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Physical Depreciation and Discounting in Dynamic Markets with Adverse Selection.

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446
ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
¹4
ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
Physical Depreciation and Discounting
in Dynamic Markets with Adverse Selection
M. Janssen, V. Karamychev
We investigate the nature of the adverse selection problem in a
market for a durable good where trading and entry of new buyers and
sellers takes place in continuous time. We focus on the role of the
interest rate and physical depreciation. We show that when the physical
depreciation rate is relatively small, infinitely many equilibria exist
where all goods are traded within finite time after their appearance in
the market. In contrast, when the physical depreciation rate is relatively
large the trade of new goods will stop in any equilibrium after a finite
moment in time. At intermediate values, stationary equilibria, different
from the static equilibria, may emerge. For any given level of the
relative depreciation rate the interest rate only determines the speed of
evolution along the equilibrium path.
1. Introduction
Since the pioneering work of Akerlof (1970) and Wilson (1979, 1980), adverse
selection is regarded as one of the main sources of market failure. In a few recent
articles Janssen and Roy (1999a, 1999b) and Janssen and Karamychev (2000) have
considered the adverse selection problem in a dynamic perspective. They ask the
question whether the market mechanism, by changing prices over time, provide
adequate incentives for sellers of different qualities to sort themselves over time.
This question is relevant in markets where goods have a use value that extends over
some time periods and where high quality goods have a higher use value than low
quality goods. They show in a variety of settings that there exist dynamic competitive equilibria where all goods are traded after a finite number of periods. The
main idea behind this result is that given a sequence of prices high quality sellers
have more incentives to wait (and enjoy a higher use value before selling) than low
quality sellers do. Once certain (low) qualities are sold, only relatively high qualities
remain in the market. Risk-neutral consumers can predict that sellers of different
qualities will sort themselves into different time periods and, hence, they are willing
to pay higher prices in later periods. The equilibria are thus such that higher
qualities are sold in later periods at higher prices.
_____________________
Maarten C. W. Janssen – PhD in economics, Erasmus University Rotterdam.
V.A. Karamychev – PhD in economics,Tinbergen Institute Rotterdam.
Ñòàòüÿ ïîñòóïèëà â Ðåäàêöèþ â îêòÿáðå 2002 ã.
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ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
447
In these dynamic models, two exogenous factors account for the decision of an
individual seller with a given quality to sell in a certain time period: the rate of time
preferences (discount factor) and the rate at which goods depreciate. Given the
complexity of the models, it is, however, very difficult (if not impossible) to evaluate
the role of each of these factors in determining the dynamic equilibrium path. The
discount factor plays a role, but because of the discreteness of the time intervals it is
difficult to analyze whether the qualitative properties of equilibria are affected by
changes in the discount factor. Physical depreciation of the goods, on the other hand,
is assumed to be absent in the models considered by Janssen and Roy (1999a, b) and
Janssen and Karamychev (2000). In this paper we analyze the role of both factors in
a continuous time version of the model. As the continuous time version is easier to
analyze we are, at the same time, able to generalize the demand side of the model
considerably by allowing consumers not to be risk-neutral.
We consider a competitive market for an imperfectly durable good where potential sellers are privately informed about the quality of the goods they own. Each
moment in time a constant flow of sellers with an identical but arbitrary distribution
of quality enters the market. All buyers are identical, have unit demand and for any
given quality, a buyer’s willingness to pay exceeds the reservation price of a seller
for that quality. As buyers do not know the quality, their willingness to pay equals
the expected valuation of goods traded at a certain time. The flow of such buyers
into the market is larger than the flow of sellers so that, in equilibrium, prices are
equal to the expected buyers' valuation. Once traded, goods are not re-sold in the
same market1).
A first result says that only the relative depreciation rate, defined as the ratio
of the physical depreciation rate and the interest rate, determines the qualitative
properties of the equilibrium path. In particular, changes in the interest rate and the
absolute physical depreciation rate that do not affect the relative depreciation rate
do not affect the nature of equilibria in any way. The interest rate itself only determines the speed of evolution along an equilibrium path and in particular, higher interest rates implies a higher volume of trade at each moment as it is easier to separate goods of different quality.
Next, we analyze the role of the relative depreciation rate. We first consider
the limit case where goods are perfectly durable so that the relative depreciation
rate equals to zero for any positive value of the interest rate. The main result here
says that there exist an infinite number of equilibria where every potential seller
entering the market trades within a finite time after entering the market. When the
quality distribution is such that there are relatively few sellers around the static
equilibrium quality such equilibria only exist when we allow price to be a discontinuous function of time before all goods are sold. We then consider the case where
goods are imperfectly durable and depreciate over time. A first observation is that
with depreciation the infinitely repeated version of a static equilibrium outcome is
not an equilibrium anymore in our dynamic model. Apart from this observation we
have three types of results. A first result is that if the depreciation rate is small
enough, i.e., if goods are "almost perfectly durable", the qualitative properties of the
dynamic equilibria without physical depreciation hold true. A second result is that if
1) For example, in car markets, it is publicly observable how many owners a car has had
up to particular point in time. Hence, second hand markets may be distinguished from thirdhand markets, and so on.
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¹4
the depreciation rate is relatively large, all equilibria result in no trade of new goods
after a finite point in time. Finally, at intermediate levels of the depreciation rate,
stationary equilibria, different from the static equilibria, may then emerge where
low quality "new" goods and depreciated goods that originally were of high quality
are traded at the same time. A full characterization of the uniform case is given for
different values of the depreciation rate.
The paper is organized as follows. Section 2 sets out the model and the equilibrium concept. Section 3 shows that only the relative depreciation rate has an effect on the qualitative properties of equilibria. Sections 4 and 5 discuss the role of
the physical depreciation rate by analyzing the case of perfectly durable goods and
imperfectly durable goods, respectively. Section 6 concludes. Proofs are contained in
the Appendix.
2. The Model
Consider a Walrasian market for a good whose quality, denoted by q , depreciates over time with a rate d Î [ 0, ¥ ). Time, denoted by t, is continuous and
t Î [ 0, ¥ ). For every time moment t a constant flow of sellers I enters the market.
Let ti be the entry time of seller i and let q i be the quality he is endowed with at
the time of entry. This implies that the quality owned by a seller i at time t is a
function q (t ) = q e -d (t -ti ) . We assume that q varies between q and q , where
i
i
i
0 < q < q < ¥ . The set of all sellers, therefore, is I = { i} = {(q i , ti )} . We denote by m (q )
the Lebesgue measure of sellers in the flow I who own a good of quality less than or
equal to q . We assume that m (q ) is strictly increasing, absolutely continuous with
respect to the Lebesgue measure and constant over time.
Each seller i knows the quality q i of the good he is endowed with and derives
flow utility from ownership of the good until he sells it. Therefore, the seller's reservation price is the present discounted value of the flow of gross utility and we normalize this to be equal to q i . This implies that the gross utility flow is (r + d )q i (t ) ,
where r is the interest rate:
q i = òt e - r (t - t i ) (r + d ) q i (t ) dt .
i
¥
On the demand side there is an inflow of new buyers at every time moment,
which is larger in size than m q . All buyers are identical and have unit demand and
quasi-linear preferences. A buyer's valuation of quality q is v(q ) , where v(q ) > q
and v¢ > 0 . Thus, under full information, first, there are always gains from trade
and, second, higher quality goods are valued more than lower qualities. Having
bought a good of quality q at time t ³ ti against price p( t ) , a buyer derives utility
u (v(q ) - p ) , where u ¢ > 0 , and we normalize u (0 ) = 0 .
()
All buyers know the ex ante quality distribution m (q ) but do not know the
quality of the good offered by a particular seller. Goods that are once bought are not
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re-sold in the same market. Buyers and sellers discount the future at the common
interest rate r and maximize their expected utility.
Before proceeding, we introduce some additional notation. Let p (I ¢, t ) be the
price that makes a buyer indifferent between buying a good from a seller i that belongs to a certain subset I ¢ Ì I at time t and not buying at all. It follows that
-d (t - t i )
q i - p (I ¢, t ) dm = 0 . Adverse selection implies that p (I, t ) < q , i.e., the
òiÎI ¢, t £ t u v e
((
i
)
)
willingness to pay for the average quality in the population is lower than the reservation price of the seller of the best quality. Thus, the static Akerlof–Wilson version
of the model has a largest equilibrium quality, which we denote by q S , satisfying
p ({ i q i £ q S } ,0) = q S . The measure of sellers in the set
m (I ¢) = m ({i i Î I ¢}) such that m (q ) = m ({i i Î [q ,q ], ti = t}) for all t.
I¢
is
denoted
as
To simplify our analysis we introduce the following regularity assumptions.
