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

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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 inefficiently low level

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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 inefficiently low level Protect consumer well-being versus not to distort production

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Regulating monopolies is challenging

The challenge could be solved if the regulator had complete information

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Regulating monopolies is challenging

The challenge could be solved if the regulator had complete information

– let the firm price at marginal cost

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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

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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?

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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?

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What we do

Regulator’s payoff consumer surplus + φ firm’s profit, φ ∈ [0, 1]

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What we do

Regulator’s payoff consumer surplus + φ firm’s profit, φ ∈ [0, 1] He can regulate firm’s price and quantity, give a subsidy, charge a tax

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What we do

Regulator’s payoff 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 = payoff if he had complete information − what he got

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What we do

Regulator’s payoff 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 = payoff if he had complete information − what he got Optimal policy: minimize

policy

max

demand,cost regret

• worst-case regret

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What we find

φ = 0 consumer surplus (consumer well-being)

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What we find

φ = 0 consumer surplus (consumer well-being) impose a price cap

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What we find

φ = 0 consumer surplus (consumer well-being) impose a price cap gain from lower price loss from underproduction

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What we find

φ = 0 consumer surplus (consumer well-being) impose a price cap gain from lower price loss from underproduction φ = 1 consumer surplus + firm’s profit (efficiency)

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What we find

φ = 0 consumer surplus (consumer well-being) impose a price cap gain from lower price loss from underproduction φ = 1 consumer surplus + firm’s profit (efficiency) encourage production with capped subsidy

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What we find

φ = 0 consumer surplus (consumer well-being) impose a price cap gain from lower price loss from underproduction φ = 1 consumer surplus + firm’s profit (efficiency) encourage production with capped subsidy loss from underproduction loss from overproduction

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What we find

φ = 0 consumer surplus (consumer well-being) impose a price cap gain from lower price loss from underproduction φ = 1 consumer surplus + firm’s profit (efficiency) encourage production with capped subsidy loss from underproduction loss from overproduction φ ∈ (0, 1) combination of price cap and capped subsidy

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What we find

more surplus for consumers mitigate under- production mitigate over- production φ ∈ (0, 1) combination of price cap and capped subsidy

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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¨

• m (1977, 1984)

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Roadmap

Environment Main result

Environment

A mass one of consumers

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Environment

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

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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)

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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

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Environment

Maximal total surplus is OPT = max

q∈[0,1]

q v(z) dz

• total value to consumers

− c(q)

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Environment

Maximal total surplus is OPT = max

q∈[0,1]

q v(z) dz

• total value to consumers

− c(q) If the firm produces q, the distortion is DSTR = OPT − q v(z) dz − c(q)

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Environment: two examples

v(q) 1 ¯ v q

2 3

z

2¯ v 3z

c(q) = 0

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Environment: two examples

v(q) 1 ¯ v q

2 3

z

2¯ v 3z

c(q) = 0 If q = 2

3, DSTR = 2¯ v 3

1

2 3

1 z dz

= − 2¯

v 3 log 2 3

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Environment: two examples

v(q) 1 ¯ v q

2 3

z

2¯ v 3z

c(q) = 0 If q = 2

3, DSTR = 2¯ v 3

1

2 3

1 z dz

= − 2¯

v 3 log 2 3

v(q) 1

1 2

¯ v

¯ v 2

q

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Environment: two examples

v(q) 1 ¯ v q

2 3

z

2¯ v 3z

c(q) = 0 If q = 2

3, DSTR = 2¯ v 3

1

2 3

1 z dz

= − 2¯

v 3 log 2 3

v(q) 1

1 2

¯ v

¯ v 2

q c(q) = ¯

v 3

If q = 1

2, DSTR = ¯ v 3 − ¯ v 4

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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

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Firm’s best response and regulator’s payoff

Fix ρ, v, c If the firm sells q at price p, the firm’s profit and consumer surplus are: FP = ρ(q, p) − c(q), CS = q v(z) dz − ρ(q, p)

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Firm’s best response and regulator’s payoff

Fix ρ, v, c If the firm sells q at price p, the firm’s profit and consumer surplus are: FP = ρ(q, p) − c(q), CS = q v(z) dz − ρ(q, p) (q, p) is the firm’s best response to (v, c) under ρ if it maximizes FP among all feasible pairs

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Firm’s best response and regulator’s payoff

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

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If complete information, regulator gets OPT

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If complete information, regulator gets OPT

Claim

Suppose that the regulator knows (v, c). Then max (CS + φFP) = OPT,

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If complete information, regulator gets OPT

Claim

Suppose that the regulator knows (v, c). Then max (CS + φFP) = OPT, where the maximum is over all ρ and all firm’s best responses (q, p) to (v, c) under ρ.

