(Shephard’s Lemma and Roy’s Identity) Suppose the utility function is u(x1, x2) = x 1 2 1 x 1 3 2 and the budget constraint is p1x1 + p2x2 = m. (a) Solve utility maximization problem for Mashallian demand and indirect utility function. (b) Solve Expenditure Minimization problem for Hicks demand and Expenditure function. (c) Use this example to verify Shephard’s Lemma. (d) Use this example to verify Roy’s Identity
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Akash M.
Assume that utility is given by U(x, y) = x^0.3 * y^0.7 and Income = I, price of good x = px and price of good y = py. Show your work for each of the following parts: (a) Use the uncompensated demand functions to compute the indirect utility function and the expenditure function (E) for this case. (b) Use the expenditure function calculated in part (a) together with Shephard's lemma to compute the compensated demand function for good x. (c) Use the results from part (b) together with the uncompensated demand function for good x to show that the Slutsky equation holds for this case.
A consumer has the following utility maximization problem: Max 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 lambda* = dU*/dB. 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. dE*/dpx and dE*/dpy, respectively.
Supreeta N.
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