Quantitative Comparative Statics for a Multimarket Paradox Philipp von Falkenhausen Technische Universität Berlin July 11, 2013 Joint work with Tobias Harks
Agenda Comparative Statics 1 Quantitative Comparative Statics 2
Agenda Comparative Statics 1 Quantitative Comparative Statics 2
Comparative Statics System at Marginal change marginal − − − − − − − − − − → equilibrium of equilibrium parameter change
Comparative Statics System at Marginal change marginal − − − − − − − − − − → equilibrium of equilibrium parameter change Examples Introduction of export taxes/subsidies (Brander and Spencer 1985, Eaton and Grossman 1986) Demand or cost shift (Quirmbach 1988, Février and Linnemer 2004) Forced reduction of produced quantity (Gaudet and Salant 1991)
Multimarket Cournot Oligopoly Oligopoly Finite number of competing firms i ∈ N producing some good. Firm i has cost c i ( q i ) for producing quantity q i . Multimarket markets m ∈ M served with the good. Cournot Each firm i produces quantity q i , m in market m , the market price is p m ( � i q i , m )
Multimarket Cournot Oligopoly Oligopoly Finite number of competing firms i ∈ N producing some good. Firm i has cost c i ( q i ) for producing quantity q i . Multimarket markets m ∈ M served with the good. Cournot Each firm i produces quantity q i , m in market m , the market price is p m ( � i q i , m ) revenue of firm i : � m ∈ M p m ( � j q j , m ) q i , m profit of firm i : revenue - cost marginal revenue of firm i on market m : π i , m ( q i , m , q − i , m ) := p m ( q i , m + q − i , m ) + p ′ m ( q i , m + q − i , m ) q i , m
Multimarket Cournot Oligopoly Oligopoly Finite number of competing firms i ∈ N producing some good. Firm i has cost c i ( q i ) for producing quantity q i . Multimarket markets m ∈ M served with the good. Cournot Each firm i produces quantity q i , m in market m , the market price is p m ( � i q i , m ) revenue of firm i : � m ∈ M p m ( � j q j , m ) q i , m profit of firm i : revenue - cost marginal revenue of firm i on market m : π i , m ( q i , m , q − i , m ) := p m ( q i , m + q − i , m ) + p ′ m ( q i , m + q − i , m ) q i , m Definition (Cournot Equilibrium) Given choices q − i of other firms, firm i produces q i such that marginal revenue on market m = marginal cost for all m ∈ M π i , m ( q i , m , q − i , m ) = c ′ � i ( q i , m ) m
Paradox: Price increase reduces profit of monopolist Example by Bulow, Geanakoplos, Klemperer (1985) Market 1 Market 2 Firm a Firm a , firm b Price: p 1 ( q 1 ) = 50 Price: p 2 ( q 2 ) = 200 − q 2 Both firms have cost: c i ( q i ) = 1 2 q 2 i
Paradox: Price increase reduces profit of monopolist Example by Bulow, Geanakoplos, Klemperer (1985) Market 1 Market 2 Firm a Firm a , firm b Price: p 1 ( q 1 ) = 50 Price: p 2 ( q 2 ) = 200 − q 2 Both firms have cost: c i ( q i ) = 1 2 q 2 i When the price on market 1 increases by 10%, the profit of firm a decreases by 0.76%.
Paradox: Price increase reduces profit of monopolist Example by Bulow, Geanakoplos, Klemperer (1985) Market 1 Market 2 Firm a Firm a , firm b Price: p 1 ( q 1 ) = 50 Price: p 2 ( q 2 ) = 200 − q 2 Both firms have cost: c i ( q i ) = 1 2 q 2 i When the price on market 1 increases by 10%, the profit of firm a decreases by 0.76%. Definition (Strategic Substitutes, Bulow et al. 1985) "Less ‘aggressive’ play (e.g., [...] lower quantity) by one firm raises competing firms’ marginal profitabilities."
Profit loss of 0.76% - so what?! Comparative statics studies marginal changes of a parameter. Open questions significance: are changes in a given parameter worth considering? 1 robustness: how sensitive is the game to changes of a parameter? 2 ⇒ Quantitative approach
Agenda Comparative Statics 1 Quantitative Comparative Statics 2
Model Assumptions Market 2 Market 1 Firm a Firm a Firms i � = a Price is affine, decreasing function of quantity Cost is convex, differentiable function of quantity Objective What is max. impact of price shock δ on market 1 on profit of firm a ? γ := eq. profit after shock eq. profit before shock
Main result Theorem (Main result) For an instance with n firms and a positive price shock γ ≥ ( 3 n − 1 )( n + 1 ) ≥ 3 4 . 4 n 2 The profit loss is at most 25%.
