What is Screening? Economics of Information and Contracts Screening: General Models. Monopolistic Screening: A More General Model.

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1 Economics of Information and Contracts Screening: General Models Levent Koçkesen Koç University Levent Koçkesen (Koç University) Screening 1 / 58 What is Screening? A contracting problem with Hidden Information Uninformed party (principal) offers contract to informed party (agent) Examples Insurance Insuree knows her risk, insurer does not Insurer offers several packages with different premiums and deductibles Finance Borrower knows the risk of project, lender does not Lender offers several packages with different interest rates and collateral requirements Hiring Applicants know their ability, employer does not Employer offers different packages, varying wages, bonuses, etc. Pricing Buyer knows her valuation of the product, seller does not Seller offers different qualities at different prices, or quantity discounts First analyzed formally, within the context of optimal income taxation, by Mirrlees, J. (1971) Levent Koçkesen (Koç University) Screening 2 / 58 Monopolistic Screening: A More General Model A monopolist is uniformed about the buyers characteristics What is the best price schedule according to quantity, quality, etc.? Early models: Mussa, M. and S. Rosen (1978) Maskin, E. and J. Riley (1984) Although we will use monopoly as an example similar results hold in other screening models We will still use a relatively simple model but we will be more rigorous than before in our analysis Two-Type Model Monopolist produces quantity (or quality) q 0 at cost c(q) Cost function is twice continuously differentiable with c(0) = 0,c 0,c 0 If monopolist sells quantity q for t dollars, his payoff is w(q,t) = t c(q) Customer has two possible taste parameters: θh > > 0 probability of is p (0,1) If type θi buys quantity q and pays t, her payoff is u(q,t,θi) = v(q,θi) t v is twice continuously differentiable with respect to q with v(0,θi) = 0,v1 > 0,v11 0 v(q,θh) > v(q,) for any q > 0 Levent Koçkesen (Koç University) Screening 3 / 58 Levent Koçkesen (Koç University) Screening 4 / 58

2 Two-Type Model Monopolistic Screening: The First Best Total surplus available is S(q,θi) = v(q,θi) c(q) Assume that some trade is desirable: S1(0,θi) > 0 and there can be too much of a good thing: there exists q s. t. This is the case when the monopolist knows the type of the customer In this case monopolist s problem: for each i {L,H} qi,ti 0 ti c(qi) s.t. v(qi,θi) ti 0 S(q,θi) < 0 for all q q Customer does not have to pay if she does not buy Monopolist chooses how much to produce and a (possibly nonlinear) price schedule Levent Koçkesen (Koç University) Screening 5 / 58 Levent Koçkesen (Koç University) Screening 6 / 58 Monopolistic Screening: The First Best Monopolistic Screening: The First Best The constraint must hold with equality. Therefore, problem becomes qi 0 v(qi,θi) c(qi) Since S(0,θi) = 0 > S(q,θi) for all q q, this is equivalent to the problem v(qi,θi) c(qi) 0 qi q qi The objective function is continuous and the constraint set is compact. Therefore, by Weierstrass s theorem, a solution to this problem exists. Let q f i be a solution. Since S1(0,θi) = v1(0,θi) c (0) > 0, qi = 0 cannot be a solution. Similarly, qi = q cannot be a solution Therefore, the solution must be in the interior, and hence v1(q f i,θi) = c (q f i ), for i {L,H} If v is strictly concave or c is strictly convex, then the solution is unique We can solve for the transfers from the constraint as t f i = v(qf i,θi) for i {L,H} Levent Koçkesen (Koç University) Screening 7 / 58 Levent Koçkesen (Koç University) Screening 8 / 58

