Find the demand functions xlowastp m and ylowastp m find

Question

Consider the utility maximization problem

max  x^a + y          subject to px + y = m

for x, y ≥ 0 and where all constants are positive and a ∈ (0, 1).

The optimal choices of x and y are denoted x^∗ and y^∗ and can be expressed as demand functions of the variables p and m; that is, x^∗ = x^∗ (p, m) and y^∗ = y^∗ (p, m). Similarly, the maximal utility function (often called the value function or indirect utility function) is given by

U^∗ (p, m):= U(x^∗(p, m), y^∗(p, m))

(a) Verify that any stationary point of the Lagrangian L will yield a global maximum of the optimization problem (Hint: Use the convexity/concavity of the Lagrangian).

(b) Find the demand functions x^∗(p, m) and y^∗(p, m).

(c) Find the partial derivatives of the demand functions with respect to p and m, and check their signs.

(d) How does the optimal expenditure on the good x vary with p? (That is, compute the elasticity of px^∗ (p, m) with respect to p).

(e) Interpret your work above when a = 1/2. What are the demand functions in this case? Verify that, in this case,

∂U^∗/∂p = -x^∗(p, m).