5.2.1 Model
The agricultural household is assumed to solve the following utility maximization problem.
T C T ThMax, , f, m U = U(Th , C; J) (5-1)
subject to:
C = g (Tf ;p, Zf ) + b (Tm ; Hm, Zm ) + V (5-2)
T = Th + Tf + Tm (5-3)
Tm ≥ 0, (5-4)
where y is off-farm earning function and
other variables are as defined in Chapter 4.
This model differs from the one in Chapter 4 in that it assumes a general off-farm earning function in the form of b (Tm; Hm , Zm ) instead of wm (Hm, Zm ) Tm. As mentioned in the end of Chapter 4, institutional conditions can make the form of y to differ from the simple form of
‘constant wage rate multiplied by work time’(wm Tm ). Through a general earning function, we can develop a more general analysis about the labor supply and demonstrate what restriction is imposted by the assumption of constant wage rate on the labor supply behavior.
5.2.2 Participation Condition
The optimality conditions can be obtained by constructing Lagrangian function5:
5 Kuhn-Tucker conditions are sufficient for optimality if the restrictions are quasiconvex in the choice variables. It requires that the off-farm income function is concave or is not ‘extremely’ convex. See Intriligator (1981), p.70. We assume that this curvature condition is met.
L = U(Th,C) + τ ( T - Th - Tf - Tm ) + λ ( g (Tf; Zf ) + b(Tm; Hm, Zm ) + V - C) + θ Tm
(5-5)
and applying Kuhn-Tucker condition to it.
∂
∂L τ
T U 0
h
= 1− = (5-6)
∂
∂L λ
C =U2− =0 (5-7)
∂
∂L τ λ
T g
f
= − + 1 =0 (5-8)
∂
∂ L τ λ θ
T b
m
= − + 1 + =0 (5-9)
∂∂θL
= Tm ≥ 0, θ ≥ 0, ∂
∂θL • θ = 0 (5-10)
in addition to (5-2) and (5-3)
These conditions are exactly the same as the optimality conditions in Chapter 4 except that wm
is replaced with b1 (Tm; Hm, Zm ). By applying the same logic as in Chapter 4, we can see that whether off-farm labor supply is positive or not depends on whether b1 (0;Hm,Zm ),
i.e. b1 evaluated with Tm = 0, is greater than the shadow price of time w0 or not. As it is shown in Chapter 4, w0 is obtained from the solution to the maximization problem in which the off-farm work is restricted to zero. In economic terms, the household decides to supply off-off-farm labor if and only if the initial marginal off-farm income is higher than the shadow price of time from ‘full-time farming’. Therefore,
Tm > 0 if i* >0 and Tm = 0 if i* ≤ 0, (5-11) where i*≡ b1 (0; Hm, Zm ) - w0 (V, T, Zh, p, Zf ) (5-12)
The variables Hm and Zm would affect the initial marginal off-farm earning in the same way as they were assumed to affect wm in Chapter 4. Thus, the effects of exogenous variables, summarized in Table 4-1, hold also in this chapter.
5.2.3 Farm Work Decisions in Case of No Off-farm Work
In case of no off-farm work, optimality conditions are simplified to:
g1 ( Tf ) = w 0 (5-13)
U U
1 2
( , ) ( , ) T C T C
h h
= w0 (5-14)
C + w0 L = w0 T + [ g(Tf ) - w0 Tf ] + V. (5-15)
T = Th + Tf (5-16)
which are identical to the system of (4-24). Therefore, the discussion in Chapter 4 about the
‘full-time farmer’ holds here, too. The economic price of time (w0) is a function of all exogenous variables except Hm and Zm, i.e.;
w0 = w0 (V, T, Zh, p, Zf ) (5-17)
Thus, the farm work time (Tf ) is also a function of all exogenous variables except Hm and Zm . The reaction of farm work time to the changes in exogenous variables, which is the main concern of this chapter, can be analyzed on the ground of the determination of farm work time as a derivative of profit function:
Tf = - π*w (w0 (V, T, Zh, p, Zf ) , p, Zf ) (5-18-a)
= Tf (V,T, Zh, p, Zf ) (5-18-b)
Differentiating (5-18-a) with respect to an exogenous variable, we get:
∂
∂ π ∂
∂ π
T k
w k
f
ww wk
= −( * 0 + * )
= - π π
Note that the result of the comparative static analysis on the shadow price of time (∂
∂ w
k
0 ) in (4-34) in the subsection 4.2.5 is used. It was also shown in the same subsection that the first term within the parenthesis in the second line is the compensated change in the shadow price.