3 v(q,θ H) t = 0 v(q,θ L) t = 0 Monopolistic Screening: The First Best Single Crossing Condition We can say more about the solution to the problem when we assume v1(q,θh) > v1(q,) The monopolist imizes the total surplus and captures it all Efficient quantities are produced v1(q f i,θi) = c (q f i ), for i {L,H} Price is set so as to capture the entire consumer surplus t f i = v(qf i,θi) for i {L,H} This is first degree price discrimination Called single crossing since v1(q,θi) is the slope of the indifference curve of type i customer in (q,t) plane Each indifference curve is given by for some constant k Therefore, at any (q,t ) v(q,θi) t = k dt dq = v1(q,θi) An equivalent condition is Increasing Differences: q > q and θ > θ implies v(q,θ) v(q,θ) > v(q,θ ) v(q,θ ) Levent Koçkesen (Koç University) Screening 9 / 58 Levent Koçkesen (Koç University) Screening 10 / 58 Single Crossing Condition v(q,θh) v(q,) implies that for any q type-h s indifference curve is above type-l s v11(q,θi) 0 implies that the indifference curves are concave Single crossing condition implies that at any point type-h indifference curve has a higher slope than type-l two indifference curves intersect at most once t First Best Solution with Single Crossing Claim If single crossing condition holds, then q f H qf L and tf H tf L. Proof. Suppose q f H < qf L. Then c (q f H ) = v1(qf H,θH) > v1(qf H,) v1(qf L,) = c (q f L ) which implies q f H qf L, a contradiction. Therefore, t f H = v(qf H,θH) v(qf L,) = tf L. q Levent Koçkesen (Koç University) Screening 11 / 58 Levent Koçkesen (Koç University) Screening 12 / 58

4 First Best Solution Graphically Types are unobservable t v(q,θ H) t = 0 Monopolist has to offer a common price schedule to both types and let them select What happens if he offers the first best solution? t c(q) = S(q f H,θ H) (q f H,tf H ) t c(q) = S(q f L,θ L) v(q,θ L) t = 0 q f H at price tf H q f L at price tf L Since q f L > 0, High type prefers (qf L,tf L ) v(q f L,θH) tf L = v(qf L,θH) v(qf L,) > 0 = v(qf H,θH) tf H (q f L,tf L ) q Everybody will buy low quantity and profit will be t f L c(qf L ), which is less than the first best profit (verify) What is the best the monopolist can do in this case? Levent Koçkesen (Koç University) Screening 13 / 58 Levent Koçkesen (Koç University) Screening 14 / 58 Types are unobservable Monopolistic Screening: The Second Best t The monopolist s problem now is to choose a price schedule t(q) to imize expected profits v(q,θ H) t = 0 t(q) p(t(ql) c(ql)) +(1 p)(t(qh) c(qh)) (q f H,tf H ) v(q,θ L) t = 0 qi argv(q,θi) t(q) for i = L,H q 0 v(qi,θi) t(qi) 0 for i = L,H (q f L,tf L ) q This is not an easy problem as the choice set is a set of functions Revelation Principle simplifies the problem tremendously We can restrict the choice set to menus {(ql,tl),(qh,th)} Levent Koçkesen (Koç University) Screening 15 / 58 Levent Koçkesen (Koç University) Screening 16 / 58

