introduction to auction theory
TRANSCRIPT
Introduction to Auction Theory
Lecture Slides for Auction Theory
Yosuke YASUDA
Osaka University, Department of Economics
Last-Update: September 27, 2016
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Announcement
Course Website: You can find my corse websites from the link below:https://sites.google.com/site/yosukeyasuda2/home/lecture/auction16
Textbook & Survey: VK is a comprehensive advanced textbook onauction theory. TB covers wide range of topics on mechanism design,most of which is directly related to auctions. MM is the most-cited, highlyreadable survey article on economics of auction.
VK Vijay Krishna, Auction Theory: 2nd, 2009.
TB Tilman Borgers, An Introduction to the Theory of MechanismDesign, 2015.
MM McAfee, R. P., and McMillan, J. (1987). Auctions and Bidding.Journal of Economic Literature, 25(2), 699-738.
Symbols that we use in lectures:�� ��Ex : Example,�� ��Fg : Figure,
�� ��Q : Question,�� ��Rm : Remark.
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Independent Private Values Model
According to Milgrom and Weber (1982):
Much of existing literature on auction theory analyzes theindependent private values model.
A single indivisible object is to be sold to one of several bidders.
Each bidder is risk-neutral and knows the value of the object tohimself, but does not know the value of the object to the otherbidders. (← the private values assumption).
The values are modeled as being independently drawn from somecontinuous distribution.
Bidders are assumed to behave competitively; therefore, the auctionis treated as a noncooperative game among the bidders.
At least seven important conclusions emerge from the model.
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Seven Important Conclusions by Milgrom-Weber
1. The Dutch auction and the first-price auction are strategicallyequivalent.
2. The second-price sealed-bid auction and the English auction areequivalent, although in a weaker sense than the strategicequivalence of the Dutch and first-price auctions.
3. The outcome (at the dominant-strategy equilibrium) of the Englishand second-price auctions is Pareto optimal; that is, the winner isthe bidder who values the object most highly.
4. All four auction forms (English, Dutch, first-price, and second-price)lead to identical expected revenues for the seller.
5. Revenue equivalence result (Theorem 1 in next slide).
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Seven Important Conclusions by Milgrom-Weber
Theorem 1
Assume that a particular auction mechanism is given, that theindependent private values model applies, and that the bidders adoptstrategies which constitute a noncooperative equilibrium.Suppose that at equilibrium the bidder who values the object most highlyis certain to receive it, and that any bidder who values the object at itslowest possible level has an expected payment of zero. Then theexpected revenue generated for the seller by the mechanism is preciselythe expected value of the object to the second-highest evaluator.
6. For many distributions including the normal, exponential, anduniform distributions, the four standard auction forms with suitablychosen reserve prices or entry fees are optimal auctions.
7. In a variation of the model where either the seller or the buyers arerisk averse, the seller will strictly prefer the Dutch or first-priceauction to the English or second-price auction.
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Asymmetric Information
The timing when asymmetric information occurs matters.
1 There arise asymmetric information after somebody taking action:
Moral hazard (hidden action).
2 There exists asymmetric information from the beginning:
Adverse selection (hidden information).
1 Agents move simultaneously through the market: Lemon Market2 Those who have private information move first: Signaling3 Those who do not have private information move first: Screening→ In auctions, seller moves first and buyer(s) moves second.
If there is only one buyer, auction is just a screening problem!
Asymmetric information is often analyzed by the simplified model calledthe principal-agent model.
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Principal-Agent Model
In the principal-agent model,
There are two economic agents:
Informed party who has relevant private information.Uninformed party who does not possess private information.
Allocate all bargaining power to one of the parties:
The principal will propose “take it or leave it” contract to the agent(who cannot propose another contract).The principal-agent game is can be seen as a Stackelberg game:leader = principal, follower = agent.�� ��Rm The set of (constrained) Pareto efficient solutions can always be
obtained by maximizing utility of one player while the other is held to agiven utility level.
→ If we are interested in identifying or characterizing the set of Paretoefficient solutions, P-A model brings no loss of generality.
