Answer to Question #295577 in Microeconomics for Zack

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

determine quantities Q₁ and Q₂ that the consumer should have to maximize utility if the consumer budgeted Kshs 350

We will determine the marginal utility functions for Q₁ and Q₂ as:

MU(Q₁) = $\dfrac{∂U}{∂Q1}$ = $\dfrac{15Q2^{\smash{0.5}}}{Q1^{\smash{0.5}}}$

MU(Q₂) = $\dfrac{∂U}{∂Q2}$ = $\dfrac{15Q1^{\smash{0.5}}}{Q2^{\smash{0.5}}}$

The condition for optimal consumer choice:

$\dfrac{MU(Q1)}{MU(Q2)}$ = $\dfrac{P1}{P2}$

Therefore, $\dfrac{15Q2^{\smash{0.5}}}{Q1^{\smash{0.5}}}$ ÷ $\dfrac{15Q1^{\smash{0.5}}}{Q2^{\smash{0.5}}}$ = $\dfrac{10}{2}$ ⇒ $\dfrac{Q1}{Q2} = 2 ⇒ Q2 = 2Q1$

From the budget constraint:

10Q₁+5Q₂=350

But Q₂=2Q₁

Therefore 10Q₁+5(2Q₁) =350

⇒20Q₁=350 

Therefore; Q₁=17.5 and Q₂=35 are quantities Q₁ and Q₂ that the consumer would have maximized utility.