VIEWS: 17 PAGES: 14 POSTED ON: 10/12/2011
Stratégies des entreprises Course outline I Introduction Game theory Price setting – monopoly – oligopoly Homogeneous Quantity setting goods – monopoly – oligopoly Process innovation 1 Monopoly (quantity setting) Inverse demand function Revenues, costs, profits Profit maximizing quantity – Basic model – Price discrimination – Several factories Double marginalization Welfare analysis Executive summary 2 Decision problem X Π 3 Harald Wiese 1 Stratégies des entreprises Inverse demand function p demand p function inverse demand p( X ) function X X ( p) X 4 Revenue, costs and profit Revenue: R( X ) = p ( X ) ⋅ X Costs: C(X ) Profit: Π ( X ) = R( X ) − C ( X ) 5 Marginal revenue with respect to quantity dR dp( X ) = MR X = p( X ) + X dX dX When a firm increases the quantity by one unit, revenue goes up by p (the price of the last unit), but goes down by dp/dX X (the quantity increase diminishes the price and this price decrease is applied to all units) ⎛ X dp ( X ) ⎞ Amoroso-Robinson MR X = p ( X ) ⎜ 1 + ⎟ relation: ⎝ p( X ) dX ⎠ ⎛ 1 ⎞ ⎛ 1 ⎞ = p ⋅⎜ 1+ ⎟ = p ⋅⎜ 1− ⎟ ⎜ ε ⎟ ⎜ ε ⎟ ⎝ X ,p ⎠ ⎝ X ,p ⎠ 6 Harald Wiese 2 Stratégies des entreprises Marginal revenue = price? dR dp ( X ) = p( X ) + X dX dX dp ( X ) 1. =0 ⇒ i.e. horizontal demand curve, perfect dX competition 2. X=0 ⇒ sale of first unit ⇒ first degree price discrimination 7 Linear demand curve in a monopoly Demand: p( X ) = a − bX Revenue: R ( X ) = aX − bX 2 Marginal revenue: MR = a − 2bX 8 Exercise (Depicting the linear demand curve p( X ) = a − bX ) Slope of demand curve: .... Slope of marginal revenue curve: .... The ............................ has the same vertical intercept, ..., as the demand curve. Economically, – the vertical intercept is ................., – the horizontal intercept is ................. . 9 Harald Wiese 3 Stratégies des entreprises Depicting demand and marginal revenue p a 1 b 2b 1 p( X ) MR X a/(2b) a/b 10 First order condition dΠ ( X ) dp ( X ) dC ( X ) ! = p( X ) + X − =0 dX dX dX MR X MC X Notation: MR := MRX MC := MC X 11 First order condition, alternative formulations ! ⎛ 1 ⎞ MC = MR = p ⋅⎜ 1− ⎟ ⎜ ε ⎟ ⎝ X ,p ⎠ ! 1 ε X ,p p= MC = MC 1− 1 ε X ,p − 1 ε X ,p p − MC ! 1 = (price-cost margin) p ε X ,p 12 Harald Wiese 4 Stratégies des entreprises Depicting the Cournot monopoly p MC Cournot M p point p( X ) MR X XM 13 Exercise (Quantity) Consider a monopolist facing the inverse demand function p(X)=24-X. Assume that the average and marginal costs are given by AC=2. Find the profit-maximizing quantity! S. : X M = 11 14 Profit in a monopoly Marginal point of view: Average point of view: p p p(X) M pM p MC MC p(X) AC MR X X XM XM MR 15 Harald Wiese 5 Stratégies des entreprises Exercise (monopoly) Consider a monopoly facing the inverse demand function p(X)=40-X2. Assume that the cost function is given by C(X)=13X+19. Find the profit-maximizing price and calculate the profit. S. : Π M = 35 16 Price discrimination First degree price discrimination: Every consumer pays a different price which is equal to his or her willingness to pay. Second degree price discrimination: Prices differ according to the quantity demanded and sold (quantity rebate). Third degree price discrimination: Consumer groups (students, children, ...) are treated differently. 