The Value and Liquidity Effects of a
Change in Market Conditions
By Paul M. Anglin*
University of Windsor
May 2003
Abstract
Papers studying the liquidity of a market tend to focus on decisions involving the
trade-off between the selling price and the time-till-sale for a given set of market
conditions. This paper characterizes the effect of a change in market conditions on a
single seller where either a “price-increasing” or a “time-decreasing” change in market
conditions can help a seller. Shamelessly stealing from demand theory, I show how the
effect of either type of change on price and on the probability-of-sale can be decomposed
into a “value effect” and a “liquidity effect.”
The analysis focuses on a famously-illiquid market, real estate, but shows how the
same ideas can be applied to markets where other selling mechanisms dominate. Two
adding-up conditions restrict the set of possible predictions regardless of the mechanism.
In this way, I offer a general theory of how markets adjust to changing conditions.
JEL: C70, D40, R31
Keyword: liquidity, market conditions, time-till-sale, probability of sale, hazard
rate, matching models, market frictions, real estate, Slutsky Equation
*
I am grateful for comments from Richard Arnott and Nancy Wallace. They are not responsible
for any errors. Further comments will be welcomed and can be sent to Department of
Economics, University of Windsor, Windsor, ON, Canada N9B 3P4 or to panglin@uwindsor.ca.
The latest version of this paper can be found at http://www.uwindsor.ca/PaulAnglin.
Value and liquidity effects
1
Consider the problem facing someone who has decided to sell their house. The selling
price depends on the type of house, on market conditions and on luck. A seller can affect the
outcome by carefully choosing the list price that is seen by potential house buyers but that choice
also affects the probability of sale and the expected time-till-sale. Researchers are beginning to
form a consensus on the best way to estimate the determinants of this second process at any point
in time.1 But there has been little analysis of the trade-off per se and no analysis of the more
fundamental question of how individuals or markets react to a change in market conditions.
The trade off between the selling price and the probability of sale is presumed to exist
because the transaction costs associated with buying and selling real estate make it an illiquid
asset. This fact should make a real estate market a good choice to study the effects of transaction
costs except for two puzzles. First, there is a popular misconception about the degree to which
real estate is illiquid. If liquidity is measured by the time taken to raise cash to pay for a random
liquidity shock, then that time should not be measured by the many months it takes to sell a
house. Even in the extreme case of a seller who owns no other liquid asset, the time may be no
more than the few days needed to qualify for a mortgage.2 Similarly, some common definitions
of liquidity appear to be confused when applied to a real estate market. For example, most
houses are sold using a matching process and, with a sequential transaction process, any attempt
to measure the liquidity of a real estate market in terms of a trade off between selling price and
the probability of sale is flawed; there is no direct trade off. More specifically, it is probably true
that a lower list price attracts more potential buyers, but the selling price is fixed only after the
buyer has shown up and only after that buyer and the seller complete their bargaining.3
1
See, for example, Anglin, Rutherford and Springer, 2003; Lu and Yu, 2001; Ong and
Koh, 2000; Forgey, Rutherford and Springer, 1996; Springer, 1996; Kalra and Chan, 1994;
Kluger and Miller, 1990; and Trippi,1977. Few papers specifically focus on liquidity: Forgey,
Rutherford and Springer, 1996; Kluger, and Miller, 1990. Most studies use data sets which span
a few months or a year.
2
I am indebted to Uday Rajan for this insight.
3
One problem with studies of liquidity is that they may require that a measurable
“market” price always exist. In a real estate market, a price may not exist for long periods of
time because no buyer has been identified. Lin and Vandell’s study (2001) of liquidity-adjusted
rates of return showed one way to overcome this obstacle. Anglin (2002) argued that the seller is
Value and liquidity effects
2
A second problem with using a real estate market to study the effects of transaction costs
is that they are difficult to quantify directly. Theoretical links between studies of a real estate
market and the concept of liquidity are usually weak. Some, e.g. Glower, Haurin and
Hendershott (1998), focussed on the effects by focussing on the actions of sellers with different
tastes but, since economists usually place few restrictions on tastes, few general predictions were
possible. In addition, the transaction costs which make the perfectly competitive model
inappropriate have led economic theorists to consider a large number of possible trading
mechanisms, e.g. using search and matching combined with bargaining over prices or using an
auction or posting a price. Each of these mechanisms has characteristics that are not completely
understood. Formally, this paper studies the effects of market conditions but, since market
conditions change more often than transaction costs, it is more important to find features that are
common to many mechanisms.
