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