Moral Hazard and Marshallian Ine ciency: Evidence from Tunisia

Transcription

1 Moral Hazar an Marshallian Ine ciency: Evience from Tunisia Jean-Louis Arcan CERDI-CNRS Université Auvergne, an EUDN Chunrong Ai Department of Economics University of Floria François Ethier Caisses Populaires Desjarins Novembre 7, 2005 forthcoming, Journal of Development Economics Abstract We formalize the link between optimal cost-sharing contracts an the prouction technology in the presence of moral hazar by appealing to several well-known results from uality theory. Builing on intuitions from the interlinkage literature, we show that optimal contractual structure is etermine by the (i) substitution possibilities that exist between i erent observable factor inputs, as well as (ii) between these inputs an unobservable e ort. We enogenize contractual choice using lanlor characteristics as instruments, exploiting the fact that, in our ataset, lanlors interact with several tenants an vice versa. The approach is applie to an unbalance plot-level panel of cost sharing contracts in a Tunisian village, using a translog representation of the restricte pro t function. Contractual terms are foun to be a signi cant eterminant of input use an therefore lea to Marshallian ine ciency, while the optimality of the unerlying contractual structure is rejecte. Keywors: moral hazar, pro t functions, sharecropping, Marshallian ine ciency JEL: O12, O13, D21, D82, Q12, C33 We thank three referees, as well as the eitor, for insightful comments that le to substantive improvements in the paper. Fiel work that le to the ata use in this stuy was generously supporte by the PARADI program, nance by the Canaian International Development Agency (CIDA). Elisabeth Saoulet an Alain ejanvry provie encouragement in the early stages of the project. We are especially grateful for the cooperation of the inhabitants of El Oulja who mae this stuy possible. We also thank seminar participants at the Université e Montréal, Simon Fraser University, the University of British Columbia, CERDI (Clermont Ferran), the Universities of Rabat an Casablanca, an DELTA (Paris) for useful suggestions. The usual isclaimer applies. Corresponing author: Jean-Louis Arcan, CERDI-CNRS, Université Auvergne, 65, boulevar François Mitterran, Clermont Ferran, FRANCE, arcanjl@alum.mit.eu, http: tel: , fax:

2 ...when the cultivator has to give to the lanlor half of the returns to each ose of capital an labour that he applies to the lan, it will not be to his interest to apply any oses the total return to which is less than twice enough to rewar him. If, then, he is free to cultivate as he chooses, he will cultivate far less intensively... so that his lanlor will get a smaller share even of those returns than he woul have on the plan of a xe payment... The position of a peasant proprietor has great attractions. He is free to o what he likes, he is not worrie by the interference of a lanlor, an the anxiety lest another reap the fruits of his work an self-enial... He is scarcely ever ile, an selom regars his work as mere rugery; it is all for the lan that he loves so well. Alfre Marshall (1920) Principles of Economics, 8th en, pp Introuction 1.1 Marshallian ine ciency Testing for the Marshallian ine ciency of agricultural contracts in less evelope countries has become something of a cottage inustry (see., e.g., Otsuka an Hayami (1988) for one among several excellent surveys). Ami the mass of empirical contributions, two papers stan out because of the innovative methoology they propose: Bell (1977) an especially Shaban (1987). In these papers, the funamental problem of assessing the prouctivity i erential that may exist between plots uner sharecropping an plots uner owner-operatorship while maintaining the ceteris paribus assumption is aresse by consiering househols that farm more than one plot, an in particular househols that are simultaneously owner-operators an sharecroppers. The use of householspeci c xe e ects then allows one to compare the prouctivity of the two classes of plots while at least maintaining constant the ientity of the househol engaging in the farming activity. While this approach, couple with Shaban s focus on cost-sharing contracts which yiel a much greater egree of contractual heterogeneity than the simple owner-operator / sharecroppe ichotomy has shape our unerstaning of the incentive e ects of tenancy contracts, several issues remain open. 1.2 Contracts, proucers theory an interlinkage First, contractual structure still remains largely exogenous in this literature, espite the fact that an impressive corpus of theory (an empirical results from other areas of economics) suggests otherwise. Secon, espite the fact that factor eman equations are usually being estimate, no thought appears to be given to the unerlying theoretical hypotheses: these factor eman equations presumably stem from some well-pose optimization problem, which may impose nontrivial restrictions on the form that the estimate relationship takes. Thir, espite the seminal contributions by Braverman an Stiglitz (1982) an Braverman an Stiglitz (1986) regaring the theoretical interpretation of cost-sharing contracts in terms of interlinkage, no attempt has been mae to relate optimal contractual structure to the prouction technology. 1 The purpose of this paper, at least in the context of the ebate surrouning the (in)e ciency of sharecropping contracts, is to show: (i) that the enogeneity of contractual structure oes matter; 1 See also Barhan an Singh (1987). 2

