Domar, Evsey D.

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working paperdepartmentof economicsTHE EFFECT OF A PROPORTIONAL SUBSIDY OR A TAXON THE QUALITY AND QUANTITY OF OUTPUT *^Evsey D. DomarNumber 156 May, 1975massachusettsinstitute oftechnology50 memorial driveCambridge, mass. 02139THE EFFECT OF A PROPORTIONAL SUBSIDY OR A TAXON THE QUALITY AND QUANTITY OF OUTPUT *Evsey D. DomarNumber 156 May, 1975THE EFFECT OF A PROPORTIONAL SUBSIDY OR A TAXON THE QUALITY AND QUANTITY OF OUTPUT *.Evsey D. DomarMassachusetts Institute of TechnologyIn an earlier paper I suggested that a bonus expressed as a linearfunction of profit and sales can induce a socialist (or any) manager toforgo his monopolistic powers.— Here I propose to investigate whether sucha bonus might also prompt him to change, and particularly to lower, thequality of his product: the latter effect would be most unwelcome insocialist countries. But because, as was shown earlier, this bonus isequivalent to a subsidy proportional to price, this investigation canbe broadened to include the effects of such a subsidy, not restrictedto its optimal value, on the quality and also on the quantity of the2/product.— And finally, since the subsidy can be less than unity, it canbe interpreted as a proportional tax as well. It will be shown that thissubsidy (or tax) can indeed affect both quality and quantity (and of coursethe price) of output, and sometimes in a rather unexpected manner.LIST OF SYMBOLS(in order of appearance)£ = price of outputX = quantity of outputk = index of quality of output£ = total costH = profit (including the subsidy or the tax) = zR - CR = revenue (without the subsidy or the tax) = px~2-z_ = subsidy (if z^ > 1) , or tax (if _z < 1)2 2H TI tl — " "Tc °° 8x ' ^ "^ ^ 2 ' ^k " 9x8k '3xSimilar notation is used for other derivatives.ItE„ = elasticity of R in respect to k.i\. XXSimilar notation is used for other elasticities.We assume that both the price and the cost of output are functionsof quantity and of quality:(1) p = p(x, k), C = C(x, k),where k is some quality index. It can stand for some easily quantifiablecharacteristic, such as the strength of a material or the longevity of amachine, or for something more elusive, like the variety of dresses orthe taste of wine. Quality is of course multidimensional, but no attemptof dealing with the general complex case will be made here.The firm (or the manager) will maximize(2) H = zR - Cin respect to x and to k:(3) H = zR - C = 0,X XX(4) H^ = zR^ - C^ = 0.The second order conditions are:(5) H = zR - C < 0,XX XX XX(^> \k = ^\k - ^kk < 0=-3-and(7) 4 = (^\k - ^xk^' < <^^xx - ^xx><^\k - ^kk^dx dkWe now want to find the signs of -r— and -r— . The differentiation ofdz dz(3) and (4) results in the following system of equations:(8)(9)H i2E + H^4^ = -RXX dz xk dz XHdxxk dzdk"^ \k dz " '\with the determinant(10) D =HXXHxk«xk\k= H H,,- H,>XX kk xkby second order conditions (7)(11)(12)dxdzdkdz-\\k + \«xk-R, H + R H,K XX X xkSince we can readily assume that the firm operates in the region wherefix HkR > 0, R, > 0, while H < 0, H,, < by (5) and (6) , ^ > 0, ^ > ifH , > 0. Thus it only remains to explore the case when H < 0. A changein the signs in (8) and (9) gives us two equations with all positive co-efficients. We can immediately conclude that:(12) if f > 0.dz ' ^'<0:(13) if dz f>o^(14) if f < 0.dz f>°.-4-and by symmetry the same relations hold for -r— as a function of -—.dz dzdx dkWe should note that for either -^— < or -r— < the other derivative mustdz dzbe positive and large .Propositions (12) - (14) can also be established by examining thesecond order conditions. This method would give us more restricted results,but on the whole it would hardly justify the effort and the space. Itdxmay be worth while to examine in detail just two cases, say when — < 0,dkor when -r- < though the results will be expressed in such unfamiliarelasticities that, I suspect, they will add little to our understandingof the problem.dxIf 3^ < 0, then from (11)dz(15) -^ < -^ .From (15), (6) and (7)^\k ^xk ^\k ^kkX X IC KFrom (16), (3) and (4)/ -t-J \ jCK. jCK. iCJv iCeCX X K kIntroducing the elasticity of R in respect to k(^«)^R = -Bk^ ' f = -R^ 'X XXand using similar definitions for the other elasticities, we can express(17) as-5-X X K Kor as(20) E^ - E^ < e!^ - e!