World Bank Reprint Series: Number Forty-five

Clive Bell and Pinhas Zusman

- argaining-heoretic pproach to

Cropsharing Contracts

Reprinted from The American Economic Review 66 (September 1976)

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(continued on inside back covcr)

A Bargaining Theoretic Approach toCropsharing Contracts

By CLIVE BIELL AND PINHAS ZUS%f N.%*

Sharecropping is a geographically wide- under sharecropping departs from thatspread and historically tenacious torm of ruling in a competitive system with fixedlease. The questions of how the rental rent leases, it being well known that theshare is determined, and how such a form latter is Pareto optimal. As Newberyof lease affects resource allocation and the (1974) has shown, the fund(lameintal diffi-distribution of income pose a challenge to culty with B-S's formulation is that thereeconomic theory which has evoked two will be excess demanid for land for allnotable recent attempts to provide an rental sbares less than unity, unless theexplanation. First, within what may be marginal productivity of land is zero. Bycalled the Marshallian tradition, P. K. contrast, in the C-N scheme of things,Bardhan and T. N. Srinivasan. (1971) the eqluilibriumn which obtains happens to(hereafter B-S) have developed a fairly be identical with tliat ruling in a com-complete scheme of analysis in which the petitive fixed-rent systemni an(l is thereforerental share is regarded by the parties to Pareto efficient. This formulation, how-the contract "as a price-like variable" ever, has two basic and related (liffictulties(1974, p. 1068), i.e., it is parametrically of its own. First, there is the (lubiousgiven to all agents. For an exogenously enforceability of the minimum labor-given real wage rate, the supply of and intensity provision of the contract. Sec-demand for leases determine the rental ondly, if land has to be rationed, bothshare, A second approach is that originally agents are l)lacel in a bargaining situation.formulated by S. N. S. Cheung and subse- Hrence, one would expect the rental sharequently extended and generalized( by which emerges from the bargaining processD. M. G. Nevbery (1973a, b), (hereafter to reflect the agents' respective bargainingC-N). In their system, also claimied to be strengths.competitive, both the rental share and the Once the overriding importance of theminimum labor input per unit of land are last point is recognized, it is plain thatstipulated in all contracts, their values the problem must be reformulated andbeing jointly dletermiiined by the landlord's analyzed within the frameworl of bargain-desire to maximize his income subject to ing theory. This is the salient feature ofthe condition that the tenant's income our approach. XNow as Newbery anddoes not fall below his alternativc earnings Joseph Stiglitz haMve showVn1, the choiCUe ofin a perfectly competitive labor mnarket. rental contract is intimately relatted to the

True to the spirit of Atarshall's analysis allocation of risk bearing aLnd the l)rovision(p. 644), B-S show that r(source a1llOGctiffll of incentives in an uLncertaini wvorld. Ifow-

* Economists, Development Research Center, World ever, in order to keep the bargainingBank. Bell is on leave from the Institute of Development aspects of the l)roblem in sharp focus, weStudies, Sussex, and Zusman from the Hebrew Univer- confine ourselves hie-re to pure cropsharingsity, Jerusalem. We would like to acknowledge helpful u-der certainty.comments by Bela Balassa and an anonymous referee. In addition to these theoretical con-Errors of analysis are ours alone.

578

VOL. 66 NO. 4 BELL AND ZUSMAN: CROPSHARING CONTRACTS 579

siderations, the foregoing competitive for- tural work-er and tenant status is probablymulations pay little attention to the fact asymmetric. It is always open to thethat in addition to his labor, the tenant tenant to work for oLiiCr as well as (oroften possesses land of his own, some capi- instead of) on the land he has leasedl in;tal, and entrel)reneurial and management but the landless laborer cannot just electskills. In this connection, it is widely ob- to become a teniant if that status requiresserved that landlord and tenant split the capital and husbandry skills to which heproduce in proportions which are close to, can get access only with great difficulty,if not exactly, 50:50. Any analysis which if at all. In this connection, it should beseeks to explain the latter must take noted that in both India as a whole anldaccount of the former. All this suggests Bihar, householcl operational holdingsthat the assumptions on which buth which are partly owned and partly leasedtheories rest must be reconsidered, and gteatly outnumber those which are whollythat the analysis of the problem must be leased in, and that the fovmer accotunt alsotailored to the characteristics of the agrar- for the lion's share of all land lease(d in.1ian system in question. Thus, tenants are drawin mainly not from

