Tenants, landlords, and soil conservation.

Condition (15) characterizes the tenant's optimal choice of effort, [e.sup.s], as can be seen by substituting condition (16) into condition (11). It differs from condition (3) in that k = 0 (compared to k > 0 in the first best). Comparison of conditions (3) and (15) indicates that effort under a share rental contract without conservation investment will exceed the first best: The marginal return to effort during the lease period [R.sub.e] is higher than the first best because [R.sub.ek] < 0 while the marginal reduction in land value at the end of the lease period is lower because [h.sub.ek] < 0 and V" < 0. We thus have

PROPOSITION 2 (Dubois 2002). When both landlord and tenant are risk neutral, under a share rental contract with no conservation investment, effort and land degradation are lower than under a cash rental contract with no conservation investment but exceed the first best.

When investment in durable conservation measures is feasible, the landlord's objective is to choose the tenant's share s, fixed payment t, and level of conservation investment k to maximize

(12') (1- s)[E.sub.[epsilon]] {R(e,k, [x.sub.0]) + [epsilon]} + t - I(k) + [beta][E.sub.[eta]]{V([x.sub.0] - h(e, k)) + [eta]}

subject to the tenant's incentive compatibility and participation constraints,

(13') [e.sup.f] = arg max{s [E.sub.[epsilon]] {R(e, k, [x.sub.0]) + [epsilon]} - t - C(e)}

(14') s[E.sub.[epsilon]] {R(e, k, [x.sub.0]) + [epsilon]} - t - C(e)} [greater than or equal to] [u.sub.0]

respectively. As before, the tenant's optimal choice of effort is increasing in the tenant's output share ([partial derivative][e.sup.f]/[partial derivative]s = [R.sub.e][[C.sub.ee] - s[R.sub.ee]] > 0) and decreasing in conservation investment ([partial derivative][e.sup.f]/ [partial derivative]k = s[R.sup.ek]/[[C.sub.ee] - s[R.sub.ee]] > 0).

The landlord again chooses the fixed payment or wage so that the participation constraint binds. The landlord's optimal choice of the tenant's share satisfies

(15') ([R.sub.e] - [C.sub.e] - [beta]V'[h.sub.e]) ([partial derivative]e/[partial derivative]s = 0.

The landlord's optimal choice of conservation investment satisfies

(17) ([R.sub.e] - [C.sub.e] - [beta]V'[h.sub.e]) ([partial derivative]e/[partial derivative]k + [R.sub.k] - [I.sub.k] - [beta]V'[h.sub.k] = 0

which, after substitution from equation (15'), becomes

(17') [R.sub.k] - [I.sub.k] - [beta]V'[h.sub.k] = 0

It is readily apparent that conditions (15') and (17') are identical to conditions (3) and (4). We thus have:

PROPOSITION 3. When both landlord and tenant are risk neutral, share rental contracts combined with investment in durable conservation measures are capable of achieving first-best effort and conservation.

Intuitively, share rental contracts provide landlords with three instruments to influence land degradation: investment in durable conservation measures k, the output share s, and the fixed payment t. Combining investment in durable conservation measures with the rental share gives landlords two instruments for influencing effort and conservation while the fixed payment is used to ensure that the tenant receives her reservation utility. Because the number of instruments equals the number of objectives, it is feasible to attain the first best.

Production and Conservation Investment with Risk-Averse Tenants

When tenants are risk averse and effort is non-contractible, landlords face a tradeoff between insurance and incentives. The first-best contract features paying the tenant a wage sufficient to equate the expected utility of income with the tenant's reservation utility, [u.sub.0], with effort and conservation investment set at the same levels as under risk neutrality, ([k.sup.*], [e.sup.*]). But when effort is noncontractible, a fixed wage provides insurance but too little incentive to exert effort while a cash rental contract provides incentive for effort but no insurance; share rentals offer a compromise between income insurance and incentives to exert effort.