17 Monopolistic price discrimination (two markets) p p1 p2 x1 (p1 ) x 2 (p 2 ) MR 1 MR 2 MC * x1 x1 total output x* 2 x2 18 Harald Wiese 6 Stratégies des entreprises Inverse elasticities rule for third degree price discrimination Supplying a good X to two markets results in the inverse demand functions p1(x1) and p2(x2). Profit function: Π ( x1 ,x2 )= p1 ( x1 )⋅x1 + p2 ( x2 )⋅x2 −C ( x1 + x2 ) 4 4 4 3 1 2 3 1 24 R1 (x1 ) R2 (x2 ) ∂ Π ( x1 ,x2 ) ! First order conditions: =MR1 ( x1 )− MC ( x1 + x2 )=0 ∂ x1 = ∂ Π (x1 ,x2 ) ! = MR2 ( x2 )− MC ( x1 + x2 )=0 ∂x 2 Equating the marginal revenues (using the Amoroso-Robinson relation) leads to: ⎛ 1 ⎞! ⎛ 1 ⎞ ⎜ ε ( x ) ⎟= p2 ( x 2 )⋅⎜1− ε ( x ) ⎟ p1 ( x1 )⋅⎜1− ⎟ ⎜ ⎟ ⎝ 1 1 ⎠ ⎝ 2 2 ⎠ ε 1 ( x1 ) < ε 2 ( x2 ) ⇒ p1 ( x1 ) > p2 ( x2 ) 19 Exercise (two markets or one) A monopoly sells in two markets: p1(x1)=100-x1 and p2(x2)=80-x2. a) Calculate the profit-maximizing quantities and the profit at these quantities, if the cost function is given by C(X)=X2. b) Calculate the profit-maximizing quantities and the profit at these quantities, if the cost function is given by C(X)=10X. c) What happens if price discrimination between the two markets is not possible anymore? Consider C(X)=10X. Hint: Differentiate between quantities below and above 20. S. : a) Π M = 1400, b) Π M = 3250, c) Π M = 3200 20 Solution III (one market) p 100 90 80 50 MR p( X ) 10 MC 20 50 80 100 X 21 Harald Wiese 7 Stratégies des entreprises One market, two factories Profit function: Π ( x1 , x2 ) = p ( x1 , x2 )( x1 + x2 ) − C1 ( x1 ) − C2 ( x2 ) First order conditions: ∂Π ( x1 , x2 ) ! = MR ( x1 , x2 ) − MC1 ( x1 ) = 0 ∂x1 = ∂Π ( x1 , x2 ) ! = MR ( x1 , x2 ) − MC 2 ( x2 ) = 0 ∂x2 ! ⇒ MC1 ( x1 ) = MC 2 ( x2 ) 22 One market, two factories II factory 2 factory 1 p MC 2 MC1 x2 * * x1 x2 x1 total output 23 Double marginalization - idea Retailer, not producer sells to consumers. Assumptions: – Zero costs for retailing. – Producer decides on a quantity and charges a price ppro to the retailer. – ppro is the retailer‘s marginal cost. – The retailer‘s MR=MC condition defines the producer‘s demand function. 24 Harald Wiese 8 Stratégies des entreprises Double marginalization - linear case p(x)=a-bX MCpro=ACpro=c Second stage: First order condition for the retailer: ! MC = p pro = a − 2bX = MR First stage: First order condition for the producer: ! MC pro = c = a − 4bX = MR pro a−c a+c X = ⇒ p pro* = 4b 2 3a + c p* = 4 25 Double marginalization - depicting the solution p Producer decides a on quantity and retailer announces ppro* to retailer. p * = 3a4+ c producer Retailer decides on quantity (fore- p pro* = a 2 c + seen by producer). MR pro MR = p pro ( X ) c a −c a a X 4b 2b b 26 Double marginalization - exercise p(X)=110-X c=10 a) Calculate the price the consumers have to pay! b) What price would they pay if the producer sold directly to the consumers? S. : a) p M = 85 b) p M = 60 27 Harald Wiese 9 Stratégies des entreprises Welfare Analysis Evaluation of economic policy measures Welfare = consumer surplus (CS) + producer surplus (PS) + taxes - subsidies CS = willingness to pay - price PS = revenue - variable costs = profit + fixed costs 28 CS, PS - graphically p supply (=MC) CS ï Welfare is PS demand maximized at the equilibrium. 