World Bank Reprint Series: Number 216

Avishay Braverman and T. N. Srinivasan

Credit and Sharecroppingin Agrarian Societies

Reprinted with permission from Journal of Development Economics, vol. 9 (December1701), pp. LO:-012i.

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Journal of Development Economics 9 (1981) 289-312. North-Holland Publishing Company

CREDIT AND SHARECROPPING IN AGRARIAN SOCIETIES*

Avishay BRAVERMANWorld Bannk, Washliinigton, DC 20433, USA

T.N. SRINIVASANYale UnirersitY, New Hnaven, C'T 06520, USA

Received July 1980, final version received March 1981

A model of linkage between land, labor and credit transactions in the context of sharecroppingis considered. It is shown that regardless of the presence or absence of linkage or any othercontrol by the landlord, as long as he can vary the size of the plot given to a tenant and thereare enough potential tenants, in equilibrium contracts a teniant's utility under sharecropping willbe the same as that which he could have obtained as a full-time wage laborer. This result hasthe fundamental implication that policies other than land reform will not affect the welfare oftenants. Government subsidization of tenant's credit results only in the subsidization oflandlords. Other partial reforms by the government, however, may force the landlord to linkcredit and tenancy contracts even if the government provides the cheaper source of credits. This,while leaving the tenant's utility unaltcrcd at its pre-reform level, will affect total oultput and theextent of tenancy.

1. Introduction

A commonly observed feature of agrarian societies in their early stage ofdevelopment is sharecropping together with a credit arrangement in whichlandlords provide credit (for consumption, working capital as well as investment)to their sharecroppers.' The extensive literature on sharecropping has notsatisfactorily addressed the nature of equilibrium in land, labour and creditmarkets in such a context.2 Further, the fact of credit linkage between a landlordand his sharecropper was viewed as a form of exploitation of tenants bylandlords. 3

*The views and conclusions presented in this paper are the sole responsibility of its authorsand should not be attributed to the World Bank. We thank Clive Bell, Wilfred Candler,Gershon Feder, Luis Guasch, Pradeep Mitra and an anonymous referee for helpful commentson an earlier draft and Vivianne Lake for editorial assistance. Also comments made at seminarsat the World Bank, Harvard, Michigan and at Stanford were helpful.

'See Bardhan (1980), Bardhan and Rudra (1978, 1980a, b and 1981), Bharadwaj (1974), onIndia and, for instance, Ransom and Sutch (1978) on the Post-Bellum Southern United States.

2Bell and Zusman (1976), Cheung (1969), Marshall (1959), Newbery (1977), Reid (1973),Newbery and Stiglitz (1979) and Stiglitz (1974).

3 Bhaduri (1973, 1977). We will be using the terms 'tenant' and 'sharecropper' interchangeably,though strictly speaking, the tenant is one who leases in land at a fixed rent (cash or kind) perseason.

0304-3878/81/0000-0000/$02.75 (Uj 1981 North-Holland

290 A. Braverman and TN. Srinivasain, C'redit and sharecropping

The purpose of this paper is two-fold

(a) to derive and characterize the equilibrium in a model of a land-scarce,labour-abundant economy under sharecropping, given an infinitely elasticsupply of identical sharecroppe-.s at a reservation utility. The reservationutility may be determined either by subsistence considerations or byemployment opportunities available to a poteniial sharecropper elsewhere inthe economy.

(b) to demonstrate that in an imperfect credit market, a landlord may offer creditto his tenant, sometimes even at a subsidized rate of interest, withoutnecessarily insisting that the sharecropper borrow only from him thusprecluding an involuntary (from the point of view of the tenant) linkagebetween credit and land transactions.4 However, any legally or sociallyimposed constraints on tenant's share (as for instance, a floor) may provideincentives for a credit-tenancy linkage that may otherwise be absent.

In the following sections we concentrate on a model of linkage between land,labor and credit transactions in the context of sharecropping.5 In order to explorethe implications of policies such as land reform, subsidized credit, taxation andthe outlawing of moneylending by landlords, we take it as given that the onlyform of tenancy is sharecropping.6 Other crucial assumptions are that a potentialtenant is precluded, as part of the tenancy contract, from working outside thefarm as a part-time wage laborer and that there are imperfections in the capitalmarket in the form of differing costs of capital to the landlord and to the tenant.

One major conclusion of the paper is valid both in context of credit-cum-tenancy contracts and in that of sharecropping contracts alone. It states that, aslong as the landlord can vary the size of the plot given to a tenant and there areenough potential tenants, tl4e equilibrium will be characterized by 'utility-equivalent' contracts even if the landlords do not possess any other instrument(e.g., share rent, interest rate). That is, in equilibrium, a tenant's utility obtainedthrough sharecropping will be the same as that which he could have obtained as afull-time wage laborer. Newbery and Stiglitz (1979) assert, without providing a

4 Bardhan and Rudra (1978).5 We do not discuss in this paper other rationales for interlinking such as uncertainty and

asymmetrical distribution of information between landlords and tenants. On these matters, seeBell and Zusman (1980), Braverman and Stiglitz (1982), Braverman and Guasch (1980) andMitra (1980).

6 One economic reason for the emergence of sharecropping contracts is the following: If onlyincentive problems exist (i.e., the landlord can neither force the worker to contribute a specifiedlevel of effort nor can he monitor it), the fixed-rent contract will be best suited to remedy them.It will, in fact, dominate a fixed-wage or a sharecropping contract. The tenant obtains all thefruits of his effort after paying the fixed rent. Fixed rents, however, imply that the tenant mustbear all risk resulting from output uncertainty due to exogenous conditions (e.g., weather,illness). If the tenant is risk averse, such a contract will be inefficient, in which case asharecropping contract will dominate it.

A. Bravermnan tici N.N Sinitvasan, C(redil 2nd .hlILIrL' in r,j' I{ 291

satisfactory proof, the same result in the context of sharecropp-ing alone, while asimilar, though not identical, conclusion has been obtained in a different settingby Cheung (1969). Our proof follows from our result that, ceteris paribuis, thetenant's optimal effort per hectare is a decreasing function of the size of the plot hecultivates. Our model excludes the possibilities of rationing equilibria in which atenant obtains a utility level exceeding his reservation utility.7

The utility equivalence result has the fundamental implication that policiesother than land reform (i.e., reform that confers ownership to the tenant of thepiece of land he is cultivating) will leave the welfare of each potential tenantunaltered while affecting the level of output, extent of tenancy and the welfare oflandlords.