Throughout this paper, we assume that these assumptions hold.
Assumption 1. The measure function m (q ) is differentiable on [ q ,q ] with
strictly positive and Lipschitz-continuous density function f (q ) = m ¢(q ) , i.e.,
f (q ) ³ e f > 0 and f (q ¢) - f (q ) < M f q ¢ - q for some e f , M f and for all q ¢,q Î [ q ,q ].
Assumption 2. The buyers’ valuation function v(q ) is continuously differen-
tiable on [ q ,q ] and there exists e v > 0 such that for all q : v(q ) - q > e v and v¢(q ) > e v .
Assumption 3. The buyers’ utility function u (m ) is continuously differentiable
() ()
on [ v(q ) - v q , v q - q ] and there exist M u > e u > 0 such that M u > u ¢(m ) > e u and
u (0 ) = 0 .
Given an evolution of market prices p (t ) , t Î [0, ¥ ) , each seller i chooses whether or not to sell and if he decides to sell, the selling time. If he chooses not to sell
his gross surplus is equal to q i and, therefore, his net surplus equals to zero. If, on
the other hand, he decides to sell at time t ³ ti his gross surplus becomes
òt i (r + d )q i (t ) e
t
- r (t - t i )
(
)
dt + e - r (t - t i ) p (t ) = q i + e - r (t - t i ) p (t ) - e -d (t - t i ) q i ,
and, therefore, his net discounted surplus is equal to
(
)
si (t ) = e - rt p (t ) - e -d (t - t i ) q i .
The set of time moments in which it is optimal to sell for a seller i is given by
Ti ( p ( t )) º arg max { si ( t ) si ( t ) ³ 0}.
t ³ti
If p ( t ) - e -d (t - t i ) q i < 0 for all t ³ ti then Ti ( p ( t )) = Æ . Each potential seller i chooses a
time t i Î Ti when to sell. Let τ = {t i }iÎI be a set of all selling decisions. This implies
that there is a flow of goods being offered for sale. We will denote this flow by J t
and it follows that J t º {i t i = t }.
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At any time moment t buyers expect a certain quality distribution to be offered for sale. This determines their subjective willingness to pay for the average
good at time t, which will be denoted by p (t ) .
A dynamic equilibrium is an equilibrium where all players rationally maximize
their objectives, expectations are fulfilled and the market always clears.
Definition 1. A dynamic equilibrium is described in terms of a path of prices
p ( t ) , buyers' willingness to pay p ( t ) and a set of selling decision τ = {t i }iÎI such
that:
a) Sellers maximize: t i Î Ti ( p ( t )) for all i Î I , i.e., every seller i chooses time
t i to trade optimally.
b) Buyers maximize and market clear: If m ( J t ) > 0 then p( t ) = p ( t ) , i.e., if
there is a strictly positive flow of goods offered for sale at time t, then each buyer
gets zero net expected utility so that he is indifferent between buying and not
buying and market clears. If m ( J t ) = 0 then p( t ) ³ p ( t ) , i.e., if there are (almost) no
goods for sale at time t then each buyer can get at most zero net expected utility.
Hence, it is optimal for him not to buy at that time as well.
c) Expectations are fulfilled when trade occurs: If m ( J t ) > 0 then p (t ) º p (J t , t ) .
d) Expectations are reasonable even if no trade occurs: If trade stops at time
tˆ and the lowest quality left in the market is qˆ then for all t > tˆ until the trade
ˆ
starts again: p (t ) ³ v e - (t -t )d q
, where q
= min q ,qˆ .
(
min
)
min
{ }
Given the set-up described above, conditions (a)-(c) are quite standard.
Condition (d) is introduced for the formal reason that the buyers’ willingness to pay
is not defined when no trade occurs. The condition says that even if the flow of
goods offered for sale is zero, buyers should believe that the lowest quality offered is
larger than the apriori lowest possible quality at that time. This condition assures
that autarky, i.e., no trade at any time, cannot be sustained in an equilibrium of the
dynamic model.
3. Equilibrium Dynamics and The Role of The Interest Rate
The model to be solved includes three exogenous functions m (q ) , v(q ) and
u (m ) , and, in addition, two exogenous scalar parameters, r and d. Obviously, all of
them could affect the model’s outcome and our aim is to understand the way they
do. In this section we first describe the system of differential equations determining
the dynamic equilibrium path. We then show that the interest rate r and the physical depreciation rate d only effect the set of dynamic equilibria of the model through
their effect on the relative depreciation rate dr .
We start our analysis by arguing that if a good of certain quality sells at time
t, then all goods with lower qualities at that time that have entered the market before (and are still in the market) will also sell at that time. Given any p (t ) a seller i
by selling at time t earns a net discounted surplus e - rt p( t ) - e -d (t - t i ) q . Maximizing
(
this expression yields the first order conditions:
i
)
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a) t (q i ) = ti ,
b) p& (t (q i )) + (r + d )q i (t ) = rp (t (q i )) ,
if p& ( ti ) + (r + d )q i (t ) £ rp (ti ) , or
if p& (ti ) + (r + d )q i (t ) > rp (ti ) .
The second order condition is simply &p& (t ) < rp& (t ) + d (r + d )q i (t ) , if t (q i ) > ti . Implicitly,
we have assumed that p (t ) is twice differentiable. As we will see, the solution we
obtain is such that this assumption is satisfied.
At first we will look for equilibria that satisfy the second order condition for
all q i . This implies that for any given q i the optimal selling time t (q i ) , if it exists, is
unique. We will see that there are equilibria such that high quality sellers will never
sell. If this is the case then the first order conditions are never satisfied for them
and the optimal selling time does not exist. Then, the first order condition (a) says
that a seller should sell immediately upon entering the market, i.e., at time ti , if the
benefit of using a good rather than selling at time ti , i.e., the use value of the good
(r + d )q i (t ) plus capital gain p& (ti ) , is smaller than the opportunity cost of owing the
good at the entry time, which is rp (ti ) . If, on the other hand, the benefit is larger
than the cost, then the seller should wait and use the good himself until the moment
they are equal to each other and sell at that time, condition (b).
It is clear now that if a seller of quality q i sells at time t then all sellers with
qualities from the range [ q ,q i ], who are in the market at time t, also prefer to sell at
that time t. This allows us to define for any t a marginal seller q (t ) as the seller of
the highest quality at that time:
q (t ) = sup{q i i Î J t } =
i
or
1
r +d
(rp(t ) - p& (t )) ,
p& (t ) = rp (t ) - (r + d ) q (t ) .
(1)
Differentiating (1) yields:
(
)
&p& - rp& - d (r + d )q = rp& - (r + d )q& - rp& - d (r + d )q = -(r + d ) q& + dq ,
which implies that the second order condition requires q& + dq > 0 and, therefore,
edt q (t ) > 0 to be an increasing function.
Now we are able to derive the main equation that must be satisfied along the
equilibrium path. Let us consider an infinitely short time interval (t , t + dt ) . All qualities that entered before and at time t from the interval [ q ,q (t ) ] have already been
traded and all qualities from ( q (t ),q ] are still in the market. Expected utility at time
t + dt from buying goods of quality from the range [q ,q (t ) ] is
q (t )
dt × òq
u (v(x ) - p ( t )) f (x ) dx .
In addition to these, “new” goods, some “old” goods will be also traded. Those
are goods that were in the market, which quality was higher than q (t ) at time t,
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and which quality becomes smaller than q ( t ) + q&( t ) dt at time t + dt . We have to calculate the expected utility of buying these goods.
Let us consider an infinitely short quality range (q (t ) , q (t ) + dq ) at time t. All
(
)
the goods of quality from the interval edt q (t ) , edt (q (t ) + dq ) that entered at time
t - t will fall in that range after depreciation. The measure of such goods is
(
) (
)
(
)
m edt (q + dq ) - m edt q or, in first-order term, dq × edt f edt q . Integrating out the cor-
(
)
f e q over t Î (0, t ) yields the following density of goods of
quality just above q (t ) at time t:
dt
responding density e
t dt
e
0
ò
(
dt
)
f edt q dt =
1
dq
t
0
ò
(
) m (e qdq)- m (q ) .
)(
dt
f edt q d edt q =
The expected utility from buying these goods is
q (t )+q& (t )dt
m edt q - m (q )
u (v(x ) - p (t ))
dx .