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If complete information, regulator gets OPT

Claim

Suppose that the regulator knows (v, c). Then max (CS + φFP) = 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

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If complete information, regulator gets OPT

Claim

Suppose that the regulator knows (v, c). Then max (CS + φFP) = 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∗)

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If complete information, regulator gets OPT

Claim

Suppose that the regulator knows (v, c). Then max (CS + φFP) = 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∗) Let ρ(q, p) = 0 for (q, p) ̸= (q∗, v(q∗))

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Simplifying regret

Fix ρ, v, c

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Simplifying regret

Fix ρ, v, c If the firm chooses (q, p), then

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Simplifying regret

Fix ρ, v, c If the firm chooses (q, p), then RGRT = OPT − (CS + φFP)

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Simplifying regret

Fix ρ, v, c If the firm chooses (q, p), then RGRT = OPT − (CS + φFP) = OPT − (CS + FP) + (1 − φ)FP

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Simplifying regret

Fix ρ, v, c If the firm chooses (q, p), then RGRT = OPT − (CS + φFP) = OPT − (CS + FP) + (1 − φ)FP = OPT − q v(z) dz − c(q)

• + (1 − φ)FP

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Simplifying regret

Fix ρ, v, c If the firm chooses (q, p), then RGRT = OPT − (CS + φFP) = OPT − (CS + FP) + (1 − φ)FP = OPT − q v(z) dz − c(q)

• + (1 − φ)FP

= DSTR

efficiency

+ (1 − φ)FP

• redistribution

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Solving for optimal policy

The regulator’s problem is minimize

ρ

max

v,c

RGRT = DSTR

efficiency

+ (1 − φ)FP

• redistribution

where maximum is over all (v, c)

– talk: the firm breaks ties against the regulator

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Solving for optimal policy

The regulator’s problem is minimize

ρ

max

v,c

RGRT = DSTR

efficiency

+ (1 − φ)FP

• redistribution

where maximum is over all (v, c)

– talk: the firm breaks ties against the regulator

minimization is over all policies ρ

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Roadmap

Environment Main result

– Lower bound on worst-case regret – Upper bound by our policy

Roadmap

Environment Main result

– Lower bound on worst-case regret – Upper bound by our policy

more surplus for consumers mitigate under- production mitigate over- production

Suppose regulator imposes a price cap k

more surplus for consumers mitigate under- production

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Suppose regulator imposes a price cap k

v(q) 1 ¯ v q k c(q) = 0

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Suppose regulator imposes a price cap k

v(q) 1 ¯ v q k c(q) = 0 DSTR = 0, FP = k RGRT = (1 − φ)k

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Suppose regulator imposes a price cap k

v(q) 1 ¯ v q k c(q) = 0 DSTR = 0, FP = k RGRT = (1 − φ)k v(q) 1 ¯ v q k

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Suppose regulator imposes a price cap k

v(q) 1 ¯ v q k c(q) = 0 DSTR = 0, FP = k RGRT = (1 − φ)k v(q) 1 ¯ v q k c(q) = k DSTR = ¯ v − k, FP = 0 RGRT = ¯ v − k

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Suppose regulator imposes a price cap k

v(q) 1 ¯ v q k c(q) = 0 DSTR = 0, FP = k RGRT = (1 − φ)k v(q) 1 ¯ v q k c(q) = k DSTR = ¯ v − k, FP = 0 RGRT = ¯ v − k Let (1 − φ)kφ = ¯ v − kφ = ⇒ kφ =

¯ v 2−φ

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Lower bound on worst-case regret

Theorem

Let L(q, p) = min { (1 − φ)qkφ − pq log q, q(kφ − p) } . The worst-case regret under any policy is at least max

q∈[0,1], p∈[0,kφ] L(q, p).