Thus, the terms within the parenthesis in the third line and, equivalently, the first term in the last line are the compensated change in farm work time. The second term in the last line can be interpreted as the reaction of farm work time to the change in full income which in turn is caused by changes in the exogenous variables k. Applying this general formulation and assuming that both home time and consumption are normal goods and also that farm work time of household is normal input as well as gross complements for other inputs, following results can be obtained.
∂
The first terms in (5-20) through (5-24) are compensated changes while the second terms are income effects. Although the signs of (5-23) and (5-24) are indefinite, they show that compensated change effects and income effects themselves have definite signs. An increase in output (input) price has a positive (negative) compensated change effect and a negative (positive) income effect on farm work time. An increase in a fixed factor, which is a complement to family labor, has a positive compensated change effect and a negative income effect on farm work time.
5.2.4 Farm Work Decisions in Case of Positive Off-farm Work
If off-farm work time is positive at the optimum, then the optimality conditions are:
g1 (Tf ) = w0 (5-25)
If b1 is independent of Tm and exogenously given as in Chapter 4, then, according to (5-26), w0 is equal to wm and the system (5-25) through (5-29) becomes identical to the system (4-13) in Chapter 4. In this case, farm work time is a function of production relevant variables (wm, p, Zf ) only and therefore, recursivity holds. However, if b1 is a function of Tm, then the equations (5-25) through (5-29) can be solved only simultaneously and, therefore, the
recursivity does not hold. In this case, optimal farm work time (Tf*) is a function of all exogenous variables.
Tf = Tf (w0 (V, T, Zh, p, Zf, Hm, Zm ), p, Zf ) (5-30-a)
= Tf (w0 (V, T, Zh, p, Zf, Hm, Zm ), p, Zf ) (5-30-b)
In the previous subsection, we applied the duality approach for comparative statics analysis. However, in this section, we employ the more ‘traditional’ approach via total differential of the system (5-25) through (5-29). This is due to the fact that if the off-farm earning function b is not assumed to be concave in off-farm work time (Tm), the condition (5-26) cannot be the first order conditions that characterizes a maximizing behavior.
5.2.4.1 Second Order Condition and Comparative Statics Analysis
If the non-negativity constraint (5-4) on off-farm work time is not binding at the optimum, the maximization problem is reduced to the one with only equality constraints (5-2) and (5-3).
Thus, Kuhn-Tucker conditions become identical to the first order conditions and the second order conditions for the maximization problem must hold. The second order conditions,6 applied to the model in this chapter, require that the sign of the border preserving principal minors of order 3 and 4 from the matrix of the second derivatives of the Lagrangian function (5-5), denoted as Lxx’ (x is the vector of the choice variables and the Lagrange multipliers (Th, C, Tf, Tm, τ, λ )), be negative and positive, respectively. It means that the two following conditions must hold.
SOC I: λ (g11 + b11 ) < 0 (5-31)
SOC II: λ2 g11 b11 + λ (g11 + b11) (U11 - 2 U U
1 2
U12 + U U
1 2
2
U22 ) >0 7 (5-32)
6 See Intriligator (1971) p.35
7 This condition is also presented in Kimhi (1989) to which a great part of the notation in this study is oriented.
It is possible to interpret the conditions SOCI and SOCII if we break down the original