ò
dq
q (t )- dq (t )dt
Taking the first order term yields
( )
dt × u (v (q (t )) - p (t ))
(
)
m edt q (t ) - m (q (t )) &
q (t ) + dq (t ) .
dq (t )
(
)
Hence, the unconditional expected utility from buying a good at time t + dt is
(
)
ìïq (t )
üï
m edt q (t ) - m (q (t )) &
q (t ) + dq (t ) ý ,
dt × í ò u (v(x ) - p (t )) f (x ) dx + u (v(q (t )) - p (t ))
dq (t )
ïî q
ïþ
which must be zero for all dt. Hence,
q (t )
m edt q (t ) - m (q (t )) &
q (t ) + dq (t ) = 0 ,
ò u (v(x ) - p(t )) f (x ) dx + u (v(q (t )) - p (t ))
dq (t )
q
(
(
)
(
)
)
or
æ
ö
F (q , p )
ç
÷
- 1÷ ,
ç u (v(q ) - p ) m edt q - m (q )
÷
è
ø
q& = dq ç
( ( )
)
where F (q , p ) º - òq u (v(x ) - p ) f (x ) dx is the expected disutility of buying goods of quaq
lity from the range [ q ,q ] against the price p in the corresponding static Akerlof model. F is differentiable in both arguments and strictly increasing in p function. By definition, F (q S ,q S ) = 0 . Together with (1) we have finally obtained the following system
ì p& = rp - (r + d )q
ïï
æ
ö
F (q , p )
(2)
÷,
íq& = dq ç
1
ç
÷
ï
ç u (v(q ) - p ) m e dt q - m (q )
÷
ïî
ø
è
which describes the evolution of price and marginal quality along an equilibrium path.
( (
)
)
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If we rescale time by the parameter r as y = rt , then the system becomes
ì dp
æ dö
ï dy = p - ç1 + r ÷q
è
ø
ï
æ
ö
ï
ç
÷
ï
ç
÷.
í
F (q , p )
÷
ï dq = d q ç
-1
ï dy r ç
æ æ dy ö
ö ÷
ç
÷
ï
u (v(q ) - p ) ç m ç e r q ÷ - m (q )÷ ÷
ç
ç
÷
ç
÷
ï
ç
÷
ø
è è
ø ø
è
î
One may easily note that the interest rate r and the physical depreciation rate
d can influence the solution only through the ratio dr , which we will call the "relative depreciation rate". This is our first result.
Proposition 1. For any given level of dr , the interest rate r and the physical
depreciation rate d only determine the speed of the evolution along an equilibrium
path. The set of dynamic equilibria only depends on dr .
Having established the way interest rate influences dynamic equilibria in what
follows we take r = 1 without loss of generality assuming that now d is a relative
rate of depreciation.
4. Dynamic Equilibria for Perfectly Durable Goods
We start investigating equilibrium properties for the case of perfectly durable
goods. In view of the fact that d = 0 is a singular point of (2) we have to take a limit
of its second equation when d ® 0 to do this. As
( )
( )
m edt q - m (q )
q edt tf edt q
= lim
= tf (q )
d ®0
d ®0
dq
q
lim
uniformly for all finite t, system (2) could be written as
ì p& = p - q
ï
F (q , p )
.
íq& =
ïî
tf (q ) u (v(q ) - p )
(3)
As
¶F
> 0 we define a function p S (q ) as F (p S (q ),q ) = 0 , which is just a
¶p
buyers’ willingness to pay for a good of quality from the range [q ,q ] in the static
model. Then, we define a (q ) as
(4)
a (q ) º
æ
dp S (q )
= -ç
ç
dq
è
¶F
¶q
¶F
¶p
ö
÷
=
÷
ø F (q ,p S (q ))= 0
Hence, p S (q ) is strictly increasing function.
u (v(q ) - p S (q )) f (q )
(v(x ) - p S (q )) f (x ) dx
q
u¢
q
ò
>0.
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Figure 1 shows the vector field of the system for some fixed t > 0 , which is
dp p&
f (q ) u (v(q ) - p )( p - q )
given by
= & =t
.
dq q
F (q , p )
p
p = v (q
)
p =q
p = p S (q
vq
q
U
qS
)
q
Fig. 1.
As p < v(q ) no dynamic path that is a solution to system (3) can be above the line
p = v(q ) . On the other hand, for any solution to be a dynamic equilibrium it must
satisfy p ³ q , i.e., the surplus of the marginal seller may not be negative. For all
intermediate values of prices where q < p < v(q ) ,
dp
> 0 . Finally, if F ( p,q ) = 0 then
dq
q& = 0 , p& > 0 and tangents at such points are vertical for any t > 0 .
Multiplicity
We will show that there exists a neighborhood U such that for any q 0 Î U
system (3) with initials q (0 ) = q 0 and p (0 ) = p0 = p S (q 0 ) has a solution (q (t ,q 0 ), p (t ,q 0 )) ,
like the solution denoted by the dotted line in Figure 1, with p ³ q . What we will
prove then in Proposition 2 is that for all q 0 Î U there exists T (q 0 ) > 0 such that all
equilibrium conditions are fulfilled, price and marginal quality increase over the time
interval (0, T (q 0 )) and either q (T ,q 0 ) = q or q (T ,q 0 ) = p (T ,q 0 ) > q S . In both cases
q (T ,q 0 ) is the largest quality that can be traded in that equilibrium and we can
extend (q (t ,q 0 ), p (t ,q 0 )) in a periodic way, namely by defining p (t + T ) = p (t ) and
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q (t + T ) = q (t ) , in this way we obtain a dynamic equilibrium where all goods from the
range [ q ,qˆ ] are traded, where qˆ = q (T ) .
Proposition 2. For d = 0 there exists an infinite number of dynamic equilibria
trading all goods from a certain range [ q ,qˆ ], where qˆ Î ( q ,q ].
S
Proposition 2 implies that the repetition of the static equilibrium is the only
stationary equilibrium. If we choose any arbitrary q 0 < q S , the dynamic path will be
such that eventually more than the static equilibrium amount of goods will be sold.
Equilibria Trading All Goods
So far, we have shown that for all distributions we can trade more than the
static equilibrium quality if we allow for trade to take place over time. In this
section we extend this result by showing that all goods can be traded if we relax the
assumption about continuity of p ( t ) .
In the following Proposition 3 we show that there exists an infinite number of
cyclical dynamic equilibria where all goods are traded at time T ,2T ,3T ,K
Proposition 3. For d = 0 there exists an infinite number of dynamic equilibria
(q ( t ), p(t )) such that for some T:
a) p (t + T ) = p (t ) and q (t + T ) = q (t ) ;
b) q (T ) = q ;
c) q (t ) and p (t ) are strictly increasing functions for all t Î (0, T ) except (at
most) at a finite number of points t (k ) kK=1 , where both functions are discontinuous.
{ }
Figure 2 represents a typical equilibrium path q (t ) . Within each cycle n,
where t Î ( nT , (n + 1)T ], the path is piecewise continuous, i.e., q (t ) is a solution of (3)
for every subcycle t Î ( t (k ) , t (k -1) ], k = 1,2,K, K , where K is a finite number defined in
the proof of Proposition 3. The equilibrium construction is such that all sellers of
quality q S earn the same discounted surplus by selling at t = t (k ) , k = 0,1,K, K - 1 .
Hence, they are indifferent between selling at each of these moments.
q (t )
q
qS
t
0=t
(4 )
()
t3
t
(2 )
()
t1
t
(0 )
(3 )
(2 )
T T +t T +t
Fig. 2.
456
¹4
ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
The discontinuities described in Figure 2 are used to build up enough time and
high quality goods to allow the expected quality to improve enough to trade all
goods. One may wonder whether these discontinuities are required for all distributions of quality. Next, we will show that for certain distributions we can construct
infinitely many equilibrium paths with q (t ) and p (t ) being continuous and strictly
increasing over the whole cycle (0, T ) . In the proof of Proposition 2 we have defined
a = a (q S ) . We will show that this parameter a plays a crucial role in analyzing when
continuous price equilibria exist. First, it must be that 0 < a < 1 . To this end, consider
the surplus of the marginal seller in the static model, denoted by s (S ) , as a function
of q : s (S ) (q ) º p(q ) - q = p S (q ) - q , and, then
ds (S ) (q S )
= a - 1 . Suppose then that a > 1 .
dq
This would imply that s (S ) (q ) > 0 in some right neighborhood of q S . But this
contradicts the assumption that q S is the highest static equilibrium quality. Hence,
generically, a < 1 . Lastly, a (q ) > 0 under assumptions 1, 2 and 3. The case where
a = 1 is a non-generic case.
It turns out that the value
1- a
a
determines the qualitative behavior of (q , p ) in
the neighborhood of (q S ,q S ) . Functions x̂(t ) and ŷ (t ) that are the solutions of the
corresponding linearized system behave quite differently depending on whether
is smaller or larger than 1, i.e., whether a is larger or smaller than
1
2
1- a
a
. Figure 3
shows the solution (xˆ, yˆ ) as a parametric function yˆ (xˆ ) with the parameter t, for two
different values of a, a = 0.1 and a = 0.6 . One can see that in the former case yˆ (xˆ )
oscillates around the origin so that the second order condition (q& > 0 ) is not satisfied.