In this paper, we discuss first some em- the mass of landless laborers, but from thepirical evidence concerninlg the existentce ranks of the small peasantry possessilgand nature of nontradable factors and landl of their ownl as well as skills anidmarket imperfections, wvhich play a key capital (or access to cal)ital), all of whichrole in the bargaining prncess. The evi- are traded (if, indleed, they are tradable n,tdence relates mainly to northeast Bihar, all) in inmperfect markets.but it should survive extension to much of lAoreover, the micro data analyzed bythe lower Indo-Gangetic plain. Secondly, Krishna Bharadwaj for several Indianwe employ John Nash's solution to the States and by Bell for northeast Biharbargaining p)roblem, and hence (lerive the reveal that both categories of tenantequilibrium rental share. By way of illus- househol(ds make very extensive use oftratioii, we calculate the predlicte d rental hired labor for cultural labor operationsand factor shares corresponding to some and harvesting. Thus the costs of super-sets of plausible production parameters. vising cultivation itself (though not theFinally, we analyze brieflv the situation in enforcemenit of a minimum input bundle,which the marginal productivity of land of course) fall on the tenant. Moreover, asis zero. the work is done by casual laborers, who

must be contacted in sufficient numbers atI. Assumptions and Evidence the right time, there are transactionis costs

The most natural question to begin with of labor hire which are passed on to theconcerns the specification of the labor tenant by the landlord leasing out hismarket. The assunmption of a perfectly land. Factors such as choice of crop), time-certain and elastic supply of labor and(I liness of operations, and good husbandry,work opportunities at an exogenously do- where the scop)e for them is large, indicaLtterminied and parametrically given wage that pUrely entrepreneurial skills are alsois not really acceptable at the micro- involvTd (see C. H. H. Rao). All theseeconiomic (village) level. But it will be point to a vital managerial clement in theretained on account of its simplicity, tenant's role -fer which he must bewhich eases the task of analyzing the sufficiently rewarded.effects of relaxing other less defensibleassurmotions. 1 See, for example, the Indian N'ational Salnple Stirvey

Fir, the relationiship) between agricul- (nos. 36, 113, 144) and S. K. Chakraverty, p. 99.

580 THIE AMERICAN ECONOMIC REVIEW SEPTEMBER 1976

The seconid major query surrounding the many kinds of labor and nonlabor inputsstylized competitive models is the absence present, no difficulties, provided all mar-of nonlabor inputQ other than land. IIere kets are )erfect. But once it is perceive(dthe principal dc ,; ilt) stems from the that tenants are (listingUisheUd from or(li-nature of the market for animal drauight. nary han(lIass labl)orers by their comnlVtfldpower, especially on the supply sidle. In over fatctors whose services are traded (ifIndia, there is no historical evidence of directly tradable at all) in markets whichhuman traction (or digging) for landl are inherenitly and(I highly imperfect, thenpreparation (except for small-scale vege- major revisions lbecome necessary, First,table cultivation), and there is a broad the ecuilibiium coii(litioni that un(ler I)er-social disdain for such method., (see fect certainty the tenant's income mustI'rufulla Sanghvi). Now, if the mfarkets for be e'ual to his alternative earnings as aboth this input and credit were perfect, wage laborer has to be dropped. Secondly,Cheunig's contract could be extended to it cannot be replaced by a generalizedstipuilate the minimum draught input per equivalent to include the alternative earn-acre as well as the -ental share and mini- inigs of factors other than "worker labor"mum labor intensity. However, the field owned by tenants because the marokets fordata suggest that: (a) hired draught is only s;uch facLors are not even remnotely p)erfect.a very small fraction of total draught in- Thlirdly, while it mav be safe to assumecputs, in marked contrast to labor; and (b) that the miarlket for unskilled labor is per-owning at least one pair of bullocks or fect, any folImulatliol of the I)roblem mustbuffalo is a sine (quta non for obtaining a respect the fact that the crux of the laindllease, and that this distinguishes tenants lease (lecision lies in the nontradlability ofowning rno land of their own from the managerial skill and dIraught power.2general class of landless laborers. These imperfections are intimately con-

These findings are backed up by the netetd witl the issue of whether or not thecomments of tenants during interxviews: landlord has the lability to enlforce con-although there existe(d a daily rate for the tracts specifying a set of minimum inputh.re of a bullock team season by season, it intensities on sharecropped land. In addi-was extremely difficult in practice to hire, tion to the a priori arguments advanced-No cultivator could afford to rely on the by B-S that such contracts are not en-hire market, nor, indirectly, could any forceable, there is empirical evidence tolandlord. Hence the likelihood that tenants that effect for northeast Bihar. The de-would own some land, for land and draught tailed testing of the competing hypotheseslicestock ownership are very highly corre- advanced by the two schools' has beenlated. Nor are such observations confined undclertaken elsewhere (Bell), with theto Bihar. In his study of allocative effi- -Marshallians coming off the better. Al-ciency in an Eastern Uttar Pradesh village, tlhough there is no space here to go intoDavid Hopper was unable to compare much detail, the fact that most tenantsthe calculated marginal prO(lLlct of bullock also owni land allows a search for the influ-labor with its market price because "therew;as not an active local market for bullockrentals to provide a comparison" (p. 623). 2 In this connection, it is worth noting Richard H.

