To simplify the analysis, for the rest of this lecture we assume a one person household. In this case the household’s maximization problem is: Max c, ℓ ,L f ,L m ,L h ≥ 0 U(c, ℓ ) (10) subject to: pc = pF(L f + L h , E A ) - wL h + wL m (11)
8 ℓ + L f + L m = E L (12) L m ≤ M (13) There are two possible cases. In the first case, equation (13) is not binding, so that the “market failure” in the labor market doesn’t hold. In this case we are back to the standard model (the simplified version with no land market) and separability still occurs. That is, the household maximizes profits without any consideration of its consumption decisions and then maximizes utility taking the profits as “exogenous” income. In this somewhat simplified model one can derive several testable hypotheses. First, if we assume further that the production function has constant returns to scale (CRS) then we have: F(L, E A ) = E A F(L/E A , 1) = E A f(L/E A ) where f ′ ( ) > 0 because ∂ F( )/ ∂ L = f ′ ( ) and f ′′ ( ) < 0 because F ′′ ( ) < 0. The F.O.C. for labor implies that w = pf ′ (L/E A ). Thus for all farmers facing the same wage w the labor to land ratio L/E A will be the same. The yield per unit of land will also be the same because the
9 yield is simply F(L, E A )/E A = f(L/E A ). Both of these properties of this model can be tested using data on farms and on local wages. In the second case the household wants to sell an amount of its labor in the local labor market that equals or exceeds M. In this case (13) is binding so L m = M, L h = 0 and L f = E h – M - ℓ . The maximization problem becomes: Max c, ℓ ≥ 0 U(c, ℓ ) (14) subject to: pc = pF(E L - M - ℓ , E A ) + wM (15) The first order conditions can be derived by setting this up as a Lagrangean, that is we want to maximize: ℒ = U(c, ℓ ) - λ (pc - pF(E L - M - ℓ , E A ) + wM) with respect to c, ℓ and λ . The F.O.C. are: U c ′ - λ p = 0 U ℓ ′ - λ pF L ′ = 0 pc = pF(E L - M - ℓ , E A ) + wM
10 The first two F.O.C. imply that U ℓ ′ / U c ′ = F L ′ . This means that the household’s production and consumption decisions are jointly determined, as opposed to first maximizing profits without regard to the utility function or consumption decisions and then maximizing utility. Thus household consumption and production decisions are not separable. The constrained optimization can be shown in Figure 1. It is also useful to show it in a separate figure. A B Slope = w/p c L f + M L, - ℓ M O
11 In this diagram the starting point is the constrained trade whereby the household provides M units of labor and can trade this for M(w/p) units of consumption. This is point O. After that, the household has to decide how much labor to put in the farm in order to maximize utility. For each choice of L f leisure will be E L – M – L f and consumption will be M(w/p) + F(L f , E A ). This is point A in the diagram. In general, at this point slope will not be equal to w/p. Also, at this point the consumption and production decisions are made simultaneously.
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