1. The problem is to maximize the function f(x;α) = αx2 - (20α+2)x, subject to the constraints 0 smaller or equal x smaller or equal 10, where α is an exogenous constant. For parts (d) and (e), you may want to check your answers by confirming that they reproduce your answers to (a) and (b).
(a) Assume α=-1. State the first-order condition, and find any points that satisfy it, within the constraints. Does f(x;1) satisfy the global second-order condition? What value of x solves the problem? (Justify your answer carefully.) What is the maximized value of f(x;1)?
(b) Repeat part (a), this time assuming α=1.
(c) For which values of α is there a point that satisfies the FOC, within the constraints?
(d) Calculate a function x(α), which shows the solution to the problem, given any real number α. (It may be helpful to consider the range of values of α identified by part (c), separately from other values of α.) Draw a graph of your function, showing the exact coordinates of at least three points that have different values for x(α).
(e) Calculate a function M(α), which shows the maximized value of f that can be achieved, given any real number α. Is this function continuous? Draw a graph of your function, showing the exact coordinates of M(α) for the same values of α that you used for the same purpose in part (e).
2. 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.
(a) With good X on the horizontal axis and good Y on the vertical axis, draw the indifference curve for utility level U=6. Show the exact coordinates of at least two points on the curve. On the same diagram similarly draw the indifference curves for the utility levels U=18 and U=30. For parts (b)-(d), assume px=$4.
(b) With good X on the horizontal axis and good Y on the vertical axis, draw the consumer’s budget line.
(c) Calculate the first-order condition for this problem.
(d) On your budget line, show the optimal point. Justify your answer carefully.
(e) On a new diagram, show the budget line and optimal point if px=$10.
(f) Calculate the consumer’s demand function x(px), which shows her optimal quantity of X demanded, as a function of its price.
(g) 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.
(h) 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)