3 (ii) that factor eman equations cannot be blithely formulate in open violation of the most elementary restrictions suggeste by proucers theory, an (iii) that it is possible to construct a irect structural test of the optimality of the observe contracts when one couples a simple principal-agent moel of moral hazar to the stanar results of the uality theory of the rm. 1.3 Structure of the paper The structure of this paper is as follows. In part 2, we evelop a simple moel of tenancy uner risk-neutrality that takes the form of cost-sharing contracts an show that the usual tests of Marshallian ine ciency (such as Shaban (1987)) su er from (i) enogeneity problems, (ii) several ienti able forms of speci cation error ue to their neglect of the elementary restrictions that stem from the theory of the rm. We then provie a characterization of optimal contractual structure uner moral hazar an show how this is intimately linke to the properties of the prouction technology. These restrictions are testable, given the appropriate ata. We then show that it is straightforwar to exten our moel to risk-averse tenants (thereby accounting for the existence of sharecropping contracts when xe rental is a viable alternative) an lanlor-etermine crop choice, an that the salient features of our characterization of optimal contractual structure are preserve. In part 3, we begin by applying stanar instrumental variables techniques to "Shaban" type regressions using ata from a Tunisian village. We show that it is possible to enogenize contractual choice in a plot level ataset in which tenants interact with several lanlors by using lanlor characteristics, expresse in terms of eviations with respect to operator househol-speci c iniviual means (this controls for operator househol unobserve heterogeneity), as instrumental variables. A battery of tests are implemente, incluing the recent Hahn an Hausman (2002a) test for the valiity of instrumental variables. Our results suggest that output per hectare is the same on sharecroppe plots than on those uner owner operators or xe rental contracts, ceteris paribus. These results are, however, tentative, in that they fail to impose the restrictions from the theory of the rm evelope in part 2. We then reconsier the problem using the translog parameterization of the restricte pro t function, which we aapt so that it correctly accounts for unobservable e ort. A rst pass, in which we estimate the full system constitute by the restricte pro t function an the associate factor share equations by the Seemingly Unrelate Regression (SUR) technique, con rms that there is inee enough variability in the cost-shares for one to be able to precisely ientify a signi cant number of own an cross-price e ects. We then enogenize contractual structure, again using lanlor characteristics as instrumental variables, an reject the null of no e ect of the contractual variables for all ten factor inputs, as well as for output. Finally, we explicitly test the restrictions evelope earlier that characterize an optimal cost-sharing contract uner moral hazar, an reject. This suggests, while the terms of the contract a ect the allocation of resources an thereby o yiel Marshallian ine ciency, that other mechanisms apart from the stanar Principal-Agent moel may be riving the particular form taken by cost-sharing contracts in the village uner scrutiny. Part 4 conclues. 3

4 2 Formalization Consier a simple sharecropping contract in which a lanlor rents a plot of lan of xe size T in exchange for a share 1 of output. The lanlor also pays a share 1 i of the cost of each factor input, inexe by i = 1; :::; I. The lanlor will be assume to be able to set the I +1 vector (; ). Let the prouction technology be given by the aitively separable form + q, where q is output, an is a ranom variable assume to be istribute accoring to the probability ensity function (pf) g(); 2 ; ;with E [] = 0 an var [] = 2. For simplicity, an in orer to focus on the pure incentive e ects of contractual structure, we assume for the time being that both parties to the contract are risk-neutral, an we rule out xe rental contracts. 2 Risk-averse tenants an the choice between sharecropping an xe rental will be consiere below. The lanlor s (superscript L) objective function is then given by: L = Z (1 ) ( + q) = (1 )F (T; X; e) (1 i)x i g() (1) i=1 X i=i X i=i i=1 (1 i)x i ; where, for clarity of exposition, all prices are normalize to one, q = F (T; X; e) represents the non-stochastic portion of the prouction technology, X i is the quantity of factor input i (X = (X 1 ; X 2 ; :::; X i ; :::; X I )), an e is the level of e ort furnishe by the tenant. The tenant s (superscript T ) objective function is given by: T = Z ( + q) = F (T; X; e) ix i i=1!e g() (2) X i=i X i=i i=1 ix i!e; where!e is the isutility of e ort, expresse in monetary terms. 3 Note, whether we are consiering the rst best optimum or the situation uner moral hazar, that we will assume that the lanlor is fully aware of the preferences of the tenant incluing, most importantly, the value of!. The tenant will accept a contract o ere by the lanlor as long as his participation constraint (PC) is satis e. We write this as: T T ; (3) where T represents the reservation level of the tenant s expecte net payo s measure in mon- 2 This is because, in the absence of a risk-averse tenant or market imperfections such as creit constraints, the lanlor coul achieve the rst-best optimum with a risk-neutral tenant simply by o ering a xe rental contract that woul guarantee the tenant his reservation level of welfare. In the notation use here, this woul correspon to setting = 1 = ::: = i = ::: I = 1, an aing a xe transfer to the tenant s objective function. Risk-aversion will be consiere in section 2.5. We will examine the choice between xe rental an sharecropping in the context of the present moel in section 2.6, once risk-aversion has been introuce. 3 The isutility of e ort is linear in e ort in orer to facilitate the application, in what follows, of stanar results from uality theory, since the marginal isutility of e ort,! can simply be interprete as another "price". An alternative speci cation woul involve e ort appearing in multiplicative form in the prouction function (i.e. F (T; X; e) = ef (T; X)), while its isutility woul take the form!(e);! 0 (e) > 0;! 00 (e) > 0. In mathematical terms, either speci cation woul be vali, an simply correspons to a particular normalization of the variable "e ort". In terms of analytical tractability, however, the chosen speci cation is much more convenient. 4

5 etary terms, which will be a function of a vector of the tenant s characteristics (such as outsie opportunities), as well as the lanlor s (for example, resient lanlors may be able to rive a harer bargain than those of the absentee ilk). 2.1 The rst-best optimum When the tenant s actions are observable to the lanlor, the solution to the lanlor s optimization problem is given by: ( ; ; X ; e ) = arg max L s: t: T T (4) f;;x;eg where = ( 1 ; 2 ; :::; i ; :::; I ). Substituting from the participation constraint, this problem may be expresse in unconstraine form as: X max F (T; X; e) fx;eg i=i i=1 X i!e T : (5) The vector ( ; ; X ; e ) is then implicitly e ne by the I +1 rst-orer conitions (FOCs): F Xi (T; X ; e ) 1 = 0; i = 1; :::; I; (6) F e (T; X ; e )! = 0; (7) (where the subscripts on F (:) enote partial erivatives) plus the participation constraint: T = q X i=i i=1 i X i!e = T ; (8) where q = F (T; X ; e ) is the tenant s supply function. As shoul be obvious from conitions (6) an (7), the cost share terms (i.e., the elements of vector ), as well as the output-share, play no role in terms of the incentives that etermine factor inputs an e ort, an serve merely to ensure that the participation constraint is satis e, through conition (8). One may therefore write: X = X (T;!; T ); (9) e = e (T;!; T ); (10) q = q (T;!; T ): (11) Consier the expecte net payo of the tenant expresse in monetary terms, at the optimum, as given by the left-han-sie (LHS) of (8). This can be ecompose into two parts: T = e T!e ; (12) where e T is not the usual pro t function familiar from receive microeconomic theory, whereas T is. In orer to avoi confusion in what follows, we will enote e T by the term "observable pro t function", in contrast to T which inclues the non-observable component of pro ts given by the opportunity cost of e ort. This istinction will be important in what follows because 5

