Answer to Question #307880 in Microeconomics for paul

Question

the utility function for a consumer utility is U=30Q1 1/2 Q2 1/2. if the price per unit of Q1 =shs 10 and shs 5 per unit of Q2 determine the quantities the consumer should maximize utility if the consumer budgeted shs 350

Answer 1

The utility function for a consumer utility is $U = 30Q_1^{\frac{1}{2}} Q_2^{\frac{1}{2}}$. if the price per unit of Q1 =shs 10 and shs 5 per unit of Q2. Determine the quantities the consumer should maximize utility if the consumer budgeted shs 350.

$U = 30Q_1^{\frac{1}{2}} Q_2^{\frac{1}{2}}$

The budget line is given by:

$350=10Q_1+5Q_2$

When the bundle is optimum, the slope of the budget line matches the slope of the utility function.

$Slope\ of\ BL=\frac{-price\ Of\ Q_2}{price\ of\ Q_1}=\frac{-2}{10}=-0 .5$

$Slope\ of\ U =\frac{-MUQ_1}{MUQ_2}$

$MUQ_1=15Q_2$

$MUQ_2=15Q_1$

$Slope\ of\ U=\frac{-15Q_1}{15Q_2}$

Taking the budget line's slope and applying it to the slope of U:

$\frac{-15Q_1}{15Q_2}=-0.5$

$-15Q_1=-7.5Q_2$

$Q_1=0.52Q_2$

We may achieve the following results by substituting Q₁ in the budget line:

$10×(0.5Q_2)+5Q_2=350$

$10Q_2=350$

$Q_2=35$

$Q_1=0.5Q_2$

$Q_1=17.5$

Therefore, to maximize utility, $Q_1 ​= 17.5$ and $Q_2=35$