Question

The utility function for a consumer utility is U=30Q₁^1/2Q₂^1/2 . 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.

$Slope of budget Line=\frac{-price Of Q_2}{price of Q_1}$

$=\frac{-5}{10}=-0.5$

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

$MUQ_1=15Q_2$

$MUQ_2=15Q_1$

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

Equating the slope of U and the slope of BL:

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

$-15Q_1=-7.5Q_2$

$Q_1=0.5Q_2$

Replacing $Q_2$ In the budget line:

$10×(0.5Q_2)+5Q_2=350$

$10Q_2=350$

$Q_2=35$

$Q_1=0.5Q_2$

$Q_1=0.5×35=17.5$