World Bank Reprint Series: Number Fifty-seven T. N. Srinivasan and Jagdish N. Bhagwati Shadow Prices for Project Selection in the Presence of Distortions: Effective Rates of Protection and Domestic Resource Costs Reprinted from 7ournal of Political Economy 86 (1978) Thcz most recent editions of Catalog of Puiblicatiotns, describing the full range of World Bank publications, and World Bank Research Program, describing each of the continuing research programs of the Bank, are available withouit charge from: The World Bank, Publications Unit, 1818 H Street, N.W., Washington, D.C. 20433 U.S.A. WORLD BANK BOOKS ABOUT DEVELOPMEN-T (FOR SALE BY THE PUBLISIER INDICATED) Research Publications International Comparisons of Real Product and Purchasing Powerby Irving B. Kravis, Alan Heston, and Robert Summers, published by The Johns Hopkins University Press, 1978 Experimenfts in Family Planning: Lessons from the Developing World by Roberto Cuca and Catherine S. 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Kravis, Zoltan Kenessey, Alan Heston, and Robert Summers, published by The Johns Hopkins 'University Press, 1975 (continued on inside back cover) Shadowv Prices for Project Selection in the Presence of Distortions: Effective Rates of Protection and Domestic Resource Costs T. N. Srinivasan I ,Irl. Bank and Itdian .d'zInlwi, il Initinutc Jagdish N. Bhagwati Afassachusetts Institute of' Tc /u I'he paper addresses the p)robleml of' dcit- ivim4 slhadow prices for use in jlljct evaluation hlietn the e%i,tim4 allocation is ('haracteri7eCl by ad valorern tracnid distortions. Thie anla -is is used to clarify aii(t resolve tlite long-standing debl)ate animg eff'ctive-rate-of-protection and d1oniestic- resource-cost proponents is to the resp)ective merits oi their meaiSutres as metho(ds of pt .1 etl nation. T'he derivation of shadow factor prices is then extended to three mii;hor factor miarket imperfections familiar from extensive tra(le-theoretic analysis, Until recentlv, ti t'icok of' trade and wtlfartc have, hv and lurge, ifl oredt the ever-in cerasinitc literature oni project e alt hwlitii. 'I'hiis is tIm / iii, i since thet hulk of the pioit t evaluation literattute attollplts to derive shadow prices to replace the mibirket prices that, in (distorted situ.itions, clearly wvill not reflect true opp))rtunity costs whereas the iimajor ztilvancis in the welfare theory of' international trad(le hiave consisted( precisely in the analysis of issues in trade and wvelfa.re when tlhe Imarket is cIl(t'iacwi,.vdl by a numbl)er of wri.terittive endogenous or policy-iniposed di.,,tiwtlin. I'Fli research umltivi-inQ this paper was sullIwrtedI by NSF gRant SOC7 1-1321). Thanks are due to Peter D)iamond ftor lhelpul convesatllions and1rl to Mit 111'ci 1Bruno, Henry Wan, lan Little, C :hristolAier Heady, Riclhard Av1heCr. Takashi N'tgishi, Koichi Haniada, Hiroshli Atstunii, iela 1Balassa, and Jacob Irenkvl lor valuable commCinents onC aln earlier (draft ol'tlhis l)al)C'r t' hvieNs exrexssed] are- those ofthe autlhors and shouldl Co(t Ce} ascribed to t1h World B3ank or to its ifliliated iniStitUtions. See Ithi.i tO (Ii CCCOI Rinvm%at ii 1 963 Johlnson 1t1I); B1b.ug .fi , Raniasw axii,and Srinivasan 11tt:); Ith.en oi ]IG I, a1d(l thl ICiCCteroCus wvritings ol'f Kenlm, Findla y, (Corden, Maogeo, Brecher, and(t several trade theorists. (.Iournal cJ Po t itical FI',c!U ,C. I'l ¶7, o1, tI, CC. 11 '( 19713 bv 'lhe tUnivenis C t I I, CO I. t 't0022¶ t1- 1 7 hil (OCO)CCO,5C) I *. 97 98 °JORNAL. OF POLITICAL ECONOMY Trade theorists have generally coni-si'dered second-best problems characterizing the nature of optimal policy in tervenution when the gi\(n distortions cannot be (directly) reiimove.d. Projcct analysis, on the other hand, poses a relate-d, but different, (qu.We.stioll: if' tlle giVen dIistoirtiols definiir-Lg cuirrent resouirce (andl expendituire) allocations cannot be re- moved, wxotuld the iiitrodtiction of a project wlhich withdlraws resolurces fromn this -xisting allocation for project use be welfare-ituproviiia? 