Question

Income and substitution, Compensating Variation: Show your work in the steps below. Consider the utility function u(x,y)-x"y a. Derive an expression for the Marshallian Demand functions. b. Demonstrate that the income elasticity of demand for either good is unitary 1. Explain how this relates to the fact that individuals with Cobb-Douglas preferences will always spend constant fraction α of their income on good x. Derive the indirect utility function v(pxPod) by substituting the Marshallian demands into the utility function C. d. Find an expression for the expenditure function E(p*py,u) e. Find expressions for the Hicksian Demands. f. Suppose α β and that p,-1, p,-2 and 1-8. Use your expression for indirect utility to calculate the welfare level achieved, Uo-v(px,py,/) g. Suppose now the price of good x rose from px to p, 4. Define the Compensating Variation or required increase in income needed to compensate the individual for the price rise How much does the individual need to be compensated for the rise in price from p,-1 to px =4 ? Depict the situation graphically (very approximately). 2. Homogenous and homothetic functions. a. Show that the utility function U(x,y)-x®yß is homogenous of degree α + β Show that the same utility function is homothetic b. Show that if the production function F(K,L) is homogenous of degree 1 then we can write F(K,L)- FxK+F.L