Question

Let individual i have the utility function Ui(x,y) = x^1/2 * y Suppose the initial endowment is x = 16 and y = 2. a. Determine 4 combinations (x,y) for which Ui=8 ; Ui=16. b. What is the minimum amount of x that i would have to be compensated to introduce him to give up 1 unit of y?

Answer 1

The utility function is given as .

U; = 1/2

(a) For utility level be 8, we have
8 = 1/2
or
64 = ry
or $x = \frac{64}{y^2}$ . For y=1, $x = \frac{64}{1^2} = 64$ . For y=2, $x = \frac{64}{2^2} = 16$ . For y=4, $x = \frac{64}{4^2} = 4$ . For y=8, $x = \frac{64}{8^2} = 1$ .

Hence, the required four combinations of (x,y) for U=8 is .

fr, )：{{64, 1), (16, 2), (4, 4), (1,8)}

For utility level be 16, we have
16 = 1/2
or
256 = ry
or $x = \frac{256}{y^2}$ . For y=1, $x = \frac{256}{1^2} = 256$ . For y=2, $x = \frac{256}{2^2} = 64$ . For y=4,
91 = -
. For y=8,
f %3D- 9
. Additionally, for y=16, $x = \frac{256}{16^2} = 1$ .

Hence, the required four combinations of (x,y) for U=16 is $(x,y) : \left \{ (256,1) , (64,2) , (16,4) , (4,8) \right \}$ .

(b) For the given endowment, we have . From the set of bundles
fr, )：{{64, 1), (16, 2), (4, 4), (1,8)}
, we can see that, consumer is endowed at second point. To go to the first point, thereby reducing the consumption of y by 1 unit, the consumer must be compensated with 64 minus 16, ie 48 units of x. Hence, the minimum amount of x is 48 units, which have to be compensated to induce consumer to give up 1 unit of y.