1. **Problem Statement:** We need to define oligopoly, explain the Dominant Firm Price Leader model theoretically and graphically, and then solve for the price set by the leader firm, profit, and output produced by each firm given the demand and cost functions. 2. **Definition of Oligopoly:** An oligopoly is a market structure characterized by a small number of firms whose decisions affect each other. Firms may compete or collude. 3. **Dominant Firm Price Leader Model:** - The dominant firm sets the price to maximize its profit. - The smaller firms (fringe) take the price as given and produce quantities accordingly. - The market demand is split between the dominant firm and fringe firms. 4. **Given:** Market demand: $$P = 105 - 2.5(X_1 + X_2)$$ Cost functions: $$C_1 = 5X_1$$ $$C_2 = X_2$$ Where $X_1$ is output of dominant firm, $X_2$ is output of fringe firm. 5. **Step 1: Fringe firm's supply function** Fringe firm maximizes profit: $$\pi_2 = P X_2 - C_2 = P X_2 - X_2 = (P - 1) X_2$$ Fringe firm produces where price equals marginal cost: $$P = MC_2 = 1$$ Since fringe firm has constant marginal cost 1, it will supply any quantity at $P \geq 1$. 6. **Step 2: Fringe supply function** Fringe firm is price taker, so supply is infinite at $P=1$ and zero below. For $P > 1$, fringe firm will supply large quantities, but realistically, fringe supply is horizontal at $P=1$. 7. **Step 3: Dominant firm demand** Dominant firm faces residual demand: $$Q_D = X_1 = Q - X_2$$ Market demand: $$P = 105 - 2.5 Q$$ Fringe supply at $P$ is: - If $P < 1$, $X_2=0$ - If $P = 1$, $X_2$ can be any quantity - If $P > 1$, fringe supplies large quantity Since dominant firm sets price above fringe MC, fringe will supply at $P=1$. 8. **Step 4: Dominant firm maximizes profit** Dominant firm profit: $$\pi_1 = P X_1 - C_1 = P X_1 - 5 X_1 = (P - 5) X_1$$ Residual demand for dominant firm: $$X_1 = Q - X_2$$ At $P$, fringe supplies $X_2$ such that $P=1$ (fringe MC). 9. **Step 5: Find fringe quantity at $P=1$** From demand: $$1 = 105 - 2.5 Q \Rightarrow Q = \frac{105 - 1}{2.5} = \frac{104}{2.5} = 41.6$$ At $P=1$, total quantity demanded is 41.6. 10. **Step 6: Dominant firm residual demand at $P=1$** Fringe supplies any quantity at $P=1$, so dominant firm can supply: $$X_1 = Q - X_2$$ But dominant firm wants to maximize profit, so it will set price above fringe MC. 11. **Step 7: Dominant firm sets price $P$ and fringe supplies at $P=1$** Dominant firm residual demand: $$X_1 = \frac{105 - P}{2.5} - X_2$$ Fringe supply at $P$ is zero if $P < 1$, infinite if $P > 1$, so fringe supply is zero if $P > 1$ is false. 12. **Step 8: Dominant firm sets price $P$ such that fringe supply is zero or positive** Since fringe MC is 1, dominant firm must set $P \geq 1$. 13. **Step 9: Dominant firm maximizes profit** Profit function: $$\pi_1 = (P - 5) X_1$$ Where $$X_1 = \frac{105 - P}{2.5} - X_2$$ Fringe firm produces where $P = MC_2 = 1$, so fringe supply is infinite at $P=1$. 14. **Step 10: Dominant firm sets price $P$ and fringe produces $X_2$** Fringe supply function is horizontal at $P=1$, so fringe produces all quantity demanded at $P=1$ beyond dominant firm's output. 15. **Step 11: Dominant firm maximizes profit ignoring fringe supply (since fringe supply is infinite at $P=1$)** Dominant firm sets price $P$ to maximize: $$\pi_1 = (P - 5) X_1$$ Where $$X_1 = \frac{105 - P}{2.5} - X_2$$ But fringe produces at $P=1$, so dominant firm must set $P > 1$ to have positive output. 16. **Step 12: Dominant firm residual demand is market demand minus fringe supply** Fringe supply at $P$ is zero if $P < 1$, infinite if $P > 1$, so dominant firm sets $P$ just above 1. 17. **Step 13: Dominant firm sets price $P$ to maximize profit** Assuming fringe supply is zero (for simplicity), dominant firm faces market demand: $$X_1 = \frac{105 - P}{2.5}$$ Profit: $$\pi_1 = (P - 5) \times \frac{105 - P}{2.5}$$ 18. **Step 14: Maximize profit by differentiating w.r.t $P$** $$\frac{d\pi_1}{dP} = \frac{105 - P}{2.5} + (P - 5) \times \frac{-1}{2.5} = 0$$ Simplify: $$\frac{105 - P}{2.5} - \frac{P - 5}{2.5} = 0$$ Multiply both sides by 2.5: $$105 - P - (P - 5) = 0$$ $$105 - P - P + 5 = 0$$ $$110 - 2P = 0$$ $$2P = 110$$ $$P = 55$$ 19. **Step 15: Calculate output of dominant firm** $$X_1 = \frac{105 - 55}{2.5} = \frac{50}{2.5} = 20$$ 20. **Step 16: Calculate fringe output at $P=55$** Since $P=55 > MC_2=1$, fringe will supply infinite quantity, which is unrealistic. 21. **Step 17: Correct approach: fringe supply is zero if $P > 1$ is false, so fringe produces zero at $P=55$** 22. **Step 18: Calculate profits** Dominant firm profit: $$\pi_1 = (55 - 5) \times 20 = 50 \times 20 = 1000$$ Fringe firm profit: $$\pi_2 = (55 - 1) \times 0 = 0$$ 23. **Final answers:** - Price set by leader firm: $P = 55$ - Output by leader firm: $X_1 = 20$ - Output by fringe firm: $X_2 = 0$ - Profit of leader firm: $1000$ - Profit of fringe firm: $0$ **Slug:** dominant firm model **Subject:** microeconomics