Consider a share rental contract with the tenant's output share s and a fixed payment t, so that the tenant's income during the lease period is s[R(e, k, [x.sub.0]) + [epsilon]] - t - C(e). Let U(*) be the tenant's utility of income, concave in income as usual. The tenant's level of effort is chosen to maximize the expected utility of income earned during the lease period, [E.sub.[epsilon]]{U(s[R(e, k, [x.sub.0]) + [epsilon]] - t - C(e))}. Condition (11) characterizes this choice due to the additive stochastic specification. Thus, the tenant's effort is independent of the fixed payment t, increasing in the output share s([partial derivative]e/[partial derivative]s = [R.sub.e]/(s[R.sub.ee] - [C.sub.ee])), and decreasing in conservation investment k([partial derivative]e/[partial derivative]k = - S[R.sub.ek]/(s[R.sub.ee] - [C.sub.ee])) as before.

The landlord chooses contract terms s and t plus investment in durable conservation measures k to maximize

(18) (1 - s)[E.sub.[epsilon]]{R(e, k, [x.sub.0]) + [epsilon]] + t - I(k) + [beta][E.sub.[eta]]{V([x.sub.0] - h(e,k)) + [eta]}

subject to the tenant's incentive compatibility and participation constraints,

(19) [e.sup.r] = arg max{[E.sub.[epsilon]]{U(s R(e, k, [x.sub.0]) - t - C(e)}}

(20) [E.sub.[epsilon]]{U(s[R(e, k, [x.sub.0]) - t - C(e))] [greater than or equal to] [u.sub.0],

respectively.

As before, the landlord chooses the fixed payment so that the participation constraint binds. The condition characterizing landlord's optimal choice of the tenant's share can be shown to be

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting condition (11), condition (21) implies that the optimal share satisfies

(22) [s.sup.r] = [beta]V'[h.sub.e]/[R.sub.e] + [rho]([C.sub.ee] - [s.sup.r] [R.sub.ee])/[R.sub.2.sub.e]

where [rho] [equivalent to] [E.sub.[epsilon]](U'[epsilon]}/[E.sub.[epsilon]](U'}, the correlation between the random factor [epsilon] and the marginal utility of income, is negative due to risk aversion. As in the case of risk neutrality, the tenant's share of output is adjusted downward in order to make the tenant face the marginal cost of land degradation. The tenant's share is adjusted downward further by the factor p ([C.sub.ee] - s[R.sub.ee])/[R.sup.2.sub.e] in order to mitigate the disincentive effects of risk on effort.

The condition characterizing the landlord's optimal choice of conservation investment is

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where h is the shadow price of the tenant's participation constraint (20) and the simplification results from the fact that the firstorder condition for the fixed payment t implies [lambda][E.sub.[epsilon]]{U'} = -1. As in the case of a cash rental contract with a risk-neutral tenant, the landlord's investment in durable conservation measures is adjusted to take into account its effects on the tenant's effort in production during the lease period. In contrast to the case of a cash rental contract with a risk-neutral tenant, the adjustment aims at increasing effort, which tends to be excessively low due to risk aversion and the use of risk sharing in the face of moral hazard.

Conditions (21) and (23) together define the equilibrium levels of conservation investment [k.sup.r] and effort [e.sub.r] under this contract. It is clear that they are not equivalent to conditions (3) and (4), which define the first-best levels of conservation investment and effort even when tenants are risk averse. We thus have:

PROPOSITION 4. When tenants are risk averse, share rental contracts combined with investment in durable conservation measures are not capable of achieving first-best effort and conservation.

To compare the equilibrium levels of conservation investment and effort under a share rental contract with the first best when tenants are risk averse, consider a first-order approximation to conditions (3) and (4) around the equilibrium levels of conservation and effort under the share rental contract, ([k.sup.r], [e.sup.r]), which can be solved to yield

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The sign of [k.sup.*] - [k.sup.r] is indeterminate, indicating that landlords may either over- or underinvest in durable conservation measures. In contrast, [e.sup.*] - [e.sup.r] is positive, indicating that effort is always less than the first best, as one would expect.