Question

Suppose that the utility function of the consumer is the following

U = logCt + χ 2 (1 − Nt) 2 where Nt is the amount of hours worked and we assume N¯ = 1 The budget constraint of the consumer is the one seen in class, but assuming T = 0 1. Find the first-order necessary conditions of the consumer’s problem.

Answer 1

To find the first-order necessary conditions of the consumer’s problem, we need to calculate the partial derivatives of the utility function with respect to the variables. We set up the Lagrangian function and take the partial derivatives with respect to Ct, Nt, and λ, and set them equal to 0. By solving these equations, we can find the first-order necessary conditions for the consumer's problem.

Step-by-step explanation:

To find the first-order necessary conditions of the consumer’s problem, we need to calculate the partial derivatives of the utility function with respect to the variables. Here, the utility function is U = logCt + χ 2 (1 − Nt) 2, where Nt is the amount of hours worked. Now, let's find the first-order partial derivatives:

∂U/∂Ct = 1/Ct

∂U/∂Nt = -2χ(1-Nt)

Since T=0, the budget constraint will be Ct = Nt. Next, we can set up the Lagrangian function:

L = U + λ(Ct - Nt)

To find the first-order necessary conditions for the consumer's problem, we need to take the partial derivatives of the Lagrangian function with respect to Ct, Nt, and λ, and set them equal to 0:

∂L/∂Ct = ∂U/∂Ct + λ = 0

∂L/∂Nt = ∂U/∂Nt - λ = 0

∂L/∂λ = Ct - Nt = 0

By solving these equations, we can find the first-order necessary conditions for the consumer's problem.