Exercises for week 6: Sharecropping
1)If tenant”s payment is w = a0 + a1Y and tenant pays to landlord (b0 + b1Y), then
w = a0 + a1Y = Y –(b0 + b1Y)
a0 = -b0
a1= -b1
2)Landlord: risk neutral
Farmer: risk averse
Landlord is maximizing profit to the tenant getting utility lever.
Y = e + η
E(η) = 0
U0 = Eu(w) – C(e) = E[u(a1Y + a0)] – C(e) = Eu(a1e + a1η + a0) – C(e)
Max E(Y – w(Y)) = E(Y - (a0 + a1Y) = E(Y(1 – a1) – a0)
a0, a1
s.t.: Eu(a1Y +a0) - C(e) = Eu(a1e + a1η + a0)– C(e)
L = (1 – a1) e(a1) - a0 + λ [Eu(a1e(a1) + a1η + a0)– C(e) – U0]
FOC:
(1)∂L
∂a1 = (1 - a1) e’(a1) - e(a1) + λ E[u’(w)(e(a1) + a1e’(a1) + η] –λ C’e’(a1) = 0
(2)∂L
∂a0 = - 1 + λ Eu’(w) = 0 → λ = 1
Eu’(w)
Put into (1):
Eu’(w)( e(a1) + a1e’(a1) + η] C’e’(a1)
→ (1 - a1) e’(a1) - e(a1) + Eu’(w)- Eu’(w) = 0
The contract requiers an effort level e choosen by the landlord:
π = E (Y – w(Y)) = E(Y – (a0 + a1Y) = (e + η – (a0 + a1(e + η)) = e + η- a1e -a1η - a1
3)Landlord cannot observe e, uses a1 and a0 to increase e.
If the effort of the tenant cannot be controlled by the landlord, the tenant has an incentive to undersupply his effort, because under SC he should pay some parts of output to the landlord.
Max U = E[U(a0 + a1Y)] - C(e) = E[u(a0 + a1(e + η) - C(e)
= E[u(a0 + a1e +a1η)] - C(e)
∂ u(w)
FOC: E ∂ w a1 – C’(e) = 0 → E u’(w)a1= C’(e)
* If: a1 = 0 , C’(e) = 0 , → e = 0
* If: a1 ↑ , → C’(e)↑ , → e ↑
* If: a1 = 1 , C’(e) = 1 , → e = 1
(Full efficiency if fixed rent).
4)
According to the Sharecropping contract tenant’s output is shared in same proportional
between the tenant and landlord (very often it’s 50%/50%).
By Marshall, the Sharecropping is inferior system (it is called Marshallian inefficiency).
If the effort of tenant can not be controlled by the landlord, tenant has an incentive to undersupply his effort, because he should pay some fraction of what he is producing to the landlord.
In the case if tenant should pay just fixed rent, he is keeping 100% of any extra output which was produced by him.
If we look at Figure1, which describes the effect of sharecropping on tenant incentives, we see that tenant receives only some part of the output, the effective return to the tenant is line OE, which is equal the production function multiplied by tenant’s output is share.
The different between this effective return and tenant’s cost OB is what the tenant is interesting in making is large as possible, because this is what the tenant receives from the deal.
Figure 1. Sharecropping contract and inefficiency.
Output
(cost)
A
C
BE
D
Labor
O
OB- production cost
OA- production function
OE- effective return to the tenant
.
When the tenant maximizes his effective return (net of cost), he generally does so at a labor input that is smaller that , the input lever that maximizes overall economic surplus.
Because the tenant receives only some fraction of output, his marginal return is smaller than the actual value of marginal product. Thus the tenant will desist from applying more labor: .
Using Sharecropping contract leads decreasing of production.
Tenant stops applying labor at a point when marginal product still exceeds marginal cost, so resulting outcomes is inefficient. And the reason is, instead of equating the marginal product of this labor to it’s marginal cost, he equates just some part ( for example half) the marginal product of his labor to it’s marginal cost.
5) Pr (e = 1 │Y)
YL / YM / YHe = 0 / 0.5 / 0.4 / 0.1
e = 1 / 0.5 / 0.1 / 0.4
Pr / 0.5 / 0.2 / 0.8
No, linear pay scheduale is not optimal. Because it is difficult for the landlord to
observe the tenants effort level.
Ex: if the outcome is M, it is only a probability that he has high effort of 0.2. The
landlord will then think that he did not make any effort, and he gets no
payment. This result that he will reduce e so he ends up with lower income.
The landlord will still choose linear pay schedule even if it isn’t optimal. He may
think that this will rise the incentives for the tenant to increase e.