With the possibility of landlords providing their tenants with credit, it is shownthat landlords will resort to that option only if their opportunity cost of capital islower than the tenants' opportunity cost of capital. If the government offers thetenant subsidized credit at a cost lower than the landlord's opportunity cost offunds, the landlord will move out of the tenant's credit market and allow thetenant to borrow from the government. The increase in surplus due togovernment subsidization of tenant's credit will filly accrue to the landlord as aconsequence of the utility equivalence result. Hence, government subsidization oftenant's credit r esults only in the subsidization of landlor-ds. Other partial reformsby the government, however, may force the landlord to tie credit and tenancycontracts (even if the governrnmeit provides the clheaper source of credit) thereby,leaving the tenant's utility unaltered at its pre-reform level while affecting totaloutput and the extent of tenancy. Our model thus provides one theoreticalexplanation for two almost opposite phenomena that are sometimes observed:low interest consumption loans from landlord to tenant and the opposite, highinterest, low volume loans.

We present the model in section 2, followed by a characterization of theequilibrium in section 3. Section 4 discusses policies of credit, land and tenancyreforms as well as the impact of taxation and technical progress.

2. The model

The tenant's choices are limited to the decision to be a sharecropper or not andthe level of his work effort, if he decides to be a sharecropper. The landlord has atleast one choice variable (plot size) and at most four choice variables: plot size,share rent, interest rate and the amount of tied credit with land contract. Tlleprincipal constraints are (i) an exogenously available level of utility for tenants atwhich the supply of tenants is perfectly elastic, and (2) tenants and landlords arenot free to mix contracts. Given the tenant's choice behavior, the landlord is a

7 0n efficiency wage and rationing equilibria see Leibenstein (1957), Mirrlees (1976) andStiglitz (1976).

292 A. Dravermlani atnd TN. Srinivasan, Credit atnd slharecropping

Von-Stackelberg maximizer of profits. Formally, we shall first describe thetenant's and landlord's problems and then the equilibrium.

2.1. The tenant

All workers are identical facing two employment alternatives: first as tenantson landlord's land, or secondly, as wage labourers elsewhere. They cannot mixcontracts.8 Each tenant is offered a plot of land, of size H hectares, in return forwhich he agrees to pay the landlord a share (1 -a) of the harv^, st. None of theworkers possess any savings at the beginning of the production period. Wageworkers are paid during the production period and, therefore, have no need toborrow for consumption. The tenant, however, borrows at the beginning of eachseason his entire consumption needs for the coming season and repays his loanwith interest at the end of the season after harvest. He does not store any grainfrom one season to the next, nor does hc have any investment opportunities.9

The tenant obtains a proportion v of his borrowings (either voluntarily or as apart of a 'tie-in' package with a tenancy contract) from his landlord at an interestrate r7. per season. He obtains the remaining proportion (1 -v) of his borrowingsfrom an alternative source (e.g., local moneylender, cooperative, governmentcredit agency) at an interest rate rA. He treats rT and )'A as parameters over whichhe has no influence. We assume that he cannot default partly to simplify theargumentation, and partly because in many areas landlords virtually hold theharvested crop as collateral, thus precluding default. Clearly, if the tenant canborrow the entire present value of his consumption at either 1-T or rA, he willchoose to borrow it from the cheaper source. However, since our discussionfocusses on tie-in contracts, we start by assuming that the tenant takes v as given,so that v >O will represent a tie-in condition over which he has no influence.

Labor provided by the tenant for cultivation (including all operations fromland preparation to harvesting) is denoted by eL, where L denotes the number ofman-years per season and e denotes the effort per man-year of labor. Thus, eLrepresents labor in efficiency units. Output Q is a concave function, homogenousof degree one in H and eL.' 0 Thus

Q = F(H, eL). (1)

Assuming the number of man-year, L (i.e., labor in natural units) to be

8See discussion of this assumption in section 3 below.9 1nvestment in a distant bank is unattractive for a poor and often, illiterate tenant.'"Bell and Braverman (1980) show that, if the production function is of constant returns to

scale and there is no uncertainty, landlords will prefer cultivation with wage labor tosharecropping. However, this result does not apply to the present analysis because we do notgive the landlord the option of self-cultivation with wage labour and because of other reasonsconcerning the modelling of tenants' effort and behaviour.

A. Bravernman and TN. Srinivasaan, Credit anid shtarecroppin.g 293

exogenously fixed, we can set (without loss of generality) L= 1. Thus we canrewrite (1) as

1f(ex)Q= F(1;ex)- , (2)

where x is man-years c- labor per hectare of land. Given that the tenant isendowed with one marl-year of labor, x represents the reciprocal of the size ofthe plot he is allotted. The function f represents the average product per hectareof land. By assumption, f' is positive and f " is negative where the primes (singleand double) denote the first and second derivatives off, respectively. The tenant'sshare of the harvest Q is a and his income is therefore aQ.

By our assumption that the tenant borrows his entire consumption needs at thebeginning of the season and has no carry-over stock or investment opportunities,it follows that his consumption c in any season equals his income csQ at the end ofihe season, discounted by (I+ i) where i is the effective interest rate on hisborrowing. Of course, i equals vrI+(l -v)rA. Thus

ctQ

+ vrT + (1 - (3)

where

l=1 += discounted share of the tenant. (4)1 +VIT +(l - V)rA

We assume that the tenant's utility function U(c, e) is strictly quasi-concave inconsumption and leisure, where leisure is defined as e - e. Furthermore, weassume that both consumption and leisure are normal goods.

