Question

Diogo has a utility function:

U = 100X^0.8Z^0.2

The price of X is Px = $10, the price of Z is Pz = $2, and his income is $800.

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

Xo = units

Zo = units

Answer 1

At the Pareto optimal point,

$$ \begin{aligned} &\frac{M U X}{M U Z}=\frac{P X}{P Z} \\ &\frac{100 \times 0.8 \times X^{0.8-1} Z^{0.2}}{100 \times 0.2 \times X^{0.8} Z^{0.2-1}}=\frac{10}{2} \\ &\frac{4 \times X^{-0.2} Z^{0.2}}{X^{0.8} Z^{-0.8}}=\frac{10}{2} \\ &\frac{4 Z}{X}=5 \\ &Z=1.25 X \end{aligned} $$

Budget constraint: \(10 \mathrm{x}+2 \mathrm{z}=800\)

Substitute equation (1)

$$ \begin{aligned} &10 x+2 z=800 \\ &10 x+2(1.25 x)=800 \\ &12.5 x=800 \\ &X^{*}=64 \text { units } \\ &Z^{*}=1.25(64)=80 \text { units } \end{aligned} $$

Thus, optimal bundle: \((64,80)\)