Price ceilings in competitive markets with variable quality

Journal of Pubhc Econormcs 22 (1983) 257-264 North-Holland

PRICE CEILINGS

IN COMPETITIVE MARKETS QUALITY

WITH VARIABLE

Nell RAY MON Unrverslty of Callforma at Santa Barbara, Santa Barbara, CA 93106, USA

Received November 1981, revised version received March 1983 This paper analyzes the nnpact of a price celling on a competltlve market for a good which yields a utlhty-beanng charactenstlc The household character&c-producing technology IS linear Firms have U-shaped cost curves m quantity and m total amount of charactenstlc produced Essentially, the market IS cleared by an (equlllbrmm) hedonic price The hedomc price rises m response to the excess demand gap induced by a cedmg on output price, brmgmg the market from an unconstramed equdlbmnn mto a constrained eqmhbnum A pnce celling also leads to a fall m quality, a fall m consumer welfare, and locally a nse m producer welfare

1. Introduction

In a competltlve market m which the only variables are pnce and quantity, a pnce celling leads m the short run to an excess demand gap, a reduction m profit for every firm, and, if the pnce celhng 1s close enough to the mltlal equlhbnum price, an increase m every household’s utility This last 1s a monopsony effect households are made better off by movement down the supply curve until margmal product benefit [measured by the supply price P”(Q)] equals marginal product cost [gven by dQP(Q)/dQ] ’ The present paper provides a simple analytlcal treatment of how this picture can change If product quahty 1s variable ’ A competltlve market 1s studied m which the product provides a utlhty-bearmg characteristic Imphatly, this charactenstlc carnes a hedomc price which 1s essentially what clears the market When an effective celling 1s imposed on output price, an increase m hedonic pnce closes the excess demand gap induced by the celling and brings the market into a price-celling-constrained (henceforth ‘constramed’) eqmhbrmm The increase m hedonic pnce outweighs any superficial pecuniary benefit of the celling on output price and makes every ‘This result on the increase m household utihtles can be proved ngorously usmg first-order condltlons and Taylor’s theorem Imphclt are the assumptions that (short-run) supply IS not perfectly elastic and that queuing costs are sufficiently small *Some aspects of the analysis presented here have been developed independently by Murphy (1980) 0047-2727/83/$3 00 0 1983 Elsevler Science Pubhshers B V (North-Holland)

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household worse off When the effective celling on output price 1s sufiiclently close to the maximum effective level, the benefit for profits of the increased hedonic price outweighs the cost of the constraint on output price so that all firms are better off The discussion above concerns the short run The long run 1s not treated separately below because the only significant change m the short-run results 1s the obvious change m the welfare result for firms (zero profit for the marginal firm) due to entry and exit 3 It should be noted that herem lies an important contrast with the long-run impact of a price celling when quality 1s not variable If product quality 1s not variable, d all firms are identical, and if there are no pecuniary externalities, then m the long run any price celling 1s either ineffective or drives the market out of existence When product quality 1s variable, there 1s a third alternative the market can be m a constrained long-run equlhbrmm 2. Equilibrium without a price ceiling Consider a single competltlve market defined by a perfectly divisible good which has a well-defined quantity unit and which provides a utlhty-bearing characteristic Followmg Lancaster (1966), a linear household technology 1s assumed, so that the ‘quality’ of a product type 1s the characterlstlc yield per unit quantity and units of different quahtles generate the characterlstlc additively Followmg Rosen (1974), the meamng of competltlon 1s that there exists a function p=p(q) relatmg quahty q 2 0 to per-umt-quantity (henceforth ‘output’) price p and that households and firms treat this function as parametric The market 1s m equlhbrmm when p(q) IS such that supply equals demand for every product type The assumption of a hnear household technology lmphes that any eqmhbrmm quahty/output-pnce function can be taken to be linear p(q)= kg, where k IS the hedomc price of the characteristic 4 Household h has a smooth utility function which, under present assumptions and submergmg prices m other markets mto the functional - kq,, Zh), where zh 2 0 1s the kVe1 Of the notation, can be Written U,, =f’(bfk characteristic consumed5 and Mk 1s income The marginal utilities f:(I, z) and ff(1, z) are positive This, together with some conventional technical assumptions on f”, guarantees that for any k>O [that IS, for any p(q) = kq with k>O], the utlhty-maxlmlzatlon problem has a unique positive solution satisfying the first-order condltlon - fik + fi = 0 This solution 1s denoted 3A formal analysis of the long run 1s avalable upon request ?hls result can be demonstrated usmg the arguments of Rosen (1974, pp 37-38) Of course, when there 1s a celhng on output price, p(q) IS lmear m CJonly where the celhng IS not vlolated %nder present assumptions, a household IS mdlfferent to the quahty/quantlty mix which provides any gven amount of the characterlstlc

