Microeconomics Monopoly and monopsony Harald Wiese Leipzig University Harald Wiese (Leipzig University) Monopoly and monopsony 1 / 53 Structure Introduction Household theory Theory of the firm Perfect competition and welfare theory Types of markets Monopoly and monopsony Game theory Oligopoly External effects and public goods Pareto-optimal review Harald Wiese (Leipzig University) Monopoly and monopsony 2 / 53 Definition monopoly and monopsony Monopoly: one firm sells Monopsony: one firm buys Monopoly: Price setting Quantity setting p X Harald Wiese (Leipzig University) Monopoly and monopsony 3 / 53 Price versus quantity setting p II I p demand p M MC AC y y M y III IV Harald Wiese (Leipzig University) Monopoly and monopsony 4 / 53 Overview Definitions Price setting Revenue and marginal revenue with respect to price Profit Profit maximization (without price differentiation) Quantity setting Revenue and marginal revenue with respect to price (?) Profit Profit maximization without price differentiation Profit maximization with price differentiation Quantity and profit taxes Welfare analysis Monopsony Harald Wiese (Leipzig University) Monopoly and monopsony 5 / 53 Revenue and marginal revenue with respect to price Revenue for demand function X (p): R(p) = pX (p) Marginal revenue (= MR, here MRp): dR dX MR = = X + p (product rule) p dp dp If the price increases by one unit, on the one hand, revenue increases by X (for every sold unit the firm obtains one Euro) dX on the other hand, revenue decreases by p dp (the price increase decreases demand that is valued at price p) Harald Wiese (Leipzig University) Monopoly and monopsony 6 / 53 Profit in the linear model Definition Let X be the demand function. Then P (p) := R (p) − C (p) | {z } | {z } | {z } profit revenue cost is profit depending on price p and P(p) = p (d − ep) − c ((d − ep)) , d c, d, e ≥ 0, p ≤ e profit in the linear model. Functions: price 7! quantity 7! cost Harald Wiese (Leipzig University) Monopoly and monopsony 7 / 53 Revenue, cost and a question I R,C cd C R p ? p Problem What is the economic meaning of the prices with question mark? Harald Wiese (Leipzig University) Monopoly and monopsony 8 / 53 Revenue, cost and a question II R,C cd C R p p0 p Rmax pmax p X CR0 Harald Wiese (Leipzig University) Monopoly and monopsony 9 / 53 Marginal cost with respect to price and with respect to quantity dC dX : marginal cost (with respect to quantity) dC dp : marginal cost (with respect to price) dC dC dX = < 0. dp dX dp |{z} |{z} >0 <0 Chain rule: differentiate C (X (p)) with respect to p means: first, differentiate C with respect to X ) marginal cost then, differentiate X with respect to p ) slope of demand function Functions: price 7! quantity 7! cost Harald Wiese (Leipzig University) Monopoly and monopsony 10 / 53 Profit maximization Profit condition dP dR dC =! 0 or − =! 0 or dp dp dp dR dC =! dp dp Problem Confirm: The profit-maximizing price in the linear model is M d+ce p = 2e . Which price maximizes revenue? Harald Wiese (Leipzig University) Monopoly and monopsony 11 / 53 Profit maximization Comparative static We have d + ce pM = . 2e How does pM change if c increases? Differentiation: dpM 1 = dc 2 Harald Wiese (Leipzig University) Monopoly and monopsony 12 / 53 Exercises Problem 1 Consider a monopolist with cost function C (X ) = cX , c > 0, and demand function X (p) = ap#, # < −1. 1 Determine price elasticity of demand marginal revenue with respect to price 2 Express the monopoly price as a function of #! dpM 3 Determine and interpret dj#j ! Problem 2 The demand function is given by X (p) = 12 − 2p and the cost function of the monopolist by C (X ) = X 2 + 3. Determine the profit-maximizing price! Harald Wiese (Leipzig University) Monopoly and monopsony 13 / 53 The linear model A reminder p a a 1 X , p 1 b a 2b 2 pX 1 p(X ) MR a a X 2b b a a X 2b b Harald Wiese (Leipzig University) Monopoly and monopsony 14 / 53 Marginal revenue Marginal revenue and elasticity (Amoroso-Robinson relation) dR dp MR = = p + X (product rule) dX dX 1 1 = p 1 + = p 1 − > 0 for j#X ,pj > 1 #X ,p j#X ,pj dp Marginal revenue equals price MR = p + X · dX = p in three cases: dp dp horizontal (inverse) demand, dX = 0: MR = p + X · dX = p =0 first small\ unit, X = 0: MR = p + X · dp = p = R(X ) " =0 dX X first-degree price differentiation, MR = p + X · dp =0 dX ) see below Harald Wiese (Leipzig University) Monopoly and monopsony 15 / 53 Profit Definition For X ≥ 0 and inverse demand function p monopoly profit depending on quantity is given by P (X ) := R (X ) − C (X ) = p (X ) X − C (X ) | {z } | {z } | {z } profit revenue cost Linear case: a P(X ) = (a − bX ) X − cX , X ≤ b Harald Wiese (Leipzig University) Monopoly and monopsony 16 / 53 Profit Average and marginal definition profit for X¯ : p P (X¯ ) = p(X¯ )X¯ − C (X¯ ) a D = [p(X¯ ) − AC (X¯ )] X¯ E G (average definition) pX F pX MR H X¯ AC Z C MC = [MR (X ) − MC (X )] dX c B A 0 X X −F (if appropriate) (marginal definition) Harald Wiese (Leipzig University) Monopoly and monopsony 17 / 53 Quantity setting with uniform price We have: inverse demand function for the monopolist: p(X ) total cost: C (X ) Monopolist's profit P : P(X ) = R(X ) − C (X ) = p(X )X − C (X ). Necessary condition for profit maximization: dP dR dC = − =! 