'rlTe sotlution to this latter problem naturally followvs fi-roni the dcrivTIatioln of the shadow p)rices of Ietors an(l outiputs for tise in project evaluation. As it happens, this proh r'n lhas been posed by Findlav ancd Wellisz (197fi) in a mIlost elegant, recent contribution, illtuminating lhow tradle- theoretie tools can be dep)loyed to aIdva.nliage in analyzing it.2 WVe follow themr in Section IL essentially takinig over their sinip)le iiodlel of trade theory, with primary factors producing tradled l Foos (inichlling the project output!, with no intermediates andl witlh fixed interniatiornal prices for the traded goods, and considering with Findlay and Wellisz the ease of a trade dktortion (i.e., a tariff or trade subsidy'). We parallel thle Findlay-Wellisz. an alysis, using somewhat dlifferent analytical tecili qe's, managing tllerefore to both ( mnipleniert antI correct it in critical ways. Next, in Section II, we relate tlehte resuilts oni the appropriate shadow prices in project evaluation to the twvo mneasures wlhlich have l)een pro- posed as 1)r(jc rt-evaliiation criteria in the developmental alld t radle literatutre : the 'll'c-ctive rate of prot'ection IERP) an( the domI('stic re- source cost (I)R( . It is shovn that the E'RP is ani itiappropriate nic.asurc for this purl)ose anid that I)R(:s, if tlhey must yield tile corre'ct social evaluation of a project., must use the second-best sliado' prices that are clerivecl in Section I that is, they must b)e appr,op1riatel\' (lefined I)RC's. Thus we succeed in caisting- light on the in()noul,;i%-e debate anuong the ERP and DRC proponents as typified, for xlmple, b)y the cont-rover sy in this Journal among Balassa andl Sclhyflowsky (1968, 1972', Bruno (1972,, and Krueger (197T2- --as to their relative imer its as itechiqtie of project appraisal.3 Finally, in Section III, wve analyze the (lderi-ation of shadow factor prices when tile given distortions arise from three alternative, polar types of factor market iiperecFetions familiar to trade. theorists, rather thlan frotn the presencie of a tra(le t.triff or sulbsidy.. 2 Nery early andl pioneering analyses bv,Joshi ? 1972 ' an(d Lal ( I97,1, attelmlpting to examine the Little-Mirriecs 19 .llanual rolb's il triade-theoretic lines must also be mentioned. (i. rdt'n ( 17-1: also has a dliscussion of these rules. 3 For a historical review of the antecedellts of the DRC concept, C sp'a illy in Israe l, see Bruno ( )72 ,. The use of ERP as a p)ojr ct criterion :pl[)p) al. on lin' itlo r hand, to have been the subject of internal W\orld Btank mitneoranda luriing the mid-l'i(ik, stem- ming prrosinabl)y from the notiorn that, in some sense, they reflected "conmparative SHADOW IRICTS FOR PROJECT SELEC.TION 99 I. The Model and Derivation of Shadow Prices As stated above, we consider the usual trade-theoretic model with two primary factors, k and 1, producing two traded outputs, X1 a.d X2, that enjoy fixed internationvxl prices p' and p'. Ihle "small" project being considerd:d will prodi: X o itv X3, at fixed in(ci-nationa! price p*. It is asso ned that the p)lanner is wvorking wvitlh a xvell-belhave(d social utility ftunction. TIhe problemi of p)roject utiandsis then is to evolve suitable prices, for the prinlary factors and output (X'3) in tle pr jc'tt wlicll woiuld enable the analvst to (leci(le whether the pro ect. sholul(d l)e accepted or rejectecl. The problem wouil(l be str;iig1itl0`brwa(rd inldeed if thcre were no clistor- tions in the sVstem: the correct valuatiorn of the primary flictois, wou ldl clearly be tlhsc in the niarkt't, as reflected by tlhe. international price-ratio p*'p*, and the correct valtuation for X3 wotul(d be tile international price pA. But the situation wte mutst now introduce is one w.here the dionmestic price-ratio) between commodities X1 and X'2 is distorted by a tariff an(i 'or tra(le subsidv an(d it is flb tllbei asstumed that this distortion must be taken as given. 