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1.a) Suppose u(x1,x2) = x1a, x2(1-a) . Given M, P1, and P2 derive the demands for the two goods: Solve for MU1, MU2 and the MRS. Now use the tangency condition MRS =-p1/p2
together with the budget line to solve for X1 (M, P1, P2) and X2 (M,P1, P2). b) Now suppose a = 1. Further, suppose M 12, P1 = 2 and P2 = 2. Draw the budget set and show the optimal point chosen by this consumer (using your demands in a)). Include a reasonable sketch of an indifference curve through the optimal point. c) Keep all parameters as in b) the same except now raise Pi to 4. Draw the new budget set and show the new optimal point chosen by this consumer. Include a reasonable sketch of an indifference curve through this optimal point. d) Now set a = 1/3 but go pack to the original prices and income of b). Draw the budget set and show the optimal point chosen by this consumer. Include a reasonable sketch of an indifference curve through this optimal point.
2. A consumer has preferences characterized by the utility function u(x1, x2) = In 21 + x2. a) What type of preferences are these? Solve for an expression for this consumer's MRS. Sketch 3 different indifference curves for this consumer.
b) Suppose M = 15, P1 = 1, P2 = 3. Use the tangency condition MRS = - to solve for the optimal amount of good 1. Given this, determine the optimal amount of good 2. Sketch this optimal choice on a graph of the budget set. Include an indifference curve through your optimal point.
c) Now increase income to M = 21. Derive the new optimal choice and show it on a graph as in b)
d) Explain any difference between the points chosen in b) and c)
1) Consider a demand curve of the form Qd = 20-2p where Qd is the quantity demanded of a good and p is the price of the good. Also consider a supply curve of the form Qs = 2p-4 where Qs is the quantity supplied. Graph these curves. At what values of P and Q do these curves intersect?
2) Now suppose at each price individuals demand four more units of output, that is the demand curve shifts to Qdd = 24-2p. Graph this new curves. Graph this new curve on the diagram drawn in part 1) At what values of P and Q does the new demand curve intersect the supply curve identified in part 1)
#340153 on Dec 2023