Question

4. Suppose that Noah's utility function is given by (a) Draw an indifference curve that goes through the bundle (2,2), where xi is on the horizontal axis and z2 is on the vertical axis. (b) Draw another indifference curve that goes through the bundle (3, 3) on the same diaram as (a). Label carefully. (c) What kind of preference does Noah have about the two goods?

Answer 1

(a) When x1 = x2 = 2,

u = (2 + 2)^0.5 = (4)^0.5 = 2

Two other bundles for which u = 2 will be (x1 = 1, x2 = 3) and (x1 = 3, x2 = 1). The indifference curve is as follows.

(b) When x1 = x2 = 3,

u = (3 + 3)^0.5 = (6)^0.5 = 2.45

Two other bundles for which u = 2.45 will be (x1 = 1, x2 = 5) and (x1 = 5, x2 = 1). The indifference curve is as follows.

(c) u = (x1 + x2)^0.5

u_x1 = u/x1 = 0.5 x (x1 + x2)^0.5

u_xx1 = /x1(u/x1) = 0.5 x 0.5 x (x1 + x2)^-0.5 = 0.25 / [(x1 + x2)^0.5] > 0, since both x1 and x2 are non-negative.

u_x2 = u/x2 = 0.5 x (x1 + x2)^0.5

u_xx2 = /x2(u/x2) = 0.5 x 0.5 x (x1 + x2)^-0.5 = 0.25 / [(x1 + x2)^0.5] > 0, since both x1 and x2 are non-negative.

Since both u_xx1 and u_xx2 are positive, preferences are convex. In addition, since indifference curves are linear, x1 and x2 are substitutes.