N Raymon, Puce ceilmgs m competztwe markets

259

z,, = d*(k) It 1s assumed that the slope of the demand curve, d*‘(k), 1s negative for all k Market demand for the characterlstlc is thus d(k) =& 8(k), with d’(k) < 0

Each firm consists of a single plant and chooses a smgle product type and quantity thereof Firm J has a smooth total cost function c”(y,, qJ) m quantity y, and quality q,, with positive marginal costs c$ >O and clq>O It 1s also assumed that the marginal cost curve, and hence also the average cost curve, m quantity are U-shaped for any fixed level of quality In producing any pair (y,, q,), firm 1 produces an amount of the charactenstlc z, =qJIJyJ With the addition of sensible boundary condltlons on cJ, there exists a well-defined, smooth cost function m character&c levels 4’(z) given by 4JM=mln{cJ(~,q)

Y.4

qy=z)

It 1s assumed that this function has U-shaped marginal and average cost functions 4”(z) and ~‘(z)/z It 1s also assumed that for any gven z, #j(z) 1s achieved by means of a unique pair (y,q)=(q’(z), j?‘(z)), and that quantity and quality are strictly complementary m the cost-mmnmzmg production of the characterlstlc (1e the derivatives ye” and BJ’ are positive) The (unconstrained) profit-maxlmlzatlon problem for firm J can be written FE;

,-

z, =kzJ- 6"(")

For any k>mmm=A,, Z 2 there 1s a unique, positive solution to this problem, denoted zJ =9”(k), where the superscript u indicates that s’“(k) IS firm J’S unconstramed supply of the characterlstlc Obviously, sJ” 1s strictly increasing m k Correspondmg to the charactenstlc level s’“(k) IS a unique choice of quantity yJ =qJ(sJ”(k)) and quality qJ = /l’(s’“(k)) Cost mmlmlzatlon implies that perceived margmal revenue equals margmal cost for quantity and quahty kqJ =c’y and ky, =c; In light of the complementarlty assumption above (q”>O, p”>O), the unconstrained profit-maxlmmng choices of quantity and quahty are strictly increasing m the hedonic price, k Fig 1 shows the derlvatlon of the unconstrained supply curve s’“(k) The firm simply sets the margmal cost of the characterlstlc equal to the hedonic price, as at k, for example The correspondmg choices of quantity and quality at k= kl are implicit m the curves MC” and AC” This can be seen from the fact that each of the average cost curves enveloped by d’(z)/z 1s