'I'le problem then, as note(d by Fin(dlay anil WVellisz (1)76, p. 545) is "an inlierntrn secowicl best one" in whlich -Llc criterioni for acceptance of the proiject is wlhetlher or0 not it xvill incrc(';ev the value of, total production at world prices as compared with tile existing itinwill, assulming that the (di,tortiona1l p)olicy Oil tile existigo roodls continiles uinclhanged'':4 this b)eing, of couirse, the pr)Io('edluIr(' SUIt, d bstr' b Little and \[irilees (1969, in their (clTLbrated M1fantial and also by Bruiiio P 1(62, 1967b) in Ilis iunportant a tical work on project evaluation. In applying this criterion for a "smiiall" project, we note first that the introdlcetion of the project will use lal)or an,i 'or capital that are with- draxvn from their present lise. As such, the answer to tile (questioll whiliter or not the project (producing X'3) ?vill increase the valtie of production at wvorlt piices is tile samnt as to the question hiethelicr the worldI price of a unit of outp)ut of tile project exceeds or falls short of its cost of pro(lductiorl as o)tainedl by evaluating the lai)or ancl cap)ital used ill p)rodulcinlg A3 at their s1hetiwoFc prices, thlat is, at i)rices that equial their inarginal coin- P Provided that iniftrior goods are tuled out, there is of course a iionotonic relationshlip betwveen welthre undI the dii tanice or the :iv;a llililv locus (at intrniational pricess fronti thes iingi,ivi tt, well-bhiiavet social lilky' iElic tinto. 'Ilius, provided the degree of protection. .licl dhence the degree of consumiption d1ismtt; mi ren1wtils tin ich.mgi,l over the entire c't- tliw betire andl after the acceptance of the pr(ject . one can ci dki g'. ai withotut eroor the fact that trade distortions Nxill also (listort cousmpllption. it tollows ininleitiatv'v, of course, that if'one is d(]'l iom wvitlh a quota re'striction, raither than ,i ;ad valorenTI litriff, so thllit ue haXve e>r'c; i i jtl a bi ble d(titgn ol') diswritni, the ilitlr'- mentioned mlonotonic relationship betwove is llare and the dlistalCe of the is tillhilll lOCus (.It iHW1t'nt.tliotil.l 1ri('c'. will btieak clown. \Moe oi this is to be found( in Bl1.ig%%:ti anid WN'ani (1977;. 100 JOURNAL OF POLITICAL ECONOMY X2 Internotional Commodity Price - Ratio A P(XI1X2) Distorted Commodity Price - Ratio ( X , 0 B X FIG. I tribution in their existing use to the value of total prodUction at world prices. Turn now to figure 1. Here AB is the production possibility curve, defined on commodities X1 and X2. At free trade, pjrodluctioni woldklI be at P* (X*, A'2) reflecting the international commodity prices. However, wvith trade distortion, the cornmnodit) price-ratio is more favorable to conm- niodity X2 and proldction is at P(X'1, X2.Now, the planner is assumllied unable to correct the situat on directly, so that the commodity price-ratio, the factor price-ratio, and factor proportions for X1 and X, are to be lheld fixecl at their respective values at P(X,, X2). Denote then the corresponcl- ing input co-Mcients as (k,, 4,) andl (k2, f2) and factor rentals as tv ancl . Now, as noted above, the second-best shadow prices of labor (zt*) and capital f^*) in this situation must equal the change in the ( 1antities of X1 and X2 output, cvaluated at inlterinatio,al prices p* and p*, rcstilting from a marginal clhanlge in labor ancl capital, respectivcly, starting at P(X1, g2) andc maintaining the distorted commondity price-ratio for production decisions.5 Thus. (fefining If = p*X ± p2X2 and the total availability of capital andI labor as K an(d L, respectively, it is clear tihat the shia(low price of labor will be d6Wdl, anid that of capital will be df1d K. wuIi-re the derivatives must be evaluiate(d fbr the distorted situation. This is readil l done as follows. First, since capitil stupply is fixe(d (K), we have: k I (dX1dL) + k2(da'X2 d) = 0, andl, for labor, the curnespondimig equation is: II (dN lIdL) + f2(dX21dI'= 1. 'llerefl)rc, dAV1/dL k (k t 2 -2f) and dX2/d( = k1i(k,i2 - k2f1). Hence, thle Slhdo 5 The notation ti*, P* is used lvre rlecauLsC th1e circumnflex rel*'rs to the distorted 'dtU1.141n and the asterisk to the evaluation of' output change at international prices. SHADOW PRICES FOR PROJECT SEIECTION 101 X2J \\Q A \ \ Distorted \ \ Price- Ratio O B B X lFiC. 2 price of labor, dlefined as: Zi2* = p* (dX1/IdL) + p* (dXV, td l, is seen to be equal to: e*=P*k, - P *kz (1 2 - I~k k12- kJ11 Similarlv, we can see that the shadlow price of capital is: - _,- pAf1 k,= 1 - k2f, ;(2) It is readily seen that these are also the values of t and 