Insert Figure 1
This paper uses a locus of feasible combinations of the expected sale price and of the
probability of sale as a general framework to identify the effects of changing market conditions
on a single seller. Figure 1 demonstrates that a change in market conditions, i.e. a change in the
locus facing an individual seller, usually has an ambiguous effect. Most observed changes in the
locus combine a “price-altering” change and a “time-altering”change and the main effect of a
price-altering change in market conditions is as the name suggests. But the effect of a timealtering change in market conditions is more subtle because of the pre-existing trade off between
price and time. Decomposing the full effect into a “value effect” and a “liquidity effect” is the
focus of this paper. Some adding-up conditions restrict the range of permissible outcomes and
can be used to test whether observed behaviour satisfies necessary conditions for optimal
decision-making. The following section offers an example showing how the most common
specification of regression equations imply restrictions on the seller’s tastes. I also show how
this general method can be applied to different types of trading mechanisms.
A real estate market offers two important advantages over another market commonly
always present and could be considered as the highest value bidder at any point in time; a sale is
recorded only when someone else beats the seller’s offer.
Value and liquidity effects
3
used to study transaction costs: a labour market. It is true that matching models are commonly
used to study both types of markets. But, whereas most labour market data sets reveal
information on the final wage and the duration of unemployment, the list price in a real estate
market should reveal information during the transaction process. Second, studies of labour
markets indicate that nearly all job offers are accepted (Devine and Keifer, 1991). Studies of
real estate markets indicate that many of the houses offered for sale do not sell (Anglin,
Rutherford and Springer, 2003) and many negotiations between a buyer and a seller do not reach
an agreement (Anglin, 1994). Thus house sellers seem to explore more margins more actively
than workers and these actions should reveal more about the market process.
Many indicators of market conditions in real estate markets have been proposed in
informal and formal studies, including
- the unemployment rate or total employment,
- the level of mortgage interest rates,
- the direction or volatility of interest rates,
- the ratio of listings to sales, also known as inventory,
- the ratio of new listings to sales, to distinguish “buyer’s market” from “seller’s market”,
- the rate of increase in prices,
- the number of sales,
and, perhaps because of a lack of testable hypotheses, trend variables. Some of these indicators
suggest that a change is primarily price-altering (i.e. unemployment rate, interest rate, rate of
increase in prices) while others suggest that a change is primarily time-altering (i.e. volatility of
interest rates, inventory, sales). It is also possible to classify indicators as internal or external.
For example, the aggregate unemployment rate may affect the behaviour and experience of a
single unemployed worker. This kind of internal indicator can be expected to change as a
market adjusts to its steady state but, ultimately, the indicator aggregates the actions of many
individuals in the market. External indicators of market conditions in a labour market could
include interest rates or government spending. Whether these classifications are correct should
be decided by empirical analysis. These classifications are not important to this paper since I
focus on a single seller who must live with the conditions however they may be determined. The
conclusion discusses how the behaviour of individuals can be integrated into an equilibrium
Value and liquidity effects
model.
4
Decomposing the Effect
Notation
As a decision problem involving a large fraction of their net wealth, most house sellers
desire to make an optimal decision. A seller’s tastes are represented by U(pS, 8) where U(.) is
assumed to be increasing in (pS, 8) and is strictly quasi-concave. For a given type of house and
given market conditions, X, the locus of feasible combinations of price-probability is defined by
(pS(X, pL), 8(X, pL))
(1)
where pS represents the expected selling price, 8 represents the probability of sale during a
period of time and pL represents the list price that can be chosen.4 By removing the common
parameter, pL, the locus can be represented as
F(pS, 8, X)= 0.
(2)
A linear approximation to F(.) simplifies the analysis: i.e.
F(pS, 8, X)= -<+ pS- J (1- 8)= 0 or
pS= <+ J (1- 8)
(3)
for some (<, J), which depend on X and J> 0 ensures that the locus is downward sloping. A
change in market conditions corresponds to a change in <, which defines a “price-altering”
change, or to a change in J, which defines a “time-altering” change, or to change in both.
