Dynamic Trading under Adverse Selection

Maarten C.W. JanssenUniversity of Vienna

Question(s)

• Adverse selection: high qualities cannot be traded in market equilibrium

• Dynamic perspective: what if trade possibilities open at different moments in time?– Choose when to sell

• When time is involved:– Given stock, or inflow of new products?– Resale allowed or not?

• Alternative view on adverse selection: high qualities need to wait (longer) to sell; what about welfare aspects of this?

Discrete Example

• Suppose there are two qualities with seller‘s reservation values θ of 10 and 20 (equal probability)

• Consumers value product vθ if they know θ. Suppose v=1.2

• Both have same discount factor δ.• Consider dynamic equilibrium where quality 10 trades

at t=1 and 20 at t=2• (Max) price p1 = 12, price p2 = 24. Upper bound on δ.

Welfare improvement• Do these results hold true in continuous model?

Walrasian market model: sellers

• Perfectly durable good, quality denoted by θ, distributedo ver, ex ante distribution F(θ)

• Time indexed by t, common discount factor 0<δ<1• Continuum of sellers, each seller owns quality θ(i).

Valuation is infinite horizon discounted sum of gross surplus. Per period (1-δ)θ

• Selling in period t yields )• Facing prices p, T(θ,p) set of periods optimal to sell

Walrasian market model: buyers

• More buyers than sellers• All buyers have unit demand and value quality θ

at vθ• No reselling (buyer leaves the market)•

Walrasian market model: equilibrium

• Dynamic eq is set prices p, selling decisions T(θ,p) and expectations p) such that– Sellers maximize– Buyers maximize and markets clear: if strictly

positive measure of trade occurs, thenp); if no trade occurs thenp)

– Expectations are fulfilled when trade occurs and reasonable when trade does not occur:• p) equals expected quality of realized trade• p) at least equal to lowest unsold quality in period t

Continuous example

• F(θ) uniform over [10,16], v=1.2, δ=.5• Static equilibrium trades [10,15]• There exists dynamic equilibrium that trades

[10,12] in t=1, (12,16] in t=3, no trade in t=2• p)=11, p)=14, p)• p))•Sellers maximize and buyers are indifferent•Expectations fulfilled or reasonable

Kind of Results

• Characterization Result: In any dynamic equilibrium, consecutive intervals of qualities traded over time

• Existence result• When is break in trading necessary• Strategic re-interpretation of results

Related Literature

• Bilateral Bargaining and durable goods monopolist (…)– Intertemporal price discrimination– Independent values– Results depend on specific formulation

• Correlated values – Strategic manipulation of beliefs

• Dynamic auctions (Vicent, 1989)– Bidders bid in different periods; he assumes specific

distribution and considers specific out-of-equilibrium beliefs• Time on the market signals low quality (Taylor, 1999)

Characterization

• Any equilibrium is of the following form: The interval is divided into a finite number of intervals such that– Lower qualities are traded earlier– Qualities at the boundary of an interval are

indifferent between trading in two consecutive trading periods

– In the interior of every interval, sellers strictly prefer to sell in “their“ period

Proof of characterization

• Define i, - = δ(v-1)– highestqualitythatcanbetraded in last period

• Welookat a sequenceoffunctions with =, andif it exists implicitly by,,

• Iftheredoes not exist such , theneitherfor all LHS < RHS, orevenif LHS > RHS. In firstcase, define

• In last case, bycontinuityofthefunctionsthere must exist an y withs.t. dynamicequilibriumtrades all qualities

When is break in trade not necessary

• Define ,– θ = δ(v-1)• Similarly define α

(v-1)]• If for all θ, then equilibria with consecutive

trading exist• Condition satisfied for uniform distribution

and decreasing distributions

Strategic Interpretations

• Suppose that sellers set prices: signaling game– Any dynamic market equilibrium is outcome equivalent

to a perfect Bayes-Nash equilibrium of the signaling model

– Reverse is not true. For example, there is a signalling equilibrium where only the qualities that are traded in a static equilibrium are traded; pessimistic out-of-equilibrium beliefs

– Intuitive criterion: who has most incentives to set price slightly above this static equilibrium price in t=2?

• Suppose buyers set prices: screening game

Final remarks

• Multiple equilibria are possible• Welfare properties– Multiple equilibria can be Pareto-Ranked– Comparison with static equilibrium (total surplus?)

• If good is not perfectly durable, but decay is exponential, then analysis goes through