Day's conclusion that mechanization contributed in aThe upshot of this examination of the major way to the decline of sharecropping in the U.S.assumptions underlying the competitive South by proviiiing an alternative to tenant-ow,iedmocdels is that the conclusions drawn from draught power.

I Four central hypothescs are set up and tested bythem may be highly misleading. In itself, Cheung (pp. 55-61), who uses evidence from prerevolu-the generalization of those analyses to tionary Chi-na, Taiwan, Knrea, and Japan.

VOL. 66 NO. 4 BELL AND ZUSMAN: CROPSHARING CONTRACTS 581

TABLE 1-MEAN INPTJTS AND OLTPUTS PER ACRE enforce contracts specifying thc mean vec-ON TENANT HOLDINGS tor of input intensities and yields for

CroppgInter- O Cowned land, the task was apparently be-mediate Hired yond them in )ractice.

Intensity Inputs" Laborm YieldaII. Bargaining Over the Rental Share

Sharecroppedplots 1.40 33.0 35.6 378.3 The foregoing discu0sson suggests a

Owned plots 1.76 64.3 46.7 561.9 number of fruitrul departures from com-petitive assumptions. We shall confine our

Soaurce: Bell. atte nltion to just two of them: the inclusionof nontradable factors and the rationing ofleases by landlords. These necessitate the

ence exerted by nontradables and other use of a quite different analytical approach,impcrfections on the allocation of re- namely, a bargaininig game formulation.sources For each such teiTant, his access to To keep matters simple, we assume thatsorcs Frecsuhtnt, hi acest tenants are landless, but end]o'ved withinputs (traded or nontraded) and hisaversion to risk are comiimoll elUlmlents in managerial and husbandry skills, and

draughlt poer TIhe introdUCtion of thehis decisions conicerniing the allocation of rtennt's own land w1ouhlthe compliresources between owned and sharecropl)Cd tiont's owfin lloc wtild p(lem the compliplots. These sources of variation across catioiis of analloation problein to thosefarmers can be elminiate(d in the following of iing a solution to the bargainingway: for each farmer, form a vector whose gtmelements are the differences between out- Suppose just one landlord and it (< x)

idn iltical tenants reside in each locality and

utarncropped land, respectivc1y. This pro- all land is homogeneous. The landlord, whoshrcrpe ln, epctvly hs r- does niot cultivate any land himnsel f. 4 pos-cedure amnumnts to a "pairing" of obsdarva-tions, a classic and powerful experimental sesss cx units of land, while each tenant

tios, clssi an poerfl epeiien the household is endlo%ved with one unit of la-method which has beeni necglected in the °thof1S'1(0Q whonuitfla

c n bor. The labor market is perfect so that thesharecropping literature. Table 1 summar- . uizes mean inpiut and output levels by the margintl utilty of leisure equals the wagetenure status of land cultivTated for a rate (w) and tenants maximize income nettenure status of land cultivainted st for of the imputed cost of labor. Contractssample of 31 farmers, cropping intensity specifying minimum intensities are un-being a proxyt for land services (aLnd St frtgmnmmitlste r nhence, indirextly, for draught services). enforceable so the bargaining process willhenc the, ind irectl,fordau service determin-iie the rental share (r) as the soleGiven the mul1.an tivariate nature of the provision of the contract. With the exis-situation, an appropriate way of testin- tenice of nontradaLhe factors, it is reason-the signiiicacew of the vector of (liffer- able toassumiethat each tenant's oUtp)Ut isences On owned and sharecropped land, a strictly concave, twice continutiouisly (lif-namely [0.36, 31.3, 11.1, 183.61, is to ferentiable function of his land holdlitg(h)

employ Jiotclling's T2 statistic. The com- antLislle labor ofputs lnl. Cp)uted value of T2 is 31.37, which is sig- and unskilled labor inputs (I)5nificant at the 0.1 percent level.

These findinigs suggest that the tenants I This assumption is mnade for simplicity's sake. Thewere highly succcssfull in "divertirig") value of the landlord's leisure is then independent of the

number and nature of the contracts he concludes, andboth nontradable and tradable resources therefore wvill not feature in thc analvsis.to the lan(d wvhich they owvned, contrary 6 For sinmplicity, fully traded intermediate inputs suchto the competitive thesory ofere(l by C X as seed, fertilizer, etc. are lumped together with un-

skilled labor. Table I indicates that this procedurc is alsoWhile landlordls may have had a mind to empirically defensible.

