Multimarket equilibrium, trade, and the law of one price.

Holt, Charles A.

1. Introduction

Although the centerpiece of any introductory economics course is the supply and demand model of perfect competition, students frequently have trouble understanding the linkages between individual markets and how differing supply or demand conditions can lead to different price tendencies between markets. Instead of implementing a single-market classroom experiment and asking students why prices converged to, say, $6, it is instructive to set up two markets and let students try to explain why prices in the two markets differ. This two-market approach also illustrates how gains from trade arise when cross-market trading is introduced. Speculators essentially unify the markets and equalize prices as arbitrage opportunities are exploited. Multimarket experiments help bridge the gap in students' knowledge that results when the leap is made from one to many markets, as is done in most microeconomics textbooks.

We use playing cards to distribute buyers' value information in classroom trading. The dual market setup can be adapted for microeconomics classes at any level, ranging in size from 10 to about 40. Somewhat larger classes can be accommodated by recruiting assistants to help pass out cards or by assigning students to work in groups. The exercise takes about an hour, including discussion. Section 2 outlines the procedures, and section 3 describes how to structure the class discussion to enhance student learning. Section 4 surveys results from (nonclassroom) research experiments in which trading occurs in interrelated markets.

2. Procedures

The basic setup involves two isolated markets, with identical demand conditions but with different supply conditions. Allowing trade only within each separate market will generate higher prices in the market with tight supply, setting the stage for the subsequent introduction of traders who can buy in one market and sell in the other. We have used two different trading institutions. In the first, we eliminate sellers by conducting auctions in which we sell a fixed number of commodity units to student buyers in each market. This institution is relatively easy to implement, especially for the instructor who has not previously conducted a classroom market exercise. In Appendix 1 we describe a second institution, a pit market, in which buyers and sellers come together in separate trading areas to call out bids and asks and to negotiate trades in a relatively unstructured situation. The pit market is more complex than the auction, but an instructor who has previously conducted a single-market classroom exercise (as described in Holt 1996) might prefer this setup.

Consider the auction setup with fixed supply and eight buyers assigned to each market. (The numbers of buyers can be increased or decreased as explained below.) Six units are available for sale in the first market, and only two units are available in the second.(1) Buyers submit sealed bids for a single unit in their own market, with the uniform purchase price determined by the highest rejected bid. Thus, if the three highest bids were $9.50, $8, and $7 in the thin (low-supply) market, the buyers with bids of $9.50 and $8 would each receive a single unit but both would pay only $7 for these units. If a tie results for the lowest accepted bid (e.g., if the bids submitted were $9.50, $8, and $8), one of the tied bids is randomly chosen, and all units are sold at the tie price ($8). Buyers are given a motive to bid by providing them with resale values that can be conveniently assigned with playing cards. We use numbered playing cards to determine resale values. The cards are shuffled, and each buyer is given a card, which determines the buyer's resale value in dollars; for example, a 6 of hearts indicates a resale value of $6. A buyer who makes a purchase at a price below the resale value earns a profit that is the difference between the resale value and the purchase price. Bids above resale value are not permitted. Thus, the buyer's value corresponds to a limit price.

The eight cards used in each market are 10, 9, 8, 7, 6, 5, 4, and 3; these values can be arrayed to form a demand function, as shown in Figure 1. The vertical lines show the perfectly inelastic supply function in each market. The demand function is not shown to students prior to trading, but if buyers bid optimally, the price will be $8 in the thin (low-supply) market and $4 in the thick (high-supply) market.(2) After buyers' resale values and the trading rules are explained by reading the instructions contained in Appendix 2, the cards are dealt, and each buyer submits a bid for a single unit by writing a bid price (identified by the student's name) on a small slip of paper given to them by the instructor. These bids are collected and ordered from high to low in each market, and the market-clearing price is announced. This price is the third-highest bid in the thin market and the seventh-highest bid in the thick market. Buyers then calculate their earnings (equal to zero for those who made no purchase) using the record sheet contained in Appendix 2. These instructions and record sheets can also be downloaded from the Web at http://theweb.badm.sc.edu/laury.

It might enhance student interest if you collect the cards between rounds, shuffle them (keeping the cards from the two markets separate), and then pass them out again to buyers before bids are made in the next period. Prices should converge to near-equilibrium levels after several periods, at which time trading between markets can be introduced.