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Single Potential Buyer
A seller seeks to sell a single indivisible good to a potential buyer.
The buyer’s utility if she purchases the good and pays a monetarytransfer t to the seller is θ − t.
If she does not purchase the good, her utility is zero.
θ is the buyer’s valuation of the good, also called her type.
The seller has a subjective probability distribution over possiblevalues of θ ∈ [θ, θ]: cdf is denoted by F with density f .
Assume positive support: f(θ) > 0 for all θ ∈ [θ, θ].�� ��Rm What is an optimal selling mechanism? Is his expected revenuemaximized by committing to a price, as well as by committing to sellingthe good at that price whenever the buyer is willing to pay the price?
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Fixed Pricing Mechanism
Pick a price p and to say to the buyer that she can have the good if andonly if she is willing to pay p. (Assume the seller can commit to it)
Note that the probability that the buyer’s value is below p is F (p).The seller’s optimization problem is just the monopoly problem withdemand function 1− F (p), that is, maxp p(1− F (p)).
In general, the seller may commit to a direct mechanism (q, t) in which
The buyer is asked to report her type, if it is θ then
Transfer the good to the buyer with probability q(θ), and
The buyer has to pay the seller t(θ).
Definition 2 (Def 2.1)
A direct mechanism consists of functions q and t where
q : [θ, θ]→ [0, 1] and t : [θ, θ]→ R.
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Revelation Principle (Single Buyer Version)
Theorem 3 (Prop 2.1 – Revelation Principle)
For every mechanism Γ and every optimal buyer strategy σ in Γ, there isa direct mechanism Γ′ and an optimal buyer strategy σ′ in Γ′ such that
(i) The strategy σ′ satisfies σ′(θ) = θ for every θ ∈ [θ, θ], that is, σ′
prescribes telling the truth.
(ii) For every θ ∈ [θ, θ] the probability q(θ) and the payment t(θ) underΓ′ equal the probability of purchase and the expected payment thatresult under Γ if the buyer plays her optimal strategy σ.
Proof.
For every θ ∈ [θ, θ] define q(θ) and t(θ) as required by (ii) in Theorem 3.The optimality of truthfully reporting θ in Γ′ then follows immediatelyfrom the optimality of σ(θ) in Γ.(← If reporting θ′ 6= θ in Γ′ is strictly better, then choosing σ(θ′) in Γmust also be better, contradicting to the optimality of σ).
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The Buyer’s Incentive Conditions (1)
Given a direct mechanism, we define the buyer’s expected utility u(θ)conditional on her type being θ by
u(θ) = θq(θ)− t(θ).
Definition 4 (Def 2.2 and 2.3)
A direct mechanism is incentive compatible if truth telling is optimalfor every θ ∈ [θ, θ], that is, if
u(θ) ≥ θq(θ′)− t(θ′) for all θ, θ′ ∈ [θ, θ].
A direct mechanism is individually rational if the buyer, conditional onher type, is willing to participate, that is, if
u(θ) ≥ 0 for all θ ∈ [θ, θ].
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The Buyer’s Incentive Conditions (2)
Lemma 5 (Lem 2.1)
If a direct mechanism is incentive compatible, then q is increasing in θ.
Proof.
Consider two types θ and θ′ with θ > θ′. IC conditions require
θq(θ)− t(θ) ≥ θq(θ′)− t(θ′) and θ′q(θ)− t(θ) ≤ θ′q(θ′)− t(θ′).
Subtracting these two inequalities, we obtain
(θ − θ′)q(θ) ≥ (θ − θ′)q(θ′) ⇐⇒ q(θ) ≥ q(θ′).
Lemma 6 (Lem 2.2)
If a direct mechanism is incentive compatible, then u is increasing in θ. Itis also convex, and hence differentiable except in at most countably manypoints. For any differentiable point θ, it satisfies u′(θ) = q(θ).
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Payoff and Revenue Equivalence
Lemma 6 and the fundamental theorem of calculus implies Lemma 7.
