1. **Stating the problem:** We have a Cobb-Douglas production function $$Q = L^\alpha K^\beta$$ with $$\alpha = \beta$$, and a total cost function $$TC = wL + rK$$. We want to: - Write the profit function. - Find the first and second-order conditions for profit maximization. - Prove that the second-order condition requires diminishing marginal returns to labor and decreasing returns to scale. 2. **Write the profit function:** Profit $$\pi$$ is total revenue minus total cost. Revenue is price $$p$$ times quantity $$Q$$. $$ \pi = pQ - TC = pL^\alpha K^\alpha - (wL + rK) $$ 3. **First-order conditions:** Find partial derivatives of $$\pi$$ w.r.t. $$L$$ and $$K$$ and set to zero. $$ \frac{\partial \pi}{\partial L} = p \alpha L^{\alpha - 1} K^\alpha - w = 0 $$ $$ \frac{\partial \pi}{\partial K} = p \alpha L^\alpha K^{\alpha - 1} - r = 0 $$ 4. **Second-order conditions:** Compute second partial derivatives: $$ \frac{\partial^2 \pi}{\partial L^2} = p \alpha (\alpha - 1) L^{\alpha - 2} K^\alpha $$ $$ \frac{\partial^2 \pi}{\partial K^2} = p \alpha (\alpha - 1) L^\alpha K^{\alpha - 2} $$ $$ \frac{\partial^2 \pi}{\partial L \partial K} = p \alpha^2 L^{\alpha - 1} K^{\alpha - 1} $$ The Hessian matrix determinant must be positive for maximum: $$ D = \frac{\partial^2 \pi}{\partial L^2} \frac{\partial^2 \pi}{\partial K^2} - \left(\frac{\partial^2 \pi}{\partial L \partial K}\right)^2 > 0 $$ Substitute expressions and simplify. 5. **Proof of diminishing marginal returns:** For the second derivative w.r.t. $$L$$ to be negative (concave), we need: $$ \frac{\partial^2 \pi}{\partial L^2} = p \alpha (\alpha - 1) L^{\alpha - 2} K^\alpha < 0 $$ Since $$p, L^{\alpha - 2}, K^\alpha > 0$$, this implies: $$\alpha - 1 < 0 \implies \alpha < 1$$ This means diminishing marginal returns to labor. 6. **Proof of decreasing returns to scale:** Sum of exponents in production function: $$ \alpha + \beta = \alpha + \alpha = 2\alpha $$ Decreasing returns to scale requires: $$ 2\alpha < 1 \implies \alpha < \frac{1}{2} $$ Hence, for the Hessian to be negative definite (profit maximum), $$\alpha$$ must satisfy $$\alpha < 1$$ and also to ensure decreasing returns to scale, $$\alpha < \frac{1}{2}$$. **Final answers:** - Profit function: $$\pi = pL^\alpha K^\alpha - (wL + rK)$$ - First-order conditions: $$p \alpha L^{\alpha - 1} K^\alpha = w$$ $$p \alpha L^\alpha K^{\alpha - 1} = r$$ - Second-order conditions require $$\alpha < 1$$ (diminishing marginal returns) and for decreasing returns to scale $$\alpha < \frac{1}{2}$$.