After reading the traders' instructions (also contained in Appendix 2), ask for one volunteer from each market to act as a trader. The lowest-numbered card in each market should be removed before passing out the remaining cards. Because these cards represent extramarginal units, this does not change the equilibrium price in either market. Each trader is given a capital endowment of $20 and is restricted to purchasing a single unit in the thick (low-price) market? Because of the price disparity between markets, this unit can be resold for a profit in the thin (high-price) market. After the introduction of traders, trading in the thick market precedes that in the thin market. Any units bought by traders are automatically added to the available supply in the next (thin) phase of the market.

With the introduction of traders, the supply in the thin market is increased by the number of units that the traders bring, and the market-clearing price is again determined by the highest rejected bid. The predicted outcome is for two units to be purchased by traders in the thick market and resold in the thin market. The addition of these two units shifts supply out to four in the thin market and lowers the price to $6. Similarly, the additional two units of demand should, in theory, raise the price to $6 in the thick market as the traders leave only four units for buyers in this market.

To accommodate additional students (i.e., for classes with more than 16 students), you can assign the extra buyers low-valued ($2-$4) units. Because these units do not trade in equilibrium, the market predictions do not change. Alternatively, you can add higher-valued buyers to each market (distributing the values among the range of demand possibilities). In this case, the number of units offered for sale and the number of traders must be increased so that prices are equated between markets when each trader purchases a unit in the thick market and sells it in the thin market. For example, you can double the number of students who participate by giving out two cards for each value, so that the resulting demand function is 10, 10, 9, 9, and so on. In this case, you should offer 12 units for sale in the thick market and four units in the thin market; when trade is introduced, you will need two volunteers from each market (four total) to act as traders. For smaller classes, the lowest-valued units ($3) can be removed or buyers allowed to purchase more than one unit by giving each student an additional playing card. The instructions in Appendix 2 show how to implement this change.

Using this design, prices should reach the predicted levels in two or three periods. After introducing traders, several more trading periods should be conducted in sequence (first trading in the thick market, then in the thin market). If time is running short, you might need to end trading before prices converge. Be sure to leave at least 15 minutes for discussion.

3. Discussion

It is useful to discuss prices in the two separate markets before considering the effects of speculative trading across markets. We will summarize a typical adjustment pattern and then consider how the subsequent questions and class discussion might be structured. Figure 2 shows the prices observed for seven auction periods in one class.(4) Recall that the predicted price is $8 in the thin market and $4 in the thick market. In each period, the average price in the thin market is shown by an open box and the average price in the thick market by an "X." Notice that prices tend to separate in periods without traders and to converge in the periods with traders. After two rounds of trading, the price was $8 in the thin market and $4 in the thick market. In round 3, only one trader submitted a market-clearing bid, so only one unit was carried over into the thin market. Notice, however, that the additional unit of demand raised prices in the thick market by $1. In this class, traders were not restricted to buy only in the thick market, and the excluded trader purchased a unit in the thin phase. This increase in demand offset the additional unit of supply and further reduced the price in the subsequent thick phase of the market, in which there were a total of seven units available. By round 5, the price was $7 in the thin market and $5 in the thick market. Prices took several periods to converge in the auction market because one trader was unable to purchase a unit until the final trading period.(5) At this time, traders purchased two units in the thick market, and prices converged at $6 in both markets.

The values, bids, and transactions prices for both auction markets are shown in Table 1. One trader displaced the buyer with a resale value of $5, except in period 4, when a seventh unit was available. Furthermore, all the gains from trade were realized in period 7, as the buyers with $6 and $5 resale values were excluded from the thick market; these units went to the "right" buyers who had previously been excluded in the other market (i.e., those with resale values of $8 and $7). Because these latter two buyers were unable to make purchases prior to the introduction of cross-market trade, the gain from the addition of these buyers is $7 + $8 = $15. This gain more than offsets the loss of buyer value of $5 + $6 = $11 in the thick market, as the price increase excluded these two marginal buyers there. There is, of course, no change in total production costs because the same units are produced as before (at a marginal cost of zero): all six units in the thick market and the two units in the thin market.

The first impulse of most instructors is to graph the supply and demand functions to show how well economic theory works. This is the wrong approach. Instead, the discussion should begin by asking why prices were higher in the thin market, which should bring the obvious response that there is a scarcity there (only two units are available for sale). Then announce the buyers' card numbers, which should be read from the actual cards in no particular order. Then ask why the prices converged to the observed levels.