The tenant's choice or control variable is e. He will not choose to work as atenant unless U(c, e) is at least as large as U, the utility he could have assuredhimself by working as a wage laborer U is exogenously given implying that thesupply of tenants is infinitely elastic at U. Thus we can solve his choice problem intwo steps. First, let the maximized value of U(c, e) subject to (3) be U*. If U* > U,he would work as a tenant, otherwise, as a wage laborer. Thus, the tenant'smaximization problem is

max U(c(e), e) subject to (2) and (3). (5)e

It is immediately apparent from (2)H(5) that the parameters a, v, rT and rA enterthe tenant's constraint set and utility function only through their effect on hisdiscounted share f,. By substituting (2), (3) and (4) in (5), maximizing with respect

294 A. Bravermnan andel TN. Srinivasan, Credit and sharecropping

to e, we get the first-order coniditioin

/3U,f'(ex)+ U2 =0. (6)

It can be shown that the second-order condition is satisfied from our strict quasi-concavity assumption on U, and the strict concavity of f (see appendix). We notealso that (6) can be solved uniquely" 1 for e to yield

e = e(x, f), (7)

Define effort per acre as z -ex. It follows (see appendix) that

az 1 flf(U2 1 U1 -U 2 U,1 ) + z(U2 2 U1 - U2 U1 2) (8)dx X 2 U1 pf3fU, +(1/X2 UU2)(Ul UI -2U 1 2 U, U2 +U 2 2 UI)

Lemma. If U is strictly quasi-con7cave in (c,-e), J is strictly concave in e, and cand - e are normal goods, then az/ax> O.

Proof. Strict quasi-concavity of U and strict concavity of f imply that thedenominator of (8) is negative. The normality conditions for c and -e are

U1 2 U 1 -U 2 U,1 <0, (9)

and

U 2 2 Ul- U2 U1 2 <0' ( 0)

They imply that the numerator is negative as well. Hence, az/ax>0. Q.E.D.

This lemma states that the tenant's effort per acre increases with a reduction in hisplot size even if the tenant's effort declines with such a reduction in plot size(increase in x).

Now,

aZ f f'Ul + (flX)((U 1 U21 -U2 U 1 l)/Ul)J1

aW~~ ~ ,Uf+(I l2 (2l-2U12U1 U2+ U22UD1

This expression cannot be signed. Hence, effort per acre may oither increase ordecrease with a ceteris paribus increase in tenant's share. Denoting by U* themaximized value of U(c, e), it can be shown [noting (6)] that

aU*/ax = -_Ul(f(z)-zf')/x 2 <0, (12)

aU*/Ofl= U1(f/x)>O, (13)

"Assume limc-, U, = J and Iime-o U 2 =O.

A. Braverman and TN. Srinivasan, Credit and sharerrop ping 295

i.e., ceteris paribus, an increase in the plot size and/or the discounted share makethe tenant better-off.

2.2. The landlord

With an infinitely elastic supply of identical tenants, and constant returns toscale in production, maximizing profits is equivalent to maximizing profits perhectare. Hence, our model yields the same results whether different landlordspossess different amounts of land or not. Therefore, without loss of generality, weassume that all landlords are identical and possess one hectare of land each,which they divide into plots of size 1/x to give each of x tenants. As stated earlier,the landlord may require that each of his tenants get a proportion v of hisborrowings from him at an interest rate r7, Assuming that an alternative use offunds would have earned the landlord an interest of )'L per season (e.g., depositsin the city's bank), his income g from each tenant is given by

g-(1 a)f(ex) +v(rT-rL)C

= x fex+v(rT-rL)cB()uig 2 n 3

x

=( f(ex) +[v(rT-rL )fl+ ) using (2)and (3)

- fe)[(1 -ca) +v(rT -rI,)/3x

- fe)[1-fi{l+vrL ±(l *-v)A} using (2), (3) and (4).

Multiplying g by the number, x, of tenants we get the landlord's income G:

G =[1-/3{1 +vrL +( - )) A3f(ex). (14)

It is clear from (14) that the interest rate r Tcharged by the landlord on his loansto his tenant affects his income only through its effect on fl, the discounted share.

The landlord maximizes G with respect to his choice variables given thetenant's effort function e(x, fl). The choice variables include the plot size l/x, andmay include the tenant's crop share a, v (if there are no laws against the landlordproviding credit) and rT, the rate of interest charged.

3. Utility equivalence and other equilibrium properties

The equilibrium presented here is a contractual equilibrium, i.e., there isdemand and supply for contracts, where a contract consists of a package

296 A. Braverman and TN. Srinitlasan, Credit anttd sharecroppinig

including plot size, crop share, interest rate and tie-in condition. It is lnot acompetitive equilibrium since the level of tenant's reservation utility isexogenously given (e.g., by subsistence factors) and, hence, the landlord is facing aprofit maximization problem subject to an inequality constraint on tenant'sreservation utility. A competitive contractual equilibrium, on the other hand, ischaracterized by landlord's profit maximization subject to equality constraint ontenant's utility, where this utility level is generated by the competitive marketforces.1 2

For the moment, let us focus only on the choice of x (the number of tenants or,equivalently, the plot size per tenant), thus keeping ,B fixed in particular. Since f isan increasing fiunction of its argument ex =z, and since z is an increasing functionof x [see (8)], the landlord's income (14) increases with x: in other words, adecrease in the tenant's plot size, which therefore leads to the hiring of moretenants, increases the landlord's profits. On the other hand, it follows from (12)that a tenant's utility U* in sharecropping decreases as x increases. Thus, if at anyvalue of x the tenant's utility exceeds his utility U in the alternative use of his labor(so that he chooses to be a tenant), the landlord, by increasing x, can increase hisincome while pushing the tenant towards U. As long as there are enoughpotential tenants, that is, as long as there is no upper limit on x, the landlord'schoice x will be to push the tenant to a utility level equalling U."3 Hence we canstate the following basic proposition:

Propositioni 1. The equtilibritum in the land-labor niiaciket wvill he character ized byutility equicll'Zt' contracts.

