Question

4. Charlie likes both apples and bananas. He consumes nothing else. Charlie consumes x bushels of apples per year and x bushels of bananas per year. Suppose that Charlie's preference is represented in the following utility function: u(x,,Xy)-x,Xy . Suppose that the price of apples is S1, the price of bananas is S2, and Charlie's income is $40. (14 points) a. Draw Charlie's budget line. Plot a few points on the indifference curve that gives Charlie a utility of 150 and sketch this curve. Now plot a few points on the indifference curve that gives Charlie a utility of 300 and sketch this curve. b. Can Charlie afford any bundles that give him a utility of 150? Can Charlie afford any bundles that give him a utility of 300? c. On your graph, mark a point that Charlie can afford and that gives him a higher utility than 150. Label that point A. d. Neither of the indifference curves that you drew is tangent to Charlie's budget line. Let's try to find one that is. What is the marginal utility for the apples? What is the marginal utility for the bananas? Using marginal utilities, find Charlie's marginal rate of substitution. e. What is the slope of Charlie's budget line? Using this and the answer to (d), write an equation, which implies the optimal choice: an equation that implies that the budget line is tangent to an indifference curve. 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 f. 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 t. 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. At which points does this happen? Verify your answer by solving the two simultaneous equations given by his budget equation and the tangency condition. g. What is Charlie's utility if he consumes the bundle (20, 10)? On the graph above, 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

a) budget line: 1*x1+2*x2 = 40

4 20 0 2

Apples on the y axis and bananas on the x axis

Utility: xi*x2 = 150

60+ 2.5,60 50 3,50 40 30- 5,30 20 10 30,5 50,3 60,2.5 10 20 30 40 50 60

Apples on the y axis and bananas on the x axis

Utility: xi*x2 = 300

60 50t 6,50 40 3010,30 20 30,10 50,6 10 10 20 30 40 50 60

Apples on the y axis and bananas on the x axis

b)

Apples on the y axis and bananas on the x axis
Utility: 150, Yes, the combination as above
Utility: 300, No as the budget constraint is breached

60 50 40 (5, 30) 30 20 (15, 10) 10 -10 10 2 30 4050 60

c)

Apples on the y axis and bananas on the x axis
A (20,10)

60 50 40 (5, 30) 30 20 (15, 10) 10 -10 10 2 30 4050 60

d) Marginal utility of apples = dU/dx1 = x2
Marginal utility of bananas = dU/dx1 = x1
Marginal rate of substitution = -x2/x1