582 THE AMfERICAN ECONOMIC REVIEW SEPTEMBER 1976

Consider the landlord's dealings with a the event of a dlisagreement, the land-single tenant in the situation in which lord's ability to distribute any tenant'scontracts with all other tenants have been parcels among the other clainmalnts makesclosed at a rental rate 7'. If these two agents his position close to what it Would be if heagree on a contract specifying a rental were to engage in culti-aLtion hinmself. In-share r, their respective incomes (includ- deed, at the margin, he wouldI be indif-ing the value of the tenant's leisure) will ferent between such a redistribution and abe: little extra self-cultivation.

(( )F(Iz, ) - ul + We now employ John Nash's methlodi toY= (1-r)FF(h, 1)-1(1) 1] arrive at a solution to this bargaining

(n = -1)?F[(ax- q)/(n -1j 1), game.- Define+ rF(I, 1l) (4) N = ( - yl) (i' 2

- y2)

where 1, is the total labor input (i.e, Then any value of r which maximizes N isfamily plus hired hands) of the teniant in a solutioni. Howeve\X r, since labor u tilizationquestioni, aind h<q, (a-q) being the area by tenants is not part of the contract,of lanid leased out to the other tenants and, tenants' bDelhaxvior, as given by equationbecause they are identical, equally (li- (2), constittes an xtrnal constraint onvided among them. Now sulppos that the the bartgaining gainm. That is, the influencemarginal product ofm- land iS poSiti v that the rental sharv exerts oni labor utili-(Ft>O) so that th1erd is excess fellmadli for zation is recognized by both larties andtnancies (r< 1), and land is fully utilized1 affects the OLutcomn of the game. Nash's(q= I) and rationecl. Maximization by theat l vs>O forsingle tenant operating ht units of land, i= 1, 2. That is, neither the tenant nor theand by all other tenants, inmplies also that landlordi will e nter into nonrrelluncrative(2) (1 - r)F2(Iz, 11)= ( onit: ra(cts. 8;ub stituLting (1) and (3) into (4),

(1 -)F 2[(U - we get the following first-ordler condition(- h)/( = * for N to be a maximum:

respectively. If, however, the two agents 0tfail to agree, the tenant will fall back on (5) --- -F(h, 1t) {rF(i, 1) + r(n - 1)wage labor and the land1lor(d will (listribute arthe ht units of land thus freed among the [FI(h, 1) - F(a/(t - 1), I')]}other (nt-1) tenants, who change their dl7labor inputs accordingly. In this situation, + [F(1, l) + r12-_the teniant's and landlord's incomes be- dr-come, respectively: [(1- r)F(z, 11) - w'l] -O

(3) y= w wvhere, fromii (2)

(s - 1 l)fF(a/(n=- 1), '- d=1 - < 0

where the labor input on each of the (t- 1) dr u,h (1 - r)F 22tenancies is now given by (1-Tr)IF.(c/ I In Nash's solution of a two-player cooperative bar-(n-1), 1') =w. The disagreement payoffs gaining game, the agreement payoffs to the parties(yv and y2) thus represent the bargaining depend on their "disagreement pa)yoffs" and the feasiblep e oset of all payoflTs. The solution is consistent with thepower of the tenant and the landlord followving four axioms: (i) "groul) rationality"; (ii) vonrespectively. The solution of the bargain- Neumann invariance of the utility inhi(ators; (iii)ing problem must reflect the respctive "independence of irrelevant alternatives"; and (iv) srni-bargaining ositionslof th parties. in metry of players. Alternatively, Nash's olution may lebargaining positions of both parties. In viewed as the outcome of some plausible bargainingthis conniection, it should be note(d that in processes (sec R.D. Luce and H. Raiffa).

VOL. 66 NO. 4 BELL AND ZUSMAN: CROPSHARING CONTRACTS 583

Thus, any r° satisfying (5) is a function menit and disagreemient pa% offs alndIof r. That is, the rental share negotiated mallin:'ilng N, it may be showni that in ain a particular contract depends on the universal cooperative solution, n=a,share stipUlated in the other (nt-1) con- hIt= l= 1, andtracts. The cluilibriulll configuration of O-

contracts is such that if any contract (7) t (1w)/t+(u1)l(1.1/u) IIexpires, all other contracts being given, which, for it stUlWficintly lar1ge(h, I)eCMeIs

renegotiation will yield the very same con- (7') (1 w)/(2-tract. Recalling that by assumption alltenants are identical, then symmetry con- Now for the lease contracts to be re-sideraitions imply r0=i. Hence, by (2), iriunerative, . <1 (the level of outpLut on11 =and h=a/ in (implying, as one would each tenancy). Thus (1-w) is the "sur-expect, that all tenants would cultivate plus" to l)e (livided blletween the tenanit'sparcels of equal area). Thus, from (5), we nontradables and the lanlo1nrd's lan(l. Ifhave there are numerous tenants an(l constaint