It should be noted that this proposition does not depend for its validity on thepresence or absence of any linkage between tenancy and credit transactions. Thelandlord's use of plot size as his sole instrument variable is sufficient to result in autility equivalent contract equilibrium, an outcome obtained by Cheung (1969)under a different structure. Our structure is that initiated by Stiglitz (1974) andutilized by Newbery and Stiglitz (1979). Assuming a separable utility function,they claimed [Newbery-Stiglitz (1979, p. 16)], that competition betweenlandlords will eliminate the less attractive contracts and will drive the inequalityU*> U to equality thereby achieving utility equivalence. As dernonstrate- inProposition 1, the utility equivalence outcome results from profit maximizationand not from competition. Nor is the proposition trivial, arising solely from thefact that there is an infinitely elastic supply of potential tenants at U, since the

' 2See Braverman and Stiglitz (1982) for discussion of competitive vs. non-competitivecontractual equilibria.

'31t can also be argued that if at an initial x, U* is less than U, the potential tenant will notchoose sharecropping. As such, in order to obtain someone to cultivate his land, the landlordwill have to increase the plot size, i.e., reduce x. We are ignoring the fact that a tenant is'indivisible' while land is divisible.

A. Bravertmlan anid TN. Srinivasan, Credit anid slhar-ecrroppingl 297

possibility of an excess applicants equilibrium at U*'> U can occur if only tileoutput share instead of the plot size is the control variable of the landlord. A well-known case of excess applicants equilibrium arose under the efficiency wagehypothesis [e.g., see Leibenstein (1957), Mirrlees (1976) and Stiglitz (1976)],primarily because the landlord is not allowed to use an instrument completelyorthogonal to effort to reduce U* to U without affecting effort. In our model, theuse of the power to vary the plot size, although non-orthogonal to effort,guarantees the utility equivalent contract result since tlle tenant's effort per acreincreases with a eeduction in his plot size.' 4 Additional instruments such ascropshare and interest rate are not needed for this purpose.

Of the two assumptions used in deriving our result, namely, that bothconsumption and leisure are normal goods, and that the tenant is prohibited, aspart of his contract, from working as a part-time laborer outside the farm, thelatter is perhaps more controversial. Its realism is primarily an empirical issue. Itis true that tenants often work as part-time laborers, but the extent of such work islimited. There is also some evidence to suggest that landlords believe that a tenantwill put greater effort into cultivation, the smaller his plot size.

From the utility equivalence

U{c(x, /X), e(x, /3)} = U, (15)

where

c(x, /) =: f f e(x, 13)x}x

we can solve for x (the inverse of the plot size) as a function x(,B) of the discountedshare, /3. By appropriate differentiation of (15) (see appendix) we obtain

dx Jx

d: 9(f-f'x) >0, (16)

i.e., in order to maintain the tenant on his iso-utility curve, the landlord mustincrease the tenant's discounted share if he reduces the plot size. Thus, from nowon when analyzing changes in /3, unless otherwise specified, we assume that thelandlord changes x along the curve x(pl) so as to maintain the tenant at a welfarelevel of U.

Now, denote c =-f'(f-zf')/ff"z as the elasticity of substitution betweeneffective labor, e, and land.

'4 The fact that the size of the plot cultivated by the tenant does not change over time doesnot contradict its use as a policy instrument by the landlord, It only means that in a stagnantsituation, once an optimal size has been determined, there is no necd to change it.

298 A. Braverman and TN. Srinivasani, Credit atnd sharecropping

It is shown in the appendix that

de{x(f3), /} ae dx +e

d, Ax d,dB +T#

/3f"U 1 +(I/xU2){Ul U2-2Ul2 U 2 U 1 +U 2 2 U1} a 1

Hence

Proposition 2. The tenant's effort e increases, stays the same, or decreases as hisdiscounted share ,B in output increases, according as the elasticity of substitution a' isgreater than, equal to or less than unity.

It is shown further in the appendix, that effort per hectare, z, satisfies

dz/d/3>0. (18)

Newbery and Stiglitz (1979) derived (17) assuming a separable utility functionin a model that did not feature credit. However, all the results derived so far donot utilize the credit features of the model.

Turning now to the other choice variables of the landlord, (oa, v, r1T), it can beshown by writing his income as

G = (1 -/30)f(ex), (19)

where 0 = 1 + vrL + vrA, that (ce, v, rT) enter G only through their effect on /3and 0, since e and x are functions of fl only. Now

aG/aO= -flf <0. (20)

This means that an income maximizing landlord will choose his optimal 0 to be

0* =rminimum feasible 0 for any given ,B, (21)

and then choose /B to maximize (1- /30*)f(ex). Since 0 depends only on v (whichlies between 0 and 1), if the given value of ,B does not restrict the choice of v, then

O*=(l+rj) and v-*=1 if L < -A,; (22)

=(1 +r ) and v=0 if 1',> rA.

Thus, to minimize 0 is to give a weight of 1 to the smaller interest rate, and aweight of 0 to the larger one.

Now, by definition, l=cc/(1G+vrT+(1-V)rA). The range for /3 for feasible(ot, v, rT) (i.e., 0 - c < 1, 0 < v < 1, rT - 0) is therefore [0, 1]. And any ,elO [0, 1] can be

A. Braverman atnd TN. Srinivasan, Credit and slharecroppintg 299

reached by a suitable choice of (a, 1'T) if v = 1. This holds true even if there isan institutionally specified floor aF on . Thus, in the case 7'L -< rA the landlord canset v* = I and 0* = 1 + r L and choose /f (that is ac and r7T) to maximize G. In essence,what is happening is that, with 1

L <5r 1 the landlord is the cheaper source ofcredit and by offering credit with tenancy (setting v* = 1) the landlord ensuresthat the tenant uses the cheaper source of credit.

If v = 0, then values of , > l/(1 + rA) are not attainable through choice of ac. Nowwith v = 0, any P in [0, 1/(1 + rA)] can be reached by a suitable choice of cc as longas there is no floor on ax. And ,B> 1/(I + rA) is irrelevant for maximizing G when1

L >rA since then 0Ž1 +rA so that f30>1 making G<0. Thus we can assert,using (22), that the landlord's optimal choice is v* = 0 if 1-L > rA- Once again,the landlord ensures that the tenant gets credit from the cheaper souice. We cantherefore state

Proposition 3. The landlord, with no restriction on his choice of crop shares, willensure that the tenant gets credit fromn the cheaper source. In the event that he is thechea per soturce (r, - r -), hle does this by offering a tenancy contract with credit. In111e case wvlhere r I>r, , he d1oes this b v not offering anzy credit to the tenant.

