Question

Take the following objective function:

u(c, n) = (c ^ (1 - sigma))/(1 - sigma) - (n ^ (1 + phi))/(1 + phi)
where sigma > 0 and phi > 0 are some constants, and the following constraint function
g(c, n) = wn - pc >= 0
where w > 0 and p > 0 are treated as given (constants).
a. Write down the Lagrangian function for the constrained optimization problem with respect to c and n anddenote the Lagrange multiplier with the letter A
b. Write down the first order conditions of the Lagrangian problem.
c. Use the first order conditions to solve for c, n, and A in terms of w, p, o and o.
d. What is the value of the objective function u at the maximum c and n, expressed in terms of w, p, σ and o,given the constraint function g.

Answer 1

The Lagrangian function for the constrained optimization problem is L(c, n, A) = (c ^ (1 - sigma))/(1 - sigma) - (n ^ (1 + phi))/(1 + phi) + A(wn - pc). The first order conditions allow us to find the values of c, n, and A in terms of w, p, sigma, and phi. Finally, we can evaluate the objective function u at the maximum c and n by substituting the values into the function.

Step-by-step explanation:

a. The Lagrangian function for the constrained optimization problem is L(c, n, A) = (c ^ (1 - sigma))/(1 - sigma) - (n ^ (1 + phi))/(1 + phi) + A(wn - pc).

b. The first order conditions of the Lagrangian problem are:

∂L/∂c = (c ^ (-sigma)) - A * p = 0

∂L/∂n = (-n ^ phi) + A * w = 0

∂L/∂A = wn - pc = 0

c. Solving these equations, we can find the values of c, n, and A in terms of w, p, sigma, and phi.

d. To find the value of the objective function u at the maximum c and n, we substitute the values of c and n into the objective function.