公共经济学4 管制2

Social welfare is the weighted sum of net consumer surplus and the firm’s profit α (S(p(θ), θ) − T (θ)) + (1 − α)π(θ), where α ∈ [0, 1/2 ]. (Again, the exact specification is up to you!)
p ¯
S(p, θ) =
p ◮
q(ˆ, θ)d p , with p ˆ
dS = −q(p, θ). dp
There is a type-dependent, lump-sum transfer T (θ) from the principal to the agent. The principal specifies a set of contracts T (θ), p(θ) .
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Consider any two types θ1 and θ2 wlog θ1 < θ2 . Calculate the first-best p(θ1 ), p(θ2 ) and T (θ1 ), T (θ2 ).
Then, in order to ensure incentive compatibility, it must be true that both π(θ1 , θ1 ) = 0 ≥ π(θ1 , θ2 ) = p(θ2 )q(p(θ2 ), θ1 ) − c(q(p(θ2 ), θ1 )) + T (θ2 ), π(θ2 , θ2 ) = 0 ≥ π(θ2 , θ1 ) = p(θ1 )q(p(θ1 ), θ2 ) − c(q(p(θ1 ), θ2 )) + T (θ1 ). Checking this algebraically is a nightmare—so let’s try graphics.
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Public Sector Economics: Lecture 4 Demand uncertainty First-best Second-best Summary
The model
Consider the following model due to Lewis & Sappington (1988): ◮ The agent’s output q ∈ [0, ∞) is unobservable. Output q is produced at cost c(q) with c ′ (q) > 0, c ′′ (q) = 0.
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Public Sector Economics: Lecture 4 Demand uncertainty First-best Second-best Summary
First-best benchmark
Substituting the binding (PCθ ), the principal’s problem is the unconstrained max α (S(p(θ), θ) + p(θ)q(p(θ), θ) − c(q(p(θ), θ))) + (1 − α)π(θ)
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Lewis, T.R. and D. Sappington (1988) “Regulating a Monopolist with Unknown Demand,” American Economic Review, 78(5), 986–98. Armstrong, M. (1999) “Optimal Regulation with Unknown Demand and Cost Functions,” Journal of Economic Theory, 84(2), 196–215. Segal, I. (2003) “Optimal Pricing Mechanisms with Unknown Demand,” American Economic Review, 93(3), 509–29.
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Price p ∈ [0, p ) is contractible and set by the principal. ¯ Consumer demand D(θ) = q(p, θ) depends on the consumer type θ ∈ [θL , θH ] with density g (θ) interpreted as lowest valuation θL and highest valuation θH . Net consumer welfare from the provided quantity q at p is
The second-best problem: c ′′ > 0, case π(θ2, θ1)
p D(θ2 )
Again, we need to check that p1 q21 ≤ c(q21 ) − T (θ1 ).
D(θ1 )
MC↑
(T < 0 with c ′′ > 0.) Green: Transfer. Red: Net loss.
As usual, the rhs inequality is the participation constraint with zero-normalised outside option. For known consumer tastes θ, the regulator lets (PCθ ) bind and we obtain π(θ) = 0 ⇔ T (θ) = c(q(p(θ), θ)) − p(θ)q(p(θ), θ). (PCθ )
The second-best problem
The principal offers a set of conditional contracts T (θ), p(θ) . (The general alternative would be to offer a ‘schedule’ T (p(θ)).) Since θ is private information to the agent, we need to ensure incentive compatibility. Denote the agent’s profits if of true type θ ˆ ˆ but claiming to be of type θ by π(θ, θ). We impose that π(θ, θ) = p(θ)q(p(θ), θ) − c(q(p(θ), θ)) + T (θ) ≥ ˆ ˆ ˆ ˆ ˆ p(θ)q(p(θ), θ) − c(q(p(θ), θ)) + T (θ) = π(θ, θ) ˆ for any θ ∈ [θL , θH ]. As usual, but stated a little more generally, the interpretation is that reporting the true type θ must constitute a Nash equilibrium for the agent.
MC↑
T (θ) < 0,
T >0 p T <0 MC↓
∂p ∂T >0& > 0. ∂θ ∂θ
If c ′′ < 0, ie. MC↓ T (θ) > 0, ∂T ∂p <0& > 0. ∂θ ∂θ
q(p)
q
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Public Sector Economics: Lecture 4 Demand uncertainty First-best Second-best Summary
Baidu Nhomakorabea
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Public Sector Economics: Lecture 4 Demand uncertainty First-best Second-best Summary
First-best: Participation
The firm’s profit π from selling to type θ-consumers is given by π(θ) = p(θ)q(p(θ), θ) − c(q(p(θ), θ)) + T (θ) ≥ 0. (PCθ )
Regulation with unknown demand: Reading
The canonical application under incomplete information on demand is the price regulation of a monopolist in the electricity or telecommunications industries. A current application is (‘last mile’) broadband provision. The reading for this second part of the regulation block comprises
p =0
with foc p(θ)qp (·) − q(p(θ), θ) + q(p(θ), θ) − cq (q(p(θ), θ))qp (·) = 0 which simplifies to [p(θ) − cq (q(p(θ), θ))] qp (·) = 0 giving the familiar interpretation of marginal benefit equals marginal cost p(θ) = cq (q(p(θ), θ)).
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Public Sector Economics: Lecture 4 Demand uncertainty First-best Second-best Summary
First-best benchmark
p p ¯ Demand q(p, θ)
The binding (PCθ ) gave us T (θ) = c(q(p(θ), θ)) − p(θ)q(p(θ), θ). Thus, if c ′′ > 0, ie. MC↑
D(θ1 )
(T < 0 with c ′′ > 0.)
MC↑ p2
Green: Transfer. Red: Net loss.
p1
q12
q1
q2
q
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Public Sector Economics: Lecture 4 Demand uncertainty First-best Second-best Summary
Public Sector Economics — MSc Econ 2011
4. Regulation II
Paul Schweinzer
November 17, 2011.
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Public Sector Economics: Lecture 4 Demand uncertainty First-best Second-best Summary
p2
p1
q1
q2
q12
q
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Public Sector Economics: Lecture 4 Demand uncertainty First-best Second-best Summary
The second-best problem: c ′′ > 0, case π(θ1, θ2)
p D(θ2 )
Switching notation, we need to check that p2 q12 ≤ c(q12 ) − T (θ2 ).
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(ICθ )
Public Sector Economics: Lecture 4 Demand uncertainty First-best Second-best Summary
The second-best problem: c ′′ > 0
There is a simple argument which shows that first-best can always be implemented