Question

Consider a static labour supply model for an individual. Assume that the person works a positive number of hours. Assume that the utility function is of the following form: U= x0.5 +δ(T-h)(0.5) where x is consumption and h is hours of work, and δ is a parameter of the utility function. The person is paid a wage of w for each hour worked. The person has no other source of income other than employment earnings. Assume that the price of x equals 1.

A. Specify the utility maximization problem, write down the Lagrange function, and solve for the first order conditions.
B. Solve for the MRS condition between consumption and hours of work.
C. Assume that δ=2 if the person has a child living at home, and δ=1 otherwise. Explain (and either show mathematically or graphically) how this will affect the optimal number of hours of work of the person.

Answer 1

The utility maximization problem in this model seeks to maximize the utility function and find the optimal values of consumption and hours of work. The marginal rate of substitution measures the rate at which the individual is willing to trade off consumption for leisure. When the person has a child at home, the optimal number of hours of work will be reduced, shifting the labor supply curve backward.

Step-by-step explanation:

The utility maximization problem in this labor supply model is to maximize the utility function U = x0.5 - δ(T-h)0.5, where x is consumption, h is hours of work, T is total available time, and δ is a parameter. The Lagrange function for this problem is L = x0.5 - δ(T-h)0.5 + λ(w(h-T)) , where λ is the Lagrange multiplier and w is the wage. The first order conditions are ∂L/∂x = 0, ∂L/∂h = 0, and ∂L/∂λ = 0. Solving these equations will give the optimal values of x and h.

The marginal rate of substitution (MRS) between consumption and hours of work is given by ∂U/∂x / ∂U/∂h. This measures the rate at which the individual is willing to trade off consumption for leisure. The MRS can be determined by taking the partial derivatives of the utility function with respect to x and h and dividing them.

If δ=2 when the person has a child living at home, and δ=1 otherwise, this means that the person values leisure more when they have a child at home. Mathematically, this will affect the optimal number of hours of work by reducing it. Graphically, it will shift the labor supply curve backward, indicating that the person will choose to work fewer hours at the same wage.