Regulation
Analyzed in two contexts:
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Procurement: The firm supplies a good to the government
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Regulation: The firm supplies a good to consumers on behalf of
government
Economics of Information and Contracts
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Defense contracts
Utilities (electricity, gas), telecommunications, railroads
There are many natural monopolies in utilities, telecommunication,
infrastructure, and transportation industries
Levent Koçkesen
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Koç University
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Due to large fixed costs, average cost is decreasing even at large
production levels
It may be socially efficient to have a monopoly
Government needs to regulate to prevent them acting as a monopolist
But it doesn’t know the relevant characteristics (e.g., technology)
and/or does not observe actions (e.g., cost reducing effort) of the firm
Levent Koçkesen (Koç University)
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Regulation
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a: a fixed fee
b: fraction of costs born by the firm (measures the power of incentives)
Possible contracts
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fixed-price (price-cap): b = 1
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incentive (cost-sharing): 0 < b < 1
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40.9%
13.6%
31.4%
cost-plus
incentive
fixed-price
Regulation:
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historically: cost-of-service
more recently: price-cap and incentive regulation
firm does not bear any cost
low powered: firm does not have an incentive to lower cost
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Procurement: 1960 US
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cost-plus (cost-of-service): b = 0
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Regulation
Typically, government uses accounting (cost) data and reimburses a
fraction of firm’s monetary expenditures c
Its aim is to obtain the highest societal welfare in providing the good
We will model it as follows: government pays c and a net transfer t to
the firm
t = a − bc
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Levent Koçkesen (Koç University)
government does not reimburse any cost; pays only a fixed fee
high powered: firm is the residual claimant for its cost savings
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Regulation
Regulation
Government: provide incentives to reduce cost and extract firm’s rent
Fixed-price
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Incentives: Good
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Rent extraction: Bad
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Early models
Firm is the residual claimant
Provides efficient effort
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Rent is non-negative when cost is high
Exogenous reduction in cost benefits the firm
The bible: Laffont, J.-J. and J. Tirole (1993), A Theory of Incentives
in Procurement and Regulation, Cambridge: MIT Press.
Cost-plus
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Baron, D. and R. Myerson (1982), “Regulating a Monopolist with
Unknown Costs,” Econometrica, 50, 911-30.
Laffont, J.-J. and J. Tirole (1986), ”Using Cost Observation to
Regulate Firms,” Journal of Political Economy, 94, 614-41.
Incentives: Bad
Rent-extraction: Good
Optimal contract
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Perfect information about technology: fixed-price
Imperfect information: incentive (cost-sharing) contract
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Regulation
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Perfect Information
Our model: a simple version of Laffont and Tirole (1986)
Regulator knows θ (therefore, can deduce e = θ − c)
The firm produces a single indivisible output
Accounting cost is c = θ − e
Its problem is:
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min{θ − e + t}
(e,t)
θ ∈ {θL , θH }, prob(θL ) = p
e is effort which costs the firm ψ(e) = e2 /2
We assume ∆θ = θH − θL ≤ 1 − p
Regulator wants to maximize S − P
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s.t. t − e2 /2 ≥ 0,
e≥0
The constraint must bind: t = e2 /2
min{θ − e + e2 /2}
S is (constant) consumer surplus from consuming the good
P = c + t is total payment, t is net transfer to the firm
e≥0
If the solution is in the interior (e∗ > 0)
Therefore, regulator wants to minimize P
Payoff of the firm P − c − e2 /2 = t − e2 /2
−1 + e∗ = 0 ⇒ e∗ = 1
Reservation payoff of the firm is zero
Comparing with the boundary (e = 0) shows that the solution is in
the interior
c is observable but not θ or e
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Perfect Information
Unobservable θ and e
Effort is efficient
What happens if regulator offers first-best?
marginal cost = ψ ′ (e∗ ) = e∗ = 1 = marginal benefit
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cL = θL − 1, cH = θH − 1 and tL = tH = 0.5
For the efficient type (θL )
and
∗
t = 0.5
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firm receives no rent
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Regulator pays the firm a fixed total payment P = θ − 0.5
Induces the firm to minimize c + e2 /2
This is a price-cap (fixed-fee) regulation
Effort required to obtain cost cL is 1
Effort required to obtain cost cH (e′ ) is 1 − ∆θ < 1:
c H = θH − 1 = θL − e ′
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Since transfers are the same, she prefers to mimic the inefficient type
Usual recipe for optimal contract:
t = a − (c − c∗ )
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Efficient type (θL ):
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Inefficient type (θH ):
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where
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a = ψ(e∗ ) = 0.5
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c∗ = θ − e∗ = θ − 1
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Unobservable θ and e
min
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1. IRL does not bind:
tL − e2L /2 ≥ tH − (eH − ∆θ)2 /2 > tH − e2H /2 ≥ 0
p(tL + θL − eL ) + (1 − p)(tH + θH − eH )
2. ICL binds, otherwise
subject to
tL − e2L /2 > tH − (eH − ∆θ)2 /2 > tH − e2H /2 ≥ 0
tL − e2L /2 ≥ 0
and the regulator can decrease tL
tH − e2H /2 ≥ 0
3. IRH binds, otherwise (by (1)) can decrease tL and tH by the same
small amount
tL − e2L /2 ≥ tH − (eH − ∆θ)2 /2
tH − e2H /2 ≥ tL − (eL + ∆θ)2 /2
4. eH ≤ eL + ∆θ or cH ≥ cL : ICL and ICH imply
Note that for type θL , effort required to achieve cost cH = θH − eH is
equal to eH − (θH − θL ). Similarly, for high type’s effort to achieve
low cost.