In this model, the choice of list price affects the seller’s utility only if it affects (pS, 8).
Thus, a seller seeks to solve
max(pS, 8) U(pS, 8)
s.t. -<+ pS- J (1- 8)= 0
(4)
with the results 8*(<, J) and pS*(<, J). The assumptions stated above ensure that the second
order conditions are satisfied and that there exists a unique solution. The dual to this primal
problem is
In a continuous time model, with a constant hazard rate, 8, and a constant discount rate
of r, the expected present value of selling a house satisfies U(pS, 8)= pS 8/(8+ r). Similarly, in a
stationary environment, the expected time-till-sale would be 1/8. The fact that market
conditions can change suggests that the environment is not stationary and that the probability of
sale at any point in time may be a more stable measure than the expected time-till-sale.
4
Value and liquidity effects
min(pS, 8 ) pS- J (1- 8)
5
s.t. U(pS, 8)= Z
(5)
with the results 8c(J, Z) and pSc(J, Z). The superscripts in these solutions indicate that the seller
is compensated for any change in market conditions to achieve the same level of utility. Define
the Probability-Adjusted Price, PAP(J, Z), as
PAP(J, Z)= pSc(J, Z)- J (1- 8c(J, Z)).
(6)
In the special case of a frictionless market, 8= 1, PAP equals the expected selling price.
Yavas (1992) offered one of many models which can generate these relationships from
first principles and, with little effort, more general models can be considered. Or, if a seller has
more than the one decision variable presumed in this simple presentation (e.g. Arnold,1999;
Yavas and Yang, 1995) then the relevant locus would be the upper envelope of a set of onevariable loci in price-probability space. If the seller cannot choose, then the effect of any change
can be found mechanically.
Relationships
Optimization creates a relationship between the solutions to the primal and dual
problems. Proposition 1 exploits this relationship.
Proposition 1.
i) PAP(J, Z)= <.
ii) 8c(J, Z)= 8 *(PAP(J, Z), J) and pSc(J, Z)= pS*(PAP(J, Z), J).
iii) M8c/MJ= -(M8 */M<) (1- 8*)+ M8*/MJ.
iv) MpSc/MJ= -(MpS*/M<) (1- 8*)+ MpS*/MJ.
v) J M8c/MJ= MpSc/MJ at an interior solution.
Proof
i) The feasible set is 0= -<+ pS- J(1- 8). Substituting the solution to the dual problem
proves that
0= -<+ pSc- J (1-8c)= -<+ PAP.
(7)
The claim follows immediately.
ii) This claim is proven by substituting part i) into the solutions to the primal problem and
noting that the solutions must be identical since the primal and dual problem are different
methods to the same answer (see, for example, Varian, 1984, ch 3).
iii) and iv): The proof uses the identities proven in ii) and the now-standard method of
Value and liquidity effects
deriving the Slutsky Equation first presented by Cook (1972).
6
v) Since (MpSc/MJ)/(M8c/MJ)= dpS/d8|u is the marginal rate of substitution, it must equal J
at an interior optimum. Q.E.D.
To complete the analogy to the Slutsky Equation, I designate (M8c/MJ, MpSc/MJ) as the
liquidity effects, and ((M8*/M<) (1- 8*), (MpS*/M<) (1- 8*)) as the value effects, of a time-altering
change in market conditions. (M8*/M<, MpS*/M<) are the value effects of a price-altering change
in market conditions. The most obvious example of a price-altering change is a change in house
type: < depends on the characteristics of the house. Even if dJ/dX= 0 and regardless of tastes, a
seller of a more valuable house can charge a higher price or sell in less time or both because their
price-probability locus is higher. A change in interest rates may also represent a price-altering
change in market conditions and evidence on any one should reveal the effect of other pricealtering changes.
Regardless of tastes, feasibility implies
0= -<+ pS*(<, J)- J (1- 8*(<, J))
(8)
and this equality leads to two conditions which are valid for all tastes. First, differentiating
equation (8) with respect to < and J proves that
Mp*/M<+ J M8 */M<= 1
Mp*/MJ+ J M8 */MJ= 1- 8*.