0* that satisfy the equations:6 p - + k,, (4) Now, it is easy to see that the sliift in otitpuits, as labor (capital) is wvith(drawn v roin P, iP naiiithining the (listortion and hiencie the (listorted commodity price-ratio, is viehled by the correspond ing R ybczynski line. So, assumiinlg that X1 is K-intensive at P (i.e., k1i1 > k2!i,), otne can see, in figuire 2, that the tcecon iiy will miove fron P dowii linle: PIB' as 6 This is also the p)roce(dtre iuggvtde loi deriviiig shadlow factor prices y l)iamond and MirrIves r1976h itn their anialysis of a uniar prol)lemn. It may he niotted hete 11t.t , in the case where the trade doistrtion is tiOt ad Nalorem viiut, say, a speci lie tariff lor suhsidvy) or a UoaMItilNltivi restriction, the coefficients 11, k,. 12, and k2 %s ill changei %u ilh the Witil- dtrawal of factors even for a \ttll"' ,itijet( l andl one cannot Use this procedure for estimating sha(loNw factor priers. N loreovvr, note also that, if the oiumber of facuors dilfihrs from the number of goodN, flihei sha(dow factor prices may not be umiquely defined for small chan-es anid, or may be nonstationary for large chfanges. On all this. see Bhagwati and Wan (1977). 102 JOtIRNAL. OF POLITICAI, ECONOMY labor is redicedi, up line PQ as labor is increasec], tup PA' as capital is recduced, and down PR as capital is increasedi. It equally follows, fronm the evaluationi of these slhifts at the interznalinn1ed (ratlher than tlhe (listorltel) comnno(lity price-ratio, that t&* will be ner-ativc if the in ternartional price line is steeper that PB' (i.e.,p*/p* > k/k,) andi$I will be negative if the international price line is flatter than P1' (i.e., p*7p* < 11'i2), andI that nonnnegative values for Z'* and I* will obtain onily -wlicn the inter- national price-ratio is in the ranige spanned by PB' and PA'. That it is possible for &* or p* to be negative woul(d al)ewar to be a paradox. For, it of course implies, for instan-.c, that wvhen (say) &* < 0, it would pay society to inplemenit a project with zero outptlt (X3) and positive labor input: in other wordIs, that if labor were withdrawnii from exis ing production, thanks to the project, this will increase the value of such production at international prices. But then this paradlox is only yet another instance of "immiserizing growtlh"; the p1restcLec of the nmarginal labor is iunimiscrizirng. given the distortion;7 and thus the paradox is readilv resolve(l. In their dlerivation of shaclow factor prices for the above problem, however, Findlay and Wellisz (1976j bypass this pcssibility of negative factor prices by deriving these p)rices instead via the solttion to a pro- gramming problem which is tantainootnt to (see fig. 2) clerhiimg the shadow factor prices correspnm(lillng to the interrnational prices but sulh ject to a 'Feasihle" production pm;sibility curve defined( by A'PB'. These Findlav-Wellisz shadlow prices (o^*, ! are clearly yielde(d by putting the international price-ratio tangent to A'PB', in the usual way, and are illustratecl to advantage in figure 3. Figure 3 is the all-too-familiar Samuelson diagram and needis no explanation. Now, nioenmelnt along the nresvtit-ldl pro(dnction possibility curve APB in figure 2 corresponds to inoveicot ailonig the curve QPR in fimrle 3, relating the commodity price-ratio to tlihe correspondlin, factor price-ratio. Similarly, movement along the restricted production possibility curve A'PB' in lfigure 2 corresponds in figure 3 to following the y-axis in the fourtlh quadrant from o up to the point S where OS = k1/k2, then along the curve SPNZ up to Z (where N is at a dlistance '1/'2 from the x-axis) ancl then following a straighlt line parallel to the x-axis. Tlihe (restrieted curve SPNZ, depicting w/y as a fullnctioll of can be shown to be incr.casing and concave, witlh a common tangenlt with tlhe (mitrestrictedl) curve QPR at P. T'hius, the Findlav- 7Ct. Bhagwzimi (196(i8; Johnsoa J9137) who deals with the precise dlistortion in our model here, and Bhagwati (1971) who states the general theory of iininkcrizing growvth that explains and ties toge(ther the ulllirerm instanIces of iinmniserizimig growth. The phenonmcnon of megative shadow