F(') returns to scale in land and un-skille(l lal)or(6) r°- [|1-02] (I +1)-(n --1) - alone (v= 1), To-i- w, which'is the (C-N

) soluition. Thlie compeltitive share of nion-< tra(lables is (1-u), so that the (onliewtitiv-e

renita! share is (v-wi), hich is less thanr° from (7') for v <l, the liffterece reflet R t-

where f02 is the point elasticity of ouitput ing the landlord's bargaininlg power.with respect to labor; F(.)= F(at/n, I) is An imiportant imnplication of a fixed( co-the output on each farm in the event of efficient technology is that there is noagreement; F(')=F(a/(n-1), I') is the incentive elemllenlt in the conitract, as laboroutput on each of the other (n- 1) ten- inputs per acre are fiXred. The landlordlancies if there is a disagreement; and has no enforcemnent. l)rol)lon and r actsE2 =F2 1'1,F2 2 (the inverse of the elasticity of purely as a distributive instrUmient. UIndlerthe marginal productivity of labor). Thus these prodluction conditions, it followsequation (6) provides a genieral determina- immediately thattion of r° once a, it, w, and the technologyare specified.

In order to illuminate the influence Hence, from (3) an(d (5), 'Y'- vt 2- i 2\,

exerte(d by these factors on the rental share, i.e., the agents' gains over their respectivewe shall Inow (lerixv soluLtions for tech- disagreement l)payos are equal.inologies in which the elasticities of sub-stitution betTeen land and unskilled labor (13) Cobb-Doutglas ftechnolo.QA: Fare zero, unity, and infinity, respectively.' Ah3'l- A constant, 0 <s3,ji. 1. Here.

consid(ler the Ilinitinig case when c' n(n- 1)(A) Zero elasticitY of suibsliltulioni teclh- -the extra area received b) each of tlle

nology:F=min [Ii-,, It], 0<v5 1. As teniantts other (1t-i) tenants if the itth fails toare niaximizers, li=l. WVithout loss of reach a contraclt-is small.9 Then, by ex-generality, let a=n so that only fanmil)y pandingF(') as a'Paylor serics about F(.)labor is used. By writing down the agree- and using the condiitionl that the othier

8 For this production technology, the utility pnssibilit'7 It turns out that the derivation using a general C1ES frontier U(i , '2) for given a, n, and w is linear.

technology does not yield simple interpretable expres- F Ior example, in a village possessing 1000 acres andsions. 100 tenant households, a/(n-1)n-O.l acres.

584 THIE AMERICAN ECONOMIC REVIEW.6{ SEPTEMBER 1976

TABL1; 2-VALUES or r° FOR NuNn:Rors TENANTS changf,s iln (x whe 1t-i ( 1) is suFflcientlyAND COBB DOUCLAS TECIiN1LOGY small, anid total outp)ut will fall if there is

o 0.2 0.4 0.6 a failure to reach agreemen1V1t with onie or_______... mrn tenants. SeCCdlV(ll, there is a Inarke(

'02 tentlenecy for r° to take values nearer to0 0.50 0.56 0.62 0.71 the miagic 0.5 thain 31 for yiruially all. bu1 t0.2 0.40 0.46 0.53 0.64

0.4 0.30 0.36 0.45 0.60 extrrne (j1,/2) )airs.Tlhirdly, for plausible0.6 0.20 0.27 0.40 - values of iX anI( f32, r° Clusters clo .... y

aroulndI 0.5. Ilol)j)er, for example, etportsvalues of ,13 in the range 0.2 0.4 an(l of 12

in the range 0-0.2 (p). 615).tenants alter their labor inputs on re- A related issue is the structure of factorceiving an extra a, 'n(n- 1) units of lanid shares correspon(ling to the ruling valueso as to maintain F 2=w/(1-r 0 ), some of r°. The share of (field) labor, whethermanipulation gives supplied b) the teinanit's owni household or(8) r° = (l- 3)/[2 - 1J(1- )] by the landless is given immediately by

c(ILiautioii (2):which is indepen(leit of factor e(lownientbs s(u Lion2(2)and the wage rate. As I decreases with r for .t ( - r0 )1F2(.)jF()given w and It, the rental share must now (1 -

play an allocati-. - as well as a distributive frole. LI S fllowing the Marsllan rule for

