Linear demand curve: Q = a – bP Elasticity: E d = (ΔQ/ΔP)/(P/Q) = -b(P/Q)E d = -1 in the middle of demand curve (up is more elastic) Total revenue and Elasticity:Elastic: Ed < -1↑P→↓R (↑P by 15%→↓Q by 20%)
Inelastic: 0 > Ed > -1↑P→↑R (↑P by 15%→↓Q by 3%)
Unit elastic: Ed = -1R remains the same (↑P by 15%→↓Q by 15%) MR: positive expansion effect (P(Q) – sell of additional units) + price reduction effect (reduces revenues because of lower price (ΔP/ΔQ)/Q) 1. Monopoly – maximizes profit by setting MC = MR
Monopolist’s Markup = price-cost margin = Lerner index: (P-MC)/P = -1/ Ed (the less elastic demand, the greater the markup over marginal cost)
2. Price Discrimination
Perfect price discrimination – the firm sets the price to each individual consumer equal to his willingness to payMR=P(Q)=demand (without the price reduction effect), no consumer surplus,find profit from graph Two-part Tariffs – a fixed fee (= consumer surplus) + a separate per-unit price for each unit they buy (P = MC) 2 groups of customers – with discrimination: inverse demand function for individual demands → MR → MR=MC * without discrimination: sum of not-inverse demand functions = one option for aggregate demand. Other option is the “rich” people demand function. Compare profits to find Qagg.
* max fixed payment F (enabling discrimination) = ∆ π; max d added to MC1 = ∆π/q1 (with discrimination) Quantity-dependent pricing – one price for first X units and a cheaper price for units above quantity X. profit function = π = Pa*Qa+Pb(Qb-Qa)-2Qb Qb includes Qa, so the additional units sold are Qb-Qa. Example: P=20-Q. Firm offers a quantity discount. Setting a price for Qa (Pa) and a price for additional units Qb-Qa (Pb). Pa=20-Qa Pb=20-Qb. Π=(20-Qa)Qa + (20-Qb)(Qb-Qa) -2Qb = 18Qb-Qa^2 – Qb^2 +QaQb derive π’a=-2Qa+Qb π’b=18-2Qb+Qa compare to 0. 2Qa=Qb. Plug into second function: 18-2(2Qa)+Qa=0. So Qa=6 Qb=12 3.
Cost and Production Technologies
Fixed costs: avoidable – not incurred if the production level = 0; unavoidable/sunk – incurred even if production level = 0, don’t exist in the long run, for the short run typical Efficient scale of production – min AC: derivative of AC = 0; MC = AC Production technologies – production method is efficient it there is no other way to produce more output using the same amounts of inputs Minimization problem – objective function: min(wL+rK), constraint: subject to Q=f(K,L) → express K as a function of L, Q (from production function)→ plug the expression into objective function (instead of K)→ derive with respect to L = 0 → express L (demand for labor) → plug demand for labor into K function → express K (demand for capital) → TC=wL+rK Marginal product ratio rule – for f(K,L)=KaLb – at the optimum: MPL/w = MPK/r : find MPL, MPK from production function → find relationship between K,L using marginal product ratio rule → plug K/L into production function → find K/L for desired level of production – for f(K,L)=aK+bL: compare MPL/w, MPK/r → use production factor with higher marginal value, if equal – use any combination
Short run – 1. Quantity rule – basic condition: MR = P = MC → 2. Shut-down rule: P(Q) ˃ AC(Q) produce MR=MC, P(Q) ˂ AC(Q) shut down, P(Q) = AC(Q) – profit = 0 for both options * shut-down quantity and price: min AC (derivative of AC = 0); AC=P=MC (profit = 0) * when computing AC ignore unavoidable/sunk fixed costs (not influenced by our decision) – market equilibrium: multiply individual supply functions (from P=MC example TC = 4q^2 so MC = 8q compare to p so 8q=p so q=p/8) by number of firms = aggregate supply function Qs → Qs=Qd (demand function) → equilibrium price and quantity Long run – profits = 0 → P=AC, equilibrium: MR=P=MC=ACmin * in the long run, unavoidable/sunk cost don’t exist → fixed costs are avoidable → take them into account – market equilibrium: find individual supply function (MC=P), quantity produced by 1 firm (MC=AC =price → plug price into demand function → total quantity demanded → number of firms in the market = total quantity demanded/quantity produced by 1 firm
5. Oligopolistic Markets
Game Theory – Nash Equilibrium: each firm is making a profit-maximizing choice given the actions of its rivals (cannot increase profit by changing P or Q); best response = a firm’s most profitable choice given the actions of its rivals Bertrand Model – setting prices simultaneously; 1 interaction: theoretically max joint profit when charging monopoly price (MC=MR) but undercutting prices → P=MC, π= 0; infinitely repeated: explicit x tacit collusion (when r is not too high) Cournot Model – choosing quantity (based on beliefs on the other firm’s production) simultaneously → market price – market equilibrium: residual demand for firm 1 from the inverse demand function → profit 1 as a function of q1, q2 → derivative = 0 → best response function for firm 1 → same steps for firm 2 → find q1, q2, market quantity → price, profits Stackelberg Model – choosing quantities sequentially; firm 1 not on its best response function → higher profit, firm 2 is – market equilibrium: find best response function of firm 2 → plug into profit function of firm 1 → derivative = 0 → q1, q2 (from BR2 function) → price, profits