Charlie likes both apples and bananas. He consumes nothing else. The consumption bundle where Charlie consumes x_A bushels of apples per year and x_B bushels of bananas per year is written as (x_A, x_B). Last year, Charlie consumed 20 bushels of apples and 5 bushels of bananas. It happens that the set of consumption bundles (x_A, x_B) such that Charlie is indifferent between (x_A, x_B) and (20, 5) is the set of all bundles such that x_B = 100/x_A. The set of bundles (x_A, x_B) such that Charlie is just indifferent between (x_A, x_B) and the bundle (10, 15) is the set of bundles such that x_B = 150/x_A.

(a) On the graph below, plot several points that lie on the indifference curve that passes through the point (20, 5), and sketch this curve, using blue ink. Do the same, using red ink, for the indifference curve passing through the point (10, 15).

(b) Use pencil to shade in the set of commodity bundles that Charlie weakly prefers to the bundle (10, 15). Use blue ink to shade in the set of commodity bundles such that Charlie weakly prefers (20, 5) to these bundles.

For each of the following statements about Charlie’s preferences, write “true” or “false.”

(c) (30, 5) ∼ (10, 15). 

(d) (10, 15) > (20, 5). 

(e) (20, 5) ≥ (10, 10). 

(f) (24, 4) ≥ (11, 9.1). 

(g) (11, 14) > (2, 49). 

(h)A set is convex if for any two points in the set, the line segment between them is also in the set. Is the set of bundles that Charlie weakly prefers to (20, 5) a convex set? 

(i)Is the set of bundles that Charlie considers inferior to (20, 5) a convex set?

(j)The slope of Charlie’s indifference curve through a point, (x_A, x_B), is known as his marginal ________of ________ at that point.

(k) Remember that Charlie’s indifference curve through the point (10, 10) has the equation x_B = 100/x_A. Those of you who know calculus will remember that the slope of a curve is just its derivative, which in this case is −100/x₂

A. (If you don’t know calculus, you will have to take our word for this.) Find Charlie’s marginal rate of substitution at the point, (10,10). 

(l) What is his marginal rate of substitution at the point (5, 20)? 

(m) What is his marginal rate of substitution at the point (20, 5)?

(n) Do the indifference curves you have drawn for Charlie exhibit diminishing marginal rate of substitution?