factor prices, in turn, is related to the empirically important phenomenon of value subtraction at international prices: the latter requires, but does not necessahily follow from, the fornmer; see Bhlagwati, Srinivasan, and NVan (1977). SHADOW PRIClES FOR PROJEC'T SELECTION 103 kJ /I 0 u r w/r P [P2 = C PIPll2a FIG. 3 Wellisz shadow price-ratio di*/1* will be infinite for p*/p* 2 k/k2 and zero for p*/p* < ,1t2, while taking positive values in the range spanned by kl1k2 and /1i12. This procedluire therefore clearly will yield slhadow prices that coincide with the correct onlCS yieldcl by our pr.oceduire only when &*M* 2 0, that is, in figures 1 andI 2, only for the parametric case where the international price-ratio lies in the range spanned by PA' and PB'. For the parametric cases 'lhere the international price-ratio lies outside of this range, the Findlay-ANellisz procedure will incorrectly yield, not negative shadow factor prices, but a shadow factor price-ratio, = 0 or oc, accordinig to whether the production specialization, corresponding to the initerinational price-ratio, occurred in figure 2 at B' (on X1) or A' (on X2)A 8 II. ERPs, DRCs, et Al. We have thus deduced, in the precedinlg sectie-n, the precise shadlow prices that must be iisedl, in a distortedI situation, for project appraisal. We are 8 An alternative analysis of the ilnapproj)rialcne5s of lihe Findlay-Wellisz procedure., in programming terms, is availalble from the authors, on request. 104 JOtRN.\l, OF POlITICAl. ECONOMY tlfcrerere now%% in a Iosition to deci(le on the couipeting claimiis of the ERP ancl DRC proponents as to their relative merits as teclniqlqles of pro jc(t appraisal. As careful readling of this debate in this Joournal (1972), already cited, will iiniinstalnalb- reveal, the first priority in this area is to clefine one's concepts ilniitl'i)i't1iiOi.JVv Since these and oilllc ('ecOIiOnfliktS (listinigoislh between (direct and ind(lirect use of faIctors, thiuis iwiltidin,t iiitcrneciatec; whlichl were niot included in the aiialvsis in Section I above, we sliotilt(l first state that o-tir p ject-acceptanice criterorn sllitalh)lv a llct ldcd, is the roll ,,in: p3 ˇ> k3 i* + 13'P +. 1AP (5) where it is niow assumned that X. is usd(1 in plroject (X3) with coeflicientfi per unit outp)ut of X3 and(I %e%icrt k-3, 13, andf1 are assunmed fixe(d so that onie is c'sii ntinily treatin,g each pi'ro'ess as a projtc(t. Wlhat the ecriterioin sav's, of couirse, is that the p)roject, to) b)e acce)te(l, mIust pl)Odli(C Outplut wvhichl. wlhen evaltuatedI at international prices, exceeds or equials the cost of prio(t iictioii evaluated at the :'eco(-ccnd-best sla(ldow factor prices. Now, Inote that the Irli' t_-hanld(j sicle of t5 is wiittei in a formi that includes the primary an(d intermnediate iniptuts. 13But, it can e'quid.elh'ntlr he written in the form inlud li(ingi direct plus in(irect pi;niury flutors, that is, l)y (leI cliposing interinedliates into primlary factors: * > ,k3 t- f, k * + t13 + .tJDlf*. (6) -Now, olotinig that the D)R(, concept ihuplies that on1e iS InCeaso ring thle donmestic resources usedu in atn activity to produce a tinit of foreign ex- c11an11L, we can (listinguish slharply among the followving, alternative concepts that corrt -ponil, in orne way' or another, to the concepts that are often apparently use(d indclistingquislhibly in the literatulre. Note, initially, that b)v first l)est we will refer to Iaetor valuiations " n* CMTesPMI(iliIng to thelfirst-best optimal sittnation at , 2 in figuire 1. BIy seconcl best, we will denote insteatl the factor valuations (¸'*, * that reflect the wecond-best optimal situation, given the distor- tion. Finally, by "'private" we will (dtenote thie factor valuations (af1 p) that acttually obtain in the distorted situation at P. Next, we slhouldl also note that the c'ThAate incld(les a(lditintally a d i... .owtiOI.I a;1in( nn11 neaSti ten-s working witlh iniem rtenlied ites or altern in tively X6i11I the iitlnl'iculinte.s dt'cotIip)osed into the priniiory fictors producing themll. I 1 ne, ve will dlistingluislh also between (lecoiuposed-iiuteriitediates anti iairst-internnnc'diats meaures.9 We will tius have