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markets

Fig 1

given by c’(z/q, 4)/z for some fixed q, thus also determmmg quantity from the relation z=qy Moving from left to right m fig 1, the enveloped average cost curves (mth concomitant margmal cost curves) correspond to ever higher levels of quality, and the tangencies correspond to ever higher levels of quantity Hence, moving upward from A, along the curve p’(z), which IS also the unconstramed supply curve, the profit-maxlmmng levels of the character&c, quantity, and quality are mcreasmg The market situation 1s shown m fig 2 As indicated above, the demand curve IS downward-slopmg The unconstramed supply curve for the charactenstlc, gven by s”(k) = 2, s-‘“(k), IS upward-slopmg since each mdlvldual firm’s supply curve IS The supply curve IS drawn only for those hedonic prices at which all firms are wllhng to produce (the marginal firm with indifference), that IS, for k 2 max, A, = A Adding plausible boundary condltlons [hm, _A+ s”(k) - d(k) -c 0 and hm, + ms”(k) - d(k) > 01, m the absence of an effective price celling there exists a unique equlhbnum hedonic price k*> A and correspondmg market amount of characterlstlc produced z* = d(k*) = s”(k*) The equlhbrlum quality/output-pnce function IS of course p(4) = k*q, and firm J produces quantity $(sJU(k*)), quality fiJ(dU(k*)), and amount of the charactenstlc s’“(k*) = j?J(P'(k*))qJ(s'"(k*)) 3. Equilibrium with a price ceiling

Suppose that a celling p has been imposed on output price Then the

N Raymon, Prtce cehngs m competltwe markets

k

261

r”(k)

Fig 2

quality/output-pnce

function has the form

kq,

for

q

such that

kq SF,

L PP

for

q

such that

kq>p

PM =

Facing a function of this sort with a particular hedonic price k, each household, bemg mMerent to the quahty/quantlty mix yielding any gven level of the charactenstlc, makes the same choice as that made facing p(q)= kq wlthout a celling p However, the presence of a price cellmg does affect the choices of firms For firm 1, maxlmlzmg profit gven a quality/output-pnce function of the form Just above is equivalent to solvmg

subject to

kq, Q

Given earher assumptions - m particular the assumption of U-shaped cost curves m quantity - a umque, stnctly positive solution to this problem exists for any pair (k, p) with k> A, and jj>hm,,,, + kj3J(sJ”(k)) Clearly, if the pnce celhng IS not binding, the firm simply makes the unconstrained choices discussed above, however, if it IS bmdmg, the firm chooses qJ =p/k

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N Raymon, Prtce cedmgs VI competttrve markets

and y, satisfying the first-order condltlon p=c’(y,, p/k) Note that If ji 1s bmdmg, p/k I?‘, the celling 1s bmdmg and the firm can no longer produce where k= 4”(z) The optimal strategy 1s now to choose q=p/k and y to satisfy j = c$(y, j/k). This last is equivalent to choosmg z so that the hedonic price equals the constrmned margmal cost

For example, when k= kl, the firm chooses q=p/k,, z at the mtersectlon point S,, and y= klz/j Contmumg m this fashion, one constructs the constrained supply curve sJ(k,p) which comades with s’“(k) for kstJ(p) In general, s’(k, p) may be either posltlvely or negatively sloped but must he to the left of s’“(k) for k>t’(p) [This follows from the fact that for any fixed z, the marginal cost (l/q)e’,(z/q, 4) rises as 4 falls, which m turn follows from the complementarlty assumption ] In the example shown m fig 1, the constrained supply curve 1s S,S,S2, which comcldes with the unconstrained supply curve over the portion S,S, The constrained supply curve sJ(k,p) for the characterlstlc 1s contmuous over the domain speclfied above and smooth over that domain except where k = t’(p) It 1s nnportant to note that for (k, p) such that p 1s bmdmg but k 1s ‘near’ e’(p), the partial sJ,(k, @) IS posltlve This follows from the global property that s’(k,p)O The market characterlstlc supply curve with a price celling 1s s(k, p) =c, sJ(k, fi) Since .$(k, j) ~0 IS possible when k > t’(p), one may have s,(k,@
N Rawon, Price cedmgs m competztlve markets

