parameters (α, βs and βf). Without loss of generality, we focus on the landlord’s choice of Hi and Hf
without consideration to how this land is allocated among tenants. As long as tenants’ time constraints
(equations [11] and [17]) are not met, the tenants are indifferent to the size of plot they obtain.
Therefore, with a large enough number of tenants, the landlord is also indifferent to how the land is
distributed among tenants of each type once the choice of Hs and Hf is made.15 The optimization problem
is,
max L = F ( Tio, T 2 l∣S ) + [ (i-a )F (t 1 „t 2 ,,h,.s ) + β sHsJ
T1 oT 2 O, T1s, HsHf.a.β ,.β f
+ βfHf + (1-T1o - T2o- T1s)u + λ[1 - Tio - T2o - T1s] +
[18]
Λ s [OF ( T1 s .t2 ,tH,.S ) -12 s V - β sHs] + Λ f [F[tif. t2f, Hf ; s] - (tif + t2f ) v - β fHf]
where H = Ho + Hs + Hf . The first four terms are the landlord’s income from owner-farming,
sharecropping, fixed-rent leasing and non-farm employment. The fifth term is the landlord’s time
allocation constraint, and the next two terms are the reservation income constraints of agents. The landlord
takes tenants’ choices per-unit labor time inputs (z2s*. zif* .z2f* ) as given.
The first order conditions with respect to βf, βs and α are
β f : Hf [ Λ f - 1] = 0
βs : Hs [Λ s - 1] = 0 [19]
a :: F ( T1 s, 12 s, Hs ; S )[ Λ s - 1] = 0
From the first two first order conditions, we see that Λf =1 for fixed-rent farming to exist (Hf ≥ 0)
and Λs =1 for sharecropping to exist (Hs ≥ 0). The first order condition with respect to the share-rent also
tells us that Λs =1 if production takes place in the sharecropped land (F>0). Since the multipliers are non-
15 This indeterminacy arises from our assumptions of a production function that is linear homogeneous in T1, T2 and
H, and a non-binding time constraint for tenants. When these assumptions are relaxed, the landlord may choose a
small enough farm size for each tenant in order to maximize their time inputs. We abstract from this because our focus
is on the allocation of contracts.
11
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