Demand and Elasticity
Demand and Elasticity
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 = -1 R 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 pay MR=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
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 = -1 R 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 pay MR=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