Chapter 3 The Dynamics of Price Adjustment in Experimental Random Arrival and
3.5 Results: Price Changes and the Dynamics of Price Movements
3.5.3 The Marshallian Nature of RA Environments
In all of the experiments discussed here, at a given moment in time, subjects typically had more than one incentive available in their private markets. When subjects accessed their private markets each of these orders were displayed in a list in order of most to least profitable. The technology also had the limitation that subjects were unable to sort the list of available incentives by other features, such as time till expiration. A natural assumption given the structure of preference inducement used here is that subjects acted on incentives in the order of most to least profitable.
Such behavior induces what has been called the Probabilistic Marshallian Path.
In a deterministic Marshallian path traders are matched according to the available gains from trade, with the highest valued buyer and the lowest valued seller trading first, the second highest buyer and second lowest valued seller trading second and so on. In contrast, in a probabilistic Marshallian path traders with higher available gains from trade are not assured of trading before lower rent traders, they merely have a high probability of doing so.
3.5.3a The Marshallian Speed of Quantity Adjustment
We can still see the Marshallian theory of quantity adjustment at work by looking at individual bids and asks and ask whether the waiting time before an offer in the public market is accepted is a decreasing function of how profitable it is for the other side of the market (as a function of the offer’s distance from the TE price).
For this analysis, we use the Cox proportional hazard model to estimate the hazard of a bid or ask being accepted as a function of its TE distance. The model
assumes that for every offer price alive in the market at a moment in time there exists a distribution of waiting times, f(P,X,t), until that offer is accepted by the other side of the market. We can define a hazard at time t to be the instantaneous probability of an offer being accepted in the next infinitesimal moment of time, conditional on it having not been accepted up to time t.
1 ( () )
(3.9))
( F t
t t f
h
We will also assume, as a consequence of Marshallian theory, that the
instantaneous probability of a bid or ask being taken is shifted either up or down by the amount of rent it offers potential traders on the opposite side of the market. That is:
( )exp
TEDistance
(3.10))
| ( 1
)
| ) (
( h t 1
X t F
X t t f
h b
Where hb(t) is the hazard rate for an incentive prices at the TE price. Cox (197?) shows that under the assumptions described above, we can estimate 1 without making any
assumptions about the underlying distribution of failure times using partial maximum likelihood. By estimating 1, we can calculate relative hazard ratios:
(3.11) )( exp 1 ) ( )
( t
hb X b t
h t HR
This ratio tells us how much higher or lower, on average, the “instantaneous transaction rate” is for an offer a certain distance away from the TE price relative to an offer priced at the TE price. Similarly, the inverse of this ratio tells us how much longer (or shorter), in seconds, one will expect to wait for an order to be taken by pricing it a given distance away from the TE price. As one might expect, the higher (lower) an order to sell (buy) is priced relative to the TE price, the longer one can expect to wait until that order is filled.
By explicitly looking at adjustment in terms of transaction rates, we also learn more about the process of adjustment than what we have already shown. In particular, we learn that supply and demand works not only on the size and direction of price movement, but also on the rates of transaction.
For the purposes of our study, an offer is “born” the moment it is listed on the book as either the best bid or ask and survives until it is either taken or censored. Often, waiting times are censored because bids and asks either expire, are canceled, or are improved by a newly placed order. Such observations are said to be right censored because they did not survive long enough as the best bid or ask for a time-till-taken to be observed. These observations nonetheless contribute to the likelihood function of the Cox model, and hence to the estimation of parameters. Censored observations
contribute to the likelihood in that the unobserved waiting time is known to have been larger than the length of time that the offer existed before it was censored.
Result 8: The speed of transaction for units at the bid and ask price is influenced by the amount of rent available to the opposite side of the market at that price. The higher (lower) a bid (ask) is, the faster a transaction will occur at that price.
Support: In Table 3.8, we estimate Equation (3.10), listed as Model 1, using partial maximum likelihood and report the hazard ratio and its level of significance. We also repeat the analysis for FCE distance in place of TE distance, reporting the results as Model 2, and as well as a combined model nesting all of the classical variables, reported as Model 3. We stratify each model by experiment and by whether the offer was a bid or ask. In essence, this allows the base hazard rate to vary across bids, asks and
experiments.
Hazard ratios less than one, are associated with increased waiting times until an offer is accepted, while hazard ratios are associated with decreased waiting times. For example, in Model 1, we estimate hazard ratios on TE distance of about 0.8 for both bids and asks. This means that for every standard deviation a bid (ask) is below (above) the TE price, that offer will be accepted by the other side of the market only about 80%
as fast as an offer placed at the equilibrium price.
Models 2 and 3 confirm that FCE distance also affects the rate at which bids and asks are accepted, although the effect of FCE distance appears to be larger in magnitude and significance for asks than for bids. The effect of TE distance on the speed at which offers are accepted also appears to be robust to the inclusion of other classical
variables. Some of the these variables, such as Excess Rent and potential gains from trade also affect transaction speeds independent of the distance to the temporal
equilibrium, although the direction of these effects is theoretically hard to interpret. For example, in theory, positive ER and/or potential gains from trade should be associated with upward price movement and thus an increased spread of offer acceptance at the ask price and decreased speed of acceptance at the bid price. Yet, what we observe is that Excess Rent increases the speed of both bid and ask taking, while positive potential gains from trade decreases the rate of bid and ask taking.
