1. **Problem Statement:** Analyze the three-stage optimization problem involving quantities $q$, $q^*$ and wages $w$, $w^*$ with objective functions and first-order conditions (FOC), second-order conditions (SOC), and comparative statics (CSA). 2. **Third Stage ($q^*$):** Objective function: $$\Pi^* = (1-\lambda)[P(q+q^*) - w^* - c^*] q^*$$ FOC: $$\frac{\partial \Pi^*}{\partial q^*} = (1-\lambda)[P + P' q^* - w^* - c^*] = 0$$ Since $P' = \frac{\partial P}{\partial Q} < 0$, SOC: $$\frac{\partial^2 \Pi^*}{\partial q^{*2}} = (1-\lambda) 2 P' < 0$$ Cross partial: $$\frac{\partial^2 \Pi^*}{\partial q^* \partial q} = (1-\lambda) P' < 0$$ Reaction function slope: $$\frac{dq^*}{dq} = -\frac{\Pi^*_{q^* q}}{\Pi^*_{q^* q^*}} = -\frac{(1-\lambda) P'}{(1-\lambda) 2 P'} = -\frac{1}{2} < 0$$ Follower’s reaction function: $$q^* = -\frac{P - w^* - c^*}{P'}$$ Comparative statics: $$\frac{\partial q^*}{\partial w^*} = \frac{1}{P'} < 0$$ 3. **Second Stage ($q$):** Objective function: $$\Pi = [P(q + q^*(q)) - w - c] q + \lambda [P(q + q^*(q)) - w^* - c^*] q^*(q)$$ FOC: $$\frac{d\Pi}{dq} = \frac{\partial \Pi}{\partial q} + \frac{\partial \Pi}{\partial q^*} \frac{dq^*}{dq} = 0$$ Expanding: $$[P + P' q - w - c] + \lambda P' q^* + q \frac{\partial P}{\partial Q} \frac{\partial Q}{\partial q^*} \frac{dq^*}{dq} + \lambda [P' q^* \frac{dq^*}{dq} + P \frac{dq^*}{dq} - w^* \frac{dq^*}{dq} - c^* \frac{dq^*}{dq}] = 0$$ Simplified: $$P - w - c + P' (q + \lambda q^*) + [P' q + \lambda (P' q^* + P - w^* - c^*)] \frac{dq^*}{dq} = 0$$ SOC: $$\Pi_{qq} = 2 P' < 0$$ Cross partial: $$\Pi_{q q^*} = P' (1 + \lambda) < 0$$ Equilibrium: $$q = -\frac{P - w - c + \lambda P' q^*}{P'}$$ Comparative statics: $$\frac{\partial q}{\partial q^*} = \lambda P' < 0, \quad \frac{\partial q}{\partial w} = \frac{1}{P'} < 0, \quad \frac{\partial q}{\partial \lambda} = -q^* < 0$$ 4. **First Stage ($w$, $w^*$):** Objective functions: $$\Omega = w q = w q(w), \quad \Omega^* = w^* q^* = w^* q^*(w^*)$$ FOC: $$\frac{d\Omega}{dw} = q(w) + w \frac{\partial q}{\partial w} = 0$$ $$\frac{d\Omega^*}{dw^*} = q^*(w^*) + w^* \frac{\partial q^*}{\partial w^*} = 0$$ SOC: $$\frac{d^2 \Omega}{dw^2} = 2 \frac{\partial q}{\partial w} + w \frac{\partial^2 q}{\partial w^2} < 0$$ $$\frac{d^2 \Omega^*}{d w^{*2}} = 2 \frac{\partial q^*}{\partial w^*} + w^* \frac{\partial^2 q^*}{\partial w^{*2}} < 0$$ Equilibrium wages: $$w = -\frac{q}{q_w}, \quad w^* = -\frac{q^*}{q^*_{w^*}}$$ **Final summary:** - The follower’s reaction function slope is negative. - Quantities $q$ and $q^*$ decrease with increases in wages $w$ and $w^*$. - The equilibrium wages are determined by the ratio of quantities to their derivatives with respect to wages. This completes the analysis of the three-stage optimization problem with all FOCs, SOCs, reaction functions, and comparative statics.