1. **State the problem:** We have a duopoly with market demand $P=100-0.5Q$ where $Q=Q_1+Q_2$, and cost functions $C_1=5Q_1$ and $C_2=0.5Q_2^2$. We want to find the Cournot equilibrium: total output, equilibrium price, and profits. 2. **Set up the profit functions:** Firm 1's profit: $\pi_1 = P Q_1 - C_1 = (100 - 0.5(Q_1 + Q_2))Q_1 - 5Q_1$ Firm 2's profit: $\pi_2 = P Q_2 - C_2 = (100 - 0.5(Q_1 + Q_2))Q_2 - 0.5 Q_2^2$ 3. **Find best response functions by maximizing profits:** For Firm 1, take derivative w.r.t. $Q_1$: $$\frac{\partial \pi_1}{\partial Q_1} = 100 - 0.5(Q_1 + Q_2) - 0.5 Q_1 - 5 = 0$$ Simplify: $$100 - 0.5 Q_1 - 0.5 Q_2 - 0.5 Q_1 - 5 = 0$$ $$95 - Q_1 - 0.5 Q_2 = 0$$ Solve for $Q_1$: $$Q_1 = 95 - 0.5 Q_2$$ For Firm 2, take derivative w.r.t. $Q_2$: $$\frac{\partial \pi_2}{\partial Q_2} = 100 - 0.5(Q_1 + Q_2) - 0.5 Q_2 - Q_2 = 0$$ Simplify: $$100 - 0.5 Q_1 - 0.5 Q_2 - 0.5 Q_2 - Q_2 = 0$$ $$100 - 0.5 Q_1 - 2 Q_2 = 0$$ Solve for $Q_2$: $$2 Q_2 = 100 - 0.5 Q_1$$ $$Q_2 = \frac{100 - 0.5 Q_1}{2}$$ 4. **Solve the system:** Substitute $Q_2$ into $Q_1$'s best response: $$Q_1 = 95 - 0.5 \times \frac{100 - 0.5 Q_1}{2} = 95 - \frac{0.5}{2}(100 - 0.5 Q_1) = 95 - 0.25(100 - 0.5 Q_1)$$ $$Q_1 = 95 - 25 + 0.125 Q_1 = 70 + 0.125 Q_1$$ Bring terms together: $$Q_1 - 0.125 Q_1 = 70$$ $$0.875 Q_1 = 70$$ $$Q_1 = \frac{70}{0.875} = 80$$ Now find $Q_2$: $$Q_2 = \frac{100 - 0.5 \times 80}{2} = \frac{100 - 40}{2} = \frac{60}{2} = 30$$ 5. **Total output:** $$Q = Q_1 + Q_2 = 80 + 30 = 110$$ 6. **Equilibrium price:** $$P = 100 - 0.5 \times 110 = 100 - 55 = 45$$ 7. **Profits:** $$\pi_1 = P Q_1 - C_1 = 45 \times 80 - 5 \times 80 = 3600 - 400 = 3200$$ $$\pi_2 = P Q_2 - C_2 = 45 \times 30 - 0.5 \times 30^2 = 1350 - 0.5 \times 900 = 1350 - 450 = 900$$ **Final answers:** - Total output $Q=110$ - Equilibrium price $P=45$ - Profits: $\pi_1=3200$, $\pi_2=900$