It is instructive to comnpte r0 from(8) the use of the variable factor not onlyfor various pairs of /1 and 032. If there are depresses employment, but also re(lucesconstant returns to scale in h and I labor's share in output. The iresidlual out-(1+0 2=v= 1), (8) reduces to r° 0-, 1 which put, of course, aCcrues to the nontradableis the C-N solution. However, resource factor. Denotingr the factor shares of land,allocation is not Pareto optimarl. for fieli la hbor, an(l nonLradables by s,, s¢,, andF2 =a /(t-r 0 )>ni when contracts stip- s,,es rlS)P'CtiVCl), the most plausible valueswhenconracs sip- of 01i anid /2 yield the sets of factor sharesulating a minimum labor intensity cannot showni in Table ld. Takinsg the competitivebe enforced and the elasticity of substitu- structures of factor shares as a benchmark,tion between It and I is inonlzero. Thus, that: (a)thard ex ih ewhile land receives its comrpetitive cost itisseenthat (a) the landlord exploits theshare, labor inputs per unit of land are tenant; (b) both the tenant (as mage)lower than they would be under fixed reits. and the lan i(lor(d exiloit labor, tlhougli only

dproduction is not efficient, in the fifth case is the landlord's exploita-andpodci is not Table tion of the tenant more than offset by theMfore genierally, from Table 2, we seefirst that r° cxceed(s the output elasticity tt's exploittion of laborof land by a good miiargini if F is higlyi) (C) Land and labor tCrf cC sUblS61i(uLs:concave. This is noteworthy b)ecauIse al- F=v(It+l)",wherev<l. T1ere, F>0 every-though the lani(llor(I is a potential mo- where, so there is excess demand for land ifnopolist, all landI is uni(ler cultivation. For r<1. Also, fromii (2), v(l-r)(Jz+li>t'=w.the assumptions on the technology ensure Thus, if land is redistributed in the eventexcess demand for land by tenants (01 and of a disagreement, the labor input em-,B2 are constanit, and F1>0 everywhere). l)lo)ycdl by each of the othler (it-1) tenantsAlso, when F is strictly concave, amaxi- will fall so as to keepl. (I+i>)- (constant.mizing lan(dlor(d will not keep land out of Jlenlce prodluction on the otlher tenanciescultivation, sinice r° is insensitive to will not change following a redistributionl

VOL. 66 NO. 4 BELL AND ZUSMAN: CROPSHARING CONTRACTS 585

TABLE 3-FACTOR SIARES

31 0.4 0.2 0.4 0.2 0.4 0.2/2 0 0 0.2 0.2 0.4 0.4

St 0.62 0.56 0.53 0.45 0.45 0.36SW 0 0 0.10 0.11 0.22 0.26St 0.38 0.44 0.37 0.44 0.33 0.38

of land, and the landlord's bargaining it follows from the terimts' miaxiniizing,position is the weaker thereby. In this behavior U,(1-r)(1+.Th1=w] that thiscase, the 'Nash function takes the simple is the limiting value of r° as w-)>. 'More-form over, because r°-(1 -v) X2 is the maxi-(4') N = rF(.)[(1 - r)F() -wl1] mum of allro on [26 [(, v], and as the C()bb-

Douglas value in (8) excee(Is this value,from which we obtain, at length, the rental share in tec nologrv C is less

1- 2(1 -Vthain that in B for all w>0. A further de-(1 - nt

(9) r° = ((uctioin from tenant' (lecisions regarding(2- - [) labor utiliztionl is thlat b)eCause r°0

where O <[2<v and is also a function of a. (1-v), (2-v) for 0 2 (W)=0 (WhiCh imp1j)lies

To round off the anial)ysis, it is instruc- 1=0), theni no labor will be used if -,v>v 1tive to compare the rental shares ruling in -(1-v)j(2-v) ] =.v (2-v) < 1. In thisthe above cases as w varies, but with given range of -, r°=(1-P)/( 2 -v). Also, wefactor endowments (a, n) and returns to note from (9) that if v= 1, then r° = 0 forscale (v). In the zero elasticity of substi- all w>0, a result also obtained by New-tution case (7'), it is immninediate that r° is her) (1974, p. 1063). 'J'he intlitiVe inter-linearly decreasing in w, with rl. 1(2-v) pretation of the behavior of the rental(<• for v<1) at w=0, and r°=0 at shares in different technologies is that for

1 ( < 1 if lease contracts are remuner- a given wage rate, the higher the elasticityalive to tnilants). Now in a Cobaeb-I)ougmlrs of substitution, the greater the incentivestechnology, r° is constant and less than (lower r) to use labor the landlord will pro-unity for v < 1. It is easily shown by com- vide to the tenant. An increase in the wageparing (7') and (8) that there exists some rate improves the bargaining strength of0 <¶L* < 1 such that the renttl share ruling the tenant and thus tends to lower thein a Cobb-Douglas teclnology is less than rental slhare. However, the associated fallthat in a Leon tief technology if w <w*, and (if any) in the latter dependcls in a compJi-onversely if w>w*. cated way on the elasticity of substitutioll.The perfect substitutes case is more and the wage rate.