Reniairk. As discussed above, in the case of rL rA where offering credit isoptimal, it rernains optimal even if there is an institutionally imposed floor on thetenant's crop share, the reason being that any given / =B/(1 + rT) (and a fortiorithe optimal /3) can be achieved with an infinite number of pairs (a, rn), of which,another infinite set will meet the required floor.

Proposition 3 is consistent with empirical observations [Bardhan and Rudra(1978)] that landlords frequently offer interest-free loans to their tenants. Forexample, in the case of rL <rA, with v* = 1, the. interest rate, rT charged by thelandlord is essentially arbitrary, and it could as well be zero. Hence, if thereis no floor on a, the situation observed is not really one of tie-in, since theparties can untie the transactions without altering the outcome. This will notbe the case, however, if the environment faced by the parties is subject tocertain constraints such as government regulations. This topic will becovered in the next section.

Returning to the case where there is no floor on a, we have seen that if rL < rA,)* = (I + rL) and with G = [1-/(1 + rL)]f(ex), the range for B is O, l/(l + L)1-l fr, > r,A 0* (1 + rA), with G = [1 - /(I + rA)]f(ex), the rangefor/3is [0, 1/'(1 + rA)].In either case, G, being a continuous function of ,B, defined over a compact set,attains its maximum. If this maximum is attained at an interior point, we have

aG d-= 0*f +(1 -* 0*)f' O (ex)=O,

Ldx de=.O*f +(1 -/3*0*)f/ e-± +x- =0,

300 A. Braver man antd TN. Srinivasan, Credit and sharecropping

or

f* L f ex +f xLf de using (16),1 /3*Q* f7L/*(f -exf')flf

or

/3*0* S +1** de

1-f*0* 1-S f d,B'

where S=exf'/f is the imputed share of labour in crop output. Using

Proposition 2, we can assert that

/*0**'S according as o 1. (23)

In the case where r L <rA, 0*=(1 +rL) and /*=a*/(I +1-7.), and in the case where

-, > r,,, t* = ( t rA) and /3* = o*/(I + r,,). Since in the first case 1-T can be chosen to

be r,, /*0* becomes the crop share a* in either case. So using (22) we can state:

Propositioin 4. If there is no restiictioii on the landlord's choice of instruments

(e, l¶ r T), and optimal strategy fbr him involves his offering his tenlant a crop share a*

stuchl that * -S accor-dinig as a g1.

Remark. In the case of rL- r 1A since /3*O*=a*(l+rL)/(1+rT), by choosing

(a*, r T) with rT sufficiently less (greater) than 1-L, the landlord can offer an a* which

is less (greater) than S, even if a is greater (less) than unity.

Newbery and Stiglitz (1979) established Proposition 4 without incorporating

credit or its linkage to tenancy. The above remark extends their result to a case

where it is optimal for the tenant to borrow from his landlord. It also implies that

it is possible to observe crop shares lower than the imputed share of labor-even for

a production function with an elasticity of substitution larger than 1.

4. Policy analysis

4.1. Tenancy r eforms

First, consider a reform which imposes a floor, aF, on the tenant's share aof the harvest. This is a common feature of many agrarian reform laws in

India. As discussed earlier in the case where rL -A4, if in an equilibrium

(oa*, 1, r*) prior to the promulgation of the reform law the landlord was

offering a crop share below the legal floor aF, he will raise the crop share

after its promulgation to aoF and at the same time raise the interest rate to

r** so that in the new equilibrium (aF,1,rT*), tx /(1+ ir)-a*/(1+rT)=,/*.

Since output depends only on /3*, it is unaffected by reform. Given utility

equivalence, the lenant's welfare is unaffected anyway.

A. Brnaver-iniani a:'l TN. Srinivasan, Cr1edit anzd sharecroppinig 301

Suppose now that the legal floor is imposed. Consider the following twoalternatives: (i) an initial equilibrium in which the landlord is not the cheapersource of credit, i.e., rL > rA so that v* = 0, 3* =-a*/(l +±r,) with c**< xa, or (ii)initially 1-L < 7r, and v* = 1, #* = o*/(l + 1T) with x* < aF. However, as part of atenancy reform, the interest rate on the tenant's alternative source of credit isbrought below rL. In other words, along with the floor aP there is a changein r14 which brings it below rL. This joint reform of tenancy and credit, couldbc viewed as two consecutive reforms, first a credit reform with no tenancyreform, so that the landlord switches to the equilibrium with one asteriskfrom one witlh two asterisks and then to a tenancy reform imposing a floor.This way, it suffices to discuss only the tenancy reform.

In such a situation the landlord can partially nullify the tenancy reform bvforcibly tying the credit and tenancy contracts. In a technical sense, even in thiscase, the tenancy reform may be made ineffective. For example, consider asequence of contracts offered to the tenant, the sequence indexed by n1: (t"= aF,

t={c= F-fJ(l + 1 ')}/1*, TrI=11). Clearly, v"O0 since in the initial equilibrium3*(A+1) )= c* <CXF and for large enough n, v" will be less than one. Thus, for

large enough n, each member of the sequence is a feasible contract. Now ,B"=c7/(1 v"r!.+(1 -v)rA,). The plot size sequtence is x(/3'). As n--oo, oc"converges to Fx v" converges to zero, #n'.*# and r1-+oc. By choosing nsufficiently large (thereby making rn large, but finite), the landlord can remainas close as he wishes to his income prior to the imposition of the floor evenafter the reform!"5 What this argument suggests is that, after the reform thereis no optimial policy for the landlord, but there exist policies that will givehim an income as he wishes to his income pr'ior' to reform. Since, prior toreform, he was maximizing his income without the floor constraint on thetenant's crop share, that income provides an upper bound to his income afterreform. Since policies exist, which get as close as one likes to this upperbound, thlis upper bound is the least upper bound.

