1. **Problem statement:** We have a firm's production function given by $$Q = A k^\alpha l^\beta$$ where $0 < \alpha < 1$ and $0 < \beta < 1$. The input prices are $w$ for labour ($l$) and $r$ for capital ($k$). We want to set up the profit maximization problem and find the firm's input demand functions for labour and capital. 2. **Profit maximization problem:** The firm's profit $$\pi$$ is revenue minus cost: $$\pi = pQ - wl - rk$$ where $p$ is the output price. Substitute production function: $$\pi = p A k^\alpha l^\beta - wl - rk$$ The firm chooses $k$ and $l$ to maximize profit: $$\max_{k,l} \pi = p A k^\alpha l^\beta - wl - rk$$ 3. **First-order conditions (FOCs):** Take partial derivatives of profit with respect to $k$ and $l$ and set to zero: $$\frac{\partial \pi}{\partial k} = p A \alpha k^{\alpha - 1} l^\beta - r = 0$$ $$\frac{\partial \pi}{\partial l} = p A \beta k^\alpha l^{\beta - 1} - w = 0$$ 4. **Solve for input demand functions:** From the first FOC: $$p A \alpha k^{\alpha - 1} l^\beta = r$$ From the second FOC: $$p A \beta k^\alpha l^{\beta - 1} = w$$ Divide the first equation by the second to eliminate $pA$: $$\frac{p A \alpha k^{\alpha - 1} l^\beta}{p A \beta k^\alpha l^{\beta - 1}} = \frac{r}{w}$$ Simplify: $$\frac{\alpha}{\beta} \cdot \frac{l}{k} = \frac{r}{w}$$ Rearranged: $$\frac{l}{k} = \frac{r}{w} \cdot \frac{\beta}{\alpha}$$ So, $$l = k \cdot \frac{r}{w} \cdot \frac{\beta}{\alpha}$$ 5. **Substitute $l$ back into one FOC to solve for $k$:** Use second FOC: $$p A \beta k^\alpha \left(k \cdot \frac{r}{w} \cdot \frac{\beta}{\alpha}\right)^{\beta - 1} = w$$ Rewrite: $$p A \beta k^\alpha k^{\beta - 1} \left(\frac{r}{w} \cdot \frac{\beta}{\alpha}\right)^{\beta - 1} = w$$ Combine powers of $k$: $$p A \beta k^{\alpha + \beta - 1} \left(\frac{r}{w} \cdot \frac{\beta}{\alpha}\right)^{\beta - 1} = w$$ Solve for $k^{\alpha + \beta - 1}$: $$k^{\alpha + \beta - 1} = \frac{w}{p A \beta} \left(\frac{r}{w} \cdot \frac{\beta}{\alpha}\right)^{1 - \beta}$$ Since $\alpha + \beta - 1 < 1$ (because $\alpha, \beta < 1$), we can write: $$k = \left[\frac{w}{p A \beta} \left(\frac{r}{w} \cdot \frac{\beta}{\alpha}\right)^{1 - \beta}\right]^{\frac{1}{\alpha + \beta - 1}}$$ 6. **Find $l$ using the relation from step 4:** $$l = k \cdot \frac{r}{w} \cdot \frac{\beta}{\alpha}$$ 7. **Summary:** The firm's input demand functions are: $$k^* = \left[\frac{w}{p A \beta} \left(\frac{r}{w} \cdot \frac{\beta}{\alpha}\right)^{1 - \beta}\right]^{\frac{1}{\alpha + \beta - 1}}$$ $$l^* = k^* \cdot \frac{r}{w} \cdot \frac{\beta}{\alpha}$$ These functions show how the firm chooses capital and labor to maximize profit given input prices and production parameters.