Final answer:
The optimality condition from the household's one-period utility maximization problem suggests that as wages increase, the household is incentivized to provide more labor up to the point where the additional utility from increased income equals the loss from less leisure, embodying the principle of diminishing marginal utility.
Step-by-step explanation:
The household's utility maximization problem involves choosing consumption, Ct, and labor, Nt, to maximize their utility, represented by the utility function U = ln Ct + θt ln (1 - Nt), subject to the budget constraint Ct = wtNt. To solve for the optimality condition, we start by setting up the Lagrangian Λ considering the constraint that consumption is equal to wage times labor:
Λ = ln Ct + θt ln (1 - Nt) + λ(wtNt - Ct)
Where λ is the Lagrange multiplier. Taking the partial derivatives of the Lagrangian with respect to Ct, Nt, and λ and setting them equal to zero gives us the first-order conditions. For Ct, we get:
∂Λ/∂Ct = 1/Ct - λ = 0
For Nt:
∂Λ/∂Nt = -θt/(1 - Nt) + λ wt = 0
And for λ, we have the constraint Ct = wtNt.
Solving the first-order conditions, we get λ = 1/Ct, and substituting into the second equation gives:
- θt/(1 - Nt) + 1/Ct wt = 0
Solving for Nt, we find the optimal labor supply function. The effect of wt on Nt can be interpreted from this condition: as wt (the wage rate) increases, the household has an incentive to work more, increasing Nt, but only to the point where the additional utility from income equals the loss of utility from reduced leisure. This results from the trade-off between earning more income and having more leisure time. This embodies the economic principle of diminishing marginal utility, as well as illustrating the shifts in the labor-supply curve in response to changes in wages.