0 dX dX dX or, equivalently, MR =! MC Harald Wiese (Leipzig University) Monopoly and monopsony 18 / 53 Quantity setting with uniform price p MC AC Cournot point p M demand MR X M X Problem Inverse demand function p (X ) = 27 − X 2. Revenue-maximizing and profit-maximizing price for MC = 15? Harald Wiese (Leipzig University) Monopoly and monopsony 19 / 53 Clever man: Antoine Augustin Cournot Antoine Augustin Cournot (1801-1877) was a French philosopher, mathematician, and economist. In his main work \Recherches sur les principes math´ematiquesde la th´eorie des richesses", 1838, Cournot presents essential elements of monopoly theory (this chapter) and oligopoly theory (next chapter) Inventor (?) of the Nash equilibrium Harald Wiese (Leipzig University) Monopoly and monopsony 20 / 53 Quantity setting with uniform price The linear model p p MC AC X , p c D a X ,p 1 a 0 E M X , p p M pX pX MR c MR A B F X M X PC X X ( 1 (a−c) , c ≤ a X M = X M (c, a, b) = 2 b 0, c > a Harald Wiese (Leipzig University) Monopoly and monopsony 21 / 53 Quantity setting with uniform price Maximum profit p MC AC Cournot point p M profit ACX M demand MR X M X Harald Wiese (Leipzig University) Monopoly and monopsony 22 / 53 Quantity setting with uniform price Comparative statics I M 1 (a−c) ¶X M ¶X M ¶X M X (a, b, c) = 2 b , where ¶c < 0; ¶a > 0; ¶b < 0, M 1 ¶pM ¶pM ¶pM p (a, b, c) = 2 (a + c), where ¶c > 0; ¶a > 0; ¶b = 0, 2 M 1 (a−c) ¶PM ¶PM ¶PM P (a, b, c) = 4 b , where ¶c < 0; ¶a > 0; ¶b < 0. Problem 2 M 1 (a−c) dPM Show P (c) = 4 b and determine dc ! Hint: Use the chain rule. Harald Wiese (Leipzig University) Monopoly and monopsony 23 / 53 Quantity setting with uniform price Comparative statics I Solution 1 (a−c)2 dPM d 4 b = dc dc 1 = 2(a − c) (−1) 4b a − c = − 2b Harald Wiese (Leipzig University) Monopoly and monopsony 24 / 53 Alternative expressions for profit maximization 1 MC =! MR = p 1 − j#X ,pj ! 1 j#X ,pj p = 1 MC = MC 1 − j#X ,pj − 1 j#X ,p j p − p 1 − 1 p − MC j#X ,p j 1 =! = p p j#X ,pj Harald Wiese (Leipzig University) Monopoly and monopsony 25 / 53 Monopoly power perfect competition: Profit maximization implies \price = marginal cost" Explanation: With perfect competition every firm is \small" and has no influence on price. Inverse demand is then horizontal, hence MR = p. Monopoly: The optimal price is in general above marginal cost. Definition (Lerner index) p − MC p Harald Wiese (Leipzig University) Monopoly and monopsony 26 / 53 Monopoly power Lerner index ! p−MC ! Perfect competition: p = MC and hence p = 0 Monopoly: MC =! MR = p 1 − 1 and hence j#X ,p j p − p 1 − 1 p − MC p − MR j#X ,p j 1 =! = = p p p j#X ,pj Interpretation: If demand reacts strongly to price increases, the monopolist wants to choose a price close to marginal cost. Harald Wiese (Leipzig University) Monopoly and monopsony 27 / 53 Monopoly power, but zero monopoly profit p AC Cournot pM point MC p(X ) MR X X M C X M p > MC, but AC X M = = pM X M Harald Wiese (Leipzig University) Monopoly and monopsony 28 / 53 Forms of price differentiation First-degree price differentiation: Every consumer pays his willingness to pay =) complete absorption of consumer surplus Second-degree price differentiation: The firm requires different prices for different quantities (e.g., quantity discount) =) different prices for high-intensity users and low-intensity users Third-degree price differentiation: Consumers are grouped in different categories. =) uniform price only within a category Harald Wiese (Leipzig University) Monopoly and monopsony 29 / 53 First-degree price differentiation Every consumer pays his willingness to pay: dp MR = p + X · = p =0 dX A price decrease resulting from an extension of output concerns only the marginal consumer, but not inframarginal consumers (those with a higher willingness to pay) Harald Wiese (Leipzig University) Monopoly and monopsony 30 / 53 First-degree price differentiation Formally: Take the derivative of revenue with respect to quantity d R y p (q) dq MR = 0 = p (y) dy Hint: Differentiating an integral with respect to the upper bound of integration yields the value of the integrand (here p (q)) at the upper bound.