Suppose that an individual’s utility function for consumption, c, and leisure, h, is given by and assume, for simplicity, . This person is constrained by two equations: (1) an income constraint that shows how consumption can be financed,, Where n is non-labor income.
(2) a total time constraint. Set up the Lagrangian expression for this utility-maximization problem, derive F.O.C, find the optimum and, and the labor supply function for this person
To set up the Lagrangian expression for this utility maximization problem, we need to introduce a Lagrange multiplier, λ, for the income constraint and another Lagrange multiplier, μ, for the total time constraint.
The Lagrangian expression is given by:
L(c, h, λ, μ) = U(c, h) + λ(I - c - T(h)) + μ(W - h)
where U(c, h) is the utility function for consumption and leisure, I is the total income, T(h) is the amount of time spent on leisure, and W is the total available time.
To find the first-order conditions (F.O.C.), we take the partial derivatives of the Lagrangian with respect to c, h, λ, and μ and set them equal to zero:
∂L/∂c = ∂U/∂c - λ = 0 (1)
∂L/∂h = ∂U/∂h - λ∂T/∂h - μ = 0 (2)
(I - c - T(h)) = 0 (3)
(W - h) = 0 (4)
From equation (3), we can determine the consumption level, c, as:
c = I - T(h) (5)
Substituting equation (5) into equation (1), we have:
∂U/∂c = λ (6)
From equation (4), we can determine the leisure level, h, as:
h = W (7)
Substituting equation (7) into equation (2), we have:
∂U/∂h - λ∂T/∂h - μ = 0 (8)
Solving equations (6) and (8) simultaneously will give us the optimal levels of consumption and leisure.
Finally, to find the labor supply function for this person, we use the fact that labor supply (L) is equal to the total time available (W) minus the leisure time (h):
L = W - h (9)