Lemma 7 (Lem 2.3 – Payoff Equivalence)
Consider an incentive compatible direct mechanism. Then for allθ ∈ [θ, θ] we have
u(θ) = u(θ) +∫ θ
θ
q(x)dx.
Since u(θ) = θq(θ)− t(θ), we also obtain the following.
Lemma 8 (Lem 2.4 – Revenue Equivalence)
Consider an incentive compatible direct mechanism. Then for allθ ∈ [θ, θ] we have
t(θ) = t(θ) + (θq(θ)− θq(θ))−∫ θ
θ
q(x)dx.
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Characterizing Optimal Mechanism
Theorem 9 (Prop 2.2)
A direct mechanism (q, t) is incentive compatible if and only if
(i) q is increasing.
(ii) For every θ ∈ [θ, θ] we have
t(θ) = t(θ) + (θq(θ)− θq(θ))−∫ θ
θ
q(x)dx.
Theorem 10 (Prop 2.3)
An incentive compatible direct mechanism is individually rational if andonly if u(θ) ≥ 0 or equivalently t(θ) ≤ θq(θ).
Lemma 11 (Lem 2.5)
If an incentive compatible and individually rational direct mechanismmaximizes the seller’s expected revenue, then t(θ) = θq(θ).
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Proof of Theorem 9
Proof.
Since necessity has been shown, we only need to prove sufficiency. Thatis, we have to show u(θ) ≥ θq(θ′)− t(θ′) for any θ, θ′ ∈ [θ, θ].
u(θ) = θq(θ)− t(θ) ≥ θq(θ′)− t(θ′)
⇐⇒∫ θ
θ
q(x)dx ≥∫ θ′
θ
q(x)dx + (θ − θ′)q(θ′)
⇐⇒∫ θ
θ′q(x)dx ≥
∫ θ
θ′q(θ′)dx ⇐⇒
∫ θ
θ′(q(x)− q(θ′))dx ≥ 0.
When q is increasing, the last inequality always holds for any θ′ 6= θ.
Using Theorem 9 and Lemma 11, the seller can restrict his attention onthe set of all increasing functions q : [θ, θ]→ [0, 1].← Note that t(·) is completely determined by q(·).
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Optimal Selling Mechanism
Extreme point theorem guarantees that the seller can restrict hisattention to non-stochastic mechanisms.
Non-stochastic mechanism is monotone if and only if there is somep∗ ∈ [θ, θ] such that q(θ) = 0 if θ < p∗ and q(θ) = 1 if θ > p∗.
The seller cannot do better than quoting a simple price p∗ to thebuyer (and the buyer either accepting or rejecting p∗).
Theorem 12 (Prop 2.5)
The following direct mechanism maximizes the seller’s expected revenuesamong all incentive compatible, individually rational direct mechanisms.Suppose p∗ ∈ arg maxp∈[θ,θ] p(1− F (p)). Then,
q(θ) = 1 if θ ≥ p∗, and q(θ) = 0 if θ < p∗,
t(θ) = p∗ if θ ≥ p∗, and t(θ) = 0 if θ < p∗.
If there are multiple potential buyers, we consider competitive bidding,i.e., strategic interactions among buyers must be incorporated.
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Bayesian Games
Following Harsanyi (1967), we can translate any game of incompleteinformation into a Bayesian game in which a Nash equilibrium isnaturally extended to a Bayesian Nash equilibrium:
(1) Nature draws a type vector t(= t1×· · ·× tn) ∈ T (= T1×· · ·×Tn),according to a prior probability distribution p(t).
(2) Nature reveals i’s type to player i, but not to any other player.
(3) The players simultaneously choose actions ai ∈ Ai for i = 1, ..., n.
(4) Payoffs ui(a; ti) for i = 1, .., n are received.
By introducing the fictional moves by nature in steps (1) and (2), wehave described a game of incomplete information as a game of imperfectinformation: in step (3) some of the players do not know the completehistory of the game, i.e., which actions (types) of other players werechosen by nature.