In case a = 0.6 x̂(t ) and ŷ (t ) are increasing functions so that in the neighborhood of
the static equilibrium quality, prices and marginal qualities are increasing functions
as well.
Proposition 4 constructs for quality distributions with a > 12 equilibria trading
all goods where price and marginal quality are continuous in every cycle.
Proposition 4. For d = 0 and a > 12 there exists an infinite number of cyclical
dynamic equilibria (q (t ) , p(t )) such that:
a) p (t + T ) = p (t ) and q (t + T ) = q (t ) ;
b) q (T ) = q ;
c) q (t ) and p (t ) are strictly increasing and continuous functions on (0, T ) .
The result obtained in Proposition 4 says that in case a >
1
2
we can choose q 0
sufficiently close to q S such that we do not need to build more than one subcycle in
order to build up enough time and high quality goods to allow the expected quality
to improve enough to trade all goods. Basically, the condition a > 12 says that in a
2002
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ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
neighborhood of q S there is a sufficient mass of goods so that at the moment when
the marginal quality becomes larger then q S , the marginal seller is able to make a
positive surplus.
a=0.1
1,2
a=0.6
Y
1
0,8
0,6
0,4
0,2
X
0
-1,2
-1
-0,8
-0,6
-0,4
-0,2
-0,2
0
0,2
0,4
-0,4
-0,6
-0,8
Fig. 3.
5. Dynamic Equilibria for Imperfectly Durable Goods
Now we turn back to analyze system (2) for the case where the good under
consideration is not durable, i.e., d > 0 . We start our analysis by considering two
extreme cases, when d is very large and when d is close to zero. Then we will
consider intermediate values of d. As the expressions to be discussed are relatively
difficult to interpret we use the linear model, where v(q ) = vq , u (m ) = m and f (q ) = 1 ,
for illustration purposes. In the linear model 1 < v < 2 and m (q ) = q - q .
High Relative Depreciation Rate
Suppose first that d > max
q
( ( ) ) º d . We will show that in this case, we get
v q -q
q
back a type of traditional "Akerlof-result": after some moment in time no new goods
that come to the market will be traded. The argument can be sketched as follows. In
any continuous equilibrium path (q (t ), p (t )) price decreases over time as
p& = p - (1 + d )q £ v(q ) - (1 + d )q < 0 . This price dynamics leads to the following: either
at a certain moment t ¢ price p (t ¢) becomes smaller than q (t ¢) , i.e., the marginal seller
earns a negative surplus p (t ¢ ) - q (t ¢) and the path (q (t ) , p(t )) cannot be a dynamic
equilibrium, or price p ( t ¢) becomes equal to v( q ) . In the latter case the marginal
458
ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
¹4
quality at that moment q (t¢) must be equal to q , in other words, only the lowest
quality q is traded at that time and there is no uncertainty about quality. Indeed, if
it had been q (t ¢) > q , then some higher quality goods would have been traded and
the price must have been higher than v(q ) .
From that moment on the price p (t ) will always be strictly smaller than v( q ) ,
hence, the marginal quality q (t ) will always be strictly smaller than q . Therefore,
the second dynamic equation of (2) is not valid any more. This equation should be
replaced by a simple dependence p = v(q ) as q £ q for all t > t ¢ and the only quality
that will be traded at time t is q (t ) , hence the price must be equal to v(q ) . The first
ì p& = p - (1 + d )q
, which
equation of (2) is still valid and we get the following system í
î p = v(q )
can be rewritten as
(5)
q& =
v(q ) - (1 + d )q
.
v¢(q )
Whatever solution with initials q (t ¢) = q the above equation has it is clear that
q < q for all t > t ¢ , i.e., no new goods that just entered the market will be traded.
One may wonder what happens if we allow q (t ) to be discontinuous. It turns
out that the results remain the same if we require that trade takes place at every
moment in time. Indeed, suppose that at a certain time moment t ¢ the marginal
quality is discontinuous. Then, the price at that time must be either continuous,
hence the first equation of (2) is valid, or the price has a strictly negative increment.
In the latter case the first equation of (2) will be working just after time t ¢ and,
again, p& = p - (1 + d )q £ v(q ) - (1 + d )q < 0 . Therefore, in both cases the above analysis
leads to no new goods trading after a certain time. The reason why price may not
have a positive increment at t ¢ is that in this case all the sellers who sell just before
t ¢ would wait the instant price increase and there would be no trade just before t ¢ .
The above argument is summarized in the following proposition.
Proposition 5. If d > d then there is no trade of new goods after a certain
time moment along any dynamic equilibrium path where trade takes place at every
moment.
For the linear model the expression for d simplifies to d = v - 1 and equation (5)
d - (v - 1)
takes the form q& = q . The solution in this case exponentially converges to
v
zero, which implies that the marginal quality traded becomes almost worthless.
Hence, when time passes, almost no gains from trade are realized as in the pure
adverse selection result in the static model analyzed by Akerlof (1970).
Low Relative Depreciation Rate
For the case of a relatively small value of d, we rewrite system (2) as
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459
ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
ì p& = p - q - dq
ï&
F (q , p )
F (q , p )
ïq =
- dq =
- dq ,
í
x1dt
m edt q - m (q )
u
(
v
(
q
)
p
)
tf
e
q
u (v(q ) - p )
ï
ïî
dq
( )
(
)
where x1 Î (0.1) . Then,
F (q , p )
(
u (v(q ) - p ) tf ex1dt q
)
=
æ
qgxt ex1x 2dt
ç
1
d
ç
u (v(q ) - p )tf (q ) çè
f exdt q
F (q , p )
(
)
ö
÷
÷÷ ,
ø
where
g=
(
)
f ex1dt q - f (q )
(
)
q ex1dt - 1
(
Î - M f ,M f
)
for all q < q - e , and x 2 Î (0.1) . Hence, system (2) can be written as
ì p& = p - q - dq
ï
æ
ö,
F (q , p )
AF (q , p )
íq& =
- d çç
- q ÷÷
ï
(
(
)
)
(
)
(
(
)
)
(
)
u
v
q
p
tf
q
u
v
q
p
tf
q
è
ø
î
where A is bounded for all q < q - e . Hence, any solution of (2), being written as
(q (t ,q 0 , d ), p(t ,q 0 , d )) , uniformly converges to the solution (q (t ,q 0 ), p(t ,q 0 )) of (3) for all
q 0 and all t such that q (t ,q 0 ) < q - e when d converges to zero. Now, let us define
~
~
~
t (e ) as q (t (e ),q 0 , d ) = q - e . It follows that lim q&(t ,q 0 , d ) = +¥ for t ³ t (e ) . Hence,
(
e ®0
)
~
$T d > t such that q T d ,q 0 , d = q . We can state this result as the following proposition.
Proposition 6. If a > 12 , then there exists d > 0 such that for all d Î [ 0, d )
there exist an infinite number of cyclical dynamic equilibria (q (t , d ), p (t , d )) such that
for some T d > 0 :
( )
b) q (T ) = q ;
(
)
a) p t + T d = p(t ) and q t + T d = q (t ) ;
d
(
)
c) q (t , d ) and p (t , d ) are continuous functions on t Î 0, T d .
Let us consider an example, where q = 10 , q = 20 , f (q ) = e0.1(q -q ) , v(q ) = 1.2q ,
u (m ) = m and d = 0.01 . Figure 4 shows that all goods are sold in finite time.
Proposition 6 generalizes this example and argues that we can extend the conclusion
of Proposition 4 to the case where d is small enough.
460
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ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
Although the result obtained in Proposition 6 is proved for the case a >
only, we think that it is valid for any value of a. As for 0 < a £
1
2
1
2
without
depreciation the equilibrium path is piece-wise continuous the proof would be much
more technical without adding new insights.
22
p
21
20
19
18
p = p S (q
17
)
16
q
15
15
16
17
18
19
Fig. 4. An example with a unique unstable steady-state
q = 20 , f (q ) = e
0.1(q -q )
20
( q = 10 ,
, v(q ) = 1.2q , u (m ) = m , d = 0.01 ).
Intermediate Range of Relative Depreciation Rate
Having established dynamic equilibrium properties for two extreme cases,
when the good is almost perfectly durable or when it depreciates quickly, we will
analyze the properties of the equilibrium path at intermediate values of the relative
depreciation rate. Having initially been time-dependent the system becomes autonomous for large t when q edt ³ q , which is guaranteed by the second order condition q& + dq > 0 , and, therefore, m q edt = m q . We consider the corresponding auto-
( ) ()
nomous system:
(6)
ì p& = rp - (r + d )q
ïï
æ
ö
F (q , p )
÷.