cornllicated. Here, note first that by (2),132(w)=vl. (1+1), where 10 = 1 (a=n). By III. Some Land Uncultivated:definition, 0 <,,(W) <v. Also : 2(w)-)0 as CausesandConsequences1-*0 and [32 (w)-7v, its upper bound, as I To complete the anal)sis, we now con-becomes large. As [32 (w)-0, r°~(l-v)/ sider the cases which may lead to some(2-P). and as [32 (w)-+v, r°(1-v)/2. By land renmaining unutilizecd. At first glance,(lifferenitiatinig r° in (9) wvith respect to [2, this may come about in two ways: eitherwe see that dr0/d132 <0 for v < 1 so that r0 all tenants are able to choose a parcel sizedecreases monotonically as [2 increases such that Fj=0 (assumiing an appropriatefrom 0 to v, Now as r0-- (1-i) 2 for large 1, productioni functioni); or the parcel size is

586 THE AMERICAN ECONOMIC REVIEW SEPTEAMBER 1976

also (etermiie(d in the bargainiing process. This despite the fact that the marginalLet us deal with each of the cases in turn, productivity of land is zero.

Treading a now familiar path, we have It is important to relate the analysis ofin the former case, if a contract is reached, this case to the discussion of the com-con(litionis described by equation (1) above. petitive formulations in the literature. Re-If thCre is a failure to reach agreement, call that, as Newbery has shown, F = 0 isthen w,ve hlave the only cond(ition under which a B-S(10) yl =-wZ,, y2 = (it-1)?F(t, I) equilibrium will exist. Given fixed factors

in limited supply (in this case, nontrada-In this case. bles) together with an abundance of land,

(11) N= [(1 - r)F(. )- ] [rF(.)] it is highly plauLsible that the marginalproduct of land will fall to zero if the parcelwhere F(.)=F(Iz, 11). Note that N is now size is ma(le sufficientl) large. This mayind(1epcndlenit of the contracts reached with well be the case in some parts of Latinthe other tenants. Recall that labor inputs America. What is striking is that ouralpl))ied. by the tenant depend on the rental analysis predicts a positive rental share inshare e-u(Lationi (2)-and that tenant this situation. That the rental share ismaximization implies Fi(I, l) = 0. Dif- positive stems directly from the landlord'sfurenitiating N partially with respect to r, power to prevent a would-be tenant fromwe get cultivating land if the landlord so chooses.

O.F The second case is one in which both the(12) rF- rl"i.)2 + + d.2'2 -] size of the parcel ancl the rental share are

ar Lri determined by bargraining. To solve this[(1 - r)F(.) - 1] new bargaining problem we ma:imize N in

- t (11) with respect to r and It, subject to theBy differentiatilng totally the tFnan t2s land availability constraint h+(n.- 1)1tca,maximizing coIndtions, viz. ld = andl (2),i aiid taking inito account the fact thatand hence solving for d11/dr, WC t,>t~iii tenants are maximizing on their own ac-

dl, 1 F2F11 count: (1-r)F 2=wZ and (1-r)F,>O. The- = conditions for a Nash solutioni (nmaximumdr (1l- r) A ) are obtained by setting0

where A=FllPF2 2- F2> (0 by the strictconcavity of F. Hence, for N to be a (14) - =0nmaxinumtni sinmultancously for all con- artracts, we have, from (12), ON

(l- 2 fl2) coniplemnentarily2- 0341 + (1 - 0)F'2 F1 1 /AI ' + (cx) 1) [ a

Now, for 5)0Wz r0 >t), but rlr)itralrily small,[(1-,ro)Ir( -)n-<1X >0t if lbndr is at all Suppose the land availability constraintcultivatd(. Also from (12), does not bind and (1-r)Fi>O. We have,

from (15),

= F(.)[(1 - r)F(-) - wli] > 0 (16)d (1 -l' F(h l) + F +dli

for the samne reason. Then, sinice 0<r0 <1I -rvZ(11, 11)-w1)J-Othe maximumn value of N, and thus thesolution to the bargaining game, exists and Tenanits' enil)loyn-jent decisions implyis associated with a positive rental share. dli/dli, F-IiM/IF 2 2 > 0 for well1-behaved pro-