6 the lanlor an, more importantly for our purposes, the econometrician, oes not observe T but is able to observe e T, ex post. Then straightforwar i erentiation, since X an e are inepenent of (; ), yiels: T = e T = q ; T i = e T i = X i ; i = 1; :::; I: (13) Note that the Envelope Theorem oes not nee to be invoke here in carrying out these comparative statics given the inepenence of factor emans with respect to (; ). 2.2 Moral hazar with elegation When the lanlor cannot observe the tenant s actions, he faces aitional constraints stemming from the tenant s optimal choice of factor inputs an e ort. 4 In the traition of Holmström (1979) an Harris an Raviv (1979), write the solution to the lanlor s optimization problem in this case as: ( ; ; X ; e ) = arg max L (14) f;;x;eg s:t: 8 < Consier the incentive compatibility (IC) constraint: : T T (P C) (X ; e ) = arg max T fx;eg (IC) (X ; e ) = arg max fx;eg T = arg max fx;eg F (T; X; e) X i=i i=1 ix i!e: (15) This is characterize by the following I + 1 FOCs: 5 F Xi (T; X ; e ) i = 0; i = 1; :::; I; (16) F e (T; X ; e )! = 0: (17) In contrast to the case without moral hazar, the solution to equations (16) an (17), along with the participation constraint, yiels: X = X (T;!; T ; ; ); (18) e = e (T;!; T ; ; ); (19) q = q (T;!; T ; ; ); (20) 4 In what follows, we assume that the tenant chooses e an X. Formally, in terms of the well-known elegation argument suggeste by Braverman an Stiglitz (1986), this will obtain only when the tenant possesses an informational avantage over the lanlor thus making elegation by the latter to the former optimal. The results that follow o not i er appreciably in qualitative terms if the lanlor is assume to be able to choose X. The etaile characterization of the optimal contract furnishe in Proposition 1 is, however, slightly moi e. Note that we abstract from the important insight of Barhan an Singh (1987) concerning the absence of incentive e ects of cost sharing at the margin, when the lanlor chooses X. 5 For simplicity, we shall assume that the rst-orer approach is vali; see Jewitt (1988). 6

7 where q = F (T; X (T;!; T ; ; ); e (T;!; T ; ; )). The optimal levels of physical inputs an e ort chosen by the tenant are now functions of the terms of the cost-sharing contract, as is output. The i erence between equations (9) an (18) constitutes the basis for the classic tests of Marshallian ine ciency carrie out by Bell (1977) an Shaban (1987). In the absence of moral hazar, optimal input use (as well as output) shoul be inepenent of the terms of the contract (; ). 2.3 Why the stanar approach is wrong A rst important weakness of such tests, as shoul be obvious from the optimization problem pose in (14), is that estimating factor eman equations of the form Xi = X i (T;!; T ; ; ) leas one straight into a potentially serious enogeneity problem, because an are not ranomly chosen by nature an correspons to part of the solution to the problem. In other wors: carrying the reasoning begun with the lanlor s optimization problem through to its logical conclusion, it shoul be obvious that = (T;!; T ) an = (T;!; T ). Substituting these last expressions into (18) thus leas to the full solution in terms of exogenous variables, which takes the form: X = X (T;!; T ): (21) In analytical an, most importantly, in econometric terms, equations (9) an (21) are inistinguishable. The reuce form representations of the factor eman equations o not allow one to test whether moral hazar is present. The upshot shoul be clear: a factor eman equation which takes the form Xi = Xi (T;!; T ; ; ) shoul be estimate as a structural equation in which appropriate instruments must be foun for the terms of the contract. 6 A secon weakness of existing tests for Marshallian ine ciency stems from their neglect of the elementary restrictions on functional form that stem from receive proucers theory. In concrete terms, regressing Xi on the vector (; ) (as well as plot-level controls an househol-speci c e ects, if possible) is neither here nor there in the sense that such a linear speci cation oes not correspon uner any circumstances to a well-behave factor eman equation. In particular, given that one shoul interpret i as the "e ective" factor price, face by the tenant, for input i, an as the "e ective" output price, the equation being estimate shoul at least satisfy the homogeneity property of a factor eman equation (see, e.g., Varian (1978)). Moreover, the factor eman equations shoul be estimate as a system using the SUR approach, as is one in conventional microeconometric work associate with the theory of the rm (see, e.g., Fuss an McFaen (1978)), an the stanar symmetry restriction on the cross partial erivatives shoul also be impose. To the best of our knowlege, these restrictions have never been impose in the context of the Marshallian ine ciency ebate. Carrying out the stanar regression of Xi on (or ( 1 =; :::; i =; :::; I =)) an plot characteristics therefore constitutes a gross error in speci cation, an any conclusions one may try to raw from the estimate coe cients associate with may simply be riven by what amounts to enogeneity or speci cation bias. 6 The same is true of the output supply equation, as given in (20). To be precise, one nees at least I instrumental variables in that the I physical factor eman equations X (T; ; ;!), are functions of the ratios 1 ; :::; i ; :::; I : 7

