe equilibrium umber of rms i the idustry is less tha or equal to C 0. Outlie swer 1. We begi by computig the equilibrium for a typical rm i. Pro ts for the rm are where i = q i K K := X j=1 C 0 cq i (3.1) The rst-order coditio for maximisig (3.1) with respect to q i (takig all the other q j as give) is q i = q 1 i K q 2 1 i K 2 c = 0 (3.2) If all rms are idetical, the i equilibrium all rms must produce the same amout ad so K = q i (3.3) cfrak Cowell

3 Microecoomics Substitutig (3.3) i (3.2) we get 2 cq i = 0 (3.4) from which the result follows immediately. To d the elasticity of demad for rm i take logs of the iverse demad curve (i the questio) ad di eretiate with respect to q i q i p i = 1 + q i K (3.5) To d the elasticity i the eighbourhood of the equilibrium substitute (3.3) i (3.5) ad take the reciprocal. 2. The case = 1 represets a situatio where the goods are perfect substitutes. We the d that rm i s pro ts are i = q i K = C 0 cq i (3.6) C 0 [ 1] 2 = 2 C 0 (3.7) Requirig that the right-had side of (3.7) be o-egative implies r (3.8) C 0 cfrak Cowell

4 Microecoomics CHPTER 3. THE FIRM ND THE MRKET Exercise 3.3 rm has the cost fuctio F aq2 i where q i is the output of a sigle homogeous good ad F 0 ad a are positive umbers. 1. Fid the rm s supply relatioship betwee output ad price p; explai carefully what happes at the miimum-average-cost poit p := p 2aF I a market of a thousad cosumers the demad curve for the commodity is give by p = bq where q is total quatity demaded ad ad b are positive parameters. If the market is served by a sigle price-takig rm with the cost structure i part 1 explai why there is a uique equilibrium if b a =p 1 ad o equilibrium otherwise. 3. Now assume that there is a large umber N of rms, each with the above cost fuctio: d the relatioship betwee average supply by the N rms ad price ad compare the aswer with that of part 1. What happes as N! 1? 4. ssume that the size of the market is also icreased by a factor N but that the demad per thousad cosumers remais as i part 2 above. Show that as N gets large there will be a determiate market equilibrium price ad output level. Outlie swer 1. Give the cost fuctio F aq2 i margial cost is aq i ad average cost is F 0 =q i aq i. Margial cost itersects average cost where i.e. where output is ad margial cost is aq i = F 0 =q i aq i q := p 2F 0 =a (3.9) p := p 2aF 0 (3.10) For p > p the supply curve is idetical to the margial cost curve q i = p=a; for p < p the rm supplies 0 to the market; at p = p the rm supplies either 0 or q. There is o price which will iduce a supply i the iterior of the iterval 0; q. Summarisig, rm i s optimal output is give by 8 p=a; if p > p >< qi = S(p) := q 2 f0; qg if p = p (3.11) >: 0; if p < p cfrak Cowell

5 Microecoomics 2. The equilibrium, if it exists, is foud where supply=demad at a give price. This would imply p = p a b a p = a + b which would, i tur, imply a equilibrium quatity q = a + b but it ca oly be valid if a+b q. Notig that q = p=a this coditio is h i equivalet to a p 1 b. 3. If there are N such rms, each rm respods to price as i (3.11), ad so the average output q := 1 P N N i=1 q i is give by 8 p=a; if p > p >< q = q 2 J(q) if p = p (3.12) >: 0; if p < p where J(q) := f i N q : i = 0; 1; :::; Ng. s N! 1 the set J(q) becomes dese i [0; q], ad so we have the average supply relatioship: 8 p=a; if p > p >< q = q 2 [0; q] if p = p >: 0; if p < p (3.13) 4. Give that i the limit the average supply curve is cotiuous ad of the piecewise liear form (3.13), ad that the demad curve is a dowwardslopig straight lie, there must be a uique market equilibrium. The equi- librium will be foud at p; p b Usig (3.9) this ca be writte p2af0 which, usig (3.10) is ; p 2aF 0 b p; q where := p bp=a. I the equilibrium a proportio of the rms produce q ad 1 rms produce 0. of the cfrak Cowell