altoglether six measures of I)R( s an(d one for ERP'. We niauv therefore now Utate tllhese alterniative ('O 01( )tS:'iiteallires in ' regrd to the Plci( jet prod icig W3, with W We could also, in l)rineil)e, have distinguished htcs(eean "gross" and "'n' values, as explaiunel in the text pwm IlIowtver, there is no evidence that gross measures have been computed so that we confine ourselves in the texL to only the net measures. SHADOW PRICES FOR PROJECT SErLECTION 105 brevity, noting that, in the (lenominator of all the measures set out below, commodities (Xl, X2, and X3) are always valued at their international prices. I o DRC,: First-best, Direct-Intlirlzluwdintes lMeasutre Here, we have the evahlation of the primnary factors at first-best slhadow wage and rental (zz*, y*), correspondciing to the sittuation where the international commodity prices obtain donmcstically and therefore the distortions have been eliminated. These are also the shadow prices suggested by Bacha and Taylor (1971). In this case, we define: LRC1- k3Y* + 13t' - (7) p3 -f P.1 for the project, using the "direct" formulation:f1p* in the (lenonminator, rather than decomposing that into [(f,k*)y* + (f,1"l)w*] as in the next imieasuire DRGj1 (where k* and 1* are the coefrlcicnts corresponding to first-best shadow prices). DRCOJ: Firsi-best, Dter(-owupo.sed- Intetruwtdiat/t.s Mfeaslure Here, DIRC1 mlodlifies eqtuivalently therefore to: DRC11 k3Y* ± 13W* *8 A [(f,k A)Y* + (fV11)w*] (8) (Now, note that we have been referring onlly to formulations that deal with value added in the denomiinator. TIwesC DRG' ineasuws are tlhereforc 'remm se~ctors are those by Tgvie ( I 9i5l.,johnsonn 1l ( Bhrngwati and Srinivasan (1971), Herberg and Keenip r19¶71),J jcnes (1971a), a;nd INLgec (1 9761 the welfareanalysesare by higvii (1958) antd 13h;igwati and l;;n li I ¶IUM). Pearce andl Muncllak have madte valuable coni ribuwi-;i; also. 18 The nmodel as set out in Harris anti Todaro f ¶17(1 is mlisslptirlvd on the demand side. See fliverefore the correct spcific.ulitimi, as set out in Blagwati and Srinivasan (1974) and followed here. IIO JOURNAL OF POLITICAL ECONOMY Since the availabilitv of foreign exchange in this model is given by Z = F2 + (P*/P*) F1, the second-best shadow price of labor is clearly: dF 1 -1.' (17) dL p 'L ; ( LD)F"' With F" < 0 by strict concavity of F, ancl L > L1, we then see that the second-best shado'.i wage for labor is less than the agricultural wage which, in turn, is less than the manufacturing wage. Note also that the shadow wage is positive, instead of zero, despite the unemployed labor; this is because any withdrawal of labor from the labor force (L), while initially reducing uneiiiployiont ne will simultaneously raise the expected wa,e in manufacturing andl hence result in IcdtiictioIn of agricultural employment and output. The foregoing analysis assumes that the employment (at whatever wage rate) in the project has no impact on the expectcd wage in the inanu- factulring sector except insofar as it affects the manufacturing labor force. Thuts writing s as the employment in the project and il as the resulting migration from avgricu1ltuLre, the expected wage in tlle manufactuiing sector after migration is iTL2i(L - r) - (LI - 1i) whiclh is equiated in turn to the agricultural wage p*F(L, - 1). IHowcver, if we were to assume that the pr oject laborers are emiployed at sonie wage, ze P, a n(l that project emnployinent at this wvage afTects tlle expected wage in the manu- facturing sector, thle latter wouldl be wL, + wPe"L - (L1 - l) wlhiclh again is equated to p*F (Ll - ii). Solving the latter for )I and noting that the shadow wage is the loss in agricultural output per unit of project employment, that is, p*F(ql/s), we get slhadow wage = (u,P * F [F' - F-(L - L,)], In the case where F' = 0, this r(eluces to w"P, the wage paid to the project laborer. If we make the Fuirther asslummptioll that wP = w, that is, the project employs labor at the 1namiiifhctiring wvage, the shadoxv wage equals the manuLfactuLrilng wage: a highlY special case, as we have just shown, bIut one which has been focused upon in the standard cost-benefit analysis of the Harris-Todaro model. Generalized Slick, llage Shiuft now to the inodel where the wage is sticky across the two sectors at the level zw. Assminigic then that commodity X2 is capiLal-inten.sive, that is, (K2/L2 > K/L,), we now get: F2 K2 * ' z2 (18) 2.2 L2 L2F2 L2 L1F - KJ (l9) SHADOW PRICES FOR PROJECT SELECTION I I XI a) -s A .Y a,\ .