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PROGRESSIVE SCREENING: LONG-TERM CONTRACTING WITH A PRIVATELY KNOWN STOCHASTIC PROCESS Maher Said with Ralph Boleslavsky (University of Miami) January 2013

LONG-TERM CONTRACTS

Long-term contracts are employed widely across many settings: ▶ Trade; ▶ Employment; ▶ Supply chain management; ▶ Financial contracts; ▶ Insurance; ▶ ….

Boleslavsky and Said (2013): Progressive Screening

LONG-TERM CONTRACTS

But in all these settings, important features of the economic environment change over time: ▶ Trade: Consumers’ values for goods; ▶ Employment: Workers’ productivity; ▶ Supply chain management: Supplier production costs; ▶ Financial contracts: Borrower’s wealth; ▶ Insurance: Chance of claim; ▶ ….

Boleslavsky and Said (2013): Progressive Screening

LONG-TERM CONTRACTS

Changes are not necessarily unanticipated: all parties may be aware the environment is dynamic. But these changes need not be observed by all parties. Optimal long-term contracts need to incorporate new info over time. And to do so, need to incentivize information revelation!

Boleslavsky and Said (2013): Progressive Screening

THIS PAPER

These are the sorts of settings we’re interested in. But we also want to look at the efgect of an additional source of private information: ▶ Agents have private information about the state of the economic environment. ▶ And also about the process governing its evolution.

Boleslavsky and Said (2013): Progressive Screening

EXAMPLE

A manufacturer is introducing a new product to a market. But sale occurs via local retailers. The manufacturer has a good sense of broad national trends. The retailer may have better projections of how demand will evolve in its local market.

Boleslavsky and Said (2013): Progressive Screening

QUESTION AND PREVIEW

So in settings like this, what should the optimal contract look like? In particular, how do we incentivize revelation of both process and states? We characterize a class of environments in which the optimal contract takes an especially simple form: ▶ The optimal dynamic contract can be implemented by a deterministic sequence of static contracts. ▶ The seller screens stochastic processes with a menu of such sequences. ▶ Each sequence “progressively” screens buyer’s values. ▶ Rents reduced by increasingly restricting supply.

Boleslavsky and Said (2013): Progressive Screening

BASIC MODEL

Buyer has single-unit demand in each of T ≤ ∞ periods. Flow utility in period t: vtqt − pt. Marginal utility vt evolves over time: vt = αtvt−1, where v0 := 1. In each period t = 1, . . . , T, buyer privately observes αt ∈ {u, d}, where ∆ := u − d > 0. αt is iid in each period: Pr(αt = u) = λ and Pr(αt = d) = 1 − λ. At time 0, buyer privately observes λ ∈ [0, 1]. λ is distributed according to cdf F (with density f).

Boleslavsky and Said (2013): Progressive Screening

BASIC MODEL u u2 u3 d ud d2 d3 ud2 u2d

λ 1-λ λ λ λ λ λ 1-λ 1-λ 1-λ 1-λ 1-λ

λ

Boleslavsky and Said (2013): Progressive Screening

BASIC MODEL

In each t ≥ 1, the seller can produce 1 unit at 0 cost. Both seller and buyer discount future by δ ∈ (0, 1]. ▶ Additional restriction: if T = ∞, then δu < 1. In period 0, the seller ofgers a long-term contract. ▶ If buyer accepts: sales/consumption occur in t ≥ 1. ▶ If buyer rejects: both parties receive 0 payofg. One-sided commitment: ▶ Seller has full commitment power. ▶ Buyer has outside option with 0 utility in every period.

