We begin again with Charlie of the apples and bananas. Recall that Charlie’s utility function is U(x_A, x_B) = x_Ax_B. Suppose that the price of apples is 1, the price of bananas is 2, and Charlie’s income is 40

(a) On the graph below, use blue ink to draw Charlie’s budget line. (Use a ruler and try to make this line accurate.) Plot a few points on the indifference curve that gives Charlie a utility of 150 and sketch this curve with red ink. Now plot a few points on the indifference curve that gives Charlie a utility of 300 and sketch this curve with black ink or pencil.

(b) Can Charlie afford any bundles that give him a utility of 150? 

(c) Can Charlie afford any bundles that give him a utility of 300? 

(d) On your graph, mark a point that Charlie can afford and that gives him a higher utility than 150. 

(e) Neither of the indifference curves that you drew is tangent to Charlie’s budget line. Let’s try to find one that is. At any point, (x_A, x_B), Charlie’s marginal rate of substitution is a function of x_A and x_B. Infact, if you calculate the ratio of marginal utilities for Charlie’s utility function, you will find that Charlie’s marginal rate of substitution is MRS(x_A, x_B) = −x_B/x_A. This is the slope of his indifference curve at (x_A, x_B). The slope of Charlie’s budget line is −1/2 (give a numerical answer).

(f) Write an equation that implies that the budget line is tangent to an indifference curve at _____________. There are many solutions to this equation. Each of these solutions corresponds to a point on a different indifference curve. Use pencil to draw a line that passes through all of these points.