We implicitly assume eH − ∆θ ≥ 0, which we will need to verify once
we solve the problem.
Levent Koçkesen (Koç University)
provides less than first best effort (to reduce efficient type’s rent)
obtains no rent
Solving the Problem
Regulator chooses a contract {(tL , cL ), (tH , cH )} to minimize
p(tL + cL ) + (1 − p)(tH + cH ) subject to IR and IC constraints.
Since c = θ − e, the problem can be written as
(ti ,ei )
provides first best effort
obtains positive rent
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e2H /2 − (eL + ∆θ)2 /2 ≤ tH − tL ≤ (eH − ∆θ)2 /2 − e2L /2
Result follows from simple algebra
5. ICH can be ignored: follows from steps (2) and (4)
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Solving the Problem
Solving the Problem
Therefore, the problem is equivalent to
min
(ti ,ei )
Substituting in the constraints, the problem is equivalent to
p(tL + θL − eL ) + (1 − p)(tH + θH − eH )
min {p(e2H /2 + e2L /2 − (eH − ∆θ)2 /2 − eL ) + (1 − p)(e2H /2 − eH )}
subject to
eL ,eH ≥0
tH − e2H /2 = 0
or
tL − e2L /2 = tH − (eH − ∆θ)2 /2
max −{p(e2H /2 + e2L /2 − (eH − ∆θ)2 /2 − eL ) + (1 − p)(e2H /2 − eH )}
eL ,eH ≥0
Note that
tL − e2L /2 = ∆θ(eH − ∆θ/2)
eH ≥ ∆θ implies that type θL obtains a rent
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Solving the Problem
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Solving the Problem
Critical points are given by
The objective function is strictly concave and the constraint functions
eL and eH are concave
Therefore, if a solution exists Kuhn-Tucker conditions will uniquely
identify it
−p(eL − 1) + λL = 0
−(p∆θ + (1 − p)(eH − 1)) + λH = 0
λL , λH , eL , eH ≥ 0,
λL eL = λH eH = 0
First equation and λL ≥ 0 ⇒ eL > 0 ⇒ λL = 0, which implies
The Lagrangean is given by
eL = 1
L = −{p(e2H /2 + e2L /2 − (eH − ∆θ)2 /2 − eL ) + (1 − p)(e2H /2 − eH )}
+ λL eL + λH eH
Similarly the assumption ∆θ ≤ 1 − p implies that
eH = 1 −
p
∆θ
1−p
and eH ≥ ∆θ, as required.
Levent Koçkesen (Koç University)
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Properties of the Solution
Financial Contracts and Credit Rationing
Familiar “no distortion at the top” and “under-provision for inefficient
types” result
Optimal regulation: offer a menu so that
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Efficient firm chooses a price-cap regulation
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A constant payment PL = tL + θL − eL
tL = aL − bL c, aL = tL + θL − eL and bL = 1
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tH = aH − bH c, aH = tH + eH (θH − eH ) and bH = eH = 0.5
Levent Koçkesen (Koç University)
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If there is excess demand of funds, the interest rate should increase to
equate demand with supply
The reason might be adverse selection
As the interest rate increases only riskier projects apply
Most popular article:
Inefficient firm chooses a cost-sharing arrangement
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Why are certain worthy projects not funded?