Boleslavsky and Said (2013): Progressive Screening

RELATED LITERATURE

Part of literature on optimal dynamic mechanism design: ▶ Baron and Besanko (1984); Battaglini (2005); Besanko (1985); Courty and Li (2000); Deb (2009, 2011); Esö and Szentes (2007); Kakade, Lobel, and Nazerzadeh (2011); Pavan, Segal, and Toikka (2011); and others…. Most closely related paper: Courty and Li (2000). ▶ Essentially our setting but with T = 1. ▶ With T > 1, incentive compatibility is tricky. ▶ Also, T > 1 lets us consider long-term characteristics of optimal contract. Also: Pavan, Segal, and Toikka (2011). ▶ Their paper is more ambitious/general. ▶ In some settings, they have similar results. ▶ Our paper: suffjcient conditions on primitives of the environment.

Boleslavsky and Said (2013): Progressive Screening

RELATED LITERATURE

Information acquisition in dynamic mechanisms: ▶ Bergemann and Välimäki (2002); Gershkov and Szentes (2009); Krähmer and Strausz (2011); Shi (2011). ▶ Information arrival is exogenous in our world. Multidimensional screening: ▶ Matthews and Moore (1987); Rochet and Stole (2003); others…. ▶ Key distinction: IC is easier here because of timing—with dynamics, single dimension

• f (new) private info in each period.

And our results are reminiscent of priority pricing/option contracts as in Harris and Raviv (1981) or Wilson (1993).

Boleslavsky and Said (2013): Progressive Screening

CONTRACTS

Some notation: ▶ History of shocks: αt := (αt, αt−1, . . . , α1). ▶ History of shocks after s: αt

−s := (αt, αt−1, . . . , αs+1).

▶ Value given sequence of shocks: v(αt

−s) := t τ=s+1 ατ.

Revelation principle holds, so (wlog) consider direct mechanisms. A contract is then: ▶ A sequence of payments p = {pt(αt, λ)}T

t=0.

▶ A sequence of allocation probabilities q = {qt(αt, λ)}T

t=1.

Boleslavsky and Said (2013): Progressive Screening

SELLER'S PROBLEM

The seller wants to maximizes profjts: max   

1

• p0(λ) + E

T

• t=1

δtpt(αt, λ)

• λ
• dF(λ)

   s.t. IC and IR constraints.

Boleslavsky and Said (2013): Progressive Screening

INCENTIVE COMPATIBILITY

On path, buyer must have no incentive to misreport new private information. Truth has to be better than lying and optimal continuation reporting. But optimal continuation reporting may involve additional misreports! So unlike a static mechanism design problem, we have to worry about compound contingent deviations. ▶ Large and intractable set of IC constraints that must be satisfjed. ▶ The “brute force” approach of imposing these constraints directly is generally useless (unless we impose further assumptions).

Boleslavsky and Said (2013): Progressive Screening

"STANDARD" APPROACH

Some dynamic mechanism design work (e.g., Baron and Besanko 1984; Courty and Li 2000; Krähmer and Strausz 2011) avoids this by looking at two-period models. ▶ Relatively easy to solve second period for any history. ▶ Then backward induction is straightforward.

Boleslavsky and Said (2013): Progressive Screening

"STANDARD" APPROACH

Most others (e.g., Esö and Szentes 2007; Pavan, Segal, and Toikka 2011; Kakade, Lobel, and Nazerzadeh 2011) use “independent shocks.” ▶ Period-t value is a continuous function of period-(t − 1) value and an independent shock. ▶ This representation always exists via probability integral transform. Also typically impose a Markov structure on “true” types. Combine with a “strong monotonicity” condition. Jointly: optimal strategy after a misreport is a “correction.” ▶ Our context: if I misreported αt−1 as α′

t−1, will then report αt as (αt−1αt)/α′ t−1.

This permits a one-shot deviation approach to IC.

Boleslavsky and Said (2013): Progressive Screening

"STANDARD" APPROACH

This doesn’t work in our setting. Period-t value is not a suffjcient statistic for t + 1 value. ▶ αt+1 is correlated with λ. And full independent shock approach is still non-Markov in values: ▶ Defjne {ξt}T

t=1 to be a sequence of iid U[0, 1] RVs.