8 2.4 Contractual choice an the prouction technology The main original contribution of this paper, apart from proviing an empirical application in which Marshallian e ciency is teste while enogenizing contractual structure an using the restrictions that ow from the apparatus of proucers theory (more on the mechanics of this in part 3 of the paper), is to formally test the optimality of contractual structure, as it is set out in a stanar principal-agent framework, by focusing on the lanlor s optimal response to the non-observability of tenant e ort. Such optimizing behavior, if it is present, shoul lea to an intimate structural link between the optimal contract an the unerlying prouction technology A characterization of the optimal contract The point of eparture is the constraine optimization problem face by the lanlor in (14). The following Proposition, which is riven in part by the linear homogeneity property of the restricte pro t function, provies a characterization of the optimal cost-sharing contract in terms that shoul be easy to interpret from the stanpoint of receive uality theory. Proposition 1 (i) The optimal cost-shares, enote by the 1 I vector, are characterize by the following restrictions: q i which can be re-expresse as (ii): X j=i Xj j=1 i! e i = 0; i = 1; :::; I; (1 ) q i X j=i j=1 1 j X j i = 0; i = 1; :::; I; (iii) the optimal output-share is characterize by the restriction: which can be re-expresse as (iv): q X j=i j=1 X j! e = 0; (1 ) q X j=i j=1 1 j X j = 0; (v) if the lanlor chooses plot size, its optimal value is characterize by: q T X j=i j=1 X j T! e T = 0: Proof: see Appenix. It shoul be obvious that other motivations, asie from a response by the principal to the agent s opportunistic behavior with respect to e ort, may also be riving contractual choice. As such, we shall go to consierable pains in the empirical portion of the paper to provie instrumental variables estimates that o not put all of the onus of the proof on the restrictions provie by Proposition 1. On the other han, the explicit characterization of the optimal contract given 8

9 by Proposition 1 potentially provies one with a structural test of the empirical valiity of the principal-agent approach to sharecropping, conitional of course on the maintaine hypothesis that the chosen parameterization of the prouction technology is correct. The characterization of optimal contractual structure given by Proposition 1 is easy to unerstan in terms of the stanar uality theory of the rm (see, e.g., Diewert (1982)). First, note that the elements in the summation in parts (i) an (ii) of Proposition 1 are given by expressions of the form X j which, by Hotelling s Lemma, correspon to the elements of a portion of i the Hessian matrix of secon erivatives of the restricte pro t function of the tenant (evaluate at ( ; )), with typical element: X j i = 2 T i j = X i j : (22) Secon, the erivatives of the supply function (q ) with respect to the cost-shares, again in parts (i) an (ii) of Proposition 1, correspon to elements of the Hessian matrix of the form: q i = 2 T i = X i ; (23) while, in part (i): 7 e i = 2 T! i = X i! : (24) Parts (ii) an (iv) of Proposition 1, which stem from Euler s Theorem as applie to the restricte pro t function of the tenant, are particularly clear (an initially somewhat counterintuitive) in terms of the incentives put in place by the lanlor: at the optimum, the lanlor sets ( ; ) so that the marginal impact of each i on the value of the share of output he receives is just o set by the sum of the marginal impacts of i on the share of the cost of each input he bears, while seemingly ignoring the marginal impact of the contract on the opportunity cost of e ort (a similar line of reasoning applies to ). Proposition 1 shows that it is possible, given an appropriate parameterization of the prouction technology through its ual representation in terms of the restricte pro t function (an thus the corresponing factor eman an supply equations), to construct a full empirical characterization of optimal contractual structure. technology itself. The latter will be intimately linke to the prouction In particular, those elements stemming from (22) correspon to substitution possibilities between physical factor inputs, whereas those elements given by (24) correspon to substitution possibilities between physical factor inputs an e ort. For those familiar with the literature on interlinkage (Braverman an Stiglitz (1982) an Braverman an Stiglitz (1986)), the intuitive appeal of this characterization shoul be obvious On the importance of elasticities of substitution What are the implications of Proposition 1 in terms of the form taken by optimal cost-sharing contracts, an can the form of the optimal contract be relate to a given characteristic of the 7 Note also, in parts (iii) an (iv) of the Proposition, that: e = 2 T! = q! : X j = 2 T j = q j, q = 2 T 2, 9

10 prouction technology? As an illustration, consier a simple Cobb-Douglas prouction technology with two physical inputs plus e ort: q = AX 1 1 X2 2 e!. Ignoring multiplicative constants, the restricte pro t function (for the tenant) associate with this prouction technology is given by: T = h i !! 1 1 2! A : (25) For this simple example, it is straightforwar to show that the two conitions given by Proposition 1 (i) that characterize the optimal cost-sharing contract reuce (iviing one conition by the other) to 1 = 2. The level of the cost-shares is then euce from the bining participation constraint. The Cobb-Douglas prouction technology thus results in optimal cost-shares that are equal across factor inputs. The same is true for a secon commonly use prouction technology, the CES prouction function, which suggests that the conitions given in Proposition 1 will yiel heterogeneity in optimal cost-shares only in the presence of heterogeneity in elasticities of substitution. In orer to aress this issue, the choice of an appropriate functional form for the pro t function is crucial. We turn to this question in section Risk-aversion We now relax the assumption of a risk-neutral tenant. optimization problem in the presence of moral hazar is given by: In this case, the solution to the lanlor s ( ; ; X ; e ) = arg maxe L (26) f;;x;eg s:t: 8 < : E U( T ) T (P C) (X ; e ) = arg maxe U( T ) (IC) fx;eg where U(:) represents the tenant s increasing an concave utility function, T shoul now be interprete as the tenant s reservation level of welfare, while the lanlor is assume to remain risk-neutral. Consier the incentive compatibility (IC) constraint more explicitly: Z (X ; e ) = arg max fx;eg U ( + F (T; X; e)) P i=i i=1 ix i!e! g(): (27) The associate FOCs are given by: Z [F Xi (T; X ; e ) i ] Z [F e (T; X ; e )!] U 0 (:) g() = 0; i = 1; :::; I; (28) U 0 (:) g() = 0; (29) 10