- p CD C: 0 X2 (K-intensive Manufactures) FIG. 4 where FK, FW, FL, and FL are the partial der ivatives with respect to K and L, respectively; that is, they are marginal prod.ucts of capital and labor; and F21L2 and F1/L1 are the average 1)toltltCtS of labor in production of X2 and X1, respectively. We can then see that, in terms of figure 4, the production possibility curve is APB, P representing the point at which (F2/L2 - K2/L2 F) = w. At points to the left [right] of P, (F2/L2 - K2/L2 * F2) > [<] w It is evident then that, with the minimum wage constraint, the feasible production possibility curve will be APQ where PQ is the Rylbczynski line (for variations in labor) and, at points on PQ other than P, tlhere is un- employed labor. Let the capital-labor ratios at P then be .K2/L2 and .K1/L1. Now, when the international price-ratio p*/pp yields tangency along AP, the market and shadow wages will be naturally identical, and will exceed zZ, if the tangency is offP. For the price-ratio tangent to APB at P, the production equilibriuLin however may be anywvhere between P and Q, the different production equilibria implying clifferent labor arvailabilitics. Therefore, for this tangenitial price-ratio, the shadow and actual wages will be .Z for production at P, whereas the actual wage will be wv but the shadow vage will be zero for points other than P on PQ.' 9 Finally, for all commodity price-ratios steeper than the price-ratio tangent at P, there 19 At points other than P on PQ, furtherni.ore, the shadow rental of capital will be the average product of capital in X2 at P along the curve APB, rather than its market value which will equal the marginal product. 112 JOU R N,I. OF POLITICAL, ECONOMI'Y will be complete specialization on X2 at Q and the correspondiing actual wage will be zT) while the shadow wvage wvill be zero.20 Hence, unlike in the sector-specific wage stl;incnss case, the unemploy- ment of labor can indeedI he taken to imiply a zero slhadow wvage for labor. Howvever, associatt-d witlh this, the slhado%v rental of capital will (exceerd its market rental, so that the stadar(ird pre- cription of putting the wa1ge of unemploye(d labor equal to zero lbut usingq the imarket rental of capital is wrong. The WVage-DifereNIial Case Take, finally, the distii iol Nvwhere the wNage in X2 is a mulltiple i. of tllat in X1. In this case, it is well known that tile p)Lod(lIctio!n possibility curve will shrink to AQB, in figure 5. F'urthermore, AQB n need not be coneave- to the origin, the nlarket equilibrium need not be unique for any coIm- modity price-ratio, and the conllnndity price-ratio will not equal the marginal rate of tlanSforolllatiunl along AQB.21 Let the arzM,'keL equilibrium in the initial, distortedI situation be at Q. Tlhen, we can derive the two R)bcz.yusMi lines, QB' (for variations in labor availability) and Q.' (for variations in cap)ital ava;bb:ibility), assuming as earlier that X2 is capital intensive. Now, the international price-ratio equals the ratio of mw,ignal )rodllcts of capital in p)roducing X, an( X,1 with the techniques corresponding to Q (i.e., pI2 = FK/ Jf the latter dlerivatives as at Q). On tlle otlier hanld, the slope of QB' (ML'eaS1re]d against the vertical axis) will equcal the ratio of the corresponding avera~ge pioduets. It follows then that the international price-line would be flatter tllan QB' and steeper tilan QA', given the capital intensity of X2 rclltfiv". to A'1, provided tlhere were no wage dlifferential ;,. 1l lw cvr, in tile pi(Seflsce of the wage differential, the international price-linie may wvell be steeper (flatter) than QB'(QA'1, with the watr, in X2 exceeding tllat in X1 by factor 2.( > 1). the crundition for this "reversal" of r-lative slopes of the price-ratio an-d the Rybczyrlsl;i line being that X2 ceases to be capital intensive relative to N1 if tile factor intensities are compared on a dijtcrilial-cttg h/led basis. 