1
Price Stability in Positive and Negative Expectations
Feedback Experimental Markets
Peter Heemeijer, Cars Hommes, Joep Sonnemans and Jan Tuinstra
Center for Non-linear Dynamics in Economics and Finance, Department of Economics
and Econometrics, Universiteit van Amsterdam, Roetersstraat 11, 1018 WB Amsterdam,
The Netherlands
The evolution of many economic variables, such as stock or commodity prices, is
determined by expectations that market participants have with respect to the
development of these variables. Neo-classical economic theory assumes that people
form expectations rationally.1-2 This implies that market participants make correct
price forecasts and that prices converge to their market clearing equilibrium
values swiftly, thereby leading to an efficient allocation of resources.3-4 However,
large fluctuations of prices on financial markets have fueled the debate whether
this is indeed a good description of economic behavior. 5-10 Here we show, by means
of experiments with human subjects, that market behavior depends to a large
extent on how the realized market price responds to an increase in average price
expectations. If the price responds by decreasing, which is typical for commodity
markets, prices converge quickly to their equilibrium value, confirming the
rational expectations hypothesis. If, on the other hand, the realized price increases
after such an increase of expectations, as is typical for financial markets, large
fluctuations in realized prices are likely.
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A key difference between natural and social sciences is that expectations play an
important role in the latter. An investor buys a stock that he expects to go up in the
future, a chip-manufacturer builds a new production facility because she expects that
demand and therefore prices will be high when the goods are produced. Expectations
determine behavior of economic agents and the actual market outcome (i.e., price and
traded quantity, following from demand and supply) is an aggregation of that behavior.
Simultaneously, economic agents form their expectations on the basis of market history.
A market therefore is, like many other economic environments, an expectations
feedback system: past market behavior determines individual expectations which, in
turn, determine current market behavior and so on. The structure of an expectations
feedback system can be characterized as positive or negative. In demand-driven
financial markets the feedback is positive and self-confirming: if many agents expect
the price of an asset to rise and therefore start buying the asset, aggregate demand for
the asset will increase, and so, by the law of supply and demand, will the price. When
investors generally expect markets to go down, the market will actually go down. In
supply-driven commodity markets the feedback is negative: if many producers expect
future prices to be high they will increase production. This leads to low prices and firms
will (disappointed) decide to decrease production only to find that they were wrong
again. To determine the influence of the direction of the feedback, we designed
experimental markets that differ only in that respect and are equivalent along all other
dimensions. The participants in the experiment are students, who earn more money if
they predict market prices more successfully.
Firstly, the negative feedback treatment derives from the well-known cobweb
model11-16 which represents a standard commodity market with a production lag.
Individual price expectations determine aggregate supply which, together with demand,
determines the realized price. Demand for the commodity is linearly decreasing in its
price and supply is linearly increasing in the expected price. Moreover, the market price
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increases if demand exceeds supply and decreases when demand falls short of supply.
For a typical choice of parameters the price-adjustment formula is given by:
pt =
)
(
e
20
123 − p t + ε t ,
21
(1)
where pt is the market price in period t, p he,t is the market price expected by participant
e
h for period t, p t =
1 6 e
∑ ph,t is the average prediction of the six participants in the
6 h=1
experimental market and ε t represents a small demand shock dealt to the price in period
1
t , where ε t ~ N (0, ) .
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Secondly, the positive feedback treatment represents a simple model of a demanddriven financial market. Demand of speculators for a risky asset depends positively
upon the asset’s expected price increase.17-19 As before, the realized asset price moves in
the direction of excess demand. Parameters of the model can be chosen such that the
evolution of prices is given as:
pt =
e
20
(3 + p t ) + ε t ,
21
(2)
e
where pt , p t and ε t are defined as above.