11 which are ientical to (16) an (17) uner the relatively mil assumption of non-satiation (U 0 (:) > 0). 8 The characterization of the optimal contract that woul follow is, however, less than transparent, an is therefore less amenable to structural testing. For this reason, we assume that (i) the tenant s utility function isplays constant absolute risk-aversion (CARA), an that (ii) g() correspons to the normal ensity ( N(0; 2 )). 9 Then the usual properties of the characteristic function, or Fourier transform, of the normal istribution, along with some straightforwar algebraic manipulations, imply that the tenant s objective function can be rewritten as: 10 ln E U( T ) = F (T; X; e) ix i!e i=1 X i=i ; (30) where ln ( (:)) is a monotonically increasing transformation. The restrictiveness of these assumptions concerning tenant preferences an the form taken by the stochastic prouction technology may be open to ebate, but it is clear that they allow us to maintain our use of the well-evelope corpus of proucers theory when it comes to characterizing optimal contractual structure. Consier once more the lanlor s optimization problem given in (26), in light of (30). then have the following Proposition. Proposition 2 Assume that the tenant s preferences satisfy CARA an that the aitive stochastic shock to output is istribute N(0; 2 ). We Then (i) the optimal cost-shares are characterize by the same restrictions as in Proposition 1, an the optimal output-share is characterize by (ii): q X i=i i=1 X i! e 2 = 0: or (iii): (1 ) q X i=i i=1 (1 i ) X i 2 = 0; where is the tenant s Arrow-Pratt coe cient of absolute risk-aversion. Proof: see Appenix. The main change in the characterization of the optimal sharecropping contract brought about by the introuction of tenant risk-aversion is, unsurprizingly, that the optimal share of output becomes a function of 2 an. On the other han, the interpretation of the optimal contract in the language of stanar uality theory, as set forth after Proposition 1, is preserve. 8 The same is true of the rst best optimum, where the introuction of risk aversion, in the presence of an aitive stochastic, shock yiels the same characterization of (X ; e ) as in (6) an (7). 9 Ignoring multiplicative constants, a CARA utility function correspons to U(z) = exp f zg. 10 For x N(; ); the characteristic function is ' x (t) = exp t t0 t where = p 1. It is this functional form that then allows one to write R +1 1 exp f xg N(; 2 )x = exp ; see, e.g. any stanar Mathematical Statistics text, such as Roussas (1997). Note that a key assumption here is the aitively separable form taken by the stochastic prouction technology, since a prouction technology of the form F (T; X; e) with N(1; 2 ) woul imply that: log h i E U( T ) = F (T; X; e) X i=i i=1 i X i!e R [F (T; X; e)] 2 2 : The characterization of (X ; e ) woul then be substantially i erent, an we woul no longer be able to apply the results of receive proucers theory. Given that we have no reason to prefer the multiplicative over the aitive form of the stochastic shock to output, we prefer to remain with the simpler speci cation. 11

12 Risk-aversion on the part of the tenant, couple with the assumptions of Proposition 2, also allow one to consier cropping as a lanlor choice, in an extremely simple framework. Assume that several i erent crops are available, an that crop k is characterize by k N( k ; 2 k ) where k is the mean level of the stochastic shock to output an 2 k is its variance; the istribution of k may, of course, be a function of plot characteristics such as soil type. Then the solution to the lanlor s problem is given by: ( ; ; k ; 2 k ) = " s:t: arg max (1 ) ( k + q )) f;; k ; 2 kg # ( k + q ) P i=i i=1 ix i!e P i=i i=1 (1 i)x i k = ln T ; (31) where (X ; e ) = arg max fx;eg " ( k + F (T; X; e)) P i=i i=1 ix i!e The vector (X ; e ) therefore continues to be characterize by (16) an (17). # k: (32) It is then easy to verify that the optimal contract ( ; ) that emerges is a straightforwar extension of the characterization provie in Proposition 2, although it will now also epen on k, which is itself enogenously etermine. 2.6 Sharecropping, xe rental contracts, an the ecision to rent out Now that risk-aversion has been introuce, it is possible to formally aress the issue of the choice between xe rental an sharecropping contracts. In the case of a xe rental contract, the tenant is resiual claimant, with = i = 1; i = 1; :::; I. The bining participation constraint implies that the optimal rental payment R is given by: R = q X i=i i=1 X i!e ln T ; (33) where the tenant chooses the rst-best values (X ; e ). Uner sharecropping, the lanlor s objective function evaluate at the optimum (using the manipulations etaile in the proof of Proposition 1) is given by: E L = q X i=i i=1 X i!e ln T : (34) Since the tenant is by construction ini erent between the two contractual forms because his participation constraint is bining in both cases, the lanlor will prefer a sharecropping contract to a xe rental contract when 0 < E L R. By two trivial appliciations of the Envelope Theorem, the following Proposition is then immeiate: Proposition 3 The lanlor s preference for a sharecropping contract over a xe rental contract is: (i) increasing in the variance 2 of the stochastic shock a ecting output, (ii) increasing in the egree of absolute risk-aversion of the tenant. Proof: see Appenix. 12

