Question

Given the following: 25 тах ¤1 > 0, x2 > 0 < 2000 5 * x1 + 20 * x2 s.t. • MU, = .75 * MU, = .25 * Equation Description: A consumer is attempting to maximize utility subject to a budget constraint. Utility equals the product of two terms: the first term is units of good 1 raised to the.75 power; and, the second term is units of good 2 raised to the .25 power. The price per unit of good 1 is $5; the price per unit of good 2 is $20; and, consumer income is $2000. Marginal utility of good 1 equals the product of three terms: the first term is.75; the second term is units of good 2 raised to the .25 power; the third term is units of good 1 raised to the -25 power. Marginal utility of good 2 equals the product of three terms: the first term is .25; the second term is units of good 1 raised to the.75 power; the third term is units of good 2 raised to the -75 power. Question: How many units of consumption of good 1 maximizes utility? O 400 units of good 1 O 25 units of good 1 O 300 units of good 1 O 75 units of good 1 O 150 units of good 1 O None of the above O 100 units of good 1

Answer 1

At utility maximizing equilibrium ,

MU1 P1 MU2 P2

this implies that,

$\frac{0.75[\frac{x_{2}^{0.25}}{x{_{1}}^{0.25}}]}{0.25[\frac{x{_{1}}^{0.75}}{x{_{2}}^{0.75}}]}= \frac{5}{20}$

now on solving further

3 %3D

12

$12x_{2}=x_{1}$.................................equation (1)

substituting this equation (1) in the budget constraint

budget constraint is , $5x_{1}+20x_{2}=2000$

after substitution, $5(12x_{2})+20x_{2}=2000$

$60x_{2}+20x_{2}=2000$

$80x_{2}=2000$

$x_{2}=25$.............................(2)

Now, substituting this value of x₂ in equation (1), we get:-

$x_{1}=12(25)$

$x_{1}=300$...................................(3)

now substituting this value of x's in equation (2) and (3) in the utility form we get

$U=300^{0.75}*25^{0.25}$

$U=72.084*2.236$

$U=161.179$$U=161.179\approx 161.19$

This implies that 300 units of good 1 and 25 units of good 2 maximizes the utility and maximum utility achieved can be 161.19 utils.