íq& = dq ç
1
çç
÷÷
ï
è u (v(q ) - p ) m q - m (q )
ø
îï
(()
)
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461
ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
(
)
We will show that there exists a steady state q * , p * , not necessarily unique,
(
such that (q (t ) , p(t )) = q , p
*
*
)
is a solution of system (6) for all t ³
1
d
ln
q (t )edt ³ q . Indeed, solving for q&(t ) = p& (t ) = 0 yields:
q
q*
, when
p = (1 + d )q
ì
.
í
î F (q , (1 + d )q ) = u (v(q ) - (1 + d )q ) m q - m (q )
( ()
(7)
Suppose that d <
v (q )-q
q
)
(()
(
)
(( )
and, on the other hand, F q , (1 + d )q > 0 = u v q - (1 + d ) q
)( m (q ) - m (q )), there exists at
) that solves (7). Hence, there is at least one steady state.
For the linear model F (q , p ) = (q - q ) ( p - 12 v(q + q )) and system (7) simplifies to
*
(
)
º d £ d . As F (q , (1 + d )q ) = 0 < u (v(q ) - (1 + d )q ) m q - m (q )
least one point q Î q ,q
p = (1 + d )q
ì
.
í
1
î(q - q ) (1 + d )q - 2 v(q + q ) = (vq - (1 + d )q ) q - q
(
(
)
)
Having been considered as a function of q the left-hand side of the second
equation is a quadratic function decreasing at q where it equals to zero and
increases at q where it is positive. Then, the right-hand side is a quadratic function
as well, which has zeros at 0 and q and is positive between them. Hence, the
system has two solutions, the first one between 0 and q , which is not feasible, and
the second, which is being searched for, between q and q . So, for the linear model
the steady state is unique.
The natural question about the local stability of the singular point q * , p * for
the general model can be resolved by taking a linear analysis of (6) in the neighborhood of q * , p * . The corresponding linear system is:
(
(
)
ì y& = y - (1 + d ) x
ï
dq *
ï x& =
´ - xu ¢ v q * - p * m q - m q * v¢ q * +
ï
u v q * - p* m q - m q *
,
í
ï
ï
*
*
*
* öü
æ q*
ï + y çè òq u ¢ v(x ) - p f ( x ) dx + u ¢ v q - p m q - m q ÷øý
þ
î
(( )
{ (( )
)( ( ) ( ))
(
)
(( )
(
)( ( ) ( )) ( )
)( ( ) ( ))
)
where p = p* + y and q = q * + x . q * , p * is stable if
1+
(( )
*
u vq - p
- dq *
*
) (m (q ) - m (q ))
*
(( )
)( ( ) ( )) ( )
u ¢ v q * - p * m q - m q * v¢ q * < 0 ,
)
462
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ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
which can be stated as d >
(( )
)
u ¢(v (q ) - (1 + d )q )v¢(q ) q
u v q * - (1 + d )q *
*
*
*
*
. The latter inequality cannot
be solved at the current level of generality. For the linear model, the inequality
v -1
simplifies to d >
ºd*.
v +1
For the general model we can show by means of examples the richness of
qualitatively new phenomena that may emerge along possible equilibrium paths. The
examples are presented in Figure 4 and Figure 5, where a cross denotes a steady
state. Figure 4 shows a case where a steady state exists, but is unstable. An
equilibrium path is depicted where all goods are eventually traded, even new goods
of relatively high quality. Figure 5 sows that a stable steady state can be either
below the static equilibrium as in the left graph, or above it as in the right graph.
Unlike the static equilibrium in these stationary equilibria all qualities are eventually
traded in the market. However, owners of qualities q i > q * first wait until their good
has depreciated to q * before selling.
13,1
(q
p
*
, p*
)
20
(q
p
*
, p*
)
19
13
18
12,9
17
16
12,8
p = p S (q )
15
p = p S (q )
12,7
12,6
11,3
11,4
11,5
11,6
11,7
14
q
11,8
11,9
q
13
12
12
13
14
15
16
17
Fig. 5a.
Fig. 5b.
q = 10 , q = 20 , f (q ) = e -(q -q ) , v (q ) = 1.2q , u (m ) = m ,
q = 10 , q = 30 , f (q ) = e -0.01(q -q ) , v (q ) = 1.2q ,
d = 0.1 , q » 11.82 and q S » 13.01
*
u (m ) = m , d = 0.1 , q » 17.32 and q S » 14.94
*
18
Figure 6 finally shows an example of unstable steady state with a periodic
cycle. Here d is "slightly" below d * . Then q * , p * is not stable but there exists a
cycle, i.e., a “closed loop” or a periodical solution of the corresponding autonomous
system. In the long run price as well as marginal quality fluctuate with an
asymptotically constant period.
The three cases discussed above are summarized for the simple case of the
linear model in Figure 7, where the system (2) takes the form
(
)
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463
ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
ì p& = p - (r + d )q
ïï
æ
ö
v
íq& = dq ç (q - q ) p - 2 (q + q ) - 1÷ .
çç
÷÷
ï
dt
ïî
è (vq - p ) e - 1 q
ø
(
p
13,2
(
)
)
cycle
(periodic solution)
13
12,8
12,6
p = p S (q )
12,4
q
12,2
10,5
11
11,5
12
12,5
Fig. 6.
q = 10 , q = 20 , f (q ) = e - (q -q ) , v(q ) = 1.2q , u (m ) = m
Unstable steady state
and
d = 0.09
Stable steady state
No trade of new
goods after some
time in any
equilibrium
Infinite number of equilibria
0
v -1
v -1
v +1
δ
δ
Fig. 7.
The figure shows the different possible equilibrium paths for different values
of the relative depreciation rate. There are four relevant regions. First, for all d
larger than v - 1 trade of new goods stops after some time. Second, if d smaller than
v - 1 bus still larger than
v -1
v +1
there exists a steady state, which attracts the evolu-
tion path, and in the long run price as well as marginal quality are constant. Third,
for all d smaller than vv+-11 the steady state is not stable but, and this is the fourth
region, if d is small enough than there are infinite number of dynamic equilibria
where marginal quality increases upFigure
to the6. highest quality q .
,
,
,
,
and
464
ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
¹4
Conclusions
In this paper, we have studied the role the interest rate and the physical
depreciation rate play in dynamic competitive equilibria under adverse selection.
Without physical depreciation, the infinite repetition of the static (Akerlof) equilibrium is one of the equilibria in the dynamic model. There are, however, infinitely
many other equilibria where all goods are sold within finite time after entering the
market. These results change when physical depreciation is taken into account.
When the depreciation rate is small, repetition of the static equilibrium stops being a
dynamic equilibrium, but there are still infinitely many other equilibria where all
goods are sold. When the depreciation rate is relatively high, all equilibria exhibit no
trade of new goods after some finite moment in time. At intermediate values of the
depreciation rate, new stationary equilibria may emerge where all goods are eventually traded. In this type of equilibrium, owners of high quality goods sell only after
the good has depreciated enough. All these results are independent of the value of
the interest rate.
*
*
*
REFERENCES
1.
Abramowitz M. and Stegun I. A. Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables. New York: John Wiley & Sons, 1972.
2.
Akerlof G. The Market for Lemons: Qualitative Uncertainty and the Market
Mechanism // Quarterly Journal of Economics. 1970. ¹ 84. Ð. 488–500.
3.
Inderst R. and Muller H. Competitive Search Markets for Durable Goods
University of Mannheim, 2000.
4.
Janssen M. and Roy S. Trading a Durable Good in a Walrasian Market with
Asymmetric Information // International Economic Review (forthcoming), 1999a.
5.
Janssen M. and Roy S. On the Nature of the Lemons Problem in Durable Goods.
Markets // Florida International University Working Paper, 1999b.
6.
Janssen M. and Karamychev V. Cycles and Multiple Equilibria in the Market for
Durable Lemons // Tinbergen Institute Discussion Paper TI 2000-025/1, 2000.
7.
Taylor C. Time-on-the-Market as a Signal of Quality // Review of Economic
Studies. 1999. ¹ 66. Ð. 555–578.
8.
Wilson C. Equilibrium and Adverse Selection // American Economic Review.
1979. ¹ 69. Ð. 313–317.
9.
Wilson C. The Nature of Equilibrium in Markets with Adverse Selection // Bell
Journal of Economics. 1980. ¹ 11. Ð. 108–130.