The two treatments are nearly perfectly equivalent. The equilibrium price for both
treatment is equal to 60; that is, if all participants predict a price of 60, the realized price
will, on average, be 60. Both prices series are generated as a linear function of the
average predictions of six participants, the realization of the random shocks is exactly
e
the same and the absolute value of the slope of the relation between pt and p t is equal
to 20/21 for both treatments. The only difference between the treatments is the sign of
this slope.
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Thirteen experimental markets of 50 periods were created, six with negative and seven
with positive feedback. In each market six students participated (and each student only
participated in one market). Results are shown in Figures 1 (negative feedback) and 2
(positive feedback). Each individual panel shows, for one experimental market, the
realized prices and the six time series of individual predictions. Two characteristics of
the data catch the eye immediately. First, in the negative feedback markets prices tend
to go through an initial phase of high volatility, neatly converging afterwards to the
equilibrium price, only to be disturbed occasionally by the impact of a mistake by one
of the group members. Allowing for an initial learning phase, average prices and
volatility are not significantly different at a 5% level from what the rational expectations
hypothesis predicts, for a majority of these markets. In the positive feedback markets,
although the heterogeneity of predictions lasts for a much shorter period, it is not
followed by a quick convergence to the equilibrium price. Rather, most groups
demonstrate a slow oscillatory movement around the equilibrium price of 60, which
seems to come close to it only in the very long run. Second, in both treatments there is
little dispersion between individual predictions within experimental markets, which is
particularly remarkable for the nonconverging positive feedback treatment. Participants
in the positive feedback treatment coordinate on a common nonequilibrium prediction
rule.
Convergence of prices and coordination of expectations is demonstrated in more detail
in Figure 3. The upper panel shows the median of the absolute difference between the
market price and the equilibrium price of 60 for both treatments. We find a higher
degree of convergence in the negative feedback treatment after period two (statistically
significant at 5% in 44 of the 48 periods, Wilcoxon test). Coordination of expectations
is measured by the standard deviation of the expectations of the market participants. The
lower panel shows the median of these (6 or 7) standard deviations for each period. A
low standard deviation implies a high level of consensus among the participants about
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the future price. We find that the standard deviation is higher (and therefore
coordination less) for the negative feedback treatment in the early periods 2-7
(statistically significant at 5%, Wilcoxon test). After period 7 coordination is very high
in both treatments. Note that, outside of equilibrium, it pays off for participants in the
negative feedback treatment to ‘disagree’ with the majority: if the average prediction is
high, the realized price will be low. This drives the heterogeneity in predictions in the
early periods and the fast convergence to the equilibrium price. In the positive feedback
treatment, on the other hand, ‘agreeing’ with the majority pays off since the market
price will be close to the average price prediction. This coordination of price
predictions in the positive feedback treatment is surprising, since participants were not
able to observe each others' predictions during the experiment, making the coordination
itself "blind". Price predictions and market prices can concisely be summarized as
exhibiting "slow coordination and fast convergence" in the negative feedback treatment,
and "fast coordination and slow convergence" in the positive feedback treatment.
3
3
i =1
i =1
Linear prediction rules of the form p h,et = c + ∑ oi pt −i + ∑ s i p he,t −i + ν t were
estimated for each individual participant. Predictions of 71 out of the 78 participants
could be described succesfully this way, which suggests that participants only use recent
information to form predictions. Participants from the positive feedback treatment tend
to have more coefficients significantly different from zero and therefore need more
sophisticated prediction rules to capture the more complicated market price evolution.
Many prediction rules are actually quite simple. In particular, for 40 of the 78
participants, predictions can be described as
p h,et = α 1 pt −1 + α 2 phe,t −1 + (1 − α 1 − α 2 )60 + β ( pt −1 − pt −2 ) + ν t .