0 X 29 The deadweight loss of a monopoly p Without price discrimination a monopoly realizes a deadweight loss. MC pM Perfect p∗ p = MC competition MR = MC p( X ) MR XM X∗ X 30 Harald Wiese 10 Stratégies des entreprises Exercise (deadweight loss) Consider a monopoly where the demand is given by p(X)=-2X+12. Suppose that the marginal costs are given by MC=2X. Calculate the deadweight loss – without price discrimination, – with perfect price discrimination. S. : DLwopd = 2 DLwpd = 0 31 Exercise (price cap in a monopoly) p How does a price cap influence the demand and the marginal revenue curves? MC pM p cap p( X ) MR X M X 32 Right or wrong? Why? p MC cap p pM p( X ) MR XM X 33 Harald Wiese 11 Stratégies des entreprises Price cap and welfare p additional welfare MC pM p cap =p M ,cap p( X ) X X M X M ,cap MR 34 Taxes on profits p C( X ) R( X ) Π(X) MR pM MC Π(X)(1−t) p( X ) XM X 35 Taxes on profits and welfare Quantity / price unchanged CS is constant PS decrease; in the same extent net public revenues increase deadweight loss is zero 36 Harald Wiese 12 Stratégies des entreprises Additional deadweight loss due to quantity tax p consumer’s surplus: A+B+C→A producer’s surplus: T+E+F →E+B additional net public revenue: 0 →T deadweight loss A pT C p B MC + t MC E F T p( X ) MR X XT X 37 Exercise (quantity taxes) A monopolist is facing a demand curve given by p(X)=a-X. The monopoly’s unit production cost is given by c>0. Now, suppose that the government imposes a specific tax of t dollars per unit sold. a) Show that this tax would raise the price paid by consumers by less than t. b) Would your answer change if the market inverse demand curve is given by p(X)=-ln(X)+5. c) If the demand curve is given by p(X)=X-1/2, what is the influence on price? 1 Industrial Organization ; Oz Shy S. : a) , b) 1, c) 2 2 38 Illustrating the solutions p a) b) p pM(c+t) pM(c+t) pM(c) c+t c+t p( X ) p (c) M p( X ) MR c c X MR X 39 Harald Wiese 13 Stratégies des entreprises Lerner index of monopoly power First order condition: ! dp ⎛ 1 ⎞ MC ( X ) = MR ( X ) = p( X ) + X = p⎜1 − ⎟ dX ⎜ ε X ,p ⎟ ⎝ ⎠ Lerner index: ⎛ 1 ⎞ p − p ⎜1 − ⎟ p − MC ⎜ ε ⎟ ! = ⎝ X,p ⎠= 1 p p εX ,p 40 Monopoly profits and monopoly power p AC pM = AC ⇒ ΠM = 0 Cournot pM point MC p( X ) MR X XM 41 Executive summary A profit-maximizing monopolist always sets the quantity in the elastic region of the demand curve. Monopolistic power: price will be set above the marginal cost by a profit maximizer. If the demand curve is tangent to the average cost curve, the profit-maximizing price is set above marginal cost and equal to average cost. ⇒ monopolistic power and zero profits Monopolistic quantities without price discrimination (!) lead to a welfare loss. A quantity tax leads to a welfare loss, a tax on profits does not. 42 Harald Wiese 14