Robust Monopoly Regulation Yingni Guo, Eran Shmaya Northwestern - PowerPoint PPT Presentation

Robust Monopoly Regulation Yingni Guo, Eran Shmaya Northwestern University CCET, Sep 2019

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Regulating monopolies is challenging A regulator may want to constrain a monopolistic firm’s price 2 / 27

Regulating monopolies is challenging A regulator may want to constrain a monopolistic firm’s price Price-constrained firm may fail to cover its fixed cost, ending up producing at an ine ffi ciently low level 2 / 27

Regulating monopolies is challenging A regulator may want to constrain a monopolistic firm’s price Price-constrained firm may fail to cover its fixed cost, ending up producing at an ine ffi ciently low level Protect consumer well-being versus not to distort production 2 / 27

Regulating monopolies is challenging The challenge could be solved if the regulator had complete information 3 / 27

Regulating monopolies is challenging The challenge could be solved if the regulator had complete information – let the firm price at marginal cost 3 / 27

Regulating monopolies is challenging The challenge could be solved if the regulator had complete information – let the firm price at marginal cost – subsidize the firm for its other costs 3 / 27

Regulating monopolies is challenging The challenge could be solved if the regulator had complete information – let the firm price at marginal cost – subsidize the firm for its other costs What shall the regulator do when he knows much less about the industry than the firm does? 3 / 27

Regulating monopolies is challenging The challenge could be solved if the regulator had complete information – let the firm price at marginal cost – subsidize the firm for its other costs What shall the regulator do when he knows much less about the industry than the firm does? If he wants a policy that works “fairly well” in all circumstances, what shall this policy look like? 3 / 27

What we do Regulator’s payo ff consumer surplus + φ firm’s profit , φ ∈ [0 , 1] 4 / 27

What we do Regulator’s payo ff consumer surplus + φ firm’s profit , φ ∈ [0 , 1] He can regulate firm’s price and quantity, give a subsidy, charge a tax 4 / 27

What we do Regulator’s payo ff consumer surplus + φ firm’s profit , φ ∈ [0 , 1] He can regulate firm’s price and quantity, give a subsidy, charge a tax Given a demand and cost, regret to the regulator: regret = payo ff if he had complete information − what he got 4 / 27

What we do Regulator’s payo ff consumer surplus + φ firm’s profit , φ ∈ [0 , 1] He can regulate firm’s price and quantity, give a subsidy, charge a tax Given a demand and cost, regret to the regulator: regret = payo ff if he had complete information − what he got Optimal policy: minimize demand,cost regret max policy � �� � worst-case regret 4 / 27

What we find φ = 0 consumer surplus (consumer well-being) 5 / 27

What we find φ = 0 consumer surplus (consumer well-being) impose a price cap 5 / 27

What we find φ = 0 consumer surplus (consumer well-being) impose a price cap gain from lower price loss from underproduction 5 / 27

What we find φ = 0 φ = 1 consumer surplus consumer surplus + firm’s profit (consumer well-being) (e ffi ciency) impose a price cap gain from lower price loss from underproduction 5 / 27

What we find φ = 0 φ = 1 consumer surplus consumer surplus + firm’s profit (consumer well-being) (e ffi ciency) encourage production impose a price cap with capped subsidy gain from lower price loss from underproduction 5 / 27

What we find φ = 0 φ = 1 consumer surplus consumer surplus + firm’s profit (consumer well-being) (e ffi ciency) encourage production impose a price cap with capped subsidy gain from lower price loss from underproduction loss from underproduction loss from overproduction 5 / 27

What we find φ = 0 φ = 1 consumer surplus consumer surplus + firm’s profit (consumer well-being) (e ffi ciency) encourage production impose a price cap with capped subsidy gain from lower price loss from underproduction loss from underproduction loss from overproduction φ ∈ (0 , 1) combination of price cap and capped subsidy 5 / 27

What we find more surplus for consumers mitigate under- mitigate over- production production φ ∈ (0 , 1) combination of price cap and capped subsidy 6 / 27

Closest literature Monopoly regulation: Baron and Myerson (1982) Mechanism design with worst-case regret: Hurwicz and Shapiro (1978), Bergemann and Schlag (2008, 2011), Renou and Schlag (2011) Delegation: Holmstr¨ om (1977, 1984) 7 / 27

Roadmap Environment Main result

Environment A mass one of consumers 8 / 27

Environment A mass one of consumers v : [0 , 1] → [0 , ¯ v ]: a decreasing u.s.c. inverse demand function 8 / 27

Environment A mass one of consumers v : [0 , 1] → [0 , ¯ v ]: a decreasing u.s.c. inverse demand function – ( q , p ) is feasible if p � v ( q ) 8 / 27