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A Simple Model
Stiglitz, J. and A. Weiss (1981)
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Symmetric Information
There is a population (unit mass) of risk-neutral borrowers
Each borrower owns a project that requires an investment of 1
Two types of borrowers: risky (θr ) and safe (θs )
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The problem of the bank facing borrower of type i is
max
prob(θs ) = q
Di ≥0
Each project has an uncertain return X ∈ {0, Ri }, Ri with prob. pi
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pi (Ri − Di ) ≥ 0
Assume ps > pr and Rr > Rs
We will look into two scenarios:
For now assume pi Ri = m > 1
1. pi Ri = m > 1: Both projects worth investing
2. pr Rr < 1 < ps Rs : Only safe project worth investing
Solution is simple: lend all α and set
Single bank with total amount of funds max{q, 1 − q} < α < 1
Net expected payoff of the bank if it lends to type i and requires a
repayment of Di is given by pi Di − 1
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pi Di − 1
subject to
θr = (pr , Rr ) and θs = (ps , Rs )
Di = Ri
Expected payoff of the bank is α(pi Ri − 1) = α(m − 1) > 0
Assumes limited liability: if the return is zero, the borrower defaults
and bank gets zero repayment
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Adverse Selection
Adverse Selection
Assume that a bank contract is simply D
D > Rs ⇒ only risky apply ⇒ best to set D = Rr
D = Rs < Rr ⇒ risky borrowers would like to pay more D to get a
loan
But as D increases borrower pool becomes riskier and the bank’s
payoff decreases
payoff = (1 − q)(pr Rr − 1) = (1 − q)(m − 1)
D ≤ Rs ⇒ both apply ⇒ lend all α at D = Rs
payoff = α[q(ps Rs − 1) + (1 − q)(pr Rs − 1)] = α[qm + (1 − q)pr Rs − 1]
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Adverse selection
D = Rs
some borrowers cannot get credit
credit rationing
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Second Best
Levent Koçkesen (Koç University)
max
The bank’s problem is
(xi ,Di )
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This suggests the reduced problem:
xi is probability of getting a loan
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Solving the Problem
A contract is {(xs , Ds ), (xr , Dr )}
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Can the bank do better by designing more sophisticated contracts?
If (1 − q)(m − 1) < α[qm + (1 − q)pr Rs − 1]
(xi ,Di )
xs ps (Rs − Ds ) ≥ 0
xr pr (Rr − Dr ) ≥ xs pr (Rr − Ds )
subject to
xi pi (Ri − Di ) ≥ 0,
0 ≤ xi ≤ 1,
xr ≥ xs
i = s, r
xi pi (Ri − Di ) ≥ xj pi (Ri − Dj ),
0 ≤ xi ≤ 1,
i, j = s, r
This should be familiar by now
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If bank offers the first best menu
{(xs , Ds ), (xr , Dr )} = {(α, Rs ), (α, Rr )}
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Risky type mimics the safe type
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i = s, r
qxs + (1 − q)xr ≤ α
i = s, r
qxs + (1 − q)xr ≤ α
Levent Koçkesen (Koç University)
qxs (ps Ds − 1) + (1 − q)xr (pr Dr − 1)
subject to
qxs (ps Ds − 1) + (1 − q)xr (pr Dr − 1)
IR constraint of the high type (θr ) does not bind
IC constraint of the low type (θs ) does not bind
Monotonicity in the allocation xr ≥ xs
The original problem (P) is equivalent to the reduced problem (RP)
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1. Constraints of (P) imply those of (RP): From the two IC constraints:
xr Rs − xs Rs ≤ xr Dr − xs Ds ≤ xr Rr − xs Rr
max xs [q(ps Rs − 1) − (1 − q)pr (Rr − Rs )] + (1 − q)xr (pr Rr − 1)
which implies xr ≥ xs
2. At the solution to (RP) IC constraint holds as equality: otherwise can
increase Dr
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Here we use xr > 0. If xr = 0, then by Step (1) xs = 0, which cannot
be a solution to (RP)
IR and IC constraints of (RP) imply IR for type θr
xr = 1 > α
xs > 0 implies
Together with monotonicity and budget constraint
α − (1 − q)
xs =
<α
q
IC for risky type (Rr − Dr = xs (Rr − Ds )) and Ds = Rs imply
xr pr (Rr − Dr ) ≥ xs pr (Rr − Ds ) ≥ xs pr (Rs − Ds ) ≥ 0
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xs ,xr
Since we assumed pi Ri = m > 1
q(ps Rs − 1) − (1 − q)pr (Rr − Rs ) ≥ 0
3. Solution of (RP) satisfy the constraints of (P)
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Suppose xs > 0. Then, Ds = Rs
Substitute the IC and IR into the objective function
Step (2) and xr ≥ xs imply IC for type θs
ps [xs (Rs − Ds ) − xr (Rs − Dr )] = ps [xs Rs − xr Rs + xr Rr − xs Rr ]
= ps (xr − xs )(Rr − Rs )
≥0
Dr = xs Rs + (1 − xs )Rr > Rs = Ds
If q(ps Rs − 1) − (1 − q)pr (Rr − Rs ) < 0
xs = 0
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Interpretation
The bank trades off the rents extracted from risky borrowers with
ability to lend all its funds
When safe borrowers are numerous enough, wants to lend to them,
which leads to rent for risky borrowers
But rather than financing everybody by setting D = Rs , the bank
gives risky borrowers preferential access by setting xr > xs in
exchange for higher repayment
Credit rationing disappears: The only ones not fully funded are safe
borrowers, but they are indifferent about being funded
Credit rationing reappears if we instead assume pr Rr < 1 < ps Rs
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Bank wants to turn away risky borrowers (i.e., set xr = 0), but cannot
do it without also denying the safe ones (since xr ≥ xs )
Either everybody is funded (if (qps + (1 − q)pr )Rs ≥ 1) or nobody, i.e.,
there is a financial collapse
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