First, students will have to figure out that the optimal bid is one's own resale value.(6) Noting that bids above value are not permitted, ask whether it is optimal to bid below value. Someone will say yes, then ask the students for an example in which more money is earned by bidding below value. The discussion of examples will make it clear that bidding below value does not lower the price that this person pays, which is the highest rejected bid. Then ask whether bidding below value can ever cause regret. The answer is that this bidder might lose a purchase that would have been profitable with a bid at resale value. It helps to write the numbers being used in the discussion on a price line on the blackboard and to mark the buyer's reservation value clearly. Finally, the relationship between bidding at value and the market-clearing (competitive) price needs to be brought out by questions. Here you might need to lead somewhat to help students discover how to apply supply and demand in this particular case with discrete demand and fixed (inelastic) supply. Having data for two different market structures can be helpful for the evaluation of ad hoc explanations of what determines price levels. These stories can generally be made to fit what happened in one market but not in both. For example, a conjecture that price should be equal to the average of the demand values would work for the market with traders but not for either of the isolated markets.

Table 1. Transactions Record for Two Auction Markets

 Thick Market Thin Market
Period Value Bid Price Value Bid Price

1 10 8 3 10 10 6
 8 7 3 8(*) 6 6
 9 6 3
 7 6 3
 5 5 3
 4(*) 4 3

2 10 10 4 10 9 8
 9 9 4 9 9 8
 8 8 4
 7 7 4
 6 6 4
 5 5 4

3 10 10 5 10 10 8
 9 9 5 Trade(*) 10 8
 Trader 7 5 9 8 8
 8 6 5
 6(*) 6 5
 7 5 5

4 10 10 4 10 10 7
 Trader 10 4 9 9 7
 8 8 4 8 8 7
 7 7 4
 9 6 4
 5(*) 5 4
 4(*) 4 4

5 Trader 10 5 10 10 7
 8 8 5 9 9 7
 7 7 5 8 8 7
 10 6 5
 9 5 5
 6(*) 5 5

6 Trader 10 6 10 10 7
 9 9 6 9 9 7
 8 8 6 8 8 7
 7 7 6
 10 6 6
 6(*) 6 6

7 Trader 10 6 10 8 6
 Trader 10 6 9 8 6
 9 9 6 8 8 6
 8 8 6 7 7 6
 10 7 6
 7 7 6

An asterisk (*) denotes an inefficient buyer inclusion.

After students understand how prices are determined in the isolated markets, you can turn to the effects of trade. Follow-up questions can address the reasons for trade across markets in this context. By now, students should realize that profits are available to traders who buy in the low-priced market and sell in the high-priced market. Therefore, the subsequent discussion can emphasize how trade affects prices and the masons for this. Try to get your students to relate their intuitive answers to shifts in demand in the low-priced market and the resulting increase in supply when traders enter the high-priced market. Questions about whether profit opportunities persist should make it clear that prices will tend to equalize. Next, ask what buyers in each market will think about the advent of speculation. The answer should pick up the fact that buyers benefit when prices fall and suffer when prices rise. The obvious question, then, is whether the gains of one group more than offset the losses of another. This is a much more subtle issue for students, and you will have to focus the discussion on total earnings of the group as a whole. Recall that the same units are bought as before trade. However, including the high-value buyers ($7 and $8) in the thin market more than offsets the exclusion of the low-value buyers ($5 and $6) in the thick market for a net gain of $7 + $8 - $5 - $6 = $4. This might seem like a small amount to the students, so express it as a percentage of the total earnings (buyer surplus) and discuss what this percentage might mean at the economy-wide level when, for example, trade barriers are removed. Finally, the gains from trade should be related to the surplus area in the supply and demand figure.

You should point out that the increased efficiency comes at the expense of buyers in the thick market, who pay a higher price or are shut out of the market entirely. This is one reason that free trade can be controversial, especially when it is difficult to see other gains from trade (e.g., reduced prices in other markets or gains due to comparative advantage). Finally, note that the speculators make profits without producing any real goods, and ask how efficiency can arise as a result of such "unproductive" activity.

The tendency for prices to equalize is sometimes referred to in textbooks as the "law of one price." Although this exercise demonstrates its predictive power in a simple market, you might ask for conditions under which this law might not hold. For example, does California wine cost the same in California as it does in Japan? This should induce people to mention transportation costs. Understanding can be improved by considering the effect of a $1 cost of moving a unit from the thick market to the thin market.