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Bayesian Nash Equilibrium
Definition 13
In a Bayesian game, the strategies s∗ = (s∗1, ..., s∗n) are a (pure-strategy)
Bayesian Nash equilibrium (BNE) if for each player i and for each ofi’s types ti in Ti, s∗i (ti) solves:
maxai∈Ai
∑t−i∈T−i
ui(s∗1(t1), . . . , s∗i−1(ti−1), ai, s
∗i+1(ti+1), . . . , s∗n(tn); t)
×pi(t−i|ti).
The central idea of BNE is both simple and familiar:
Each player’s strategy given her type must be a best response to theother players’ strategies (in expectation).
A BNE is simply a Nash equilibrium in a Bayesian game when eachtype of every player is treated as separate player.
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Simple Example
�� ��Ex The nature selects state A with prob. 1/2 and B with prob. 1/2.Before the players select their actions, player 1 observes nature’s choice,but player 2 does not know it. Then, what is the BNE?
1�2 L RU 1, 1 0, 0D 0, 0 2, 2
A
1�2 L RU ′ 0, 1 1, 0D′ 2, 0 0, 2
B
There is a unique BNE in which player 1 chooses DU ′ and player 2chooses R. The best reply function for each player is derived as follows:
R1(L) = UD′, R1(R) = DU ′.
R2(UU ′) = L,R2(UD′) = R,R2(DU ′) = R,R2(DD′) = R.
Clearly, (DU ′, R) is a unique combination of mutual best responses, i.e.,a (Bayesian) Nash equilibrium.
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Revelation Principle (General Version)
The revelation principle, due to Myerson (1979) and others is animportant tool for designing games (or mechanisms) when the playershave private information.
Definition 14
A direct mechanism is a static Bayesian game in which each player’sonly action is to submit a message (mi ∈Mi) about her type. That is,strategy space satisfies Mi = Ti for every player i.
Theorem 15 (Revelation Principle)
Any BNE (of any Bayesian game) can be attained by a truth-telling BNEof some direct mechanism.�� ��Rm When no direct mechanism can achieve some outcome in atruth-telling BNE, then there exists no mechanism (no matter how itwere general or complicated) that can achieve the outcome.
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Proof of Theorem 15
Proof.
Let s∗ : T → A be the BNE of the original Bayesian game. Consider thedirect mechanism which selects the corresponding equilibrium outcomegiven reported types.
The outcome of the direct mechanism is set equal to s∗(m) for anycombination of revealed types of the players m ∈M .
Then, it is easy to show that truth-telling, mi = ti for all i, must bea BNE of this direct mechanism.
Suppose not, then for some i, there exists an action a′i = s∗i (t′i) 6= s∗i (ti)
such that∑t−i∈T−i
ui(a′i, s∗−i(t−i); ti)pi(t−i|ti)
>∑
t−i∈T−i
ui(s∗i (ti), s∗−i(t−i); ti)pi(t−i|ti),
which contradicts to that s∗ is a BNE of the original game.
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Simple Auction Model with 2 bidders
Imagine that there is a (potential) seller who has a painting that is worthnothing to him personally. He hopes to make some money by selling theart through an auction.
Suppose there are two potential buyers, called bidders 1 and 2.
Let x1 and x2 denote the valuations of the two bidders.
If bidder i wins the painting and has to pay b for it, then her payoffis xi − b.
where x1 and x2 are chosen independently by nature, andeach of which is uniformly distributed between 0 and 1.
The bidders observe their own valuations before engaging in theauction.
The seller and the rival do not observe a bidder’s valuation; theyonly know the distribution.
In what follows, we study two prominent sealed-bid auctions:a first-price auction and a second-price auction.
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First-Price Auction (1)
Bidders simultaneously and independently submit bids b1 and b2.
The painting is awarded to the highest bidder i∗ with max bi ,
who must pay her own bid, bi∗ .