= ⇒ ˜ vt(ξt, λ) =

• d˜

vt−1(ξt−1, λ) if ξt < 1 − λ, u˜ vt−1(ξt−1, λ) if ξt ≥ 1 − λ. ▶ λ still matters!

Boleslavsky and Said (2013): Progressive Screening

OUR APPROACH

We take a slightly difgerent approach. Solve a relaxed problem; impose only a subset of IC constraints. ▶ Consider “one-time” deviations from truth-telling: buyer must report truthfully after a deviation. The solution to this relaxed problem is “path independent”: it depends only on αt only through v(αt). We can pair this solution with a static pricing rule. The resulting mechanism decouples incentives from one period to the next, yielding “full” IC.

Boleslavsky and Said (2013): Progressive Screening

IC CONSTRAINTS

At time t ≥ 1, for all histories (αt−1, λ) and all αt, α′

t: T

• s=t

δs−tE

• qs(αs

−t, αt, αt−1, λ)v(αs −t, αt) − ps(αs −t, αt, αt−1, λ)

• αt, λ
• ≥

T

• s=t

δs−tE

• qs(αs

−t, α′ t, αt−1, λ)v(αs −t, αt) − ps(αs −t, α′ t, αt−1, λ)

• αt, λ
• .

This is essentially a static IC constraint. Note: ▶ Payofgs are quasilinear. ▶ v(αs

−t, αt) = αtv(αs −t, αt−1) =

⇒ single-crossing.

Boleslavsky and Said (2013): Progressive Screening

IC CONSTRAINTS

Standard machinery = ⇒ IC in t ≥ 1 if, and only if, ▶ Downward IC constraint is satisfjed; and ▶ Expected discounted probability of allocation is montone. IC in t = 0 is less straightforward: ▶ λ does not enter fmow utilities; impact is purely informational. ▶ Changes in λ yield difgerent evolution of preferences. ▶ So no single-crossing condition to help us out.

Boleslavsky and Said (2013): Progressive Screening

IC CONSTRAINTS

But we can write ˆ U0(λ′, λ) = −p0(λ′) + δE [U1(α1, λ′)|λ] + (λ − λ′)

T

• t=2

δtE

• Ut(u, αt−1, λ′) − Ut(d, αt−1, λ′)
• λ
• .

And then we resort to an envelope theorem: U ′

0(λ) = T

• t=1

δtE

• Ut(u, αt−1, λ) − Ut(d, αt−1, λ)
• λ
• .

Higher λ = ⇒ higher probability of benefjting from u–d payofg gradient. Note: necessary condition, but not suffjcient. And also useful for showing downward IC constraints (for t ≥ 1) must bind.

Boleslavsky and Said (2013): Progressive Screening

VIRTUAL SURPLUS

With some algebra and integration by parts, the seller’s objective becomes

T

• t=1

δt

1

• E
• v(αt)
• 1 −

t

• s=1

1d(αs) ∆/d 1 − λ 1 − F(λ) f(λ)

• q(αt, λ)
• λ
• dF(λ).

The term v(αt)

• 1 −

t

• s=1

1d(αs) ∆/d 1 − λ 1 − F(λ) f(λ)

• is the virtual value, refmecting distortions due to asymmetric information.

Boleslavsky and Said (2013): Progressive Screening

VIRTUAL VALUE

ϕ(αt, λ) = v(αt) − v(αt)

t

• s=1

1d(αs) ∆/d 1 − λ 1 − F(λ) f(λ) First term: contribution to social surplus. Second term: information rents. ▶ Inverse hazard rate (1 − F(λ))/f(λ): rents to buyer with type λ paid to all higher types. ▶ Remaining part: “responsiveness” of period-t value to changes in λ. ▶ v(αt)∆/d is change in value from a high instead of low shock; 1/(1 − λ) is “likelihood”

• f change.

