When Jasons utility is maximized how many hours does he spen

When Jason’s utility is maximized, how many hours does he spend watching TV each day?

What is the marginal rate of substitution between hours at the gym and hours watching TV?

(a)

U = (XY)^0.5

X + Y = 8

U = [X. (8 - X)]0.5 = [8X - X²]^0.5

Utility is maximized when dU / dX = 0

0.5. [8X - X²]^-^0.5 [8 - 2X] = 0

Since X is not 0, we simplify this as

0.5 [8 - 2X] = 0

8 - 2X = 0

2X = 8

X = 4 [Hours of TV]

So, Y = 8 - X = 8 - 4 = 4

(b) MRS = MUX / MUY

MUX = dU / dX = 0.5 (Y / X)^0.5

MUY = 0.5 (X / Y)^0.5

So, MRS = Y / X = 4/4 = 1