(9)
(10)
Thus,
Corollary 1.
i) MpS*/M<= 1 only if M8*/M<= 0 or there is a corner solution 8*= 1.
ii) M8*/MJ= 0 if and only if MpS*/MJ= 1- 8.
A weakness of Proposition 1 is that it approximates the price-probability locus with a
straight line. Intuitively, the locus should have an asymptote since the distribution of the
willingness-to-pay of buyers is likely to be bounded above. Proposition 2 implies that
interaction between the curvature of the price-probability locus and a change in market
conditions exaggerates differences between different types of sellers.
Proposition 2
Suppose that F8> 0, Fp> 0 and F88 Fp- Fp8 F8> 0 (i.e. the price-probability locus is an
decreasing convex function) and consider an increase in the flow of buyer: i.e. Fk(pS, 8)= F(pS,
Value and liquidity effects
7
k8)= 0 for some k> 1. The magnitude of the time-altering change is lower for sellers charging a
higher list price.
Proof
F(.) implicitly defines the price-probability locus, pS(8). Computation shows that the
assumptions in the proposition imply that dpS/d8< 0 and d2pS/d82< 0. Sellers who offer a higher
list price expect a higher selling price and a lower probability of sale. For given market
conditions, different sellers who choose different points on the locus face different linear
approximations to F(.), where the relevant value of J is the slope of F(.) at that point. Obviously,
a change in the flow of buyers has minimum effect on a seller seeking the highest value bidder
and a maximal effect on a seller seeking to sell at any price.
More formally, using total derivatives, the slope of Fk(.) is
dpS/d8= - Fk8/ Fkp
and, therefore,
d(Fk8/ Fkp)/dk= (k2F88 Fp- k2 Fp8 F8)/(Fp)2> 0
(11)
for k> 1. Q.E.D.
Example and Applications
Analysts who study real estate markets often use a log-linear model to explain the selling
price and a proportional hazards model to explain the time-till-sale:
E(log(pS))= X$
(14)
8= exp(X().
(15)
where X represent the explanatory regressors, and $ and ( represent parameters to be estimated.
This simple two-equation system makes deriving a seller’s implied indifference curves easy.
To focus on (<, J), and since (<, J) changes only if X changes, it helps to restate these
equations as
E(log(pS))= A<+ BJ
(16)
log(8)= C<+ DJ.
(17)
or, expressing (<, J) as functions of (pS, 8), as
<I= B1 E(log(pS))+ B2 log(8)
(19)
JI= B3 E(log(pS))+ B4 log(8)
(20)
Value and liquidity effects
where (<I, JI) represents the values which produce (pS, 8) as the optimum5 and
8
B3= -C/(A D- B C)
(21)
B4= A/(A D- B C).
(22)
Proposition 1 shows that a seller’s indifference curve in (pS, 8) space satisfies
dpS/d8 |u = -JI(pS, 8)
(18)
Thus, solving for a seller’s indifference curve requires solving a differential equation (Varian,
1984, Sec. 3.10). If B3= C= 0, then combining equation (18) and (20) reveals that the differential
equation is
dpS= -B4 log(8) d8
(23)
and the seller’s indifference curve must satisfy
pS= L- B4 (8 log(8) - 8)
(24)
for a constant L to be determined by an initial condition. If B4= A= 0,6 then the relevant
differential equation is
dpS= -B3 log(pS) d 8
(25)
and the associated indifference curve satisfies
-B3 8= L+ log(log(pS))+ log(pS)+ (log(pS))2/(2 2!)+ . . .
(26)
Different tastes produce different expansion paths in (pS, 8) space and, if variation in A,
B, C, D spans an incomplete range of tastes, equations (14) and (15) may impose unintended
assumptions. It may be better to start a flexible functional form, such as an Almost Ideal
Demand System or a Generalized Leontief, which produce regression equations that are easy to
implement and represent specific tastes.
One common use of equations (14) and (15) is to study whether different types of sellers
choose different list prices and whether different list prices affect the probability of sale. For
example, Anglin, Rutherford and Springer (2003) include a measure of the Degree of Over5
This system differs slightly from the system which is usually estimated because this
system excludes a measure of the degree of over-pricing or the selling price discount; a change
in tastes should have the obvious effect of changing (pS, 8) for any given (J, <). As well, these
equations assume that the solution is unique.
