Question

Diogo has a utility function: U = 100X^0.75Z^0.25

The price of X is Px = $2, the price of Z is Pz = $4, and his income is $1,000.

What is Diogo's optimal bundle? (round your answer to one decimal place)

X0= _ units

Z0= _ units

Answer 1

U = 100 X^0.75Z^0.25

A consumer maximizes his utility at the point where marginal rate of substitution is equal to the price ratio of the two goods. The optimal bundle is also calculated by using this only.

So, marginal rate of substitution = marginal utility of good X / marginal utility of good Z

= [ 100 x 0.75 x X^-0.25 x Z^0.25] / [ 100 x X^0.75 x 0.25 x Z^-0.75]

= 0.75Z / 0.25X

Equating this with the price ratio of two goods, 0.75Z / 0.25X = price of good X / price of good Z

   0.75Z / 0.25X = 2/4

3Z / X = 1/2

X = 6Z

The equation of the budget line can be written as,

(price of good X x quantity of good X) + (price of good Z x quantity of good Z) = Income

2X + 4Z = 1000

Putting the value of X = 6Z in the above equation to get the optimal bundles,

2(6Z) + 4Z = 1000

12Z + 4Z = 1000

16Z = 1000

Z = 62.5 units is the answer.

X = 6Z

X = 6(62.5) = 375 units is the answer.