4. Further Reading

Miller, Plott, and Smith (1977) conducted the first laboratory experiment in which traders or speculators were allowed to buy in one market and sell in a subsequent market. Unlike the exercise described here, there were sellers with a stationary supply curve, and the demand shifted in and out in alternating "seasons." In each session, some subset of participants were given the right to purchase in the initial low-demand (low-price) season and to sell in the high-demand season.(7) Speculation caused seasonal price differences to diminish rapidly over time, with a consequent increase in market efficiency. Similar results are reported by Williams (1979) and, for markets with different locations, by Plott and Uhl (1981). Convergence to an intertemporal competitive equilibrium was less dramatic when both supply and demand shifted between seasons (Williams and Smith 1984). Some of this literature is surveyed in Plott (1989) and Holt (1995).

Appendix 1: Pit Market Procedures

With larger classes, it is possible to use students as buyers and sellers who negotiate prices in a trading pit. Detailed instructions for conducting a classroom pit market are given in Holt (1996). Because this is more complicated than the auction design, we suggest that the instructor conduct a single pit market exercise before attempting to conduct this exercise, in which two markets are conducted either simultaneously or sequentially. There must be at least 25 students (9 sellers and 16 buyers) and several assistants (possibly students) to help verify contracts and record the transactions.

The demand functions are the same as those described above, but use only red playing cards (Hearts and Diamonds) to assign buyers' value numbers. Supply is determined by sellers' costs, also specified using playing cards. We use numbered black cards (Clubs or Spades) to indicate costs; for example, a 3 of Clubs represents a cost of $3 for producing a single unit. Production costs are not incurred unless a unit is actually sold, in which case the seller earns the difference between the sales price and the unit cost. A good design to use involves nine sellers with costs of 6, 7, and 10 in the thin market and 2, 2, 2, 2, 3, and 3 in the thick market. These costs. arranged from low to high, determine the supply functions, much the way the demand function was created using value numbers. The market-clearing prices match those of the auction markets described above: $4 to $5 in the thick market and $8 to $9 in the thin market.

The trading is initiated by asking students to keep their cards private and come to two corners of the room, one for each market. Before traders are introduced, the two markets can open at the same time to save time. There is a separate reporting table for each market and the instructor or an assistant at each table to report and announce transaction prices as they occur. Once in the trading pit, buyers can call out bids to purchase a single unit, and sellers can call out asking prices to sell a single unit. In this way, students negotiate freely as they stand around in a group. When a buyer and a seller have agreed on a price, they come to the table designated for their market to report the price, waiting in line in pairs if there is a queue. The instructor or an assistant should check to ensure that the price is no greater than the number on the buyer's card and no lower than the number on the seller's card (otherwise, the buyer and seller are sent back to the trading pit). Once a trade is verified, it is announced loudly and written on the blackboard next to the name of the market (Blue or Green). Then the person at the reporting table takes the cards and sends the buyer and seller back to their seats to record their earnings. This recording is facilitated by the record sheet contained in Holt (1996).

Speculation across the pit markets can be allowed by letting two observers come in as traders or by assigning the low-valued buyer in each market to be traders. As in the auction market, traders are given a capital endowment of $20 and instructed to purchase no more than a single unit. Instructions for these traders are provided at the end of this appendix. A trader might fail to sell a unit purchased in the previous phase of the market. In this case, the unit cannot be carried over into the next period, and the trader incurs a loss equal to the purchase price of the unit. When intermarket trade is allowed, each of the two traders should purchase one unit in the thick market, which opens first, and sell it in the thin market. This speculation shifts out demand in the thick market and supply in the thin market, equating the competitive prices in the range of $6 to $7.

The discussion should be conducted as described for the auction market. Because each unit can trade at a different price (unlike the uniform price auction), average prices can differ by a small amount between the thick and thin markets, even after the introduction of traders. Also, the effects of trade on sellers are the reverse of those experienced by buyers: Sellers in the thick market benefit from the increase in prices, whereas sellers in the thin market suffer from the decrease in price. However, because the same number of units are produced as before. the net gain in value ($4) is identical to that experienced in the auction market. How this gain is divided between buyers and sellers depends on the actual transactions prices.

For the pit market, use the instructions included in Holt (1996) together with the trader instructions below. The trader record sheet shown in Appendix 2 can also be used for the pit market.

Trader Instructions (Pit Market)

For the next several periods, one buyer from each market (Blue and Green) will act as a trader. These traders are able to both buy and sell units, as described below.

Blue and Green trading will now be conducted in sequence (first trading in the Blue market and then in the Green market). Traders will be given an initial capital endowment of $20 and can use all or part of this money to purchase units of the good.