13 The intuition behin these two results shoul be obvious. As the variance of risk increases, the transfer that the tenant furnishes the lanlor uner the xe rental contract falls in orer for his participation constraint to remain satis e. Similarly, in the case of the sharecropping contract, the magnitue of the transfer achieve through the share of output an the share of costs borne by the tenant also fall. However, the rst fall (uner xe rental) is greater than the secon (uner sharecropping), since the cost to the tenant in terms of his welfare inuce by a given increase in the variance of risk is greater uner the xe rental contract, where the tenant is the resiual claimant, than in the sharecropping contract, where he only bears a fraction of the risk. Similarly, the more risk-averse the tenant, the greater the likelihoo that the lanlor will o er him a sharecropping contract because the costlier it will be (to the lanlor) to inuce him to accept resiual claimancy. The preceing setup is reaily extene to allow for the lanlor s ecision to rent out or cultivate the plot himself as an owner-operator (whence the superscript O). When the lanlor cultivates the plot himself, his objective function, evaluate at the optimum, is given by: E O = q X i=i i=1 X i e ; (35) where is the lanlor s opportunity "price" of e ort (as with! for the tenant), (X ; e ) = arg maxf (T; X; e) fx;eg X i=i i=1 X i e; (36) an q = F (T; X ; e ). In light of our iscussion concerning the choice between sharecropping an xe rental contracts, it is then obvious that the lanlor ecies to farm the plot himself when: E O = max E O ; R ; E L : (37) 3 Empirical Implementation 3.1 A rst pass The ata The ata use in this paper stem from elwork unertaken by two of the authors in the village of El Oulja, Tunisia. See Matoussi an Nugent (1989), La ont an Matoussi (1995) an Arcan, Ai, an Ethier (1998) for other escriptions of the village. Summary statistics for the full sample are provie in Tables 1 an 2 an will be iscusse as neee in what follows. At the outset, it is important to note that, since all of the ata stem from househols that live in one village, there is no cross-sectional variation in input prices. However, roughly one quarter (120) of the 455 plots in our sample are farme uner either sharecropping (45 plots) or xe rental contracts (75 plots) with the sharecroppe plots involving cost-sharing. 11 This allows us to ientify the parameters of interest of the prouction technology through its usual ual representation in terms of the restricte pro t function, because heterogeneity in cost-shares inuces variation in 11 We con ne our attention to plots on which non-tree crops (olive an fruit trees are therefore exclue) are grown. Tree crops are subject to i erent, often intertemporal, contractual structures that are beyon the scope of this paper. 13

14 the e ective price face by the househols cultivating the plots of lan. Slightly more than one half of the cost-shares ( i ) are equal to 50%, one thir 100%, one sixth 0%, with the remainer being equal to 75, 70 or 66%. That there is su cient variation in cost-shares for one to be able to ientify the relevant price e ects in our empirical application is mae clear in the statistics presente in Table 2. Here, it becomes obvious that there is substantial variability in the cost-shares, even when they are expresse as eviations with respect to operator househol-speci c means. 12 In the last three columns of Table 1, we report escriptive statistics for the three categories of plots. Sharecroppe plots are, on average, larger than plots uner xe rental, which are in turn larger than owneroperator plots. Another important istinguishing feature of sharecroppe plots is that they are less likely to be irrigate (the i erence is signi cant at the 2% level). The only crop that is signi cantly less likely to be grown on sharecroppe plots is foer (p-value= 0:03). In Tables 3 an 4, we ivie lan-owning an tenant househols into i erent categories corresponing to their position on the lan rental market. Lan owners come in three basic varieties: pure owner-operators, pure lanlors (who o not farm any of their lan themselves), an mixe owner-operator/lanlors (who farm some of the plots in their possession themselves an rent out the rest). We also single out lanlors who are not full-time resients of the village. Since the lanlors in question often live in a nearby village (Mejez-El-Bab), the term "non-resient" shoul be unerstoo in this sense: many of these lan owners are not absentee lanlors by any means, though some, who resie in Tunis, are. Similarly, lanlors who are not peasants themselves (again, this means that these lanlors o not consier agriculture to be their principal activity, though it may occupy a goo eal of their time) are consiere separately. For tenants, we istinguish between those who o not farm any lan of their own, an those who are owner-operators an simultaneously rent in. The main features istinguishing pure lanlors from pure tenants are househol size an lan ownership (see Table 4). Pure tenants correspon to large househols of ten members (mixe operator tenants also correspon to larger househols), while pure lanlors correspon to househols less than half this size. Conversely, pure tenants own very little lan (4:33 hectares), which they o not farm themselves (often this is grazing lan left fallow or olive trees, which are exclue from our sample), while pure lanlors own four times as much lan an simultaneously possess very little agricultural machinery (one of the reasons they o not operate the plots themselves) an are poorer. The pure lanlors in El Oulja are therefore not the agricultural capitalists sometimes epicte as constituting the lanlor sie in the sharecropping literature; they are also more epenent on non-agricultural income (an therefore more likely to have a principal activity other than agriculture) than pure owner operators an mixe operator lanlor househols. The role of agricultural capitalists is playe in El Oulja by non-resient lanlors, who also farm a substantial number of plots as owner-operators, because of their important stock of agricultural machinery. They also have very important lan holings an are signi cantly more eucate on average (this i erence in years of eucation stems from a number of lanlors who we tracke own in Tunis an who are also urban professionals). As shown in Table 3, the mix of their rente out plots between sharecropping an xe rental is slightly more skewe towars xe 12 Approximately one half of the stanar eviation of a given cost share parameter stems from within-operator househol heterogeneity. For example, for chemical fertilizer, the total stanar eviation of the associate cost share parameter is 14.32, with the within operator househol portion being equal to