^,R R, C C,X Tc X kif 4^ < 0.dzFollowing the same procedure we can find that(21) E^ - E^ < E^ - KfR^ R C, CX k Xif 1^ < 0.dzkWhile it is probable that E^ > (because an improvement in qualityXshould raise the marginal cost) » E^ > (because further quality improvementsVshould be more expensive, and Ej, < (if demand elasticity declines to theKXright) , I would not venture to predict on a. priori grounds the signs of theother elasticities and particularly of the differences between them.dx dkIf we recollect that a H , > always yields — > 0, — > 0, and thatonly a large negative H,(subject to the second order restrictions) candx dkgive us -r- < or -r— < 0, a negative effect of the subsidy either on quantityuz — dzor on quality seems unlikely. But since we do not know the probabilitiesof each configuration it is best to leave the question open. It is possiblethat a subsidy can improve the quality to such an extent as to reduce thequantity, and vice versa (while exactly the opposite would be true of a3/tax). All this can happen, but I wonder if it has ever happened in reality?—It is also possible that a subsidy may have an unexpected effect onprice. For-6-/oo\ dp dx.dk(22) d5 = Px d^ ^ Pk d^ •Assuming, as usual, that P < and that £, > 0, we find that only if-r— > 0, -y— < (one of the "less probable" cases) will -r^ be definitelydz dz - dznegative. If -j- < 0, -r- > 0, then -r'- > 0, while in the supposedly mostdx dk"common" case when -r— > 0, -r— > the result is uncertain.dz dzThe introduction of quality as a decision variable may also cast somedoubts on the welfare effects of our bonus scheme. No longer can we assert(abstracting from the complex general equilibrium considerations) that thebonus, even if set correctly, will increase social welfare by inducing themanager to move from the usual monopolistic position, point D on the diagramsof the earlier paper, to point A (of the same demand curve) where marginalcost equals price. All we can now claim is that, if quality changes, themanager will move from point D on the demand curve for products of onequality to point A on the demand curve for products of another quality.4/It is plausible that social welfare will increase, but it is not certain.—-7-NOTES* This paper was prompted by a question raised by one of my graduatestudents in the Seminar on Economic Development at La Trobe Universityin Melbourne (June-August, 1974) whether my bonus scheme (see note 1)might not lower the quality of output. Unfortunately, I do not rememberwho of the several students should be thanked for that.I am very grateful to my colleague Professor Peter A. Diamond for hisgentle guidance through some labyrinths of welfare economics. He is notto be held responsible, however, for any of my remaining mistakes.I am also grateful to the National Science Foundation (Grant NSF-GS-2627)for its financial support.1. "On the Optimal Compensation of a Socialist Manager ", The QuarterlyJournal of Economics , LXXXVIII (February 1974), 1-18.2. That optimal value of z_ was the one which induced the manager tomove to the point where marginal cost equaled price. Ibid. , pp. 10, 16-17.3. In the last few years there have appeared a number of articles onthe effects of monopolization on the quality of output, most of them dealingwith the durability of capital goods. For a bibliography see Richard W.Parks, "The Demand and Supply of Durable Goods and Durability", The AmericanEconomic Review,LXIV (March 1974), 37-55. It seems that the results ofthat discussion have been rather inconclusive. See also an unpublishedpaper by Michael Spence, "Product Selection, Fixed Costs and MonopolisticCompetition" (1974).4. I have not proved yet that changes in quality will not affect theconvergence of the iterative process (the "Simple Rule") described in theMathematical Appendix of the earlier paper.-^/^tN^ssrDate DueI >l "MS?£CIWAX 20 77BEG 2 S TT4 >8jmf^=^il^Oi86l 9AG 1 0*89-i9aoA?^fliTl'?^Lib-26-67MIT LlBRflflieSno. /^^3 TDfiD DD3 TbD DflBHB31.M415 no.153Fisher, Frankl/The stability of genera724523_D*BKS 00174197,9..D*BaaO ODE Sbfl 712T-J5 E19 w no.154Klndleberger, /Commercial expansion an724542 D*BKS.00.0.0.97.6.1...3 TDflD DDO 523 bhTHB31,IV1415 no.l55Joskow, Paul L/The future of the U.S.3 TDflD ODD bflT 37THB31.IVI415 no.156Domar Evsey^D/The effect of anLuv^ ^"cui r prP,*BKS,0.0023137oport3 ^Oao ODD bfiT mLoHB31.IVI415 no.157Bh^awati, Jagd/Devaluation in a smallI llii;i iPi*PI^S 000231343 TDfiO ODD bfii 3TSHB31.M415 no.158Bhagwati, Jagd/Welfare-theoretical ana724545III iiiiliiiiii2313 TOaO ODD bflT MHB31.IVI415 no.l59Bhaqwati, Jagd/Education in a "job ladIjiiiiiliiiiiiii*3 TOfiO ODD bMb M37HB31.M415 no.160Bradbury, Kath/Simultaneous estimation725446 0*BKS 00023132liiiiiilil ill3 TOAD ODD bST 33fl

The effect of a proportional subsidy or a tax on the quality and quantity of output

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