The implication of the above discussion is that, if tying is permitted, thelandlord can reduce the tenancy and credit reform to insignificance. Suppose nowthat the government bans tying, along with tenancy and credit reforms. Clearlythe landlord's income will decline, while the tenant's welfare continues to be at thelevel he could have achieved while working as a wage labourer. What about theeffect on output? Since the landlord no longer has the instrument by which he canmaintain the pre-reform discounted share, #*, of the tenant, the reform will raise/3. Since we know from (18) that d(z)/d/ > 0, we can assert that output f(z) will goUp.1 6 Thus

5 This is perhaps a rationale for empirical observations of tenants being charged high interestfGr rather small loans.

"6Some care is needed in interpreting this result. An increase in P raises the number ofefficiency units of labour, i.e., ex supplied by eacih tenant, and increases the nitlmber of tenantsthrough a reduction in plot size. If the elasticity of substitution is less than unity, effort pertenant will decline, so that output per tenant will decline. But the increase in the number oftenants more than offsets this decline.

302 A. Bravermani anid TN. Srinivasan, Credit and shlarecroppinzg

Proposition 5. A tenancy refor m which imposes a floor on the tenanit's share of'thecr'op with or without credit reform7l (to make credit available to the tenant at a ratelower than the landlord's opportun2ity cost of capital), will have no effect on output. Ifit is coupled with a ban on tying of credit and tenancy transactions, it will raiseoutptut, reduce the teniant's plot size and increase the number of tenlants.

Now consider only a ban on tying of credit and tenancy. This is, of course,meaningless when the landlord is not the cheaper source of credit, since no tyingwill be observed anyway. Suppose the ban is imposed when the landlord isoffering credit, i.e., when rL <7 -A and v* 1. Clearly, this immediately raises thecost of credit to the tenant to rA, In the landlord's income maximization problem,fixing v at zero (i.e., preventing linking), fixes 0 at (I + rA), i.e., raises 0 from itsoptimal value of I + rL prior to the ban to (1 + rA). Since G is a monotonicdecreasing function of 0, at any value of ,B, G is lower than before. Clearly, evenwith the optimal value of ,B, G is lower. This means that landlord's incomedefinitely goes down. What about output? As long as f(z) as a function of / isconcave, optimal ,B for any specified 0 is a decreasing function of 0. Hence, as 0 isincreased from (1 +rL) to (1 4-rA), optimal ,B goes down. This means that firstly,the optimal plot size increases thereby reducing the number of tenants andsecondly, output goes down since f(z) is an increasing function of ,B.

4.2. Land r eformn

Suppose starting from an initial equilibrium [a*, v*, r*] and x(#*), each tenantis given the ownership of the plot he cultivates and has to forego the opportunityto borrow from one landlord. Clearly, the tenant's welfare improves, for if 7-L > rA,

J,*=O and P*=ja*/(I + r/). With reform ca becomes unity, r,A remains unchangedso that the tenant's (now a landowning peasant's) discounted share /3 increases,while the size of the plot remains the same. Hence, without changing, his effort e(and its disutility) he will gain in consumption and, hence, total utility. Byoptimally adjusting his effort to the changed /3, he can raise his utility even further.

Now if 7-L <-A initially v* = 1. Since, the landlord is indifferent in this casebetween alternative combinations of (a, rT) which result in his optimal ,3*, we canview the land reform, as if it first changed the interest rate charged by the landlordto r, with a corresponding change in a to maintain the same /3* and then raised thetenant's crop share to unity. The two moves together imply that the tenant's post-reform discounted share is higher. From this point, the argument is the same as inthe previous case.

What about the effect of land reform on output? Land reform increases thediscounted share /3 while keeping the plot size fixed. Thus, output is f [e(/3)x],where x is fixed. Since the former tenant will choose e to maximize his utility.given any : and x, we know from eq. (A. 13) in the appendix that

A. Braverman and TN. Srinivasan, Credit and sharecroppin,g 303

= f'Ul + (f/xU1 ){U2 1 Ul - Ul 1 U 2}ii/ /3Xf"Ul +(1/(xU 1 )2 ){U2 Ul1 -2U 1 2 U 1 U2 ± U2 2 U1}

- f'U{l+Pf(fl u22 1)}l (24)

where A denotes the negative denominator. Now, /3f/x is the consumption of thetenant. Hence

ae/laflO according as -c )-- 1. (25)U1T U2 ,

@e/f3 - 0 implies that output increases, remains unchanged or decreases as ,Bincreases. Thus

Proposition 6. A land reform which confers ownership to the plot of land that atenanit used to cultivate in a sharecropping contract with a landlord willincrease, niot change, or decrease otutput, according as -c(U 1 1/U1-U 2 1 /U 2 )5 1.

(25) represents the elasticity of the marginal rate of substitution betweenconsumption and leisure with respect to consumption. The tenant maximizesU[c, e] subject to c=/3(f(xe)/x). Now the marginal rate of substitution (MRS)between c and e in U is - Uc/Ue and the,marginal rate of transformation (MRT)between c and e through production is dc/de=,ff'(xe).

At given x and e, dlog (MRT=1,

and

d log (MRS) d log (MRS) d log cdlog/, dlogc dlog:

d log (MRS) (U 1 l U2 )1dlogc cU 1 U 2 ]

Since for optimality MR T = MRS, the impact of a change in ,B on e is obtained bya comparison of the two elasticities.

Furthermore, consider the case' of a separable utility function, i.e., U(c, e) = u(c)-v(e). Then (25) becomes

a:O according as -- , , 1. (26)

304 A. Bravermani and TN. Srinivasani, Credit anid shaarecroppinig

The negative of the elasticity of marginal utility (u"c/u') is defined by Arrow(1971) as the measure of relative risk aversion. The intuitive explanation for thevalue of this elasticity to be of relevance in our case, even though there is nouncertainty, is the following: On the one hand, an increase in ,B increases tenant'sincome; hence, the marginal utility of income declines relative to the marginaldisutility of effort, and ceteris pa1ribtus, the new landowner would like to reduce hiseffort. On the other hand, his share in the marginal productivity of effortincreases, with increasing /, thus creating an incentive for more effort. Whetherthe income effect or the marginal productivity effect is the dominant forcedepends solely on the elasticity of the marginal utility.