5 Revelation Principle Revelation Principle In general the monopolist need not constrain himself to nonlinear price schedules t(q) He can set up a mechanism in which the customer chooses a strategy s from some set S and the monopolist assigns the outcome g(s) = (q(s),t(s)) A mechanism is a pair (S,g), where g : S D, and D is the set of outcomes A mechanism implements an outcome f : Θ D if there exists s : Θ S such that for all θ Θ, g(s (θ)) = and u(g(s (θ)),θ) u(g(s ),θ), for all s S A direct (revelation) mechanism is (Θ,g), i.e., the customer (agent) reports a type θ Θ and the monopolist (principal) assigns outcome g(θ) A direct mechanism is incentive compatible (truthful) if reporting the true type is optimal, i.e., for all θ Θ u(g(θ),θ) u(g(θ ),θ), for all θ Θ Theorem If there exists a mechanism that implements an outcome, then there exists a truthful direct mechanism that implements the same outcome. Levent Koçkesen (Koç University) Screening 17 / 58 Levent Koçkesen (Koç University) Screening 18 / 58 Proof. Suppose the mechanism (S,g) implements f. Then there exists s : Θ S such that for all θ Θ, g(s (θ)) = and which implies that u(g(s (θ)),θ) u(g(s ),θ), for all s S u(g(s (θ)),θ) u(g(s (θ )),θ), for all θ Θ Revelation Principle θ f = g s s g s (θ) Let the direct mechanism be (Θ,f). Then g(s (θ)) and hence u(,θ) = u(g(s (θ)),θ) u(g(s (θ )),θ), for all θ Θ u(,θ) u(f(θ ),θ), for all θ Θ Thus (Θ,f) is a truthful direct mechanism that implements the same outcome. u(g(s (θ)),θ) u(g(s (θ )),θ) u(,θ) u(f(θ ),θ) Levent Koçkesen (Koç University) Screening 19 / 58 Levent Koçkesen (Koç University) Screening 20 / 58

6 Monopolistic Screening: The Second Best Revelation principle implies that the monopolist s problem can be reformulated as (P) (ql,tl),(qh,th) p(tl c(ql)) +(1 p)(th c(qh)) (1) v(ql,) tl v(qh,) th (2) v(qh,θh) th v(ql,θh) tl (3) v(ql,) tl 0 (4) v(qh,θh) th 0 (5) There are also non-negativity constraints: ql,qh 0 Constraints (2) and (3) are called Incentive Compatibility (IC) constraints Constraints (4) and (5) are called Individual Rationality (IR) constraints Levent Koçkesen (Koç University) Screening 21 / 58 Relaxed Program This problem is equivalent to a simpler one in which constraints are IC constraint of type H binds IR constraint of type L binds qh ql We have seen this in the two simple examples we analyzed before (RP) (ql,tl),(qh,th) p(tl c(ql)) +(1 p)(th c(qh)) (6) v(qh,θh) th v(ql,θh) tl (7) v(ql,) tl 0 (8) qh ql (9) Claim Solution sets to the programs (P) and (RP) are the same. Levent Koçkesen (Koç University) Screening 22 / 58 Proof 1. The constraint set of (P) is a subset of that of (RP): (2) and (3) imply that v(qh,) v(ql,) th tl v(qh,θh) v(ql,θh) Single crossing condition implies that qh ql. 2. At the solution to (RP), (7) holds as equality: Otherwise can increase th a little bit. 3. Solutions to (RP) satisfy the constraints of (P): Since (7) holds with equality and qh ql, single crossing implies th tl = v(qh,θh) v(ql,θh) v(qh,) v(ql,) which implies (2). Also, (7), (8), and θh > imply (5): v(qh,θh) th v(ql,θh) tl v(ql,) tl 0 Proof (cont.) 1. Let a P solve (P) but not (RP) Step (1) implies a P satisfies the constraints of (RP) At any solution a R to (RP) value of the objective function is strictly greater than its value at a P Then, a R is not in the constraint set of (P), contradicting Step (3) 2. Let a R solve (RP) but not (P) Step (3) implies a R is in the constraint set of (P) There is an a P in the constraint set of (P) that gives a strictly higher value for objective function Step (1) implies a P is in the constraint set of (RP) Contradicts a R solves (RP) This method is quite common: 1. Find an easier relaxed problem by omitting difficult constraints 2. Verify that any solution to the relaxed problem satisfies the omitted constraints Levent Koçkesen (Koç University) Screening 23 / 58 Levent Koçkesen (Koç University) Screening 24 / 58