6
I ignore the detail that E(log(pS)) usually differs from log(E(pS)). When this difference
is caused by variables omitted from a data set but observed by a buyer and a seller, this
difference has no effect.
Value and liquidity effects
Pricing, DOP, in the regressions equations:
9
E(log(pS))= AJ+ B<+ E DOP
(27)
log(8)= CJ+ D<+ F DOP.
(28)
for some constants E and F. If the model is consistent, a seller of type DOP should have an
indifference curve that satisfies
dpS/d8|u = -B3 (E(log(pS))- E DOP) - B4 (log(8)- F DOP)
(29)
and it might be possible to rank these tastes according to a single crossing property. There exists
a ranking where an increase in DOP increases the selling price and decreases dpS/d8|u if and only
if E> 0 and B3 E+ B4 F= (A F+ C E)/(A D- B C)< 0.
This example focusses on a trade off between the expected selling price and the
probability of sale but a seller may also be concerned about price risk or time risk. Solving for a
seller’s utility function to this broader problem would require solving some partial differential
equations (Mas-Colell et al, 1995, Sec. 3.H). It is still possible to find general predictions about
behaviour since a Slutsky-like Equation remains valid regardless of the number of arguments in
the utility function. For example, consider a seller who values the higher moments of the
distribution of selling price because of risk aversion or loss aversion (Genesove and Mayer,
2001) and suppose that this risk depends on pL. The price-probability locus described above
would be extended to a price-probability-risk locus in the sense that a risk-neutral seller would
be indifferent between any extension of the price-probability locus which produces the same
projection into price- probability space. In contrast, a risk averse seller’s optimal solution would
depend on the restrictions imposed by a particular extension of the price-probability locus.
Thus, application of the Le Chatelier Principle shows that sellers who are concerned about
higher moments of the selling price distribution should be less sensitive to changes in market
conditions. More formally,
Proposition 3.
i) M8*/M<$ M8*/M<|R and M8*/MJ$ M8*/MJ|R
ii) MpS*/M<$ MpS*/M<|R and MpS*/MJ$ MpS*/MJ|R
where |R denotes the restriction that the solution must belong to the price-probability-risk locus.
Value and liquidity effects
10
Other Types of Markets
I used the example of selling a house to motivate the value and liquidity effects of a
change in market conditions. Different markets use different mechanisms and many papers have
studied the design of those mechanisms for given market conditions. This section notes how
ideas that were developed for the search and bargaining mechanism common in housing markets
might be applied to other types of selling mechanisms. I show that each mechanism is the same
in the sense that a seller’s choice implies a certain probability of sale and a certain expected
selling price. Thus, a change in market conditions change that choice in easy-to-understand
ways and, so, value and liquidity effects offer a unified approach to studying many types of
market.
Auctions
Auctions have been advocated as ideal mechanisms to sell a good whose value is
uncertain and existing theory has sought to understand the properties of auctions. A few papers,
e.g. Quan (2002), have compared different types of selling mechanisms. A few papers
(McAfee,1993; Peters, 1997) have studied the properties of auctions in a market equilibrium.
To illustrate the price-probability locus relevant for an auction, consider a second-price
auction. In this type of auction, a seller chooses a reserve price, B, which is assumed to be
publicly known and assumed to be binding, and a time to hold the auction, T. The item is sold to
the highest bidder at a selling price determined by the greater of the reserve bid and the second
highest bid. It is well-known that a buyer’s dominant strategy is to bid their willingness-to-pay.
More formally, suppose that potential buyers arrive independently at a constant rate of 8
and that their willingness-to-pay is represented by the cumulative distribution function G(p). By
selling at the second highest price, the cumulative distribution of selling prices is
n(G(max(pS, B)))n- 1- (n- 1) (G(max(pS, B)))n
where n is the number of bidders at the time of auction (Phlips, 1988). n is distributed according
to a Poisson distribution with mean equal to 8 T.7 The ex ante probability that a buyer is willing
to bid at least B is
7
If the number of bidders is not random, then the Revenue Equivalence Theorem reveals
when the expected selling price is independent of the type of auction.
Value and liquidity effects
1- E((G(B))n; n~ Poisson(8 T)).