Main result Theorem (Main result) For an instance with n firms and a positive price shock γ ≥ ( 3 n − 1 )( n + 1 ) ≥ 3 4 . 4 n 2 The profit loss is at most 25%. Corollary (Dual result) For an instance with n firms and a negative price shock 4 n 2 ( 3 n − 1 )( n + 1 ) ≤ 4 γ ≤ 3 .
Main result Theorem (Main result) For an instance with n firms and a positive price shock γ ≥ ( 3 n − 1 )( n + 1 ) ≥ 3 4 . 4 n 2 The profit loss is at most 25%. Corollary (Dual result) For an instance with n firms and a negative price shock 4 n 2 ( 3 n − 1 )( n + 1 ) ≤ 4 γ ≤ 3 . Complementing Bound Large class of instances where 25% profit loss is attained.
Proof Overview Establish basics 1 ◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution Series of simplifications 2 ◮ Given instance G , construct simplified ˜ G with ˜ γ ≤ γ Proof theorem for simplified game 3
Proof Overview Establish basics 1 ◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution Series of simplifications 2 ◮ Given instance G , construct simplified ˜ G with ˜ γ ≤ γ Proof theorem for simplified game 3
Paradox: Explained Firm a Firm b c ′ c ′ b ( q b ) a ( q a ) q a q b Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π a , 2 ( q a , 2 , x b , 2 ) c ′ c ′ b ( q b ) a ( q a ) π b 2 ( , q b 2 , , x a 2 ) , q a x b , 2 q b Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π a , 2 ( q a , 2 , x b , 2 ) c ′ c ′ b ( q b ) a ( q a ) π b 2 ( , q b 2 , , p 1 x a 2 ) , q a x b , 2 q b Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π a , 2 ( q a , 2 , x b , 2 ) c ′ c ′ b ( q b ) a ( q a ) π b 2 ( , q b 2 , , p 1 x a 2 ) , x a , 2 q a x b , 2 q b x a , 2 + x a , 1 Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π a , 2 ( q a , 2 , x b , 2 ) c ′ c ′ b ( q b ) a ( q a ) π b 2 ( , p 1 + δ q b 2 , , p 1 x a 2 ) , x a , 2 q a x b , 2 q b x a , 2 + x a , 1 Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π a , 2 ( q a , 2 , x b , 2 ) c ′ c ′ b ( q b ) a ( q a ) π b 2 ( , p 1 + δ q b 2 , , p 1 x a 2 ) , q a x b , 2 q b Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π a , 2 ( q a , 2 , x b , 2 ) π b , 2 ( q b , 2 , y a , 2 ) c ′ c ′ b ( q b ) a ( q a ) p 1 + δ q a y b , 2 q b Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π a , 2 ( q a , 2 , x b , 2 ) π b , 2 ( q b , 2 , y a , 2 ) c ′ c ′ b ( q b ) a ( q a ) p 1 + δ q a y b , 2 q b Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π b , 2 ( q b , 2 , y a , 2 ) c ′ c ′ b ( q b ) a ( q a ) π a , 2 ( q a , 2 , y b , 2 ) p 1 + δ q a y b , 2 q b Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π b , 2 ( q b , 2 , y a , 2 ) c ′ c ′ b ( q b ) a ( q a ) π a , 2 ( q a , 2 , y b , 2 ) p 1 + δ y a , 2 q a y b , 2 q b y a , 2 + y a , 1 Initial equilibrium x → Price shock δ → New equilibrium y
Paradox: Explained Firm a Firm b π b , 2 ( q b , 2 , y a , 2 ) c ′ c ′ b ( q b ) a ( q a ) π a , 2 ( q a , 2 , y b , 2 ) p 1 + δ y a , 2 q a y b , 2 q b y a , 2 + y a , 1 Initial equilibrium x → Price shock δ → New equilibrium y
Proof Overview Establish basics 1 ◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution Series of simplifications 2 ◮ Given instance G , construct simplified ˜ G with ˜ γ ≤ γ Proof theorem for simplified game 3
Proof Overview Establish basics 1 ◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution Series of simplifications 2 ◮ Given instance G , construct simplified ˜ G with ˜ γ ≤ γ Proof theorem for simplified game 3
Competitors Are Most Aggressive With Linear Cost Lemma For given G, let ˜ G be similiar to G except that ˜ c i ( q i ) = c ′ i ( x i ) q i for all firms i � = a. Then, ˜ γ ≤ γ .
Competitors Are Most Aggressive With Linear Cost Lemma For given G, let ˜ G be similiar to G except that ˜ c i ( q i ) = c ′ i ( x i ) q i for all firms i � = a. Then, ˜ γ ≤ γ . Proof (by picture). linear cost strictly convex cost π π i i c ′ 2 i ( q i ) ( , 2 ( , q q , i i , y y − − π ) π i ) i i i 2 ( , 2 ( q , q , i i , x x − − ) i i ) c ′ ˜ i ( x ) x i , 2 q i ˜ x i , 2 y i , 2 q i y i , 2 More strategic substitution: y i , 2 < ˜ y i , 2 .
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