▶ Sum over s ≤ t to account for all shocks by which λ afgects vt.

Boleslavsky and Said (2013): Progressive Screening

VIRTUAL SURPLUS

Back to seller’s objective function:

T

• t=1

δt

1

• E
• v(αt)
• 1 −

t

• s=1

1d(αs) ∆/d 1 − λ 1 − F(λ) f(λ)

• q(αt, λ)
• λ
• dF(λ).

Pointwise maximization = ⇒ allocate ifg

t

• s=1

1d(αs) ≤ 1 − λ ∆/d f(λ) 1 − F(λ). Distortions depend only on λ and the number of “bad” d shocks received. And this happens in a path independent way—we add up the number of d shocks, but their

• rder doesn’t matter.

Boleslavsky and Said (2013): Progressive Screening

PRICES

So for each λ, there is a threshold k(λ) of d’s allowed: k(λ) := max

• k ∈ Z+ : k ≤ 1 − λ

∆/d f(λ) 1 − F(λ)

• .

So how do we implement this allocation rule? We can simply set a sequence of prices: ▶ Set a price in period t equal to the lowest value “allowed” to buy: Pt(λ) := ut−k(λ)dk(λ). ▶ If (for instance) k(λ) = 3, prices are d, d2, d3, ud3, u2d3, . . . .

Boleslavsky and Said (2013): Progressive Screening

PRICES

Key feature: evolution of prices does not depend on the buyer’s reports. ▶ This is what rationalizes looking at a restricted set of IC constraints! ▶ If what I do today has no bearing on the prices I face in the future, I can’t manipulate them by lying. ▶ And with these prices, telling the truth is myopically optimal. So for any reported λ, we have “full” IC in periods t ≥ 1. Also note that payments are always less than value = ⇒ we also have IR in each period t ≥ 1.

Boleslavsky and Said (2013): Progressive Screening

PERIOD-0 IC

Is the initial report of λ guaranteed to be truthful? Suffjcient condition: 1 − F(λ) (1 − λ)f(λ) (weakly) decreasing. Similar to Matthews and Moore (1987) “attribute ordering” condition for multi-dimensional screening. This condition implies ϕ(αt, λ) is nondecreasing. ▶ This guarantees threshold k(λ) is increasing. ▶ Expected allocation (from perspective of period 0) is then monotone, and IC at time 0 follows.

Boleslavsky and Said (2013): Progressive Screening

ENTRY FEES

p0(λ) pinned down by revenue/payofg equivalence; (weakly) increasing. This gives us period-zero “entry fees.” Thresholds generate a fjnite partition of [0, 1] into intervals Λn := [λn−1, λn) such that k(λ) = n for all λ ∈ Λn. ▶ Allocations, future prices constant on Λn = ⇒ so are entry fees. Straightforward to show that p0(λn+1) > p0(λn). ▶ Buyer is willing to pay a high entry fee in return for low future prices only if she expects her value to be high enough to benefjt. ▶ This occurs only if λ is suffjciently large.

Boleslavsky and Said (2013): Progressive Screening

OPTIMAL CONTRACT

The optimal contract is very simple. ▶ Let the buyer choose among a set of “price plans.” ▶ Each plan corresponds to a fjxed sequence of future price paths. ▶ Each plan has a difgerent length “honeymoon period” with slower price growth: Plan 1: d, ud, u2d, u3d, . . . Plan 2: d, d2, ud2, u2d2, . . . Plan 3: d, d2, d3, ud3, . . . Plan 4: d, d2, d3, d4, . . . …. ▶ Longer honeymoon phases have higher entry fees in the initial period.

