Question

a) The utility function for a consumer utility is U=30Q11/2Q21/2 . If the price per unit of Q1 is Kshs 10 and Kshs 5per unit of Q2, determine quantities Q1 and Q2 that the consumer should have to maximize utility if the consumer budgeted Kshs 350

Answer 1

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

Budget line:

$350=10Q_1+5Q_2$

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

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

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

$MUQ_1=15Q_2$

$MUQ_2=15Q_1$

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

Equating the slope of the budget line with the slope of U:

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

$-15Q_1=-7.5Q_2$

$Q_1=0.52Q_2$

Replacing $Q_1$ In the budget line:

$10×(0.5Q_2)+5Q_2=350$

$10Q_2=350$

$Q_2=35$

$Q_1=0.5Q_2$

$Q_1=17.5$