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465
ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
Appendix
Proof of Proposition 2
Under assumptions 1, 2 and 3 for any t0 > 0 and (q 0 , p0 ) such that v(q 0 ) ¹ p0
system (3) with initial conditions q (t0 ) = q 0 and p (t0 ) = p0 has a unique solution
(q (t ,q , p , t ), p(t ,q , p , t )) , which is continuous w.r.t. q and p . Considering q& as a
0
0
0
0
0
0
0
0
F (q , p )
function of q , p and t, i.e., q& = Q(q , p, t ) º
, allows us to write:
tf (q ) u (v(q ) - p )
lim(tQ(q , p, t )) = lim
t ®0
t ®0
F (q , p )
F (q (0 ), p (0 ))
=
<¥.
f (q ) u (v(q ) - p ) f (q (0 )) u (v(q (0 )) - p (0 ))
Hence, system (3) has a solution even for t0 = 0 , but not necessarily unique. The
uniqueness is guaranteed by the fact that tQ(q , p, t ) º/ q + const . Finally, that solution
is differentiable at t = 0 as long as lim(tQ(q , p, t )) = 0 , i.e., p0 = p S (q 0 ) , and we will
t ®0
denote it as (q (t ,q 0 ), p (t ,q 0 )) º (q (t ,q 0 , p S (q 0 ),0 ), p (t ,q 0 , p S (q 0 ),0 )) . Indefiniteness of q&(0,q 0 )
is resolved by continuity:
F (q , p )
1
F (q , p )
=
.
lim
t ®0 tf (q ) u (v (q ) - p )
f (q 0 ) u (v(q 0 ) - p0 ) t ®0
t
q&(0,q 0 ) = limq&(t ,q 0 ) = lim
t ®0
Taking the latter limit explicitly yields:
F (q , p )
d
¶F
= lim F (q , p ) =
(q 0 , p0 ) q&(0,q 0 ) + ¶F (q 0 , p0 ) p& (0,q 0 ) =
t ®0
t ®0 dt
t
¶q
¶p
lim
= - f (q 0 ) u (v(q 0 ) - p0 ) q&(0,q 0 ) + p& (0,q 0 )òq 0 u ¢(v( x ) - p0 ) f (x )dx.
q
Hence, q&(0,q 0 ) =
1
2 a (q 0 )
p& (0,q 0 ) , where a (q 0 ) is defined in (4) and, therefore,
p&
æ dp ö
æ dp ö
ç
÷ = &0 = 2a(q 0 ) > a (q 0 ) = ç S ÷ .
q
d
q
è
øt = 0
è dq ø t = 0
0
This implies that for small t > 0 p (t ,q 0 ) > p S (q (t ,q 0 )) as long as p& 0 = p S (q 0 ) - q 0 > 0 .
(
Now we define q 0 as q 0 = inf { q ¢ "q Î (q ¢,q S ) : p S (q ) > q } and Û as Uˆ = q 0 ,q S
)
such that for all q 0 Î Û p S (q 0 ) > q 0 .
Finally, we will show that there exists a neighborhood U Ì Uˆ such that for
any solution (q (t ,q 0 ), p (t ,q 0 )) of (3), where q 0 Î U , there exists a time T (q 0 ) > 0 such
466
ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
(
¹4
)
that for all t Î (0, T ) p > q and ( p(T ) - q (T )) q - q (T ) = 0 , i.e., either all goods are sold
or the marginal surplus is zero at time T. If this were not the case then there would
have been lim p = lim q = q ¢ as p and q are increasing and bounded: q < q ,
()
t ®¥
t ®¥
p < v (q ) £ v q . But then the equation q& =
F (q , p )
for large t becomes
tf (q ) u (v(q ) - p )
ö
F (q ¢,q ¢)
1æ
+ e (t )÷÷ ,
¢
¢
¢
t è f (q ) u (v(q ) - q )
ø
q& = çç
F (q ¢,q ¢)
ln t + const for sufficiently
2 f (q ¢ ) u (v(q ¢) - q ¢ )
large t. Hence, lim q (t ) = ¥ unless F (q ¢,q ¢) = 0 , i.e., q ¢ = q S .
where lim e (t ) = 0 , and, therefore, q (t ) >
t ®¥
t ®¥
In order to rule out the possibility that q ¢ = q S (and, hence, that q (t ,q 0 ) and
p(t ,q 0 ) converge to q ¢ < q ) we rewrite system (3) as follows
ì p& = ( p - q S ) - (q - q S )
ïï
¢
æ
ö
íq& = 1 ç ( p - q ) - a (q - q ) + æç (q - q S ) ö÷ B(q , p ) æç (q - q S ) ö÷ ÷ ,
S
S
ç ( p - q )÷ ÷÷
ç ( p - q )÷
ï
at çç
S ø
S ø
è
è
è
ø
îï
where a = a (q S ) and B(q , p ) < ¥ uniformly in a certain neighborhood U Ì Uˆ of q S .
Thus, for all q 0 Î U the solution (q (t ,q 0 ), p (t ,q 0 )) can be written as
ìq (t ,q 0 ) = q S + (q S - q 0 ) xˆ (t ) + o(q S - q 0 )
,
í
î p (t ,q 0 ) = q S + (q S - q 0 ) yˆ (t ) + o(q S - q 0 )
where (xˆ (t ) , yˆ (t )) solves the corresponding linearized system
ìï y& = y - x
í
1
ïî x& = at ( y - ax )
(A.1)
y (t )
with initials x(0 ) = -1 , y (0 ) = - a . Defining k (t ) = x (t ) allows us to rewrite (A.1) as
k& = - 1 - k - k a - k . As q < q , p < q and p > p (q ) for all t Î (0, ¥ ) then x < 0 , y < 0
(
at
)
S
and y > ax . Hence, k Î (0, a )
S
S
and k& < - 1-a
< 0 for sufficiently large t. Therefore,
2
lim k (t ) = -¥ , which contradicts k Î (0, a ) .
t ®¥
Hence, for any q 0 Î U
$T (q 0 ) > 0 such that either q (T ,q 0 ) = q
or
q (T ,q 0 ) = p(T ,q 0 ) . In both cases we extend (q , p ) in a periodic way, namely
2002
ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
467
p (t + T ) = p (t ) and q (t + T ) = q (t ) . In order to show that q (T ) > q S when q (T ) = p (T ) < q
let us consider two cases.
a) q (T ) = q S . This contradicts with the uniqueness of the solution with initials
q (T ) = q S . Indeed, we always have a static solution q (t ,q S ) = p(t ,q S ) = q S and we have
found another, namely (q (t ,q 0 ), p (t ,q 0 )) , such that q (T ,q 0 ) = p (T ,q 0 ) = q S .
b) q (T ) < q S . This implies that p S (q (T )) > q (T ) , which can never happen as for
small t p S (q (t )) < q (t ) and no solution may cross the curve p = p S (q ) from above at
q < qS .
Hence, q (T ) > q S .
As we have obtained a discontinuous function p (t ) , we lost the sufficiency of
the first and second order conditions. So, we must check the optimality of the
stipulated sellers' behavior directly.
Let us take a seller i with quality q i and entry time ti Î [nT , (n + 1) T ) , where n
is the entry cycle's number. If q i > q ( T ) then he will never sell, which is clearly
optimal for him. If, on the other hand, q i < q (T ) then there are two possibilities.
a) q i > q (ti ) . In this case he maximizes his surplus by selling in the current
cycle n at time t i (q i ) , where q (t i ) = q i , and getting si (t i ) = s (t i ) , see Figure A.1 (a),
and it follows that t i > ti . Indeed, within a cycle the first and second order conditions
still work so there is a unique optimal selling time t i . If he, however, had been
waiting for the next cycle (n + 1) he would have chosen time t i¢ (q i ) = t i + T to sell,
where q (t i¢ ) = q i , and got si (t i¢ ) = s (t i¢ ) . But p (t i ) = p (t i + T ) = p (t i¢ ) and the seller i will
certainly choose the earlier time t i .
b) q i < q (ti ) . In this case let us first investigate the marginal surplus function
ds s&
s(t ) º e -t ( p - q ) . Differentiating yields s& = - e -t q& and, finally,
= = - e-t .
dq q&
Although the above expression has been obtained only for t Î [ 0, T ] it holds for any
t > 0 . To see this, suppose t Î ( nT , (n + 1) T ]. It then follows that
(A.2)
ds
(t ) = - e - nT ds (t - nT ) = - e- nT e - (t - nT ) = - e -t for t > 0 .
dq
dq
Hence, s (t ) is a positive, decreasing and convex function on (nT , (n + 1) T ) . These
properties allow us to validate the maximum principle across different continuous
segments of an equilibrium path.