7 Original problem: x P f(x) Relaxed problem: x RP f(x) f(x) Finally, note that (7) and (8) must hold with equality in the solution. Therefore, the problem is equivalent to: (ql,tl),(qh,th) p(tl c(ql)) +(1 p)(th c(qh)) (10) v(qh,θh) th = v(ql,θh) tl (11) v(ql,) tl = 0 (12) qh ql (13) P RP x Note that tl = v(ql,), i.e., the low type receives zero surplus and th = v(qh,θh) [v(ql,θh) v(ql,)] i.e., the high type obtains a surplus, known as information rent Levent Koçkesen (Koç University) Screening 25 / 58 Levent Koçkesen (Koç University) Screening 26 / 58 Ignore (13) and solve the new relaxed program Verify that any solution to the new relaxed program satisfies (13) Substitute in the other constraints and the problem becomes ql 0,qH 0 p[v(ql,) c(ql)]+(1 p)[v(qh,θh) c(qh)] Compare this to the first best problem: (1 p)[v(ql,θh) v(ql,)] ql 0,qH 0 p[v(ql,) c(ql)]+(1 p)[v(qh,θh) c(qh)] First best imizes expected surplus Second best imizes expected surplus minus expected information rent Second Best Solution We can write the problem as 1 p v(ql,) c(ql) ql 0,qH 0 p [v(ql,θh) v(ql,)] + 1 p p [v(qh,θh) c(qh)] We assume the following (known as virtual surplus) is strictly concave S v (q,,θh) = v(q,) c(q) 1 p p [v(q,θh) v(q,)] Therefore, objective function is strictly concave This implies that Kuhn-Tucker conditions are necessary and sufficient for global imum. Levent Koçkesen (Koç University) Screening 27 / 58 Levent Koçkesen (Koç University) Screening 28 / 58

8 Second Best Solution The Lagrangean is L(qL,qH,λL,λH) = v(ql,) c(ql) 1 p p [v(ql,θh) v(ql,)] + 1 p [v(qh,θh) c(qh)]+λlql +λhqh p A critical point must satisfy the following conditions: 1 p p [v1(qh,θh) c (qh)]+λh = 0 v1(ql,) c (ql) 1 p p [v1(ql,θh) v1(ql,)]+λl = 0 λl,λh,ql,qh 0, λlql = λhqh = 0 Second Best Solution qh = 0 v1(0,θh) c (0) 0, contradicting S1(0,θH) > 0. Therefore, qh > 0 λh = 0, and hence Similarly, if ql > 0 v1(qh,θh) c (qh) = 0 v1(ql,) c (ql) 1 p p [v1(ql,θh) v1(ql,)] = 0 and if ql = 0 v1(0,) c (0) 1 p p [v1(0,θh) v1(0,)] 0 Levent Koçkesen (Koç University) Screening 29 / 58 Levent Koçkesen (Koç University) Screening 30 / 58 Second Best Solution Summarizing, the unique global solution (ql s,qs H ) is given by If S v 1 (0,,θH) 0 otherwise q s L = 0 v1(q s H,θH) = c (q s H) v1(ql s,) = c (ql s 1 p )+ p [v1(qs L,θH) v1(qs L,)] Levent Koçkesen (Koç University) Screening 31 / 58 Second Best Solution We have to check if the solution satisfies constraint (13): qh s qs L If ql s = 0, this is satisfied trivially So suppose ql s > 0. Then v1(q s L,) = c (q s L)+ 1 p p [v1(qs L,θH) v1(q s L,)] > c (q s L) Remember v1(q f L,) = c (q f L ). We claim that qs L qf L. Suppose, for contradiction, ql s > qf L. Then, convexity of c implies that v1(q s L,) > c (q s L ) c (q f L ) = v1(qf L,) Therefore, v11 0 implies q s L qf L, a contradiction Therefore, we have q s H = q f H qf L qs L and hence (13) is satisfied. Levent Koçkesen (Koç University) Screening 32 / 58