Boleslavsky and Said (2013): Progressive Screening

OPTIMAL CONTRACT u u2 u3 d ud d2 d3 ud2 u2d

λ 1-λ λ λ λ λ λ 1-λ 1-λ 1-λ 1-λ 1-λ

λ∊Λ1 p0(Λ1)

Boleslavsky and Said (2013): Progressive Screening

OPTIMAL CONTRACT u u2 u3 d ud d2 d3 ud2 u2d

λ 1-λ λ λ λ λ λ 1-λ 1-λ 1-λ 1-λ 1-λ

λ∊Λ2 p0(Λ2)

Boleslavsky and Said (2013): Progressive Screening

OPTIMAL CONTRACT

For all λ < 1, prices eventually start trending upwards at rate u. But values only increase probabilistically… ▶ As T grows, early termination becomes almost certain. This is what we refer to as progressive screening: ▶ Screen across λ in the initial period. ▶ In future periods, screen across shocks αt, excluding more and more buyers. By increasingly restricting supply, able to extract additional rents from “lucky” (high-value) buyers with many u shocks.

Boleslavsky and Said (2013): Progressive Screening

OPTIMAL CONTRACT

Optimal contract has “Generalized No-Distortion at the Top” property: ▶ First noted by Battaglini (2005). ▶ If λ = 1, no distortions. ▶ If αs = u for all s ≤ t, no distortions at t. But unlike Battaglini’s two-type Markov, we don’t obtain asymptotic effjciency. ▶ Informational linkage between initial type (λ) and future information is never broken. ▶ Therefore there are always gains to distorting/restricting supply.

Boleslavsky and Said (2013): Progressive Screening

FRANCHISING?

A manufacturer is introducing a new product to a market. But sale occurs via local retailers. ▶ The manufacturer has a good sense of broad national trends. ▶ The retailer has better projections of demand evolution in local market. In our model: ▶ Let u = 1 > d, so market shrinks over time. ▶ And λ ∈ [0, 1] is anticipated rate of decline. Then: ▶ Interpret p0(λ) as a franchising fee. ▶ And {qt, pt} is a commitment to a sequence of prices, where the path imposes performance targets. ▶ The manufacturer allows downstream resale as long as the local market remains suffjciently profjtable.

Boleslavsky and Said (2013): Progressive Screening

NON-ZERO COSTS

Suppose cost of provision in each period is c > 0. Then optimal allocation is no longer time independent: ▶ Allocate to buyer only if ϕ(αt, λ) ≥ c = ⇒ thresholds are now kt(λ) = max

• k ∈ Z+ : ut−kdk
• 1 − ∆/d

1 − λ 1 − F(λ) f(λ) k

• ≥ c
• .

▶ If u < 1, then kt(λ) ≥ kt+1(λ): all buyers eventually excluded. ▶ If u > 1, then kt(λ) ≤ kt+1(λ): a buyer may be excluded in t, but a single u shock can lead to re-inclusion. The effjcient policy is like this, too. But there exists ¯ k(λ) < ∞ such that kt(λ) ≤ ¯ k(λ) for all t. ▶ Price will (eventually) grow deterministically at rate u, so even high-valued buyers are (in the limit) excluded.

Boleslavsky and Said (2013): Progressive Screening

ROBUSTNESS?

How “special” are these results? Is it just the discreteness and the binomial process driving things? Let’s generalize the model a bit… ▶ Recombinant binomial tree = ⇒ continuous distribution of shocks. ▶ Indivisible good = ⇒ divisible good with convex costs.

Boleslavsky and Said (2013): Progressive Screening

CONTINUOUS SHOCKS

In period 0, buyer privately learns λ ∈ Λ := [λ, ¯ λ]. ▶ λ is drawn from a distribution F (with density f). In t ≥ 1, privately observes shock αt ∈ A := [α, ¯ α]. ▶ αt is an iid draw from G(·|λ) with density g(·|λ). ▶ Assume that {G(·|λ)}λ∈Λ is ordered by FOSD. ▶ Also, G is twice continuously difgerentiable. Seller can produce q units at cost c(q) = q2

2 .

Otherwise, everything is as before….