11
By considering different values of B and T, one can trace out a locus showing the trade off
between the expected selling price and the ex ante probability. Thus, the conclusions of
Proposition 1 apply. If no buyer is willing to buy, then the seller must consider the value of not
selling. The next section discusses how dynamic aspects might be added to the model.
Posted Price Mechanism
Selling goods by posting a price is special because the posted price is both the list price
and the selling price. The literature on menu costs, e.g. Barro (1972) or Benabou and Konieczny
(1994), studied how posted prices respond to changing market conditions. Wolinsky (1986)
studied how pricing behaviour in a monopolistically competitive market could be based on a
model of imperfect information instead of being based on product differentiation.
A “price-probability locus” exists for a posted price mechanism and is usually expressed
as the probability of finding a single buyer willing to pay the posted price. Knowing 8(pS), (<,
J) can be fitted to equation (3) as
J= 1/(d8/dpS)
pS= <+ (1- 8)/(d8/dpS).
Much of the interest in the posted price format arises from the fact that, over a long period of
time, the probability of finding a single buyer is less important than the flow of sales during a
period ). If buyers act independently then the average flow of sales is Q(pS)= )/8(pS) and the
most obvious measure of a seller’s utility is the flow of profit: (pS- MC) ()/8) where MC
represents the marginal cost.
Financial Markets
Financial markets are a special example of a market where auctions are common. In this
type of market, the goods being sold are divisible, the auction operates continuously and the
price evolves over time. The last two features combine to create a price-probability locus.
To illustrate, consider a simple binomial process where market price starts at p0 and
changes each period either by moving up by a factor u or down by a factor d, where ud= 1.
Suppose also that the per-period probability of an increase is independent across periods and
equals 2. In this environment, a seller chooses a trading rule to sell a financial asset which can
be as simple as selling at the market price on a pre-specified date, t. Or, the rule can fix a selling
Value and liquidity effects
12
price, S, and wait until the market price rises to that level. The expected selling price under the
first rule equals the expected market price after t periods: the expected price at time t is p0 dt
E(u/d)x) where x is randomly distributed according to the Binomial(t, 2) distribution. The
probability of sale under the second rule is more complicated since it must recognize the
possibility of sale before period t but, with a computer simulation, the calculation is not hard.
These rules reveal the price-probability locus because different trading rules produce different
outcomes and the price-probability locus represents the undominated outcomes. The rule chosen
by a particular seller depends on their tastes and the prevailing market conditions.
Most studies of financial markets seem to focus on the things which cause prices or the
distribution of prices to change. One implication of the analysis in this paper is that, if a market
is less than perfectly liquid, an equally important effect of a change in market conditions is a
change in the probability of trading.
The existence of market makers in financial markets represents a possibly interesting
limitation to the analysis. A market maker increases the liquidity of a market for other traders by
varying the quantity sold or by switching from the buyer’s side to the seller’s of a market and
these actions help to explain how those other traders react to changes in market conditions. To
the extent that market makers are a constant feature of a market, they provide no insight into
changes in a market. But a market maker’s strategies imply that the concept of a “probability of
sale” is not well-defined for them.
Labour Markets
While workers worry about the effect of a change in market conditions, such as a change
in monetary or fiscal policy, empirical work, Devine and Keifer’s (1991), tends to emphasize the
role of incentives in explaining unemployment search behaviour and to de-emphasize the role of
market conditions. Models of unemployment behaviour (e.g. Pissarides, 2000) and models of
the real estate selling process have both been based on search and matching models but few
people use the word “liquidity” to characterize a labour market. By focussing on payoff-relevant
aspects of behaviour, a study of value and liquidity effects should offer insights into the market
process that can be applied equally well to many types of markets. The magnitude of each effect
may vary according to the mechanism or the market but this approach should avoid results which
depend on specific and untested features of a model such as the optimality of a reservation wage
Value and liquidity effects
strategy (e.g. Morgan and Manning, 1985).