9 sb sb Second Best Compared to First Best Remember that the first best policy was given by v1(qi,θi) = c (qi) In the second best solution the high type consumes the efficient quantity but the low type s consumption is lower than the efficient level The high type obtains an information rent The problem with the efficient allocation was that high type preferred the low type s allocation. What do you do to solve this problem? Keep giving the high type the same quantity at a lower price This gives her positive surplus, i.e., information rent Reduce quantity and price for the low type You do this to keep high type s information rent at minimum Distortion in low type s consumption is decreasing in p To make (q f L,qf H ) incentive compatible monopolist has to leave some surplus to the high type Profit is t t f H t f L q f L q f H p[v(q f L,) c(qf L )]+(1 p)[v(qf H,θH) c(qf H )] (1 p)[v(qf L,θH) v(qf L,)] q Levent Koçkesen (Koç University) Screening 33 / 58 Levent Koçkesen (Koç University) Screening 34 / 58 But the monopolist can do better than that by reducing q f L by a small amount dq. Its effect on profit: p[v1(q f L,) c (q f L )]dq +(1 p)[v1(qf L,θH) v1(qf L,)]dq More Than Two Types Suppose there are n 2 types such that: This is zero t This is positive Probability of θi is pi > 0 The rest of the model is the same: θn > θn 1 > > θ1 t f H t f L q f L q f H q v(q,θi+1) v(q,θi),v(0,θi) = 0,v1 > 0,v11 0,v1(q,θi+1) > v1(q,θi) Results generalize No distortion at the top Inefficiently low consumption for lower types No rent for the lowest type, higher types receive information rent Downward incentive compatibility constraints bind Levent Koçkesen (Koç University) Screening 35 / 58 Levent Koçkesen (Koç University) Screening 36 / 58

10 General Model General Model The monopolist s problem is: n pi(ti c(qi)) {(qi,ti)} n i=1 i=1 v(qi,θi) ti v(qj,θi) tj, i,j (14) v(qi,θi) ti 0, i (15) qi 0, i (16) We will show that this is equivalent to the following relaxed problem: n pi(ti c(qi)) {(qi,ti)} n i=1 i=1 v(qi,θi) ti v(qi 1,θi) ti 1, i = 2,...,n (17) v(q1,θ1) t1 0 (18) qi qi 1, i = 2,...,n (19) qi 0, i (20) Levent Koçkesen (Koç University) Screening 37 / 58 Levent Koçkesen (Koç University) Screening 38 / 58 Proof We first show that when qi qi 1 (14) hold if and only if (17) and v(qi,θi) ti v(qi+1,θi) ti+1, i = 1,...,n 1 (21) Clearly, (14) imply (17) and (21). To prove the other direction fix i = 2,...,n and take any j < i. Note that v(qi,θi) v(qj,θi) = i v(qk,θi) v(qk 1,θi) k=j+1 i v(qk,θk) v(qk 1,θk) k=j+1 i tk tk 1 = ti tj k=j+1 Proof 1. The constraint set of (P) is a subset of that of (RP): (14) implies that v(qi,θi 1) v(qi 1,θi 1) ti ti 1 v(qi,θi) v(qi 1,θi) Single crossing condition implies that qi qi At the solution to (RP), (17) holds as equality for all i: Otherwise can increase all tj, j i, by some ε > Solutions to (RP) satisfy the constraints of (P): Since (17) hold with equality and qi qi 1, single crossing implies ti ti 1 = v(qi,θi) v(qi 1,θi) v(qi,θi 1) v(qi 1,θi 1) which implies (21). Our result from previous slide implies (14). Showing that (14) and (18) imply (15) is left as an exercise. Similarly, for any j > i (verify). Levent Koçkesen (Koç University) Screening 39 / 58 Levent Koçkesen (Koç University) Screening 40 / 58