Where land. reform distributes the land to more owners than the originalcultivators, it may increase total output even if -c(UMl/U 1-U 2 1 /U2)> 1 sinceceteris paribus, output per hectare increases with reductions in plot size.

4.3. Taxation and techlniological progress

Suppose the government imposes a proportional output tax at the rate t ontenants and landlords (i.e., the rural community) in order to raise food to feed theurban workers. Since for any P this tax is equivalent to reducing the discountedshare of the tenant firm f3 to ji-=( - t), the tenant's decision function e(x,l3)becomes e(x, /t). It is also easily seen that the landlord's choice set x(p) becomesx(u). Thus, for any given ,B (i.e., before tax share of the tenant), the aftertax incomeof the landlord is

G = (I1-t)(1 - 0*)f (z),

where

z = e [x (l), 1l] X(/1) = z (p),

and

0*=I+rL if 7L7rA

= I +rA if rL >71r

The landlord chooses /3 to maximize G, implying that

dG F ~dzGP= d,- 1-t) [-O*f + ±( - jO*)f (z) Tj= . (27)

Now by total differentiation of (27) at the optimum we obtain

dj/dt<0 (see appendix 2). (28)

A. Bravermant and TN. Srinivasan, Credit and sharecropping 305

Furthermore,

df (z(M)) dz du (29)dt d

by (18) and (28), i.e., output declines due to the imposition of a proportionaltax. The implied decline in the aftertax share, ,u, necessitates an increase inthe tenant's plot size in order to maintain the tenant on his reservationutility U. The increase in plot size implies both a reduction in the number oftenants, x, and a decline in output. We thus obtain the following proposition:

Proposition 7. The imposition of a proportional output tax on landlords andtenants will cut the aftertax sha.re of the tenant, increase the plot size per tenant, andreduce the number of tenants as well as total output.

Modelling a Hicks neutral technical change is equivalent to modelling aproportional output tax, i.e., a Hicks neutral technological change is a shift in Awhere the production function is Af(ex). The only difference is the direction of theimpact. Hence, considering a Hicks neutral technical change and applyingProposition 7, we obtain

Proposition 8. A Hicks neutral technical change will increase the aftertaxdiscounted share of the tenant, decrease the plot size per tenant and increase thenumber of tenants as well as total output.

Now, consider the case of a Cobb-Douglas production function. Given theunit elasticity of substitution, the tenant's effort is independent of =,B#(1 - t) [see(17)], i.e., the decline in the aftertax share is totally compensated by the increase inplot size so as to leave The tenant's effort unaltered. Furthermore, it is easily seenusing (23) that the optimal ,B is u uaffected by the tax or technical changes. For theCobb-Douglas case, all factor-augmenting technical changes can be viewed asHicks neutral changes. Thus considering irrigation as a land augmentingtechnical change and applying Proposition 8, we obtain

Proposition 9. If the production function is of the Cobb-Douglas type,introducing irrigation will leave the discounted share contract unaltered, decreasethe plot size for tenant and increase the number of teniants as well as total output.1 7

'7 Since in this model landlords extract all the surplus from their tenants, they have no reasonto resist technological innovations. For theoretical discussions of landlords' resistance totechnological innovation, see Bhaduri (1973, 1979), Newbery (1975), Srinivasan (1979),Braverman and Stiglitz (1981).

306 A. Bratvermnan andl TN. Srinivasan, Credit and sharecropping

4.4. Increcise in the tenant's utility level in an alternative occupationl

Suppose, for example, that through an increase in the non-agricultural wagerate, the utility that the tenant could obtain (i.e., U) in an alternative occupationincreases. Assuming once again a Cobb-Douglas production function, so that thetenant's effort is independent of,B, it is clear that the landlord can meet the higherU only by raising the plot size, therefore reducing the number of tenants andoutput. Equilibrium /3 is unchanged. Hence

Proposition 10. If the production Junction is Cobb-Douglas, any iacrease in theutility that the tenant can obtain in an alternative occupation will raise theequilibrium plot size, reduce the number oJ tenants and output, while leaving thediscounted crop share unaltered.

5. Conclusions

In conclusion, we summarize our results. Our main result is that in a world inwhich (i) production takes place under constant returns to scale in land and laborin efficiency units, (ii) a landlord can subdivide his land into as many plots as hechooses, and (iii) a tenant chooses his effort, so as to maximize his utility -

equilibrium will be characterized by utility cquivalent contracts. In other words,even if a landlord has no power over crop shares or terms of credit, by choosingthe plot size appropriately, he will force the tenant to a utility level equal to thatwhich he (tlhe tenant) could have obtained in an alternative occupation as long asthere are enough potential tenants. He is able to do this not only because there is aperfectly elastic supply of tenants at this 'reservation' utility level, but alsobecause the tenant's effort per hectare increases with a reduction in his plot size.

This result is similar to that found in Cheung's (1969) model, where the tenant'seffort per unit of raw labor is invariant. Cheung shows that landlords will provideeach tenant a plot of land on which the tenant can earn no more than he couldhave earned in an alternative occupation. Whereas enforcement of the tenant'slabor input is necessary in a Cheungian world, it takes a different form inour model: it ensures that the tenant does not split his working time betweensliarecropping and an alternative occupation.

In this world of utility equivalent contracts, it will be in the interest of thelandlord to ensure that the tenant gets his credit from the cheapest source. If thelandlord's opportunity cost of capital is lower than that charged by the localmoneylender, the landlord will ensure that the tenant gets credit at the cheapestinterest cost by offering him a credit contract. This often is not imposed, butchosen, only if it is optimal. The tenant is pushed down to his alternative utilitylevel, not by the credit instrument, but by plot size variations.

Finally, in our model, utility equivalence implies that nothing short of landreform will affect the tenant's welfare, as long as he is a tenant. Indeed, other

A. Braverman and TN. Srinivasan, Credit and sharecropping 307

reforms such as setting a floor on the tenant's share of the crop, making creditavailable to the tenant at a cost below the opportunity cost of capital to thelandlord or banning the tying of credit and tenancy contracts, either have noeffect on the equilibrium at all or have an effect on the number of tenants, outputand the landlord's income.