263

1

p>max k liy, k/P(s’“(k)) =p” ’ J [ -1 If pzp* zmax, k*/P(sJ”(k*)), then clearly the unconstrained eqmhbrmm remains intact If p 1s effective, that is, if p tJ(p) If a firm 1s unconstrained at (k(p), j), then its production of quantity, quality, and the characterlstlc are all higher than m the unconstrained equlhbrmm at k* If a firm 1s constrained at (k(p), p), then the quality level and characterlstlc level it produces are lower than at k= k(p) without a price celling, but may be lower or higher than what it produces at k = k*( < k(p)) without a price celling All that can be said 1s that m any constrained eqmhbrmm, at least one (constrained) firm produces a lower quahty level and a lower characterlstlc level than m the unconstrained equlhbnum at k* Of course, this implies that m the special case of identical firms, every firm produces a lower quality level and a lower characterlstlc level than m the unconstrained equlhbrmm at k* Finally, consider the welfare impact of an effective price celling It 1s easy to see that every household 1s worse off m any constrained eqmhbnum than in the unconstrained eqmhbrmm The indirect utlhty function of household h 1s V(k) =f*(M,, - kdh(k), dh(k)), with derivative Vh’(k)= -f$P(k) ~0 Since k(p) > k* for any effective j, clearly Vh(k(p)) < Vh(k*) With regard to firms, one has m general only a local result For any IsE (p’, p*), let n-‘(p) be the profit of firm 1 at (k(p)), p) Then one can show that there exists a p”<‘(p)), have been smoothed

264

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Price cedmgs m competillve

markets

output price at which no firm is constrained (p*), every firm 1s better off A sketch of the proof of this fact 1s as follows The assumptions above imply that k(p) 1s differentiable on (p”,p*) except at isolated points [l e where k(p) = t’(p) for some 51, and that (where it exists) k’(p) < 0 for p below but sufflclently close to p* For any such p where k’ exists and any J, one has

nJ’(F) = k’@b’Mi%PI+ ~‘(k% 8 [ 1-

Isk-1, (PI

ko

where A’(k(j), p) 20 IS the Lagrange multlpher emergmg from firm J’S constrained profit-maxlmlzatlon problem If k(p) < c’(fi), 1’ =0 and l7J’ ~0 If k> t’(p), 1’ >O and flJ’ 1s m general of ambiguous sign However, as j approaches p* from below, 2’ approaches zero while k’ remams bounded away from zero (The proof that k’ remams bounded away from zero makes use of the fact, which 1s not hard to demonstrate, that llmk,B(Bj+ si>O) Hence, there exists a FE [p’,p*) such that for every J, II’(j) IS strictly decreasing on (fl, p*] In summary, under the assumptions made above about mdlvldual behavior and market supply and demand functions, the followmg are true (1) for any effective price celhng fi m the mterval (p’, p*), the market 1s m a constrained eqmhbrmm with hedonic price k(j) greater than the (unique) unconstramed-equlhbrmm hedomc price k*, (2) every household 1s worse off m any constrained equlhbrmm than m the unconstramed eqmhbrmm, and (3) if j is sufficiently close to p*, every firm 1s better off m the resulting constrained eqmhbrmm than m the unconstramed equlhbrmm Mamtammg the assumption of a hnear household technology, these same conclusions hold under alternative plausible assumptions on cost functions, and m an important class of cases mvolvmg multiple charactenstlcs, each bearing a hedonic price [This 1s for the short run, recall that for the long run, the conclusion m (3) changes] Extensions m which the assumption of a linear household technology 1s dropped must await further research In this case, the equlhbrmm quality/output-pnce function 1s non-hnear m the characterlstlcs rather than bemg essentially equivalent to a vector of hedonic prices One fruitful approach may be to derive supplies and demands as measures on the space of characterlstlcs [see Mas-Collel (1975)], and then to employ an appropriate sensltlvlty analysis to investigate the impact of an effective price ceiling

Lancaster,Kelvin J, 1966,A new approach to consumertheory, Journal of 14, 132-156

Pohtxal

Economy

Mas-Collel, Andreu, 1975, A model of eqmhbnum with chfferentlated commodlhes, Journal of Mathematical Economics 2, 263-295 Murphy, Mxhael M , 1980, Price controls and the behavior of the firm, International Econonuc Review 21,285-291 Rosen, Sherwm, 1974, Hedonic prices and Imphat markets Product differentiation m pure competition, Journal of Pohtlcal Economy 82, 34-55