Table 3.8: Cox Proportional Hazard Model Results
Variable Model 1 Model 2 Model 3
Bids Hazard Ratio Hazard Ratio Hazard Ratio
TE Distance 0.80*** -NA- 0.79***
FCE Distance Excess Demand Excess Rent
Modified Excess Rent Potential Gains from Trade
-NA- -NA- -NA- -NA- -NA-
0.83***
-NA- -NA- -NA- -NA-
0.95*
0.98 1.28*
1.05 0.89*
Asks Hazard Ratio Hazard Ratio Hazard Ratio
TE Distance 0.83*** -NA- 0.81***
FCE Distance Excess Demand Excess Rent
Modified Excess Rent Potential Gains from Trade
-NA- -NA- -NA- -NA- -NA-
0.83***
-NA- -NA- -NA- -NA-
0.84***
1.03 1.17***
0.87**
0.83***
Regression Statistics Observations
Offers Accepted -2 Log Likelihood
11152 6776 -37419.678
11152 6776 -37433.466
11150 6775 -37369.312
* p<.1, ** p<.05, *** p<.01
Source: using data from experiments 070208 through 071004
While Result 8 is consistent with Marshallian behavior on the part of individuals, we are unable to relate the rent of individual incentives to the speed with which they transact in markets where subjects are allowed to hold more than one unit of inventory.
This is because inventory is fungible. For the purposes of addressing the Marshallian nature of our experimental environment, experiment 080201 was designed with the restriction that traders could hold at most 1 unit of inventory, allowing experimenters to match transactions in the public market to individual incentives in traders’ private markets. Result 9 states that individuals in RA environments do exhibit characteristics creating a probabilistic Marshallian path. In Section 5.3b, we look closer at the process of limit order placement to see how this probabilistic Marshallian path, combined with limit order book structure helps to stabilize trade prices close to the temporal and flow competitive equilibria.
Result 9: Incentives with higher temporal equilibrium rents were 1) accepted faster in traders’ private markets 2) had higher probability of being transacted in
traders’ private markets, and 3) transacted faster in the public market than lower rent incentives.
Support: Support for Result 8(1-2) come from data from all experiments listed in Table 3.1, while support for Result 8(3) comes only from experiment 080201.
On the left y-axis, Figure 3.7 plots the waiting time between when incentives arrived in a traders’ private market (for buyers and sellers) and when each incentive was accepted by the subject. Included in Figure 3.7 is a piecewise linear fit of waiting times as a function of the available rent of an incentive at the current temporal equilibrium
price. On the right y-axis, Figure 3.7 also plots uniform Kernel estimates of the
probability that a trader acts on an incentive as a function of its temporal equilibrium rent.
A general pattern can be seen in the scatter plot and the piecewise linear fit plotted in Figure 3.7. Incentives with large amounts of rent, in francs, at the temporal equilibrium are acted upon in subject’s private market much faster than those with small rent. Units with negative amounts of rent (those that would be unprofitable if all trading were to occur at the TE price) that are still close to the TE are sometimes acted upon, but with much less frequency and typically after a longer amount of time. Also seen in Figure 3.7 is the fact that negative-rent incentives far from the equilibrium, those with less than -200 francs rent, are never accepted.
While Results 8(1) and 8(2) say that high rent incentives are more likely to enter the market before lower rent and/or non-profitable incentives, Result 8(3) says that these higher rent incentives are actually transacted faster in the public market.
Admittedly a good portion of Result 8(3) may be due to the single-unit inventory restriction in market 080201, but 1) we suspect that this result is true of markets in general and 2) without the restriction of a single unit of inventory, we would be unable to measure transaction waiting times since once incentives are accepted by sellers as inventory, they become indistinguishable from one another.
Figure 3.8, shows both a scatter plot of transaction waiting times against incentive rents, as well as a piecewise linear fit. Similar to Figure 3.7, we see a general downward sloping fit curve, individuals with higher incentives to trade do tend to enter
and transact in the public market faster than individuals with lower incentives. Nearly all incentives with rent above 200 francs traded in under a minute compared to an average transaction time of about two minutes for a extramarginal incentive.
Figure 3.7: Waiting Times and Acceptance Probabilities for Incentives by Rent
Source: using data from experiments 070208 through 071208 0 200 400
0 50 100 150 200 250 300 350 400
Time Until Incentive Accepted in Private Market (sec)
-5000 -400 -300 -200 -100 0 100 200 300 400 500
2000 4000
Rent of Incentive
-5000 -400 -300 -200 -100 0 100 200 300 400 500
200 400
-500 -400 -300 -200 -100 0 100 200 300 400 5000
0.5 1
Estimated Probability of Incentive Being Accepted
Figure 3.8: Waiting Times until Incentives Transacted in Public Market
Source: using data from experiment 080201 0 200 400
0 50 100 150 200 250 300 350 400
Time Until Incentive Transacted
-2000 -100 0 100 200 300 400 500
200 400
Rent of Incentive
-2000 -100 0 100 200 300 400 500
50 100 150 200 250 300 350 400
Incentives
Piecewise Linear Fit