Econ: Utility and H. K. Chen

ECON 301 — Tutorial 4
Problems for today (from the Workout Book) Problem 4.9 (except d and e) Problem 4.12 Problem 5.1 Problem 5.2 Problem 5.3

H. K. Chen (SFU)

ECON 301 — Tutorial 4

October 2, 3

1 / 15

Problem 4.9 (a)–(g)
Suppose that the utility functions u(x, y ) and v (x, y ) are related by v (x, y ) = f (u(x, y )). In each case below, write “Yes” if the function f is a positive monotonic transformation and “No” if it is not. (Hint for calculus users: A differentiable function f (u) is an increasing function of u if its derivative is positive.) (a) f (u) = 3.141592u (b) f (u) = 5000 − 23u (c) f (u) = u − 100000 (f) f (u) = 1/u (g) f (u) = −1/u Yes No Yes No Yes

H. K. Chen (SFU)

ECON 301 — Tutorial 4

October 2, 3

2 / 15

Problem 4.12 (a)–(c)
2 2 Joe Bob has a utility function given by u(x1 , x2 ) = x1 + 2x1 x2 + x2 . (a) Compute Joe Bob’s marginal rate of substitution:

MRS(x1 , x2 ) = − MU1 = −1 MU2
MU1 (x1 , x2 ) = MU2 (x1 , x2 ) =
∂u(x1 ,x2 ) ∂x1 ∂u(x1 ,x2 ) ∂x2

= 2x1 + 2x2 = 2x1 + 2x2

(b) Joe Bob’s straight cousin, Al, has a utility function v (x1 , x2 ) = x2 + x1 . Compare Al’s marginal rate of substitution. Note that MU1 = MU2 = 1, so MRS(x1 , x2 ) = −1 for Al (c) Do u(x1 , x2 ) and v (x1 , x2 ) represent the same preferences? Yes. Can you show that Joe Bob’s utility function is a monotonic transformation of Al’s? Notice that u(x1 , x2 ) = (x1 + x2 )2 = (v (x1 , x2 ))2
H. K. Chen (SFU) ECON 301 — Tutorial 4 October 2, 3 3 / 15

Problem 5.1 (a)
We begin again with Charlie of the apples and bananas. Recall that Charlie’s utility function is U(xA , xB ) = xA xB . 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