Boleslavsky and Said (2013): Progressive Screening

CONTINUOUS SHOCKS

We use the same approach as with discrete shocks. And the analysis is essentially the same: ▶ “Localized” IC constraints instead of downward IC constraints. ▶ Envelope theorem for period-0 IC. With algebra and integration by parts, the seller’s objective becomes

T

• t=1

δt

• Λ×At
• ϕ(αt, λ)q(αt, λ) − q(αt, λ)2

2

• d

t

• s=1

G(αs|λ)

• dF(λ).

Boleslavsky and Said (2013): Progressive Screening

CONTINUOUS SHOCKS

As before, the term ϕ(αt, λ) := v(αt)

• 1 +

t

• s=1

∂G(αs|λ)/∂λ αsg(αs|λ) 1 − F(λ) f(λ)

• is the buyer’s virtual value.

v(αt) is contribution to surplus. The rest is information rents paid to buyer: FOSD = ⇒

∂G(·|λ) ∂λ

≤ 0. v(αt)

αs ∂G(αs|λ)/∂λ g(αs|λ)

: ▶ Baron and Besanko (1984) “informativeness measure.” ▶ Pavan, Segal, and Toikka (2011) “impulse response.” ▶ Sum over s ≤ t to account for all shocks by which λ afgects vt.

Boleslavsky and Said (2013): Progressive Screening

CONTINUOUS SHOCKS

Pointwise maximization: q∗(αt, λ) = max{ϕ(αt, λ), 0}. But our approach relies on path independence. Need to be able to write ϕ(αt, λ) as a function of v(αt) and λ alone. Suffjcient condition: ∃ a, b ∈ R and γ : Λ → R s.t. ∂G(αs|λ)/∂λ g(αs|λ) = α(a + b log(α))γ(λ) = ⇒ ϕ(αt, λ) = v(αt)

• 1 + (at + b log(v(αt)))γ(λ)1 − F(λ)

f(λ)

• .

Boleslavsky and Said (2013): Progressive Screening

CONTINUOUS SHOCKS

∂G(αs|λ)/∂λ g(αs|λ) = α(a + b log(α))γ(λ) Obviously, not without loss. But not too unnatural: ▶ Power: α log(α)/λ. ▶ Exponential, Pareto, truncated Normal: −α/λ. ▶ Lognormal: −α. Another class of distributions: α = λz, where z ∼ H. ▶ Then ratio is again −α/λ.

Boleslavsky and Said (2013): Progressive Screening

CONTINUOUS SHOCKS

Can then write q∗(αt, λ) = ˆ q∗(v(αt), λ). ▶ Can ask buyers to report v(αt) instead of αt. Use standard pricing rule to elicit report: p∗(αt, λ) := q∗(αt, λ)v(αt) − v(αt)

αt

ˆ q∗(v′, λ) dv′. Truthful reporting is IC in t ≥ 1 ifg ˆ q∗ is increasing in v. Suffjcient condition for IC in t = 0: ϕ(αt, λ) increasing in λ. And p∗

0(λ) pinned down via revenue/payofg equivalence.

Boleslavsky and Said (2013): Progressive Screening

CONTINUOUS SHOCKS

Easy to show that exclusion is permanent. Also: set of “admissible” reports in each period is a subset of preceding period’s “admissible” reports. ▶ Continuous analog of honeymoon phases. ▶ Shrinking admissible sets = ⇒ increasing exclusion. And by “tightening screws,” can charge higher prices on remaining buyers.

Boleslavsky and Said (2013): Progressive Screening

WRAPPING UP

Model also applies immediately to an agent learning about values making a single,

• ptimally-timed purchase.

Also generalizes in a straightforward manner to multiple agents. Some comments: ▶ Specialized environment and strong suffjcient conditions. ▶ Commitment is crucial. ▶ What about multiple sellers?

Boleslavsky and Said (2013): Progressive Screening