Now let us define t i (q i ) such that τi Î ( nT , (n + 1) T ] and q (t i ) = q i , in this case
t i < ti , see Figure A.1(b). If seller i sells immediately after the entry, i.e., at time ti ,
he gets si (ti ) = e -t i ( p (ti ) - q i ) = e -t i ( p (ti ) - q (ti )) + e -t i (q (ti ) - q i ) = s (ti ) + e -t i (q (ti ) - q i ) , while if
468
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ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
he waits until the next cycle (n + 1) he will choose time t i¢ (q i ) = t i + T to sell, where
q (t i¢ ) = q i , and, therefore, his surplus becomes
¢
si (t i¢ ) = e -t i ( p (t i¢ ) - q i ) = e -t i ( p (t i¢ ) - q (t i¢ )) = s(t i¢ ) .
s(t )
Case a)
Case b)
s(t )
si (ti )
s(ti )
si (t i¢)
si (t i )
si (t i¢)
q0 q (ti )
qi
q0
q (t )
qˆ
q (t i )
q (t i¢)
qi q (t * ) q (ti )
i
q (t i )
q (t i¢)
qˆ
q (t )
Fig. A.1.
In order to show that si (ti ) > si (t i )¢ for all q i < q (ti ) let us consider D(q i ) º si (ti ) - si (t i¢ )
as a function of q i and apply the mean-value theorem:
( )
s (t i¢ ) = s (ti + T ) + (q (t i¢ ) - q (ti + T )) ddsq t i* = e -T s (ti ) + (q (ti ) - q i )e -t i
*
for some t i* Î (t i¢, ti + T ) . Then
D(q i ) = s (ti ) + e -t i (q (ti ) - q i ) - e -T s(ti ) - (q (ti ) - q i ) e -t i =
*
(
)
*
= s (ti ) 1 - e -T + (q (ti ) - q i ) æç e - t i - e -t i ö÷ > 0.
è
ø
Therefore, we have shown that for any q 0 Î U
(q ( t ,q 0 ) , p ( t ,q 0 ))
constitutes a
dynamic equilibrium trading all goods from the range [q ,qˆ ], where qˆ = q (T ) > q S .
In order to prove the following propositions we need the following lemma.
Lemma 1. For any numbers q 0 Î [ q S ,q ) and p0 Î (q 0 , v(q 0 )) there exists tˆ > 0 ,
depending on p0 - q 0 , such that for all t0 > tˆ system (3) has a unique solution with
initials q (t0 ) = q 0 and p (t0 ) = p0 . Moreover, there exists a finite time T > tˆ such that
q (T ) = q .
Proof. Under assumptions 1, 2 and 3 for any t0 system (3) has a unique
solution passing through (q 0 , p0 ) , where q 0 ³ q S and p0 Î (q 0 , v(q 0 )) . All we need to
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469
ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
show then is the existence of T if t0 is taken to be sufficiently large. We define
{
}
a º min p0 - q 0 , 12 e v and
(
( )) .
2F q S , v q
tˆ º
(A.3)
e f e ue va
As a is a function of p0 - q 0 so is tˆ . Now let us consider the solution mentioned
above when t0 > tˆ and suppose that p (t ) - q (t ) = a for some t ³ t0 . Then
&
d
( p - q ) = q& d ( p - q ) = q&æç dp - 1ö÷ = q&æç p& - 1ö÷ .
dt
dq
è dq
ø
èq
ø
Using (3) yields
d(p -q )
dt
æ e f e u (v(q ) - p ) a
ö
æ f (q ) u (v(q ) - p ) ( p - q ) ö
= q&ç t
- 1÷ > q&ç t
- 1÷ >
ç
÷
ç
÷
F (q , p )
F qS ,v q
è
ø
è
ø
( ( ))
æ e f e u (q + e v - p ) a
ö
æ e f e u (e v - a ) a
ö
> q&ç t
- 1÷ = q&ç t
- 1÷ >
ç
÷
ç
÷
F qS ,v q
F qS ,v q
è
ø
è
ø
( ( ))
( ( ))
æ e f e u e va
ö
æ e f e u e va
ö
> q&ç t
- 1÷ > q&ç tˆ
- 1÷ = 0.
ç 2F q , v q
÷
ç 2F q , v q
÷
S
S
è
ø
è
ø
( ( ))
Thus
( ( ))
p (t ) - q (t ) > a > 0 for all t ³ t0 . Now it becomes clear (see proof of
Proposition 2) that for some T we must have q (T ) = q .
Proof of Proposition 3
We first define functions (q (t ) , p(t )) for all t Î [ 0, t0 ] and some t0 such that the
condition of Lemma 1 is satisfied, i.e., t0 > tˆ(a ) , where a = p (t0 ) - q (t0 ) , and q (t0 ) = q S .
Then we show that (q (t ) , p(t )) actually is an equilibrium path. Lastly, Lemma 1 says
that all goods are traded by a certain time T.
We have already shown in proof of Proposition 2 that for any q 0 Î U and
p0 = p S (q 0 ) system (3) has a solution (q (t , q 0 ) , p (t , q 0 )) and q (T ) > q S for some T. As
q&(t ,q 0 ) > 0 , it follows that for all b Î [ q 0 ,q (T ) ] there exists an inverse t (b ,q 0 ) such
that q (t (b ,q 0 ) ,q 0 ) = b . Function t (b ,q 0 ) is continuously differentiable on b Î [ q 0 ,q (T ) ]
and continuous w.r.t. q 0 . Hence, p(t (b ,q 0 ) ,q 0 ) is continuous w.r.t. q 0 as well. We
define t (q 0 ) by t (q 0 ) = t (q S ,q 0 ) , so that q (t (q 0 ) ,q 0 ) = q S for all q 0 Î U . Note that
p (t (q 0 ),q 0 ) > q (t (q 0 ),q 0 ) = q S .
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ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
Now we will solve the linearized system (A.1). First, it can be rewritten as
Kummer's equation (see Abramowitz and Stegun, 1972, pp. 504–515):
(
)
t&x& + (2 - t )x& - x - 1-aa = 0
with initials x(0 ) = -1 , x& (0 ) = 12-aa . The unique solution is the negative to the so-called
Kummer's function M (a1 , a2 , t ) , with a1 = - 1-aa and a2 = 2 . Turning back to (A.1) we
get:
(
)
ì
¥
G n - 1-aa
ï xˆ (t ) = - M - 1- a ,2, t = -1 + 1 - a å
tn
a
a n =1 n!(n + 1) !G 1 - 1-aa
ïï
(A.4) í
,
1- a
æ
ö
¥
G
n
1
ï
a
n
1
a
d
a
ç
t ÷
ï yˆ (t ) = -a dt tM - a ,2, t = aç - 1 + a å
2
1- a
÷
n =1 (n!) G 1 ïî
a
è
ø
(
)
( (
(
))
(
)
(
)
)
where G(x ) = ò0 e -t t x -1dt is the Gamma-function, and
¥
(
) = (2 - 1 )(3 - 1 ) ´ K ´ ( n - 1 ) .
1- a
a
a
a
G( 1 - a )
G n - 1-aa
Now we define w as xˆ (w ) = 0 and it follows that w > 0 and yˆ (w ) > 0 . As
ìq (t ,q 0 ) = q S + (q S - q 0 ) xˆ (t ) + o(q S - q 0 )
,
í
î p (t ,q 0 ) = q S + (q S - q 0 ) yˆ (t ) + o(q S - q 0 )
the functions x̂(t ) and ŷ (t ) describe the behavior of the solution (q (t ,q 0 ), p (t ,q 0 )) in
the neighborhood of p = q = q S and it follows that lim t (q 0 ) = w and
q 0 ®q S
lim
q 0 ®q S
p (t (q 0 ),q 0 ) - q S
= yˆ (w ) .
q S -q0
Therefore, there exists a left neighborhood of the static equilibrium quality
U w º q 0w ,q S Î U such that for all q 0 ÎU w t (q 0 ) > 12 w .
(
)
Now we are ready to construct the pair of functions (q (t ) , p(t )) . Let us take
any q 0(1) ÎU w and define t (1) º t q 0(1) , q (1) ( t ) º q t ,q 0(1) and p (1) (t ) º p t ,q 0(1) on
t Î ( 0,t (1) ], s (1) (t ) º e -t p (1) ( t ) - q (1) (t ) . By construction we have t (1) > 12 w and
s (1) t (1) > 0 .
( )
( )
)
(
(
)
(
( )
)
Let us now consider the function r (1) (q 0 ) º p(t (q 0 ),q 0 ) - q S - s (1) t (1) as a function
of q 0 . It is continuous on [q 0(1) ,q S ]. Moreover,
( ) (( )
)
( )
(1)
( )
( )
r (1) q 0(1) = p t q 0(1) ,q 0(1) - q S - s (1) t (1) = q S + et s (1) t (1) - q S - s (1) t (1) > 0 ,
2002
471
ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
and
( )
( )
(1 )
r (1) (q S ) = p(t (q S ),q S ) - q S - s (1) t (1) = q S - q S - et s (1) t (1) < 0 .
