## Monday, January 13, 2020

### Microeconomics Ã¢â‚¬ Summary Essay

Linear demand curve: Q = a Ã¢â‚¬â€œ bP Elasticity: E d = (ÃŽâ€Q/ÃŽâ€P)/(P/Q) = -b(P/Q)E d = -1 in the middle of demand curve (up is more elastic) Total revenue and Elasticity:Elastic: Ed < -1Ã¢â€ â€˜PÃ¢â€ â€™Ã¢â€ â€œR (Ã¢â€ â€˜P by 15%Ã¢â€ â€™Ã¢â€ â€œQ by 20%) Inelastic: 0 > Ed > -1Ã¢â€ â€˜PÃ¢â€ â€™Ã¢â€ â€˜R (Ã¢â€ â€˜P by 15%Ã¢â€ â€™Ã¢â€ â€œQ by 3%) Unit elastic: Ed = -1R remains the same (Ã¢â€ â€˜P by 15%Ã¢â€ â€™Ã¢â€ â€œQ by 15%) MR: positive expansion effect (P(Q) Ã¢â‚¬â€œ sell of additional units) + price reduction effect (reduces revenues because of lower price (ÃŽâ€P/ÃŽâ€Q)/Q) 1. Monopoly Ã¢â‚¬â€œ maximizes profit by setting MC = MR MonopolistÃ¢â‚¬â„¢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 Ã¢â‚¬â€œ 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 Ã¢â‚¬â€œ a fixed fee (= consumer surplus) + a separate per-unit price for each unit they buy (P = MC) 2 groups of customers Ã¢â‚¬â€œ with discrimination: inverse demand function for individual demands Ã¢â€ â€™ MR Ã¢â€ â€™ MR=MC * without discrimination: sum of not-inverse demand functions = one option for aggregate demand. Other option is the Ã¢â‚¬Å"richÃ¢â‚¬  people demand function. Compare profits to find Qagg. * max fixed payment F (enabling discrimination) = Ã¢Ë†â€  Ãâ‚¬; max d added to MC1 = Ã¢Ë†â€ Ãâ‚¬/q1 (with discrimination) Quantity-dependent pricing Ã¢â‚¬â€œ one price for first X units and a cheaper price for units above quantity X. profit function = Ãâ‚¬ = 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. ÃŽ  =(20-Qa)Qa + (20-Qb)(Qb-Qa) -2Qb = 18Qb-Qa^2 Ã¢â‚¬â€œ Qb^2 +QaQb derive Ãâ‚¬Ã¢â‚¬â„¢a=-2Qa+Qb Ãâ‚¬Ã¢â‚¬â„¢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 Ã¢â‚¬â€œ not incurred if the production level = 0; unavoidable/sunk Ã¢â‚¬â€œ incurred even if production level = 0, donÃ¢â‚¬â„¢t exist in theÃ‚  long run, for the short run typical Efficient scale of production Ã¢â‚¬â€œ min AC: derivative of AC = 0; MC = AC Production technologies Ã¢â‚¬â€œ production method is efficient it there is no other way to produce more output using the same amounts of inputs Minimization problem Ã¢â‚¬â€œ objective function: min(wL+rK), constraint: subject to Q=f(K,L) Ã¢â€ â€™ express K as a function of L, Q (from production function)Ã¢â€ â€™ plug the expression into objective function (instead of K)Ã¢â€ â€™ derive with respect to L = 0 Ã¢â€ â€™ express L (demand for labor) Ã¢â€ â€™ plug demand for labor into K function Ã¢â€ â€™ express K (demand for capital) Ã¢â€ â€™ TC=wL+rK Marginal product ratio rule Ã¢â‚¬â€œ for f(K,L)=KaLb Ã¢â‚¬â€œ at the optimum: MPL/w = MPK/r : find MPL, MPK from production function Ã¢â€ â€™ find relationship bet ween K,L using marginal product ratio rule Ã¢â€ â€™ plug K/L into production function Ã¢â€ â€™ find K/L for desired level of production Ã¢â‚¬â€œ for f(K,L)=aK+bL: compare MPL/w, MPK/r Ã¢â€ â€™ use production factor with higher marginal value, if equal Ã¢â‚¬â€œ use any combination 4. Perfect Competition Short run Ã¢â‚¬â€œ 1. Quantity rule Ã¢â‚¬â€œ basic condition: MR = P = MC Ã¢â€ â€™ 2. Shut-down rule: P(Q) Ã‹Æ' AC(Q) produce MR=MC, P(Q) Ã‹â€š AC(Q) shut down, P(Q) = AC(Q) Ã¢â‚¬â€œ 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) Ã¢â‚¬â€œ 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 Ã¢â€ â€™ Qs=Qd (demand function) Ã¢â€ â€™ equilibrium price and quantity Long run Ã¢â‚¬â€œ profits = 0 Ã¢â€ â€™ P=AC, equilibrium: MR=P=MC=ACmin * in the long run, unavoidable/sunk cost donÃ¢â‚¬â„¢t exist Ã¢â€ â€™ fixed costs are avoidable Ã¢â€ â€™ take them into account Ã¢â‚¬â€œ market equilibrium: find individual supply function (MC=P), quantity produced by 1 firm (MC=AC =price Ã¢â€ â€™ plug price into demand function Ã¢â € â€™ total quantity demanded Ã¢â€ â€™ number of firms in the market = total quantity demanded/quantity produced by 1 firm 5. Oligopolistic Markets Game Theory Ã¢â‚¬â€œ 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 firmÃ¢â‚¬â„¢s most profitable choice given the actions of its rivals Bertrand Model Ã¢â‚¬â€œ setting prices simultaneously; 1 interaction:Ã‚  theoretically max joint profit when charging monopoly price (MC=MR) but undercutting prices Ã¢â€ â€™ P=MC, Ãâ‚¬= 0; infinitely repeated: explicit x tacit collusion (when r is not too high) Cournot Model Ã¢â‚¬â€œ choosing quantity (based on beliefs on the other firmÃ¢â‚¬â„¢s production) simultaneously Ã¢â€ â€™ market price Ã¢â‚¬â€œ market equilibrium: residual demand for firm 1 from the inverse demand function Ã¢â€ â€™ profit 1 as a function of q1, q2 Ã¢â€ â€™ derivative = 0 Ã¢â€ â€™ best response function for firm 1 Ã¢â€ â€™ same steps for firm 2 Ã¢â€ â€™ find q1, q2, market quantity Ã¢â€ â€™ price, profits Stackelberg Model Ã¢â‚¬â€œ choosing quantities sequentially ; firm 1 not on its best response function Ã¢â€ â€™ higher profit, firm 2 is Ã¢â‚¬â€œ market equilibrium: find best response function of firm 2 Ã¢â€ â€™ plug into profit function of firm 1 Ã¢â€ â€™ derivative = 0 Ã¢â€ â€™ q1, q2 (from BR2 function) Ã¢â€ â€™ price, profits