Question

8. A consumer has the following utility maximization problem Max_{x,y} U(x,y) = x(y + 1) s. t. B = p_x x + p_y y Where x, y are the two consumption goods whose prices are p_x and p_y, respectively. Her budget is B. a) From the first order conditions (FOCs) find expressions for the demand functions x* = x(p_x, p_y, B) and y* = y(p_x, p_y, B) b) Find an expression for the indirect utility function U* = U(p_x, p_y, B). Verify that ??* = ?U/?B. The consumer's utility maximization problem could be recast as the following Minimize p_x x + p_y y s. t. U = x(y + 1) c) Find the values of x and y that solve this minimization problem from FOCs d) Now suppose the SOCs are automatically satisfied so you do NOT have to check them. Derive the expenditure function E* = E(p_x, p_y, U) and show that the values of x and y that solve this minimization problem are equal to the partial derivative of the expenditure function, i.e. ?E/?p_x and ?E*/?p_y, respectively.

          8. A consumer has the following utility maximization problem
Max_{x,y} U(x,y) = x(y + 1) s. t. B = p_x x + p_y y
Where x, y are the two consumption goods whose prices are p_x and p_y, respectively. Her budget is B.
a) From the first order conditions (FOCs) find expressions for the demand functions x* = x(p_x, p_y, B) and y* = y(p_x, p_y, B)
b) Find an expression for the indirect utility function U* = U(p_x, p_y, B). Verify that ??* = ?U*/?B.
The consumer's utility maximization problem could be recast as the following
Minimize p_x x + p_y y s. t. U* = x(y + 1)
c) Find the values of x and y that solve this minimization problem from FOCs
d) Now suppose the SOCs are automatically satisfied so you do NOT have to check them. Derive the expenditure function E* = E(p_x, p_y, U*) and show that the values of x and y that solve this minimization problem are equal to the partial derivative of the expenditure function, i.e. ?E*/?p_x and ?E*/?p_y, respectively.

8. A consumer has the following utility maximization problem
Maxx,y U(x,y) = x(y + 1) s. t. B = px x + py y
Where x, y are the two consumption goods whose prices are px and py, respectively. Her budget is B.
a) From the first order conditions (FOCs) find expressions for the demand functions x* = x(px, py, B) and y* = y(px, py, B)
b) Find an expression for the indirect utility function U* = U(px, py, B). Verify that ??* = ?U*/?B.
The consumer's utility maximization problem could be recast as the following
Minimize px x + py y s. t. U* = x(y + 1)
c) Find the values of x and y that solve this minimization problem from FOCs
d) Now suppose the SOCs are automatically satisfied so you do NOT have to check them. Derive the expenditure function E* = E(px, py, U*) and show that the values of x and y that solve this minimization problem are equal to the partial derivative of the expenditure function, i.e. ?E*/?px and ?E*/?py, respectively.

Added by Adriana C.

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