(
)
( )
( )
Therefore, $q 0(2 ) Î q 0(1) ,q S
depending on t (q 0 ) such that r (1) q 0(2 ) = 0 , i.e.,
p t q 0(2 ) ,q 0(2 ) = s (1) t (1) . Again, we define t (2 ) º t q 0(2 ) , q (2 ) (t ) º q t ,q 0(2 )
and
(
2)
(
2)
(
2)
(
2)
(
2)
(
2)
p (t ) º p t ,q
on t Î 0,t , s (t ) º s t ,q
and, again, t > 1 w .
(( )
(
)
0
)
( )
]
(
(
0
( )
)
2
{ } such that lim å t ( ) = ¥ .
K ³ 1 to be the smallest number such that å t ( ) > tˆ( p ( ) (t ( ) ) - q ( ) (t ( ) )) .
Repeating this process, we get a sequence
We define
Then,
we
t (k )
k
j =1
t (k )
define
t (k ) = å Kj = k +1t ( j ) ,
as
so
k
j =1
k ®¥
j
1
t (K ) = 0 ,
that
1
j
1
1
t ( K -1) = t ( K ) ,
t ( K - 2 ) = t ( K ) + t (K -1) ,…, and t (0 ) = å Kj=1t ( j ) > tˆ . Finally, we define an equilibrium path
for t Î ( 0, t (0 ) ] as follows: p( t ,q ) = p (k ) t - t (k ) , q (t ,q ) = q (k ) t - t (k ) and, therefore,
(
)
( ) ( ) ) (
0
(
)
0
(
)
-t ( k )
s (t ,q 0 ) = e
s ( k ) t - t (k )
for t Î ( t (k ) , t (k -1) ]. For t > t (0 ) we take the solution
q t ,q (1) t (0 ) , p (1) t (0 ) , t (0 ) , p t ,q (1) t (0 ) , p (1) t (0 ) , t (0 ) of (3) as an equilibrium path.
It can be easily seen that within every interval ( t (k ) , t (k -1) ] a seller chooses the
((
( )
( ) ))
time to trade optimally. In order to check that he behaves optimally even across
those intervals (subcycles) and across cycles, again, like in the proof of Proposition 2,
we use (A.2) and considering a seller i with quality q i and entry time
t Î ( nT + t (k ) , nT + t (k -1) ]. As the arguments are quite similar to the ones given in the
i
proof of Proposition 2 we skip the details here.
Like in Proposition 2 any seller i optimally waits until the first moment after
entry when the marginal quality is larger than or equal to his own quality. Hence,
the pair of functions (q (t ) , p (t )) constructed above satisfies all equilibrium requirements. Then, it follows from Lemma 1 that q (T ) = q for some T. The constructed
equilibrium path is entirely determined by choosing q (1) which is an arbitrary point
0
w
from U . Therefore, we have obtained infinitely many (continuum of) equilibria.
Proof of Proposition 4
If a > 12 , then each term (apart from the constant) in the Tailor expansion
(
)
(A.4) is positive as G 1 - 1-aa > 0 and its radius of convergence is infinity. Hence, x̂
and ŷ are defined by (A.4) for all t Î [ 0, ¥ ),
lim yˆ (t ) = lim xˆ (t ) = lim y&ˆ (t ) = lim x&ˆ (t ) = +¥ ,
t ® +¥
and
t ® +¥
t ® +¥
t ® +¥
472
¹4
ÝÊÎÍÎÌÈ×ÅÑÊÈÉ ÆÓÐÍÀË ÂØÝ
lim
t ®+¥
yˆ
= lim
xˆ t ®+¥
yˆ&
yˆ - xˆ
yˆ& - xˆ&
= lim yˆ -axˆ = lim
= lim
t ®+¥ d yˆ -axˆ
t ®+¥
xˆ& t ®+¥ at
dt at
¥
as yˆ& - axˆ& > yˆ& - xˆ& > yˆ& - 2axˆ& = (1 - a ) å
n=2
This implies that for a >
lim lim
t ® +¥ q 0 ®q S
(
æ
yˆ& - xˆ&
(2a - 1) x&ˆ ö÷ = +¥
= lim at ç1 +
ç
y&ˆ - 2ax&ˆ ÷ø
- xˆ& 1 t ®+¥ è
y&ˆ -ax&ˆ
at
)
G n - 1-aa t n -1
t
(n - 2) !(n + 1) !G(1 - 1-aa )
> 0 for all t > 0 .
1
2
p (t ,q 0 ) - q S
yˆ (t ) + O(q S - q 0 )
yˆ (t )
= lim lim
= lim
= +¥ .
q (t ,q 0 ) - q S t ® +¥ q 0 ®q S xˆ (t ) + O(q S - q 0 ) t ® +¥ xˆ (t )
(
In other words, for any M > 0 $t ¢(M ) such that for all t > t ¢ $U a (t , M ) = q 0a (t , M ),q S
such
that
p (t ,q 0 ) - q S
>M
q (t ,q 0 ) - q S
for
all
q 0 ÎU a .
We
take
)
a = 12 e v ,
()
ìï
ìï a
üï
2 M 2 M u m q üï
M = 1 + max í
, max a (q )ý and t = max ítˆ(a ), t ¢(M ),
ý , where tˆ(a ) is as
e f e ue v (M - 1) ïþ
ïîq - q S q Î[q S ,q ]
ïþ
ïî
defined in (A.3), Lemma 1. For this t there exists a neighborhood U a (t , M ) such that
p (t ,q 0 ) - q S
> M for all q 0 ÎU a .
q (t ,q 0 ) - q S
We will show that $t ¢¢ ³ t
such that p (t ¢¢,q 0 ) ³ q (t ¢¢,q 0 ) + a . Suppose to the
contrary that
for all t ³ t . Then it must be the case that
p(t ,q 0 ) < q (t ,q 0 ) + a
p (t ,q 0 ) - q S
> M for all t > t , otherwise there would have been some t ¢¢¢ > t such that
q (t ,q 0 ) - q S
p (t ¢¢¢,q 0 ) - q S
= M and, therefore, for t = t ¢¢¢ :
q (t ¢¢¢,q 0 ) - q S
d
( )=
p -q S
q -q S
dt
p&
Mq&
q& æ p&
q&
ö
=
ç & -M÷=
q -qS q -qS q -qS è q
ø q -qS
æ f (q ) u (v(q ) - p ) ( p - q )
ö
çç t ¢¢¢
- M ÷÷ >
F (q , p ) - F (q ,p S (q ))
è
ø
æ
ö
æ p -qS q -qS ö
ç
÷
÷
e f e u (v(q ) - p ) çç
÷
÷
q -qS q -qS ø
q& ç
è
>
- M ÷,
çt
¶
F
q - q S ç ( p - p S (q )) ¶p (q ,p S (q ) + x ( p - p S (q )))
÷
ç
÷
q
q
S
è
ø
where x Î (0,1) and
q > p -a ,
F (q , p S (q )) = 0 by definition. Then, as we have supposed
2002
473
ÂÎÏÐÎÑÛ ÒÅÎÐÈÈ
( )>
d
p -q S
q -q S
dt
d
( )>
p -q S
q -q S
dt
>
(
æ
ç
ö
÷
e f e u (e v + q - p ) (M - 1)
q& ç
-M÷,
t
ç
÷
p
p
(
q
)
q
q -qS
S
u ¢(v( x ) - p S (q ) - x ( p - p S (q ))) f ( x ) dx
ç
÷
ò
q
q -qS
è
ø
q&
q -qS
æ
ç
çt
ç
è
Mq&
q -qS
æ
çt
ç
è
ö
÷
M÷>
p S (q )-q S q
p -q S
÷
- q -q
òq M u f (x ) dx
q -q S
S
ø
&
e f e u e v (M - 1)
ö
Mq æç e f e u e v (M - 1) ö÷
- 1 ³ 0,
- 1÷÷ >
t
2
ç
÷
2 M (M - a (q ¢)) M u m q
ø q - q S è 2M M u m q
ø
(
e f e u (e v - a ) (M - 1)
)
()
()
)
for some q ¢ Î q S ,q . But then p - q S > M (q - q S ) for all t ³ t that can be rewritten as
a > p - q > (M - 1) (q - q S ) and, further, as q <
lim q £
t ®¥
a
M -1
a
M -1
+ q S . Hence, on one hand,
+ q S < q , on the other hand, p& = p - q > (M - 1) (q - q S ) > 0 and lim p = +¥ ,
t ®¥
that is a contradiction with the assumption that that p < q + a for all t ³ t .
Hence, p (t ¢¢,q 0 ) ³ q (t ¢¢,q 0 ) + a for some t ¢¢ ³ t ³ tˆ(a ) and Lemma 1 applies.
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