For these participants predictions are formed as a weighted average between the
equilibrium price of 60, the last observed price and the participants last own price
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prediction augmented by a trend term β ( pt −1 − pt − 2 ) which measures how participants
respond to price changes. Figure 4 concisely represents these 40 prediction rules in a
prism of first-order heuristics. In the positive feedback environment (21 prediction
rules, dark dots) participants tend to base their prediction on the last price and
extrapolate trends in past prices from there (positive values of α1 and β ) without
taking the equilibrium price into account. On the other hand, most of the estimated
prediction rules from the negative feedback treatment (19 prediction rules, light dots) lie
along the α1 -axis implying that typically predictions in that treatment are a weighted
average between the last observed price and the equilibrium price.
Time periods in commodity markets and financial markets are typically different,
but nevertheless our findings show that commonly observed differences between
markets20-23 can be attributed to a large extent by the expectations feedback structure.
Due to the positive feedback structure, financial markets will often diverge from the
equilibrium price and be relatively unstable. Prices in a production market will be much
more stable and closer to the equilibrium value when the product (and production
technology) has been around for a while: only for relatively new products (e.g.
computer chips24-26) prices can fluctuate wildly. The fact that some established
commodity markets regularly exhibit fluctuations is consistent with our conclusions
since these fluctuations have been attributed to the presence of demand-driven
speculators.27-28 Moreover, our results are in line with some recent experiments showing
that the strategic environment is a crucial determinant of experimental outcomes.29-30
Methods
The experiments took place in the computer laboratory of CREED at the University of
Amsterdam on February 18 and 19, 2003. Every period the participants saw the
previous prices and their own previous predictions and had to predict the next price.
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They were not able to observe each others predictions. Their payment was based upon
their quadratic prediction error and was on average about 22 euro, in one and a half
hour. The instructions and the computer program are available upon request.
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Acknowledgements. We thank the Dutch Science Foundation (NWO) for financial support.
Correspondence and requests for materials should be addressed to P.H. (e-mail: [email protected]).
Figure 1: Prices and predictions in the negative feedback treatment. Each panel
contains, for one experimental market, time series for the realized price and the
time series of individual prediction of the six participants.
Figure 2: Prices and predictions in the positive feedback treatment. Each panel
contains, for one experimental market, time series for the realized price and the
time series of individual prediction of the six participants.
Figure 3: Upper panel gives the median, over the different groups, of the
absolute difference between the market price and the equilibrium price, the
lower panel gives the median, over the different groups, of the standard
deviations of predictions. Solid lines correspond to the negative feedback
treatment, broken lines correspond to positive feedback treatment.
Figure 1: Prices and predictions in the negative feedback treatment. Each panel
contains, for one experimental market, time series for the realized price and the
time series of individual prediction of the six participants.
1
Figure 2: Prices and predictions in the positive feedback treatment. Each panel
contains, for one experimental market, time series for the realized price and the
time series of individual prediction of the six participants.
2
Figure 3: Upper panel gives the median, over the di¤erent groups, of the absolute di¤erence between the market price and the equilibrium price, the lower
panel gives the median, over the di¤erent groups, of the standard deviations of
predictions. Solid lines correspond to the negative feedback treatment, broken
lines correspond to positive feedback treatment.
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Figure 4: Prism of First-Order Heuristics containing the parameter vectors of the
prediction rules of the form
p h,et = α 1 pt −1 + α 2 phe,t −1 + (1 − α 1 − α 2 )60 + β ( pt −1 − pt −2 ) + ν t . The smaller graph on the
right is a top-down view of the prism. Light dots depict prediction rules from
participants in the negative feedback treatment and dark dots rules from
participants in the positive feedback treatment. Positive (negative) values of β
correspond to a trend following (trend reversing) prediction rule. The special
cases “naivety”, “fundamentalism” and “obstinacy” correspond to p h,et = p t −1 ,
p h,et = 60 and p h,et = p he,t −1 respectively. Finally, “adaptation” refers to a prediction
rule of the form p h,et = αpt −1 + (1 − α ) p he,t −1 , with 0 < α < 1 .