Assume that a consumer has the utility function U(x,y) = 3x+y, where x and y represent the quantities of two goods, X and Y. The consumer has I=$60 to spend on the two goods, and good Y costs py=$2 per unit. The price of good x, px, is also exogenous. Assume px=$4.
(a) With good X on the horizontal axis and good Y on the vertical axis, draw the consumer’s budget line.
(b) Calculate the first-order condition for this problem.
(c) On your budget line, show the optimal point. Justify your answer carefully.
(d) On a new diagram, show the budget line and optimal point if px=$10.
(e) Calculate the consumer’s demand function x(px), which shows her optimal quantity of X demanded, as a function of its price.
(f) With px on the vertical axis, carefully graph the demand function that you found in part (f), showing the specific coordinates of the points with px=$2, px=$4, px=$6, px=$8, and px=$10.
(g) Calculate the consumer’s elasticity of demand for X, at the points where px=$4 and px=$8.
(i) Calculate a function V(px), which shows the maximized value of the consumer’s utility, given any px>0. Is this function continuous? Draw a graph of your function, showing the exact coordinates of V(px) for the same values of px that were specified in part (g).