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Proof of lower bound

1 ¯ v q p Fix q, p and some ρ If max{x, y} qkφ and x y, a firm with zero cost has FP qkφ and produces less than q: RGRT (1 − φ)qkφ + DSTR (1 − φ)qkφ + 1

q

qp z dz = q(1 − φ)kφ − pq log(q) L(q, p)

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Proof of lower bound

1 ¯ v q p z

pq z

Fix q, p and some ρ Let v(z) = ¯ v, ∀z q; qp

z , ∀z > q

If max{x, y} qkφ and x y, a firm with zero cost has FP qkφ and produces less than q: RGRT (1 − φ)qkφ + DSTR (1 − φ)qkφ + 1

q

qp z dz = q(1 − φ)kφ − pq log(q) L(q, p)

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Proof of lower bound

x y 1 ¯ v q p z

pq z

Fix q, p and some ρ Let v(z) = ¯ v, ∀z q; qp

z , ∀z > q

Let x = maxq′q ρ(q′, p′) Let y = maxq′q,q′p′qp ρ(q′, p′) If max{x, y} qkφ and x y, a firm with zero cost has FP qkφ and produces less than q: RGRT (1 − φ)qkφ + DSTR (1 − φ)qkφ + 1

q

qp z dz = q(1 − φ)kφ − pq log(q) L(q, p)

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Proof of lower bound

x y 1 ¯ v q p z

pq z

Fix q, p and some ρ Let v(z) = ¯ v, ∀z q; qp

z , ∀z > q

Let x = maxq′q ρ(q′, p′) Let y = maxq′q,q′p′qp ρ(q′, p′)

• 1. If max{x, y} qkφ, a firm with fixed cost qkφ won’t produce:

FP qkφ and produces less than q: RGRT = DSTR = q(¯ v − kφ) + 1

q

qp z dz = q(1 − φ)kφ − pq log(q) L(q, p)

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Proof of lower bound

x y 1 ¯ v q p z

pq z

Fix q, p and some ρ Let v(z) = ¯ v, ∀z q; qp

z , ∀z > q

Let x = maxq′q ρ(q′, p′) Let y = maxq′q,q′p′qp ρ(q′, p′)

• 2. If max{x, y} qkφ and x y, a firm with zero cost has FP qkφ

and produces less than q: RGRT (1 − φ)qkφ + DSTR (1 − φ)qkφ + 1

q

qp z dz = q(1 − φ)kφ − pq log(q) L(q, p)

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Proof of lower bound

x y 1 ¯ v q p z

pq z

Fix q, p and some ρ Let v(z) = ¯ v, ∀z q; qp

z , ∀z > q

Let x = maxq′q ρ(q′, p′) Let y = maxq′q,q′p′qp ρ(q′, p′)

• 3. If max{x, y} qkφ and y x, there exists q′, p′ in light-blue area

such that ρ(q′, p′) = y qkφ F Consider RHS firm: 1

q

qp z dz RGRT = DSTR qkφ − q′p′ q(kφ − p) L(q, p)

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Proof of lower bound

x y 1 ¯ v q p z

pq z

1 q′ ¯ v p′ c(q) = y

• 3. If max{x, y} qkφ and y x, there exists q′, p′ in light-blue area

such that ρ(q′, p′) = y qkφ F Consider RHS firm: 1

q

qp z dz RGRT = DSTR qkφ − q′p′ q(kφ − p) L(q, p)

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Lower bound on worst-case regret

Theorem

Let L(q, p) = min { (1 − φ)qkφ − pq log q, q(kφ − p) } . The worst-case regret under any policy is at least rφ := max

q∈[0,1], p∈[0,kφ] L(q, p).

1 ¯ v

¯ v 2

φ kφ 1 ¯ v φ

¯ v 2

rφ

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Roadmap

Environment Main result

– Lower bound on worst-case regret – Upper bound by our policy

φ = 0: regulator cares about CS only

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φ = 0: regulator cares about CS only

Theorem (φ = 0)

The worst-case regret is at most r0 given the price cap k0 .

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φ = 0: regulator cares about CS only

Theorem (φ = 0)

The worst-case regret is at most r0= ¯

v 2 given the price cap k0= ¯ v 2.

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φ = 0: regulator cares about CS only

Theorem (φ = 0)

The worst-case regret is at most r0= ¯

v 2 given the price cap k0= ¯ v 2.

Proof idea: 1 ¯ v

¯ v 2

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φ = 0: regulator cares about CS only

Theorem (φ = 0)

The worst-case regret is at most r0= ¯

v 2 given the price cap k0= ¯ v 2.

Proof idea: 1 ¯ v

¯ v 2

if q = 0, for consumers with value ¯

v 2

each adds ¯

v 2 to total surplus;

for consumers with value ¯

v 2,

average cost is ¯

v 2, so each adds ¯ v 2.

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φ = 0: regulator cares about CS only

Theorem (φ = 0)

The worst-case regret is at most r0= ¯

v 2 given the price cap k0= ¯ v 2.