6 Microecoomics CHPTER 3. THE FIRM ND THE MRKET Exercise 3.4 rm has a xed cost F 0 ad margial costs where q is output. c = a + bq 1. If the rm were a price-taker, what is the lowest price at which it would be prepared to produce a positive amout of output? If the competitive price were above this level, d the amout of output q that the rm would produce. 2. If the rm is actually a moopolist ad the iverse demad fuctio is p = 1 2 Bq (where > a ad B > 0) d the expressio for the rm s margial reveue i terms of output. Illustrate the optimum i a diagram ad show that the rm will produce q := a b + B What is the price charged p ad the margial cost c at this output level? Compare q ad q : 3. The govermet decides to regulate the moopoly. The regulator has the power to cotrol the price by settig a ceilig p max. Plot the average ad margial reveue curves that would the face the moopolist. Use these to show: (a) If p max > p the rm s output ad price remai uchaged at q ad p (b) If p max < c the rm s output will fall below q. (c) Otherwise output will rise above q. Outlie swer 1. Total costs are So average costs are which are a miimum at F 0 + aq bq2 F 0 q + a bq q = r 2 F 0 b (3.14) where average costs are p 2bF0 + a (3.15) Margial ad average costs are illustrated i Figure 3.1: otice that MC is liear ad that C has the typical U-shape if F 0 > 0. For a price above the level (3.15) the rst-order coditio for maximum pro ts is give by cfrak Cowell p = a + bq

7 Microecoomics P margial cost a+bq F/q+a+0.5bq average cost q q * = P a b q Figure 3.1: Perfect competitio from which we d see gure 3.1. q := p a b 2. If the rm is a moopolist margial reveue 1 2 Bq2 = Bq Hece the rst-order coditio for the moopolist is Bq = a + bq (3.16) from which the solutio q follows. Substitutig for q we also get c = Bq = b + Ba B + b (3.17) see gure 3.2. p = 1 2 Bq = c B a b + B (3.18) 3. Cosider how the itroductio of a price ceilig will a ect average reveue. Clearly we ow have pmax if q q R(q) = Bq if q q (3.19) 0 where q 0 := 2 [ p max ] =B: average reveue is a cotiuous fuctio of q but has a kik at q 0. From this we may derive margial reveue which is pmax if q < q MR(q) = 0 (3.20) Bq if q > q 0 cfrak Cowell

8 Microecoomics CHPTER 3. THE FIRM ND THE MRKET p p ** c ** margial cost a+bq F/q+a+0.5bq average cost q ** margial reveue average reveue Bq 0.5Bq q Figure 3.2: Uregulated moopoly otice that there is a discotiuity exactly at q 0. The modi ed curves (3.19) ad (3.20) are show i Figure 3.3: otice that they coicide i the at sectio to the left of q 0. Clearly the outcome depeds crucially o whether MC itersects (modi ed) MR (a) to the left of q 0, (b) to the right of q 0, (c) i the discotiuity exactly at q 0. Case (c) is illustrated, ad it is clear that output will have rise from q to q 0. The other cases ca easily be foud by appropriately shiftig the curves o Figure 3.3. p p ** p max margial cost c ** average reveue q ** q 0 margial reveue q Figure 3.3: Regulated Moopoly cfrak Cowell

9 Microecoomics Exercise 3.5 moopolist has the cost fuctio C(q) = q [q]2 1. If the demad fuctio is give by 1 q = 24 4 p calculate the output-price combiatio which maximises pro ts. 2. ssume that it becomes possible to sell i a separate secod market with demad determied by 3 q = 84 4 p: Calculate the prices which will be set i the two markets ad the chage i total output ad pro ts from case Now suppose that the rm still has access to both markets, but is preveted from discrimiatig betwee them. What will be the result? Outlie swer 1. Maximizig the simple moopolist s pro ts 0 = (96 4q)q q + q2 2 with respect to q yields optimum output of q 0 =10. Hece p 0 = 56 ad 0 = 350: 2. Now let the moopolist sell q 1 i market 1 for price p 1 ad q 2 i market 2 for price p 2 :The ew problem is to choose q 1 ; q 2 so as to maximise the fuctio 12 = (96 4q 1 )q 1 + (112 First-order coditios yield 4 3 q 2)q q 1 + 6q 2 + (q 1 + q 2 ) 2 2 : 9q 1 + q 2 = 90 q q 2 = 106: Solvig we d q 1 = 7; q 2 = 27 ad hece p 1 = 68; p 2 = 76 ad 12 = If we abado discrimiatio, a uiform price bp must be charged. If bp > 112 othig is sold to either market. If 112 > bp > 96 oly market 2 is served. If 96 > bp both market are served ad the demad curve is bq = 108 bp. Clearly this is the relevat regio. Maximisig simple moopoly pro ts we d bq = 34; bp = 74 ad b = Hece the total output is idetical to that uder discrimiatio, p 1 < bp < p 2 ad 12 > b : These results are quite geeral. cfrak Cowell