13
Dynamic Models
The previous discussion focussed on a static model where any change is anticipated and
permanent. Much of the interest in changing market conditions occurs because they are not
permanent and sometimes not anticipated. For example, it usually takes months to sell a house
and interest rates change during this time. During price negotiations, the decision to sell or to
wait for better conditions may vary with current conditions that evolve over time (Krainer, 1999;
Novy-Marx, 2003). In financial markets, traders try to profit from the constant flow of new
information about market conditions. These facts make a static model unrealistic. Fortunately,
the fact that a seller values only the selling price and the probabilities of sale at a sequence of
points in time allows me to analyse this problem without invoking the full complexity of a
dynamic Slutsky Equation (LaFrance and Barney, 1991; Caputo, 1990). Thus, with a few
modifications, the results derived above remain true.
Without loss of generality, suppose that time flows discretely, t= 1, 2, ..., and that market
conditions may vary with the date, (<t, Jt). Since a seller maximizes utility at each point in time,
subject to the sequence of feasible price-probability loci {(<t, Jt); t= 1, 2, ...}, a Bellman
Equation reveals the relevant utility function at any point in time:
Vt(pSt, 8t; <t+ 1, Jt+ 1)= 8t pSt+ (1- 8t) max{Vt+1(pSt+ 1, 8t+ 1; <t+ 2, Jt+ 2); (pSt+ 1, 8t+ 1) are
feasible}
(30)
With this expression and market conditions summarized by (<t, Jt), Proposition 1 can be
applied directly to find the effect of a change in period 1's market conditions on behaviour during
period 1. In the same way, Proposition 1 reveals the effect of an anticipated change in market
conditions in period t on behaviour during period t, for any period t> 1, if market conditions after
period t are fixed and if the seller survives until period t. A change in market conditions during
any period later than t affects Vt+1 and this change has an effect on behaviour during earlier
periods that would be comparable to a change in tastes in a static model. If market conditions
evolve according to a Markov process, where the distribution of (<t+ 1, Jt+ 1) is denoted by D(.;
(<t, Jt)), then modifying the right hand side of equation (30) to recognize the expectation would
produce results comparable to Proposition 1.
Value and liquidity effects
14
It may be possible to consider more complex, non-Markovian dynamic processes. For
example, a change in market conditions during one period may be correlated with changes
during subsequent periods. In such a case, the observed value and liquidity effects in any one
period can be expected to depend on the degree of correlation. Or, Taylor (1999) finds the
equilibrium to an asymmetric information game where the price-probability locus for a house
varies over time because a house which has been offered for sale for a long time suffers from a
stigma in the eyes of buyers. Quantifying these effects would require estimating the value and
liquidity effects and the dynamic process simultaneously.
Concluding Comments
From the perspective of a single seller, there are two types of changes in market
conditions: “price-altering” and “time-altering.” This paper shows that, under a variety of
different selling mechanisms, any effects on payoff-relevant dimensions of a seller’s behaviour
can be summarized by a “value effect” and a “liquidity effect”. Together with adding-up
conditions, these effects impose cross-equations restrictions that can be exploited to improve the
efficiency of a chosen econometric procedure or to test whether sellers are making an optimal
decision. I offer a simple example where a seller’s preferences can be derived from a pair of
regression equations. The proofs of these results use ideas commonly associated with demand
theory, including the Slutsky Equation. This Equation is a powerful tool because it can be the
basis for the complete set of testable predictions (e.g. Mas-Colell, Green and Whinston, 1995).
I focussed on the question of how a single seller responds to a change in market
conditions because the general idea of how an entire market adjusts to changing external
conditions is well-known: A market adjusts by
changing the price that each seller offers (or each buyer bids) and
changing the selection of participants in a market.
In an equilibrium, these changes must be compatible. With the kind of stochastic rationing
implied by an illiquid market, the first process is always relevant. For the second process and
since sellers share a pool of buyers, I conjecture that a single seller would view the changing
selection of other sellers as a time-altering change in market conditions. Thus it is important to
note how to convert this kind of change into the price changes predicted by the textbook model.
Value and liquidity effects
15
Demand theory also offers a concept, the indirect utility function, that could be used to study the
second process. If U(.; T) is the utility function of a type T seller then
W(<, J, T)= U(pS*(<, J, T), 8*(<, J, T); T)
shows the value of trying to sell a house under market conditions described by (<, J). It may be
possible to use this function, and a modified Roy’s Identity, to find a general model of selection
that is consistent with particular choice of price and probability.
Value and liquidity effects
16
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Value and liquidity effects
Figure 1: The Effect of a Change in Market Conditions on the
Price- Probability Locus
19