1. **Stating the problem:** We need to find the equilibrium price ($P^*$) and quantity ($Q^*$) without tax and then analyze the effect of tax on these values. 2. **Given functions:** - Demand function: $P = 12 - Q$ - Supply function: $P = 2 + 0.25Q$ 3. **Equilibrium without tax:** At equilibrium, quantity demanded equals quantity supplied, so demand price equals supply price: $$12 - Q = 2 + 0.25Q$$ 4. **Solve for $Q$:** $$12 - Q = 2 + 0.25Q$$ $$12 - 2 = Q + 0.25Q$$ $$10 = 1.25Q$$ $$Q = \frac{10}{1.25} = 8$$ 5. **Find equilibrium price $P^*$:** Substitute $Q=8$ into demand function: $$P = 12 - 8 = 4$$ 6. **Equilibrium without tax:** - Quantity: $Q^* = 8$ - Price: $P^* = 4$ 7. **Introducing tax:** Assume a per unit tax $t$ shifts the supply curve upward by $t$. New supply price including tax: $$P = 2 + 0.25Q + t$$ 8. **New equilibrium with tax:** Set demand price equal to new supply price: $$12 - Q = 2 + 0.25Q + t$$ Rearranged: $$12 - 2 - t = Q + 0.25Q$$ $$10 - t = 1.25Q$$ $$Q = \frac{10 - t}{1.25}$$ 9. **New equilibrium price paid by consumers $P_c$:** $$P_c = 12 - Q = 12 - \frac{10 - t}{1.25}$$ 10. **Price received by producers $P_p$:** $$P_p = P_c - t$$ 11. **Summary:** - Without tax: $Q^* = 8$, $P^* = 4$ - With tax $t$: $$Q_t = \frac{10 - t}{1.25}$$ $$P_c = 12 - Q_t$$ $$P_p = P_c - t$$ 12. **Additional functions:** - Total cost: $C = 3000 + 200Q$ - Total revenue: $R = 400Q$ These can be used to analyze profit or loss at equilibrium quantities.