Environment A mass one of consumers v : [0 , 1] → [0 , ¯ v ]: a decreasing u.s.c. inverse demand function – ( q , p ) is feasible if p � v ( q ) c : [0 , 1] → R + with c (0) = 0: an increasing l.s.c. cost function 8 / 27

Environment Maximal total surplus is � q OPT = max v ( z ) d z − c ( q ) q ∈ [0 , 1] 0 � �� � total value to consumers 9 / 27

Environment Maximal total surplus is � q OPT = max v ( z ) d z − c ( q ) q ∈ [0 , 1] 0 � �� � total value to consumers If the firm produces q , the distortion is �� q � DSTR = OPT − v ( z ) d z − c ( q ) 0 9 / 27

Environment: two examples v ( q ) v ¯ 2¯ v 3 z q 2 z 0 1 3 c ( q ) = 0 10 / 27

Environment: two examples v ( q ) v ¯ 2¯ v 3 z q 2 z 0 1 3 c ( q ) = 0 � 1 If q = 2 3 , DSTR = 2¯ v 1 z d z 2 3 3 = − 2¯ 3 log 2 v 3 10 / 27

Environment: two examples v ( q ) v ( q ) v ¯ v ¯ 2¯ v 3 z ¯ v 2 q q 2 1 z 0 1 0 1 3 2 c ( q ) = 0 � 1 If q = 2 3 , DSTR = 2¯ v 1 z d z 2 3 3 = − 2¯ 3 log 2 v 3 10 / 27

Environment: two examples v ( q ) v ( q ) v ¯ v ¯ 2¯ v 3 z ¯ v 2 q q 2 1 z 0 1 0 1 3 2 c ( q ) = ¯ v c ( q ) = 0 3 � 1 If q = 2 3 , DSTR = 2¯ v 1 If q = 1 2 , DSTR = ¯ v 3 − ¯ v z d z 2 3 4 3 = − 2¯ 3 log 2 v 3 10 / 27

Regulation policy A policy is an u.s.c. function ρ : [0 , 1] × [0 , ¯ v ] → R – if the firm sells q at price p , then it receives ρ ( q , p ) – e.g., if ρ ( q , p ) > qp , a subsidy of ρ ( q , p ) − qp – the firm is allowed to stay out of business with a profit of zero If ρ ( q , p ) = qp , ∀ q , p , the firm is unregulated 11 / 27

Firm’s best response and regulator’s payo ff Fix ρ , v , c If the firm sells q at price p , the firm’s profit and consumer surplus are: � q FP = ρ ( q , p ) − c ( q ) , CS = v ( z ) d z − ρ ( q , p ) 0 12 / 27

Firm’s best response and regulator’s payo ff Fix ρ , v , c If the firm sells q at price p , the firm’s profit and consumer surplus are: � q FP = ρ ( q , p ) − c ( q ) , CS = v ( z ) d z − ρ ( q , p ) 0 ( q , p ) is the firm’s best response to ( v , c ) under ρ if it maximizes FP among all feasible pairs 12 / 27

Firm’s best response and regulator’s payo ff Fix ρ , v , c If the firm sells q at price p , the firm’s profit and consumer surplus are: � q FP = ρ ( q , p ) − c ( q ) , CS = v ( z ) d z − ρ ( q , p ) 0 ( q , p ) is the firm’s best response to ( v , c ) under ρ if it maximizes FP among all feasible pairs The regulator’s payo ff is CS + φ FP , φ ∈ [0 , 1] 12 / 27

If complete information, regulator gets OPT 13 / 27

If complete information, regulator gets OPT Claim Suppose that the regulator knows ( v , c ). Then max ( CS + φ FP ) = OPT , 13 / 27

If complete information, regulator gets OPT Claim Suppose that the regulator knows ( v , c ). Then max ( CS + φ FP ) = OPT , where the maximum is over all ρ and all firm’s best responses ( q , p ) to ( v , c ) under ρ . 13 / 27

If complete information, regulator gets OPT Claim Suppose that the regulator knows ( v , c ). Then max ( CS + φ FP ) = OPT , where the maximum is over all ρ and all firm’s best responses ( q , p ) to ( v , c ) under ρ . Let q ∗ denote the socially optimal quantity 13 / 27

If complete information, regulator gets OPT Claim Suppose that the regulator knows ( v , c ). Then max ( CS + φ FP ) = OPT , where the maximum is over all ρ and all firm’s best responses ( q , p ) to ( v , c ) under ρ . Let q ∗ denote the socially optimal quantity Let ρ ( q ∗ , v ( q ∗ )) = c ( q ∗ ) 13 / 27