VOL. 66 NO. 4 BELL AND ZUSMAN: CROPSHARING CONTRACTS 587

luction functions. Also, (1-r)F(li, l) vated to the status of a Social norm.-w1 1 >0 if production is at all und1ertaken. It go-s without saying that SomeW Of t)heB3ut then condition (16) is not satisfied, (irastic simln)llcations in our formuLht ltiollfor all term-s on the left-liand(I side are may vitiate the anialysis. In particulalr,positive. hIence, F1= 0 if the landI avail- when a woul(l-be tenant Can (leal withability constraint is not binding. By sym- inore tlhani one landlomd his bargainingmetry, the landlord muist reach the same position is stronger than in our fOrn1li1-1conatract with all tenanits, so that t1= M-= L tion. B3ut given the spatial dlinie nSiii-, ofand r=r0 =, with Fx [Izu(rO), /0(r0 ) =0. Wre agricultural prrodtUctioll and the resulling(are then back to the first case. \lterna- costs of cultivating widely scattered par-tively, sup)po;sC that the land availability cels, our neglect of the alternative oppor-constraint binds; but then we are in our tunities stenmingig from thl existnceC oforiginal formulation of the problem, in other landlords may niot prove to be tooSection II. serious. To introduice such consi(lerirtions

WVe have thus established that if any into the anlalysis would (n1tail vI'e- C011-lanid lies unacultivatedl, then the marginal siderable complications, wxincli we haveproductivity of lan(d is zero and the rental chosen not to l)urI;ue at this stage.share is positive. This result contrasts REFERE.NCESboth with the competitive solution, inwhich the rental shlare is zero if F1=0, P. K. Bardhan and T. N. Srinivasan, "Cro-and with the monp[oly srutionl. in which sharing Tenancy in Agriculture: A The-arental share a 1 t 1 art po sitv oretical and Empirical Analysis," 4mcr.both the rental share and F, are positive Econz. Rev., Mar. 1971, 61, 48-64.in the face of land lying ur.cultivated. and - -, "Cropsharing Tenancy in

IV. Concluding Remarks Agriculture: Rejoinder," Anmer. Econ. Rev.,Dec. 1974, 64, 1067-69.In this paper, we have used bargaininig C. Bell, "Some Tests of Alternative Theories

theory to derive determinate solutions for of Sharecropping Using Evidence fromthe rental share. We have adopted this Northeast India," J. Develop. Stud., forth-ai)proach becausc both the nature of the coming.contract and the features of the agrarian K. Bharadwaj, Production Conditions ins;stem necessitate it, while comI)etitive Indian Agricultuirc, London 1974.formulations must ignole such considiera- S. K. Chakraverty, Problems of Small Farmerstions. In o)ractici , of course, the bargains of Kosi Area, Patna 1969.which are struckc needo not conform exactly S. N. S. Cheung, Tuzc Thleory of Share Ten-to tahoe Pre ditru by n ash's solution. IBut ancy, Chicago 1969.o t . R. H. Day, "The Economics of Technologicalif Nash's solution is empirically valid, and Change and the Demise of the Sharecrop-given that the bargaining process is re- per," Amer. Econ. Rcv., June 1967,57, 427-peated year after year in circumnstances 49,of unchanging technology and social W. D. Hopper, "Allocation Efficiency in aMnstitutions, then ther., will be a long0-terni Traditional Indian Agriculture," J. Farmtendency for contracts to ceniter oni SOiim Econ., Aug. 1965, 43, 611-24.average value. -Moreover, if in most sitIu;i- R. D. Luce and H. Raiffa, Games and Deci-tions the solution is not very diffe renL sions, New York 1957.from onc-hialf, agents will save themselves A. E. Marshall, Principles of Economics, Lon-the b)othler of detailed arithmetic an don 1920,settle for that magic number, whtih nas J. Nash, "Two Person Cooperative Games,"ttlefor thtaI t magic n)umber,g whuich as Econonzctrica, Jan. 1953, 21, 128-40.

oa t D. M. G. Newbery, "The Choice of RentalThus that paLrtiCUlar rental share is dcl- Contract in Peasant Agriculture," in L. Rey-

388 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1976

nolds, ed., Agric2ulture in Devclopm cnt The- ship, and Sharecropping in India," J. Polit.ory, forthcoming. Econ., May/June 1971, 79, 578-95.

, "Cropsharing Tenancy in an Equi- P. Sanghvi, Surplus Manpower in Agriculturclibrium Model," paper presented to Eu- and EC;;wcmic Developmenzt, Bombay 1970.ropean meeting of the Econometric Society, J. E. Stiglitz, "Incentives and Risk Sharing inOslo 1973. Sharecropping," Rev. Econz. Stud., Apr.

- "Cropsharing Tenancy in Agricul- 1974, 41, 219-55.ture: Comment," Amer. Econ. Rev., Dec. Government of India, The Nationzal Sample1974, 64, 1060-66. Survey, Nos. 36, 113, 144, Calcutta, various

C. H. H. Rao, "Uncertainty, Entrepreneur- years.

The Philippines: Priorities and Prospects for Development by Russell Cheetham,Edward Hawkins, and others, distributed by The Johns Hopkins UniversityPress, 1976

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