Question

Consider a Cobb-Douglas utility function given as U(H) = (24 - H)^1-a (wH)^a, where H is the number of hours spent working per day, and w is the wage rate per hour. If \(a=\frac{1}{2}\) then the corresponding labour supply (in hours) is _______( in integer).

Answer 1

Correct Answer: 12

1. Substituting \(a = \tfrac{1}{2}\): \[ U(H) = (24 - H)^{\tfrac{1}{2}} \cdot (wH)^{\tfrac{1}{2}} \] 
2. Simplify: \[ U(H) = \sqrt{(24 - H) \cdot wH} \]
3. For maximization, set derivative \(\frac{dU}{dH} = 0\): \[ \frac{dU}{dH} = \frac{w(24 - H) - wH}{2 \sqrt{(24 - H)wH}} = 0 \]
4. Simplify numerator: \[ w(24 - 2H) = 0 \]
5. Solve: \[ 24 - 2H = 0 \quad \Rightarrow \quad H = 12 \]

Final Answer:
The labor supply is: \[ \boxed{12 \;\; \text{hours}} \]