a) Assume u (x1, x2) = (x1) ^a, ((x2) ^ (1-a)). Given M, P1 and P2 deduce the demands for two goods: Solve for MU1, MU2 and the MRS. Now use tangency condition MRS=- (p1/p2) together with budget line to solve for x1(M,P1,P2) and x2(M,P1,P2)
b) Now assume a =1/2. Further, assume M=12, P1=2 and P2=2. Sketch the budget set and show the optimal point selected by this consumer (using your demands in a)). Include reasonable sketch of an indifference curve through the optimal point.
c) Keep all parameters as in b) same except now raise P1 to 4. Sketch the new budget set and show new optimal point selected by this consumer. Include reasonable sketch of an indifference curve through this optimal point.
d) Now set a=1/3 but go back to original prices and income of b). Sketch budget set and illustrate the optimal point selected by this consumer. Include a reasonable sketch of an indifference curve through this optimal point.
e) Why the values of x2 the same in b) and c)? Why are the values of x2 different in b) and d)?