22 It is then easy to see that, as in Section I, tile second-best sllhadow% wage of lalbor, that is, [PTV2/22) - p*(K1/FI)]/[(K2/1F2) (L/1F1) - (1/j1F)((L2/F2)], or thie shadow rental on capital, that is, [jP(L/1F1) - 20 (IAt Q also, the shadow price of ca.pital wvill continue to le the average product of capital in niu:iLiflictturing at p,iint P, since at Q only the nmaiulaIcturedl goodl, a, is produced using all the available capital and the same techmiqtte as at P. 21 For these and( other pahlmnlu,i,s, see Blhagu.,i and(l Srinivasan (1971) andXC M:rgee's excellent ;;urI', V i?ln. 22Jones (1971a) calls the dilr renrirl-wigilrmu irtCChiiticS the "valut-" as against the Samuelsonian "'ph)sical" lactor-intensities. SHADOW PRICES FOR PROJECT SELECTION 113 xl, X - A A 0 B' B X2 FIG. 5 p'(L2/F2)]I[(R21F2)(L1/F1) - (1C/F1)(L2/F2)J, will be negative when such reversal of relative slopes exists; and, once again, the Findlay- Wellisz procedure of deriving shadow prices would yield an incorrect zero wage (rental). IV. Concluding Remarks A few concluding observations are in order. First, while our results on project appraisal have been shown to be successfully convertible into appropriately defined DRCs, this is not the same thing of course as having shown that these were precisely the DRC definitions (as against the many others that we have distinguished) that one or more of the DRC propo- nents, in the project-appraisal debate among the DI&). and ERP proponents, had in mind. Second, while we have confined our analysis to "small" projects, drawing infinitesimal resources away from the existing distorted situation, it is equally clear from our analysis that the results will hold also for "large" projects. Given the Rybczynski-line properties of the different models, the shadow prices of factors will be identical for small and large slhifts of factors into the project.23 Third, we might as well note explicitly that our analysis could be e%:tencled to models involving nontracleci goods; this would pei-rmiit the introdtuction of the real exchange rate in a mcaninigfuLl manner into the 23 On the analysis of the possibility of such "stationarity" of the "marginal-vari- ational" shadow factor prices, in more general models with many goods and factors, see Bhagwati and Wan (1977). 114 JOURNAL OF I'OLI'liCAL FCONOMY analysis. 'I'lhe extension to ImIo(dels Nvith sectOr-slpecilic factors is not merely readily done 24 it wvill lhowever introduce no special insiglhts that qualify whliat has been lcierncd from the present pjapler. Fourtlh, note that ve are iinplicitlv assuming that, in resp)ect of r)ojects wiich will l)e 6iom.eii untder slha(low prices b)ut not unrdler acttual market prices, the resulting losses are (coveelr in some windisteriI'tion .iry way. However, if the losses cani be covered only by solte form of odisttortionarv taxation, then the slha(dowv prices (for both ietpagt' antI *tt1ttts have to be calculatedI refl('ttin this fact. lllrln'rly our ;ml mlvsis call be 1I(id to apply witlhout irotdlification to the case where the planninrg auithorities are investigating the social prohtl)ility of a 1priv'ate project which is com- merciallv' viable at imarket p)rices. In this instance, if social p)rofitability is absent, the lflannin,g autthorities can decide to climninate the activitv b)y prohibiting it, anol the revenue problemi does not arise. Fiftlh, note also that wNe evalkated the project at a given techniquie. Thus, if a project can exploit alternative techlniques, fromn which 'lntre- pren(cti>rs wotul(d choose the cost-miniiiiizing one, then the revenue )roblem will arise almio b)(ecaulse a1 sluital)le factor-use tax-cumi-slubsidv will have to be provided so that the 'correct" ie chlttimjr (i.e., that using coellicie lllsi k3, 3, andlf1 if the project AV3 hlas been shown to l)e sociahy profitable) is closeni. Finally, it is also clear that impl)licit in ouir analysis is the assitmUl)tion that p)roblenms of incoime di'trilnmitiOn anld savings can be tackled tlir(egl dep)loyt innt ol all)rol)ria(e noitldi Leloii rya itl stl ii totts. 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