To derive a Bayesian Nash equilibrium, we assume the bidding strategy inequilibrium is i) symmetric, and ii) linear function of xi. That is, inequilibrium, player i chooses
β(xi) = c + θxi. (1)
Suppose that player 2 follows the above equilibrium strategy; we shallcheck whether player 1 has an incentive to choose the same linearstrategy (1). Player 1’s optimization problem, given her valuation x1, is
maxb1
(x1 − b1) Pr{b1 > β(x2)}. (2)
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First-Price Auction (2)
Since x2 is uniformly distributed on [0, 1] by assumption, we obtain
Pr{b1 > β(x2)} = Pr{b1 > c + θx2}
= Pr{
b1 − c
θ> x2
}=
b1 − c
θ.
The first equality comes from the linear bidding strategy (1), the thirdequality is from the uniform distribution. Substituting it into (2), theexpected payoff becomes a quadratic function of b1.
maxb1
(x1 − b1)b1 − c
θ
Taking the first order condition, we obtain
du1
db1=
1θ[−2b1 + x1 + c] = 0⇒ b1 =
c
2+
x1
2. (3)
Comparing (3) with (1), we can conclude that c = 0 and θ = 12
constitute a Bayesian Nash equilibrium.24 / 40
Second-Price Auction
Bidders simultaneously and independently submit bids b1 and b2.
The painting is awarded to the highest bidder i∗ with max bi ,
at a price equal to the second-highest bid, maxj 6=i∗ bj .
Unlike the first-price auction, there is a weakly dominant strategy foreach player in this game.
Theorem 16
In a second-price auction, it is weakly dominant strategy to bid accordingto β(xi) = xi for all i.
Since the combination of weakly dominant strategies always becomes aNash equilibrium, bi = xi for all i is a BNE.�� ��Rm Note that there are other asymmetric equilibria.
For example, β1(x1) = 1 and β2(x2) = 0 for any x1 and x2
constitute a Bayesian Nash equilibrium.
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Expectation (1)
Definition 17
Given a random variable X taking on values in [0, ω], its cumulativedistribution function (CDF) F : [0, ω]→ [0, 1] is:
F (x) = Pr[X ≤ x]
the probability that X takes on a value not exceeding x.
We assume that F is increasing and continuously differentiable.
Definition 18
If X is distributed according to F , then its expectation is
E[X] =∫ ω
0
xf(x)dx
(=∫ ω
0
xdF (x))
and for γ : [0, ω]→ R, the expectation of γ(X) is analogously defined as
E[γ(X)] =∫ ω
0
γ(x)f(x)dx
(=∫ ω
0
γ(x)dF (x))
.
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Expectation (2)
Definition 19
The conditional expectation of X given that X < x is
E[X | X < x] =1
F (x)
∫ x
0
tf(t)dt,
which can be rewritten as follows (by integrating by parts):
F (x)E[X | X < x] =∫ x
0
tf(t)dt
= xF (x)−∫ x
0
F (t)dt.
The conditional expectation of γ(X) is defined as
E[γ(X) | X < x] =1
F (x)
∫ x
0
γ(t)f(t)dt.
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Order Statistics
Let X1, X2, . . . , Xn be n independent draws from a distribution F withassociated probability density function (PDF) f(= F ′).
Let Y1, Y2, . . . , Yn be a rearrangement of these so thatY1 ≥ Y2 ≥ · · · ≥ Yn.
Yk is called kth(-highest) order statistic.
Let Fk denote the distribution of Yk (with its pdf fk).
The distribution of the highest order statistic is
F1(y) = F (y)n
f1(y) = nF (y)n−1f(y).
The distribution of the second-highest order statistic is
F2(y) = F (y)n + nF (y)n−1(1− F (y))
= nF (y)n−1 − (n− 1)F (y)n.
f2(y) = n(n− 1)(1− F (y))F (y)n−2f(y).
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Expected Revenue: First-Price
In a first-price auction, the payment is max{ 12X1,
12X2} .
Recall that β(xi) = 12xi is a BNE.
max{ 12X1,
12X2} = 1
2 max{X1, X2} = 12Y1.