Proof idea: 1 ¯ v

¯ v 2

if q = 0, for consumers with value ¯

v 2

each adds ¯

v 2 to total surplus;

for consumers with value ¯

v 2,

average cost is ¯

v 2, so each adds ¯ v 2.

q if q > 0, for consumers who are served, regulator loses p ¯

v 2 each;

for consumers who are not served, regulator loses ¯

v 2 each.

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φ = 1: regulator cares about total surplus

1 ¯ v c(q) = 0

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φ = 1: regulator cares about total surplus

1 ¯ v c(q) = 0 unregulated firm serve ¯ v consumers, regulator loses surplus in light-blue area;

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φ = 1: regulator cares about total surplus

1 ¯ v c(q) = 0 unregulated firm serve ¯ v consumers, regulator loses surplus in light-blue area; If (q, p), subsidize (¯ v − p)q, q p

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φ = 1: regulator cares about total surplus

1 ¯ v c(q) = 0 unregulated firm serve ¯ v consumers, regulator loses surplus in light-blue area; If (q, p), subsidize (¯ v − p)q, q p light-blue shrinks to −pq log(q);

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φ = 1: regulator cares about total surplus

1 ¯ v c(q) = 0 unregulated firm serve ¯ v consumers, regulator loses surplus in light-blue area; If (q, p), subsidize (¯ v − p)q, q p light-blue shrinks to −pq log(q); but, subsidy (¯ v − p)q might incentivize overproduction; regulator loses (¯ v − p)q in light-gray

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φ = 1: regulator cares about total surplus

1 ¯ v c(q) = 0 unregulated firm serve ¯ v consumers, regulator loses surplus in light-blue area; If (q, p), subsidize (¯ v − p)q, q p light-blue shrinks to −pq log(q); but, subsidy (¯ v − p)q might incentivize overproduction; regulator loses (¯ v − p)q in light-gray

Theorem (φ = 1)

The worst-case regret is at most r1 given the policy: ρ(q, p) = min( q ¯ v , qp + r1 ).

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Upper bound on worst-case regret

Theorem (0 φ 1)

The policy ρ(q, p) = min( q kφ , pq + s )

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Upper bound on worst-case regret

Theorem (0 φ 1)

The policy ρ(q, p) = min( q kφ , pq + s ) 1 ¯ v

¯ v 2

φ kφ 1 ¯ v φ

¯ v 2

rφ 2kφ − 1 sφ

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Upper bound on worst-case regret

Theorem (0 φ 1)

The policy ρ(q, p) = min( q kφ , pq + s ) with sφ s rφ achieves the worst-case regret rφ. 1 ¯ v

¯ v 2

φ kφ 1 ¯ v φ

¯ v 2

rφ 2kφ − 1 sφ

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Conclusion: our advocate for non-Bayesian approach

Armstrong and Sappington (2007):

• 1. Optimal policy under Bayesian approach is sensitive to how one

models the regulator’s knowledge

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Conclusion: our advocate for non-Bayesian approach

Armstrong and Sappington (2007):

• 1. Optimal policy under Bayesian approach is sensitive to how one

models the regulator’s knowledge

• 2. Multi-dimensional screening problems are typically difficult to solve

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Conclusion: our advocate for worst-case regret

• 1. Regret has a natural interpretation:

regret = distortion

• efficiency

+ (1 − φ) firm’s profit

• redistribution

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Conclusion: our advocate for worst-case regret

• 1. Regret has a natural interpretation:

regret = distortion

• efficiency

+ (1 − φ) firm’s profit

• redistribution
• 2. Worst-case regret is more relevant than worst-case payoff

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Conclusion: our advocate for worst-case regret

• 3. Savage offers another interpretation, as observed by Linhart and

Radner (1989): Suppose the [regulator] must justify his [policy] for a group of persons who have widely varying “subjective” probability distributions. In this case, the [regulator] might want to [regulate] in such a way as to minimize the maximum “outrage” felt in the group; here “outrage” is equated to regret.

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Conclusion: three objectives

We looked for the robust policy across all v, c

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Conclusion: three objectives

We looked for the robust policy across all v, c Depending on the industry, one can do so for a smaller family of v, c

– our policy remains optimal under fixed cost plus constant marginal cost

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Conclusion: three objectives

We looked for the robust policy across all v, c Depending on the industry, one can do so for a smaller family of v, c

– our policy remains optimal under fixed cost plus constant marginal cost

more surplus for consumers mitigate under- production mitigate over- production

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Conclusion: three objectives

We looked for the robust policy across all v, c Depending on the industry, one can do so for a smaller family of v, c

– our policy remains optimal under fixed cost plus constant marginal cost

more surplus for consumers mitigate under- production mitigate over- production Organizational economics

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Thank you!