15 rental than for other categories of lanlors, though this i erence is not statistically signi cant. Non-peasant lanlors, for their part, possess little agricultural machinery. One particularity of the sharecropping contracts to which non-resient lanlors are a party is that they involve greater subsiization on their part of plowing, family labor, an especially hire labor (see Table 3). Seen from the tenant sie, mixe tenant operators bene t from a greater level of cost-sharing for plowing than their pure tenant counterparts. Mixe operator lanlors i er from pure owner operators again in terms of their relative enowments of lan an labor: the former own more lan an come from slightly smaller househols than the latter; they also possess less agricultural machinery. Finally, note that the heas of pure tenant househols are signi cantly younger than the heas of all of the lan owning categories, with mixe lan owner tenants lying somewhere in between: this might be interprete as evience of some form of agricultural "laer". Pure tenants are also much more involve in livestock than other types of househols Naive speci cations In the village, the mean i erence in the value of log output per hectare between sharecroppe an other plots is equal to 0:564 with a stanar error of 0:36 (p value = 0:118). If one compares sharecroppe plots with those uner xe rental contracts, the corresponing i erence is 1:033 with a stanar error of 0:46 (p value = 0:028). Once one controls for soil type, irrigation status, an plot size, these i erences are no longer statistically signi cant at the usual levels of con ence, though the sharecroppe- xe rental i erential ( than the sharecroppe versus all other plots i erential ( 0:187, s.e. = 0:34). 0:709, s.e. = 0:42) remains larger In column 1 of Table 5, we present an initial estimate of the i erence in the logarithm of the value of output per hectare inuce by sharecropping while accounting for operator householspeci c e ects. 13 This is essentially the proceure followe by Bell (1977) an Shaban (1987), as applie to the supply equation given in (20), where the vector (; ) has been collapse into a "sharecroppe plot" ummy, that takes on the value of one when the plot is farme uner a sharecropping contract, an zero otherwise. The speci cation inclues a set of plot controls (4 soil type ummies, an irrigate plot ummy, 8 crop ummies), an log plot area. 14 In the interests of conciseness, we limit our presentation in the two Tables that follow to the coe cient associate with the sharecroppe plot ummy. Our initial result suggests that there is no statistically signi cant i erence in the value of output per hectare cause by sharecropping (the i erential is 0:487, s.e: = 0:53), while controlling for unobserve operator househol heterogeneity. 15 Moving 13 We report results that control for househol-speci c, time varying unobservable heterogeneity (given that we have two time perios, over which we cannot, however, follow plots, since their e nition changes each year). A more restrictive speci cation in terms of time-invariant househol-speci c e ects yiels similar results here, an in all that follows. 14 The nine crops cultivate in the village are wheat, other grains, potatoes, onions, garen vegetables, tomatoes, beetroot, melon an foer. 15 We report t-statistics in parentheses in the Tables in orer to facilitate their reaing. In the absence of the crop ummies, the i erential is 0:284, s.e.= 0:55. One obtains roughly the same results for all of the columns of Table 5, when one replaces the sharecropping ummy with the average value of i over all factor inputs, each iniviual i being weighte by its share in total costs. Note that while the null hypothesis of a zero i erential is not rejecte, this is not the same as saying that the absence of a i erential is rejecte. This is because the stanar error associate with the coe cient on the sharecroppe plot ummy reporte in column 1 of Table 5 is su ciently large that an Anrews (1989) inverse power function calculation inicates that a i erence of 1:769 can be con ently rejecte (i.e., the test has high power against this alternative), while a i erence of 0:884 cannot 15

16 to househol-speci c ranom e ects (which are not rejecte by the appropriate Hausman test, 2 15 = 14:560, whose p-value is 0:483) yiels a i erential of 0:461, but which remains statistically inistinguishable from zero (p-value= 0:165) at the usual levels of con ence. In light of the theoretical iscussion of part 2, it shoul be obvious that this rst equation is grossly mispeci e, rst in terms of functional form (this is not a well-speci e supply equation), an secon in terms of the patent enogeneity of contractual choice. It is therefore not surprizing that the ensuing results are inconclusive Instrumenting contractual choice The theoretical moel presente in part 2 of the paper suggests that there are two potential sources of instruments with which to enogenize contractual choice. Recall that optimal factor inputs (in their reuce form incarnation) are given by X = X (T;!; T ), where T is the reservation utility of the tenant, assume to be a function of tenant an lanlor characteristics. While one cannot instrument contractual choice using operator (tenant) characteristics since all variables are alreay expresse in terms of eviations with respect to operator speci c means (the ensuing variables woul be equal to zero by construction), one can use lanlor characteristics, as long as we have enough rente out plots in the sample, an as long as a su cient number of tenants interact with more than one lanlor. This is essentially an empirical issue in terms of the variance of lanlor characteristics, once they are expresse as eviations with respect to operator speci c means. 16 As shoul be clear from the summary statistics presente in Table 2, there are a su cient number of tenants who engage in contractual relationships with more than one lanlor for there to be substantial within operator househol variation in lanlor characteristics. In general, the within operator househol stanar eviation in lanlor characteristics accounts for more than one half of the total stanar eviation of the variables in question. Despite this promising aspect of the ata in terms of within operator househol variation in the potential instruments, we shall be particularly sensitive to "weak instrument" concerns an subject our statistical proceures to a number of tests esigne to assess the "strength" of our propose instruments. In column 3 of Table 5 we present IV estimates of the impact of sharecropping on output per hectare, while controlling for operator househol-speci c e ects, in which contractual choice is instrumente using within operator househol variation of twelve lanlor characteristics: nonagricultural income, lanownership, resiency status, occupational status (peasant or not), years of schooling, net wealth, househol size, prime-age females in househol, value of livestock, value of agricultural machinery, age, an pension income. All other aspects of the speci cation are ientical to the within estimation results presente in column 1 of the same Table. 17 As with the within results, the point estimate an its associate t-statistic inicate that there (the test has low power against this alternative), an a i erential of 88% in the value of output per hectare as a result of sharecropping is certainly not inconsequential. The use of inverse power functions was suggeste to us by Hanan Jacoby, whom we thank. 16 This is particularly true in that, for owner operator plots, in which the tenant an the lanlor are one an the same, the propose instruments are zero by construction. 17 In our empirical work, we experimente wiely with i erent combinations of our instrumental variables, incluing much more narrowly focuse sets of IVs. We restrict our iscussion here to an instrument set that is broa enough to allow one to instrument all of the cost-sharing elements of the contract iniviually in section 3.3 of the paper, so as to rener the results here an those reporte later broaly comparable. 16