Each trader is allowed to purchase one unit in the Blue market, where traders negotiate contracts as before. However, the trader's value of the good is not yet known - it depends on the (uncertain) price at which the good is sold in the next market. If a trader purchases a unit in the Blue market, the trader has one unit of the good to sell in the Green market. The trader's capital endowment will be reduced by the purchase price of this unit. At no time is a trader allowed to buy a unit for a price greater than the amount in the current capital stock.

Traders who purchase units in the Blue market can then enter the Green market as sellers. They are free to negotiate a sales price with any of the Green market buyers. If a trader sells a unit in the Green market, the trader's capital endowment is increased by the sales price of this unit. If the sales price is higher than the amount the trader paid for the unit, the trader earns a profit on this unit. For example, if a trader buys a unit at a price of $9 and sells it at a price of $10, this trader's capital stock is increased by $1. If the sales price is lower than the amount the trader paid for the unit, the trader earns a loss on this unit. For example, if a trader buys a unit at a price of $10 and sells it at a price of $9, this trader's capital stock is decreased by $1. Traders earn a profit by purchasing at a low price and selling at a high price.

If a trader purchases a unit in the Blue market but does not sell it in the Green market, the unit expires, and the trader earns a loss equal to the purchase price of the unit. In other words, a unit purchased in one round cannot be sold in any other round.

Appendix 2: Auction Market Instructions

Buyer Instructions

Changes for the two-unit version are in parentheses and italicized in the instructions that follow.

Today we will have a market in which the people on my right are buyers in the Blue market and the people on my left are buyers in the Green market. Note that all of you are buyers. I will now give each person a numbered playing card. Please hold your card so that others do not see the number. Each card represents the value to you of one unit of an unspecified commodity that can be purchased.

You can each buy one unit (up to two units) of the commodity during a trading period (one unit for each card you hold). The number on your card is the dollar value that you receive if you make a purchase. You will be required to buy at a price that is no higher than the value number on the card. Your earnings on the purchase are calculated as the difference between the value number on the card and the purchase price that you negotiate. For example, if you have a 10 and you purchase one unit at a price of $6, your earnings are 10 - 6 = $4. (If you buy only one unit of the commodity, you receive the highest of the two values on your playing cards. For example, if you have a 10 and an 8, and you purchase one unit at a price of $6, your earnings are $4.) If you do not make a purchase, you do not earn anything in the period. Think of it this way: It is as if you knew someone who would later buy the unit from you at a price that equals your value number, so you can keep the difference if you are able to buy the unit at a price that is below the resale value.

Trading Rules

The Blue market has six units available for sale, and the Green market has two units available for sale. To purchase units, you will make written bids to purchase units of the good. The bid you submit should represent the maximum price you are willing to pay for a given unit. (You can submit different bids for each unit of the good, for example, $10 for the first unit and $9 for the second unit. Remember that the first unit you purchase is worth more to you, so the bid for the second unit should never be higher than for the first. Alternatively, you can submit the same bid for both units of the good, for example, $10 for two units of the good.)

After everyone has submitted their bids, I will place all Blue market bids in order from highest to lowest. The highest six bids will each obtain a unit of the good. The price of the good will be equal to the seventh-highest bid. Thus, no one will pay a price higher than the amount submitted in his or her bid (and the actual purchase price can be lower). In case of a tie, winners will be randomly drawn from all fled bids. The price will be equal to this (lowest) tied bid.

For example, if the bids were $10, $10, $10, $9, $9, $9, $9, $7, and $7, the highest six bids correspond to $10 through $9. However, only three of the four $9 bids will receive the good (because there are seven bids at this level). The price of the good would be $9. On the other hand, if the bids were $10, $10, $10, $9, $9, $8, $7, and $7, all those who bid $10 through $8 (the highest six bids) would receive units of the good at a price of $7 (the seventh-highest bid).

In the Green market, bids will also be ordered, but only the highest two bidders will receive units of the good. The price of the good will be equal to the third-highest bid. Thus, no one will pay a price higher than the amount submitted in his or her bid (and the actual purchase price can be lower). Ties will be handled as in the Blue market.

Some buyers with low values might not be able to purchase a unit, but do not be discouraged. New cards will be passed out at the beginning of the next period. Remember that earnings are zero for any unit not bought (buyers receive no value).

Note that you will not be earning any money in this experiment!