11 Solving the n-types Model Clearly, constraints in (17) and (18) are all binding. Therefore, the problem is n pi(ti c(qi)) {(qi,ti)} n i=1 i=1 v(qi,θi) ti = v(qi 1,θi) ti 1, i = 2,...,n v(q1,θ1) t1 = 0 qi qi 1, i = 2,...,n qi 0, i As before, ignore for now the monotonicity constraint and form the Lagrangian: n n L = pi(ti c(qi)) + λi[v(qi,θi) v(qi 1,θi) ti +ti 1] i=1 i=2 We will focus on the solution: qi > 0 for all i This implies µi = 0 for all i n +λ1[v(q1,θ1) t1] + µiqi i=1 Levent Koçkesen (Koç University) Screening 41 / 58 Levent Koçkesen (Koç University) Screening 42 / 58 Critical points of L satisfy For i < n L = pic (qi)+λiv1(qi,θi) λi+1v1(qi,θi+1) = 0 qi L = pi λi +λi+1 = 0 ti For i = n L = pnc (qn)+λnv1(qn,θn) = 0 qn L = pn λn = 0 tn So, we have no distortion at the top: Using pi λi +λi+1 = 0 and λn = pn we get λi = n k=i pk Therefore, for i < n: v1(qi,θi) c (qi) = 1 Pi [v1(qi,θi+1) v1(qi,θi)] > 0 pi where Pi = i k=1 pi Note that the principal imizes the sum of virtual surpluses: S(qi,θi,θi+1) = v(qi,θi) c(qi) 1 Pi [v(qi,θi+1) v(qi,θi)] pi This implies less than efficient qi for all i < n v1(qn,θn) = c (qn) Levent Koçkesen (Koç University) Screening 43 / 58 Levent Koçkesen (Koç University) Screening 44 / 58

12 Standard Model with Continuum of Types If the virtual surplus functions are strictly concave and (1 Pi)/pi (inverse hazard rate) decreasing in i known as monotone hazard rate property v1(qi+1,θi+1) v1(qi+1,θi) v1(qi+1,θi+2) v1(qi+1,θi+1) then qi+1 qi However, it is possible to have qi = 0 for some i < n In most papers set of types is a continuum, e.g., [,θh] A little bit more technical but cleaner results Type set: Θ = [,θh] Cumulative distribution function of θ: F, continuous density > 0 v1,v2,v12 > 0,v11 0 A contract is a pair of functions (q,t), where q : Θ R+ and t : Θ R Levent Koçkesen (Koç University) Screening 45 / 58 Levent Koçkesen (Koç University) Screening 46 / 58 Continuum of Types The monopolist s problem is: (q,t) For any contract (q,t) let θh (t(θ) c(q(θ))dθ v(q(θ),θ) t(θ) v(q(θ ),θ) t(θ ), θ,θ v(q(θ),θ) t(θ) 0, θ U(θ) = v(q(θ),θ) t(θ) We first characterize incentive compatible contracts. Theorem A contract (q,t) is incentive compatible if and only if 1. q is increasing 2. U(θ) = U(θ )+ θ θ v2(q(τ),τ)dτ for any θ,θ Proof [Only if] We can write IC constraints of types θ > θ as U(θ) U(θ )+v(q(θ ),θ) v(q(θ ),θ ) U(θ ) U(θ)+v(q(θ),θ ) v(q(θ),θ) Therefore, v(q(θ),θ ) v(q(θ),θ) U(θ ) U(θ) v(q(θ ),θ ) v(q(θ ),θ) Single crossing then implies q(θ ) q(θ) (verify as an exercise) Monotonicity of q implies that it is continuous almost everywhere Levent Koçkesen (Koç University) Screening 47 / 58 Levent Koçkesen (Koç University) Screening 48 / 58