Appendix 1: Properties of the model

Denote the tenant's utility function as U(c, e) where (c, e) denote consumptionand effort, respectively. Define leisure, = - e. Hence, U(c, e) V(c, e), whichimplies that: Vl=Ul, V 2=-U2 , V11=U11, V22=U 22, V1 2=-U1 2. Quasi-concavity of V(c, e) means that for the iso-utility V(c, e) = J c is a convex functionof0 , i.e.:

ac/= - V2/V 1, (A. 1)

and

a2Ca 2 =- V2 1D 8 + V2 2 Vl - Vl I ai + Vl 2 } 12) V1

(A.2)V-(V 2 VV2 2 2V V2 ±V2V1 1 )/Vj > °.

Hence, quasi-concavity of V(c, e) U(c, e) implies that

U1U 22-2U,U 2U12 -PUU 11 <2 - (A.3)

Now, for c and e to be normal goods the following two conditions must besatisfied:

U1U22 -U 2 U12<0 and U1U1 2-U 2 Ul 1 <O. (A.4)

We further assume that the tenant's consumption equals his income, i.e.,c =/(f(ex)/x) implying that

ac a2 ca -,Bf'> G and y-2=flxf" <O (A.5)

by the strict concavity of the production function f (ex).

308 A. Braverman atnd TN. Srinivas(an, Credit and slzarecroppinq

Let 9(e)=U(c(e),e). Hence

accp'(e)= U g+U, (A.6)

and

p'()Ul I iO +2U12 ac+U22+U ae2C(ae aec -T

Calculating the second-order conditions at the optimum [9'(e)=0] we obtain

9p"(e)= U2xf"+(U U1 1 -2UI2 U2 U1 +U 2 2UI) -Jg<O. (A.7)

By the strict quasi-concavity of U and strict concavity of f, c"(e) <O implies theexistence of a maximum to the tenant's problem.

To determine the impact of a reduction in plot size (increase in x) on tenant'seffort per acre, we denote ex z.

Thus, (A.6) can be rewritten as

u[: x Z) flf'(Z)+U2[ flfx3 =O-] (A.6')

Total differentiation of (A.6') with respect to (z, x) yields

a@r{U lf"U+fIf'(U 1 1 flj+U2)± U2flf' +U22}

+:t-Utllff _U12Z)+ l- af)+u2C Z=°-

Collecting terms and utilizing the first-order conditions (i.e., ['f U2/U1) weobtain

az _(Bf (1 J U ) Zae ( 2 (W21 Ul -U2U11)+ 2W(22U1 -U2U12)/ -9 >0.ax k\ lX2 U1

(A.8)

(A.8) is positive since p"<0 by (A.7), and the numerator is negative by thenormality conditions, (A.4).

A. Braverinan and TN. Srinivasan, Credit and sharecropping 309

Now, following the utility equivalence result (Proposition 1) the relation

U{3 e[e( /3), ]e(x, 1)}U (A.9)

determines x(,B).Applying the envelope theorem [p'(e)=O], we obtain

x'(/3) fx (A.10)Xfi(-f'z)'

The next two terms we shall calculate are de/d,B and dz/d3, where

dz az a= de

Hence,

de IlFaz a5z 1de=1 Lazx( + a -x'(l) e2 (A.12)

So in order to evaluate (A.11) and (A.12) we have only to calculate OzlaOf. [Vz/Oxis given by (A.8).]

By total differentiation of (A.6') with respect to z and ,B holding x constant weobtain

az =- (f'U 1 f (-U 1 1 U2+ U21UD)j! (p (e). (A.13)

Clearly aelafi = az/&/3 l/x.Substituting (A.13), (A.8) and (A.l0) into (A.l I) we obtain

dz f J\/df(j l{ull-XU2,x ( +(f -f,z)

+(U2 2 Ul- U2 Ul 2)--- * If-q (e). (A.14)xU1 A3f -fiz)//x

(A.14) is positive by the normality conditions (A.4) and since f/(f-f'z)> 1 dueto the strict concavity offjandJf(0)=0.

310 A. Braverman ankd TN. Sriniivasain. Cred.t and sliarecroppintc

In the same manner, calculating (A.12) and defining a -f'(f -zf')/ff"z, weobtain

de U 1 f' l-a (A.15)d/3 (1/x)qp"(e) a

Appendix 2: Comparative statics: A proportional output tax

Define the landlord's objective function as G(/3, t). [We already substituted thecondition x[,B(1 - t)] into the objective function.] Recall the first-order conditionin the text [eq. (27)] i.e.,

G,,=(l -t)[-O*f+0(1 -flO*)f'(z)zt] =0. (A.16)

By total differentiation of (A. 16) we obtain

d,B/dt =-plGpp, (A. 1 7)

and

Gfl= ( t) - 20*f'zf + (I -_ /O*)f"((zp)2 + (1 _- *)J'zafl

(A.18)

and

G-i IGP t +(I-t)[-O*f'z, + (1 - /BO*)f"zz + (1 - 0*)f /zp,].

(A.19)

Now define the aftertax share as q=_ #(l -t). Hence

zp = zm(1- t) = p ) z,(A.20)

z- t = A (A.21)

1 f3I -zS= -Zt-f2(1A-t)in= (A-t 1 bt ain (A.22)

Substituting (A.20) and (A.22) into (A.19) we obtain

A. Braverman atnd TN. Sriniuvasanl, Credit and sharecropping 311

G Gp +( rt)F*f ,f / (- 0 - 0)f I Z2' - ti 1r -t I' -t-l -S [0)f I I zp + fl-- pn (A.23)

By collecting terms and recalling (A.18) we obtain

G t=-IG t _ 1 t (A.24)

At the optimal f=,/(t), GP= 0. Hence

T d 1- + X~- fzp (A.25)

The impact of tax policy on y is

dy =-,+ + (- d:. (A.26)dt dt'

By substituting (A.25) into (A.26) we obtain

d,u f'(z) dz f'(z) dz=-(1 - t) - - A.7dt )Gpp d:l Gppf du (.7

Gp<0 by the second-order conditions for maximum, and dz/dl3>0 (see A.14).Hence, d[/dt<0. Furthermore,

df(z(Jt)) dz dtt 2 (A.28)

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