The expectation of Y1 becomes
E[Y1] =∫ 1
0
yf1(y)dy
=∫ 1
0
2y2dy =[23y3
]10
=23.
The expected revenue of the first-price auction is13
(= 12 ×
23 ).
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Expected Revenue: Second-Price
In a second-price auction, the payment is min{X1, X2} .
Recall that β(xi) = xi, i.e., trugh-telling is a BNE.
min{X1, X2} = Y2.
The expectation of Y2 becomes
E[Y2] =∫ 1
0
yf2(y)dy
=∫ 1
0
y × 2(1− y)dy =∫ 1
0
2(y − y2)dy
= 2
([12y2
]10
−[13y3
]10
)=
13.
The expected revenue of the second-price auction is13, which is identical
to the expected revenue of the first-price auction!.
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Revenue Equivalence Theorem
The two sealed-bid auctions, first-price and second-price auctions, inducedifferent equilibrium strategies but yield the same expected revenue.
Interestingly, this is not by chance; the revenue equivalence result, oftencalled as revenue equivalence theorem (RET), is known to hold inmuch more general situations.
Theorem 20
RET holds whenever the following conditions are satisfied:
Private Value: Each bidder knows her value of the object.
Independent: Bidders receives their values independently.
Symmetric: The distribution is identical among bidders.
Risk Neutral: Each bidder is risk neutral.
The above theorem does not depend on the number of bidders and thedistribution from which types of bidders are drown.
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First-Price: General Model with n bidders (1)
Consider a first-price auction with n bidders in which all the conditions inthe previous theorem are satisfied.
Assume that bidders play a symmetric equilibrium, β(x).
Given some bidding strategy b, a bidder’s expected payoff becomes
(x− b) Pr{b > Y n−11 } = (x− b)×G(β−1(b))
where Y n−11 is the highest order statistic among n− 1 random draws of
the values and G is the associated distribution.
Maximizing w.r.t. b yields the first order condition:
g(β−1(b))β′(β−1(b))
(x− b)−G(β−1(b)) = 0 (4)
where g = G′ is the density of Y n−11 .
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First-Price: General Model with n bidders (2)
Since (4) holds in equilibrium, i.e., b = β(x),
g(x)β′(x)
(x− b)−G(x) = 0 ⇐⇒ G(x)β′(x) + g(x)β(x) = xg(x),
which yields the differential equation
d
dx(G(x)β(x)) = xg(x).
Taking integral between 0 and x, we obtain(∫ x
0
d
dy(G(y)β(y))dy =
)G(x)β(x)−G(0)β(0) =
∫ x
0
yg(y)dy
⇒ β(x) =1
G(x)
∫ x
0
yg(y)dy = E[Y n−11 | Y n−1
1 < x].
�� ��Rm The equilibrium strategy is to bid the the conditional expectationof second-highest value given that my value x is the highest.
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First-Price: General Model with n bidders (3)
The expected payment (to the seller) of each bidder given x is
G(x)× E[Y n−11 | Y n−1
1 < x],
which is identical to that of the second-price auction.�� ��Rm The expected revenue is just the aggregation of the expectedpayment of all bidders, it can be derived by
n×∫ ω
0
G(x)× E[Y n−11 | Y n−1
1 < x]f(x)dx
= n
∫ ω
0
[G(x)× 1
G(x)
∫ x
0
yg(y)dy
]f(x)dx
= n
∫ ω
0
[∫ x
0
yg(y)dy
]f(x)dx = n
∫ ω
0
[∫ ω
y
f(x)dx
]yg(y)dy
= n
∫ ω
0
y(1− F (y))g(y)dy =∫ ω
0
yf2(y)dy
⇒ E[Y n2 ] since f2(y) = n(1− F (y))fn−1
1 (y).
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Appendix | Screening: Price Discrimination
Suppose there are two types of consumers, high (H) and low (L). Eachconsumer i is H with probability λ and L with probability 1− λ, and herpayoff is given as follows:
ui(q, p) = θiq − p
where q is quality and p is price of the good.