13 Proof (cont d) Also v(q(θ),θ ) v(q(θ),θ) θ θ Taking limits as θ θ U(θ ) U(θ) θ v(q(θ ),θ ) v(q(θ ),θ) θ θ θ du(θ) = v2(q(θ),θ) dθ at all points where q is continuous. If U is absolutely continuous, then θ U(θ) = U(θ )+ v2(q(τ),τ)dτ θ,θ (22) θ See Lars Stole s unpublished lecture notes on contract theory and Milgrom and Segal (2002) Econometrica paper on sufficient conditions for (22). Proof (cont d) [If] Consider any θ > θ. θ U(θ ) U(θ) = v2(q(τ),τ)dτ θ θ v2(q(θ),τ)dτ θ = v(q(θ),θ ) v(q(θ),θ) where the inequality follows from SCP and q increasing. Levent Koçkesen (Koç University) Screening 49 / 58 Levent Koçkesen (Koç University) Screening 50 / 58 Solving the Model Solving the Model Also note that θ U(θ) = U()+ v2(q(τ),τ)dτ θ U()+ v2(q(),τ)dτ = U() +v(q(),θ) v(q(),) 0 again by SCP, the fact that q is increasing, and v2 > 0. Therefore, we need only to consider IR constraint for Therefore, the problem is equivalent to θh (t(θ) c(q(θ))dθ (q,t) θ U(θ) = U()+ v2(q(τ),τ)dτ θ (23) q is increasing (24) U() 0 (25) Levent Koçkesen (Koç University) Screening 51 / 58 Levent Koçkesen (Koç University) Screening 52 / 58

14 Using (23) we can write the objective function as θh [ θ v(q(θ),θ) c(q(θ)) Integration by parts gives [Let u = θ v2(q(τ),τ)dτ] θh [ θ ] v2(q(τ),τ)dτ dθ = θh ] v2(q(τ),τ)dτ) dθ U() (1 F(θ))v2(q(θ),θ)dθ Therefore, the objective function can be written as θh [ v(q(θ),θ) c(q(θ)) 1 F(θ) ] v2(q(θ),θ) dθ U() Therefore, the monopolist s problem reduces to θh [ v(q(θ),θ) c(q(θ)) 1 F(θ) ] v2(q(θ),θ) dθ q q is increasing. First order condition for each θ: v1(q(θ),θ) c (q(θ)) 1 F(θ) v12(q(θ),θ) = 0 Note that the monopolist imizes virtual surplus From this it is clear that at the solution U() = 0 Levent Koçkesen (Koç University) Screening 53 / 58 Levent Koçkesen (Koç University) Screening 54 / 58 No distortion at the top: Lower types under-consume since v1(q(θ),θ) = c (q(θ)) 1 F(θ) v12(q(θ),θ) > 0 All types other than the lowest receive information rent If virtual surplus function v(q,θ) c(q) 1 F(θ) v2(q,θ) is strictly concave in q, then the solution is unique Monotonicity If 1 F(θ)/ is decreasing (monotone hazard rate property) v12/ θ 0 then q is increasing. To see this let θ > θ but q(θ ) < q(θ). Then, 0 = v1(q(θ),θ) c (q(θ)) 1 F(θ) v12(q(θ),θ) < v1(q(θ ),θ) c (q(θ )) 1 F(θ) v12(q(θ ),θ) v1(q(θ ),θ ) c (q(θ )) 1 F(θ) v12(q(θ ),θ) v1(q(θ ),θ ) c (q(θ )) 1 F(θ ) f(θ v12(q(θ ),θ ) ) = 0 a contradiction Levent Koçkesen (Koç University) Screening 55 / 58 Levent Koçkesen (Koç University) Screening 56 / 58

15 Summary Extensions to the Basic Model Common results for the standard model: At efficient allocation high types want to mimic low types first best is not incentive compatible They have to receive (information) rents to prevent mimicking At second best principal trades off efficiency for information rent Less than efficient for low types to reduce rent for high types Maximizes virtual surplus = surplus - information rent Efficient consumption for the highest type All types except the lowest receive information rent Higher types consume more than lower ones Multidimensional q and θ Principal s payoff directly depends on θ: common values Agent s reservation utility depends on θ Multi-principal and/or multi-agent models Dynamic models Levent Koçkesen (Koç University) Screening 57 / 58 Levent Koçkesen (Koç University) Screening 58 / 58