Question

Charlie’s utility function is U(xA, xB) = xAxB. Suppose that the price of apples is 1, the price of bananas is 2, and Charlie’s income is 40.

(a) On a graph, 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. Label that point A.

(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, (xA, xB), Charlie’s marginal rate of substitution is a function of xA and xB. In fact, 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(xA, xB) = −xB/xA. This is the slope of his indifference curve at (xA, xB).

(f) Write an equation that implies that the budget line is tangent to an indifference curve at (xA, xB). −xB/xA = −1/2. There are many solutions to this equation.

(g) The best bundle that Charlie can afford must lie somewhere on the line you just penciled in. It must also lie on his budget line. If the point is outside of his budget line, he can’t afford it. If the point lies inside of his budget line, he can afford to do better by buying more of both goods. On your graph, label this best affordable bundle with an E. Verify your answer by solving the two simultaneous equations given by his budget equation and the tangency condition.

(h) What is Charlie’s utility if he consumes the bundle (20, 10)?

(i) On the graph above, use red ink to draw his indifference curve through (20,10). Does this indifference curve cross Charlie’s budget line, just touch it, or never touch it?

Answer 1