Then, the optimization problem for the seller is described as:
max(pH ,qH)(pL,qL)
λ(pH − c(qH)) + (1− λ)(pL − c(qL))
subject to
θLqL − pL ≥ 0 (PC1)
θHqH − pH ≥ 0 (PC2)
θLqL − pL ≥ θLqH − pH (IC1)
θHqH − pH ≥ θHqL − pL (IC2)
where the cost function c(·) is convex and differentiable.
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Appendix | Taxation Principle
Thanks to the revelation principle, any incentive compatible solutioncan be implemented by a truth-telling equilibrium of some directmechanism in which the consumer reports her type.
Let (pH , qH), (pL, qL) be the corresponding contracts that each typeof the consumer will be assigned under the outcome the directmechanism.
Then, providing just these two contracts, instead of employing thedirect mechanism, must result in the identical outcomes and satisfiesIC conditions.
That is, providing a pair of contracts (non-linear tariff) (pH , qH), (pL, qL)is equivalent to using the direct mechanism.
This property is sometimes called the taxation principle.�� ��Rm The taxation principle does not exclude the possibility that(pH , qH) = (pL, qL); the principle may offer the identical contract.
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Appendix | First-Best: Perfect Discrimination
If the seller can observe the type θi of the consumer, she will solve thefollowing problem (she can disregard IC constraints):
max(pi,qi)
pi − c(qi) subject to θiqi − pi ≥ 0
Assuming c′(·) > 0 and c′′(·) > 0, the seller offers qi = q∗i such that
c′(q∗i ) = θi and p∗i = θq∗i .
Under this first-best solution, note that
Both q∗H and q∗L are the efficient qualities.
The seller extracts all her surplus from the buyer.
This type of discrimination is called first-degree price discrimination.
Forbidden by the law: “sale should be anonymous”
Infeasible if the type is not observable.
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Appendix | Second-Best Contract (1)
(PC) conditions are called participation (individually rational)constraints, and (IC)’s are called incentive compatibility constraints.
Derivation See for example, Salanie (2005).
Step 1: Drop PC2
PC2 is automatically satisfied whenever other three hold.
Note that (a) equilibrium payoff for high type is greater than (b) herpayoff if she pretends to be low type, which is greater than (c)equilibrium payoff for low type.
The difference between (a) and (c) is called information rent.
Under asymmetric information, it is impossible to extract entire surplusfrom agent since information rent inevitably arises.
Step 2: Drop IC1
Assume that IC1 is satisfied under the optimal solution.
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Appendix | Second-Best Contract (2)
Step 3: Assume PC1 and IC2 hold with equality
Given Steps 1 and 2, these two constraints must be equality.
Given Steps 1 through 3, the optimization becomes as follows:
max(pH ,qH)(pL,qL)
λ(pH − c(qH)) + (1− λ)(pL − c(qL))
subject to
θLqL − pL = 0 (PC1’)
θHqH − pH = θHqL − pL (IC2’)
Substituting PC1’ and IC2’ into the objective function, the problembecomes an unconstrained optimization problem:
maxqH ,qL
λ{θHqH − (θH − θL)qL − c(qH)}+ (1− λ)(θLqL − c(qL))
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Appendix | Second-Best Contract (3)
The FOC with respect to qH shows
c′(q∗∗H ) = θH , (5)
which implies that the quality of high-type good is optimally chosen, i.e.,first best level (q∗∗H = q∗H).
The FOC with respect to qL shows
c′(q∗∗L ) = θL −λ
1− λ(θH − θL), (6)
which implies that the quality of low-type good is too low compared tothe first best level, i.e., q∗∗L < q∗L (note c′′(·) > 0).
Finally, from (5) and (6), we conclude that qH > qL under the optimalsolution. Then, given PC1’ and IC2’, IC1 can be written as
θLqL − pL ≥ θLqH − pH ⇔ 0 ≥ θLqH − {θHqH − (θH − θL)qL}⇔ 0 ≥ (θH − θL)(qL − qH),
which is satisfied whenever qH > qL. Thus, Step 2 is verified.40 / 40