(3) For a cost-minimizing firm, when the prices of the inputs double, the optimal input combination remains the same.

6. A utility-maximizing individual allocates his monthly time of 720 hours into work (H) and leisure (N). He gains utility from his leisure hours (N) and the income (Y) he earns from his work hours (H). His utility function is $U(Y, N) = Y^\alpha N^\beta$, where $\alpha$ and $\beta$ are positive parameters. The person's hourly wage is given by w, and he has non-labor income $Y^*$. Suppose the government imposes a tax on labor income. The tax rate is t, where $0 < t < 1$. (25%) (1) Derive the individual's leisure demand and labor supply functions. (7%)

(3) For a cost-minimizing firm, when the prices of the inputs double, the optimal input combination remains the same.

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