17 is no statistically signi cant impact of sharecropping on output per hectare. This is true whether one consiers conventional 2SLS estimation, the bias-ajuste 2SLS (B2SLS) Nagar estimator suggeste by Donal an Newey (2001), limite information maximum likelihoo, or the Fuller (1977) estimator (with the "Fuller constant" set equal to 1). Although these results are broaly in line with the picture painte by the within proceure, they must be hanle with caution in that the test of the overientifying restrictions sounly rejects, be it base on conventional 2SLS or on the LIML/Fuller estimates. Since it is well-known that tests of overientifying restrictions ten to be erratic in the context of weak instruments in particular the size of the test may be signi cantly larger than its nominal value it is to this issue, an to a number of recent tests esigne to assess the valiity of instrumental variables in such a context, that we now turn. 3.2 Robustness The weak instruments null As is by now well known, weak instruments can lea to severe bias in IV estimation, an this bias oes not vanish even with large sample sizes. 18 As a result, much e ort has recently been evote to ning ways of iagnosing situations in which weak instruments may be a concern. The stanar response up until now has been to present the Shea (1997) R 2 an F statistics from the "partialle out" reuce form. As shoul be clear from the values reporte in column 3 of Table 5, our propose instruments o not appear to be weak in that both the R 2 an F statistics are above the usual critical values (the stanar cuto value of the partial F statistic is usually hel to be aroun 10). Note however, that recent work, such as Cruz an Moreira (2005), has shown that such iagnostic tests can be extremely poor inicators of instrument weakness. This leas us to another test, esigne to simultaneously assess instrument orthogonality an "strength", an which has the aitional avantage of being istributionally base The strong instruments null The spirit of the Hahn an Hausman (2002a) approach is i erent from that of the R 2 an F statistic iagnostics, which are implicitly base on the null hypothesis of weak instruments. In contrast, Hahn an Hausman (2002a) base their proceure on the null of strong instruments. Consier the B2SLS estimator, which is an example of a k-class estimator, of which conventional 2SLS, the Fuller (1977) estimator an LIML are special cases. As an illustration, consier the simple situation, as in the speci cation presente in column 3 of Table 5, in which there is one jointly enogenous right-han-sie (RHS) variable (the sharecroppe plot ummy), enote by y 2. Consier a structural equation of the form: y 1 = y 2 b + "; (38) in which all of the preetermine variables have been "partialle out," an " is the isturbance 18 See the excellent surveys by Stock, Wright, an Yogo (2002) an Hahn an Hausman (2003), an a recent very short primer on the ensuing biases by Hahn an Hausman (2002b). 17

18 term. Then the k-class instrumental variables estimator for the parameter b is e ne by: b B2SLS = y0 2P Z y 1 y 0 2M Z y 1 y 0 2 P Zy 2 y 0 2 M Zy 2 ; (39) where Z is the K imensional matrix of exclue instruments (in column 3 of Table 5, K = 12) an M Z = I Z(Z 0 Z) 1 Z 0 = I P Z is the iempotent "anihilator matrix". For = [(K 2)=n] = [1 (K 2)=n] where n is sample size, we obtain the B2SLS estimator propose by Donal an Newey (2001), whereas = 0 correspons to conventional 2SLS. 19 The Hahn an Hausman (2002a) test for the valiity of the instrumental variables is constructe by running the B2SLS regression in its usual "forwar" form, an comparing the result to that obtaine by running the "reverse" regression, in which the jointly enogenous RHS variable (the "share" ummy here) is move to the LHS, an the epenent variable (log output per hectare here) is entere on the RHS. The reverse B2SLS estimator is given by: b RB2SLS = y0 1P Z y 1 y 0 1M Z y 1 y 0 1 P Zy 2 y 0 1 M Zy 2 : (40) The basis for their test is that, if the instruments are orthogonal to the isturbance term in the structural equation an if they are "strong", stanar rst-orer asymptotics imply that there will be very little i erence between the results one obtains using the forwar (b B2SLS ) or reverse regressions (b RB2SLS ). The test, referre to as the m 2 test statistic, is stanarize by using a secon-orer expression for the variance of the i erence between the forwar an reverse estimators, an can be rea as a simple t-statistic. 20 b 2 = p n(b B2SLS b RB2SLS ), an: More formally, m 2 = b 2 = p bw 2 where 2(K 1)(n 1) 2 4 ";LIML bw 2 = h i 2 ; (41) (n 1)b 2 LIML y2 0 P Zy 2 ( K 1 n K )y0 2 M Zy 2 where b LIML is the LIML estimate of b, an 2 ";LIML is the variance of the resiuals of the structural equation estimate by LIML. 21 In column 3 of Table 5, we therefore report both the forwar an reverse B2SLS estimates of the coe cient associate with the sharecroppe plot ummy. 22 For our propose instrument set, 19 Note that B2SLS only becomes a meaningful alternative to 2SLS once the egree of overienti cation is strictly greater than 1 since B2SLS is ientical to 2SLS when K = Asymptotic properties of the test are presente in Hausman, Stock, an Yogo (2004), an the Montecarlo evience shows "that the weak-instrument asymptotic istributions provie goo approximations to the nite sample istributions for samples of size 100." The B2SLS estimator was originally propose by Nagar (1959), hence the name often ascribe to it. 21 We o not present the m 1 statistic base on forwar an reverse 2SLS. Its use is no longer recommene, following a number of Montecarlo exercises that have been performe since publication of the original paper. We are grateful to Jerry Hausman for pointing this out to us. Note also that one can replace the LIML estimates of the nuisance parameters by their Fuller estimator counterparts. This oes not lea to any appreciable quantitative i erences in our results. 22 Note that the LIML estimate is expecte accoring to theory to fall between the forwar an reverse B2SLS results because it is an optimal linear combination of these two estimators: this is not the case in the results presente in Table 5 (it will be in those presente in Table 6), which constitutes another informal inication of problems with the speci cation. The use of Bekker (1994) stanar errors, along with the Fuller estimator, oes not change the results of our statistical inference. We also estimate these equations using the Jackknife instrumental variables estimator (JIVE) propose by Angrist, Imbens, an Krueger (1999) an for which some simulation evience exists (see Blomquist an Dahlberg (1999), who also present simulation evience in the weak instruments case for LIML). 18