Question

Gilat's utility function is given by

U(x,y)=(x-2)^0.5(y-8)^0.5

for ? > 2 and ? > 8. The price of goods x and y are ?_? =$3 and ?_? =$2.

When using the cheapest bundle that will yield a utility level of ? =817, how much of good x will Gilat consume?

Enter a numerical value below. You may round to the second decimal if necessary.

Answer 1

U = (x - 2)^0.5(y - 8)^0.5

Utility is maximized when MPx / MPy = px / py

MPx = U/x = 0.5 x [(y - 8) / (x - 2)]^0.5

MPy = U/y = 0.5 x [(x - 2) / (y - 8)]^0.5

MPx / MPy = {0.5 x [(y - 8) / (x - 2)]^0.5} / {0.5 x [(x - 2) / (y - 8)]^0.5} = (y - 8) / (x - 2) = 3/2

y - 8 = (3/2) x (x - 2)

Plugging in utility function,

(x - 2)^0.5(y - 8)^0.5 = 817

Squaring,

(x - 2) x (y - 8) = 667,489

(x - 2) x (3/2) x (x - 2) = 667,489

(x - 2)² = 444,992.67

Taking square root,

x - 2 = 667.08

x = 669.08