Buyer Record Sheet

Period 1 __________ __________ _____________
 (Value) - (Price) = (Earnings)
Period 2 __________ __________ _____________
 (Value) - (Price) = (Earnings)

Total earnings all periods:

Trader Instructions (Auction Market)

For the next several periods, one buyer from each market (Blue and Green) will act as a trader. These traders are able to both buy and sell units, as described below.

Blue and Green trading will now be conducted in sequence (first trading in the Blue market and then in the Green market). Traders will be given an initial capital endowment of $20. They can use all or part of this money to purchase units of the good.

Traders are each allowed to purchase one unit in the Blue market. In this market, a trader can submit a bid to buy as before. However, the trader's value of the good is not yet known - it depends on the (uncertain) price at which the good is sold in the next market. If a trader's bid is successful the trader has one unit of the good to sell in the Green market. The trader's capital endowment will be reduced by the purchase price of this unit.

Any units purchased in the Blue market will be added to the existing supply in the Green market. If one trader purchases a unit of the good in the Blue market, there will be three units for sale in the Green market (the two units currently offered by me and one unit from the trader). The three highest bidders will purchase units at a price equal to the fourth-highest bid. If both traders purchase one unit in the Blue market, there will be two additional units available for sale in the Green market. Thus, the four highest bidders will purchase units at a price equal to the fifth-highest bid.

After sales have been made in this market, the trader's capital endowment is increased by the sales price of this unit. If the sales price is higher than the amount the trader paid for the unit, the trader earns a profit on this unit. For example, if a trader buys a unit at a price of $9 and sells it at a price of $10, this trader's capital stock is increased by $1. If the sales price is lower than the amount the trader paid for the unit, the trader earns a loss on this unit. For example, if a trader buys a unit at a price of $10 and sells it at a price of $9, this trader's capital stock is decreased by $1. Traders earn a profit by purchasing at a low price and selling at a high price.

At no time is a trader allowed to submit a bid to buy a unit for more than the amount in the current capital stock.

[TABULAR DATA OMITTED]

1 In the instructions, the high-supply market is referred to as the Blue market and the low-supply market as the Green market.

2 How these bids are determined is discussed further in section 3.

3 It might seem obvious that this is the only way to make money. However, as described below, in one market a trader bought in the "wrong" phase of the market (and lost $4) when not explicitly prohibited from doing so.

4 The auction was conducted in a class of nine undergraduate students at the University of South Carolina. Each buyer could purchase up to two units. Two low-valued ($2 and $3) units were added to demand in the thick market.

5 This illustrates the need to set the initial capital endowment high enough so that a trader who loses money on a transaction is not excluded from the market. In the auction market reported here, traders were given an initial endowment of only $10. When the trader lost $4 on an initial contract, he could not bid more than $6 (the amount of his remaining capital) in subsequent rounds. He was unable to purchase a unit with this bid, which kept prices from equalizing. This problem was noticed before period 7, and his capital endowment was increased.

6 It is possible that some buyers will discover this on their own. In the auction session reported here, after the first round one buyer asked why she would ever want to bid less than her value. It is best to defer the answer to such questions until the discussion period. at which time you can start the discussion of this issue with that student's observation.

7 The trading differed from the one-sided auction described here because sellers were also allowed to make price quotes in a two-sided (double) auction.

References

Holt, Charles A. 1996. Classroom games: Trading in a pit market. Journal of Economic Perspectives 10:193-203.

Holt, Charles A. 1995. Industrial organization: A survey of laboratory research. In Handbook of experimental economics, edited by J. Kagel and A. Ruth. Princeton, NJ: Princeton University Press, pp. 349-443.

Miller, Ross M., Charles R. Plott, and Vernon L. Smith. 1977. Intertemporal competitive equilibrium: An empirical study of speculation. Quarterly Journal of Economics 91:599-624.

Plott, Charles R. 1989. An updated review of industrial organization: Applications of experimental methods. in Handbook of industrial organization, 2, edited by R. Schmalensee and R. D. Willig. Amsterdam: North-Holland, pp. 1109-76.

Plott, Charles R., and Jonathan T. Uhl. 1981. Competitive equilibrium with middlemen: An empirical study. Southern Economic Journal 47:1063-71.

Williams, Arlington W. 1979. Intertemporal competitive equilibrium, on further experimental research. In Research in experimental economics 1, edited by V. L. Smith. Greenwich, CT: J.A.I. Press, pp. 255-78.

Williams. Arlington W., and Vernon L. Smith. 1984. Cyclical double-auction markets with and without speculators. Journal of Business 57:1-33.