on organized markets under uncertainty

ON ORGANIZED MARKETS UNDER UNCERTAINTY

NORMAN P. OBST’ University of Washington

The market adjustment mechanism usually assumed in economic literature is an artificial process designed to capture the essence of the dynamic behavior of prices.’ It is artificial because neither the institutional structure of She market nor the dynamic behavior of its partici- pants is specified. Instead, it uses a simple mathematical representation of an intuitive economic relationship: changes in price are assumed to vary with excess demand. The purpose of this paper is to derive a price adjustment mechanism for organized markets by taking into consideration each agent’s behavior and the environment to which each agent is exposed. The institutional structure of the market will be explicitly presented. A representative agent’s behavior under uncertainty will be derived. Prices will change as a result of agents’ responses in the given institutional setting.

In what follows, the behavior of those agents who wish trade will be derived first. The application of these results to the assumed specific structure of the organized market will subsequently appear. New results concerning the dynamic behavior of prices in organized markets will be obtained.

I. BEHAVIOR UNDER UNCERTAINTY

The defining characteristic of an organized market is the existence of a central agent (specialist) through whom all transactions are completed. Examples of such agents are the specialists for listed stocks and bonds and the agents for some commodity futures. This type of market is im- personal when compared to unorganized markets in which transactions at several different prices may be occurring simultaneously in local en- counters.* Because of the centralized nature of organized markets, orders from those agents who wish to trade can be stored for future reference. The “limit order,” a price-quantity pair specifying either the quantity of the good (security) to be purchased at the indicated price or below, or the quantity of the good to be sold at the indicated price or above, can

*The author is indebted to Stanley Reiter and Vernon Smith for suggestions and discussions that led to the writing of this paper, and to an anonymous referee for both stimulating and pene- trating comments.

1. See Arrow, Block and Hurwicz [3] , Arrow and Hurwicz [ S ] and Samuelson [lo] for examples. See Arrow and Hahn (41 for a recent treatment of the problem.

2. See Reiter 19) for a theory of unorganized markets.

182

OBST: ORGANIZED MARKETS 183

be sent to the specialist with the knowledge that no trading will occur until the market price falls (or rises) to the indicated set of prices.

It will be assumed that the typical trading agent (investor) decides the quantity of his income to devote to purchasing the good. Given this decision, every investor must decide at what prices and quantities to set his limit orders. In this section of the paper, results will be derived indicating when an investor will send a limit order to the specialist, and what the effect will be of an increase in wealth.

Two models will be considered. Both assume that transaction costs preclude sending more than two limit orders, although there is no theo- retical reason why the analysis could not be extended in this direction. The first model assumes purchases can be made with certainty at the current market price. The second model removes this abstraction from reality. Both, however, assume that the investor knows with certainty the future end-of-month price. I t is found that, although the investor is certain the price will rise, he may insert limit orders below the market price. The analysis could be further extended by including an uncertain end-of- month price, but it is felt that for this paper the additional complications outweigh the benefits to be gained from the more general modeL3

W,f end-of-month wealth;

WOE initial wealth;

po f last market price of preceding month;

yo pl E limit order buy price, pl < po ;

y I

r#I

The following notation will be used:

quantity of shares purchased at p o , 0 < y o < Wo/po ;

quantity of shares to be purchased at p I ;

a random variable indicating the lowest price at which a buy limit order will be executed within the month, r#I 2 0; and

end-of-month price. pF > P O . pF

In the first model it is assumed that an investor can purchase all he desires at the last market price, p o . The investor’s time horizon is the month, and the end-of-month price, p F , is known with certainty. It is assumed p F > p o , implying the investor is a buyer. The problem to be examined is under what conditions the investor will give up a certain gain by purchasing at po for only a chance of a greater gain by inserting

3. The degree of confidence about the end-of-month price is essential to the decision of which security to buy. This topic is outside the scope of the paper.

184 WESTERN ECONOMIC JOURNAL

a limit order at price p1 below po . If the order at p1 is not executed, the investor's final wealth is smaller than what it would have been if all initial wealth were invested at po .

Final wealth WF is equal to:

w F 1 ~ y o ( p F - p 0 ) + WO if # > p l

w F 2 ~ y o ( p F - p 0 ) +yI (pF-pI ) + WO if 0 ~ 4 ~ ~ 1 (1)

It is assumed the initial wealth Wo is a given fixed constant. That fixed amount of wealth is either invested at po or reserved for purchase at p l . Thus

wo = Yo Po + YI PI

If P denotes the investor's subjective probability that the order at pl is executed (the probability that OGdGp, ) , then the expected utility of final wealth can be written as

(3) E[U(W,u)]= ( I - P)u(w~') f Pu(w,u2)

Since it is assumed that pl is determined in advance, (3) may be thought of as being determined by the decision variable yo ; that is, E[U(WF)] =

The investor's objective is to maximize Z(yo). Denote the solution as yo*. Then the quantity of shares desired to be purchased at pl will be

(4)

The conditions to be derived are those under which a limit order to buy below the market price will be sent to the specialist. In notation, sufficient conditions for yl * > 0 will be derived, or, equivalently, for yo*< Wo/&. Since W o / h is the maximum value yo can take on, z'(Wo/po)<O will guarantee that the maximum value of 2 is taken on at yo*< Wo/m. Taking the derivative of 2 with respect to yo and evaluating at Wo/& yields

Z(Yol.

q * = (W* - YO*&)IPI

Assuming V(WF)>O for all WF implies

Result I: If an investor is a maximizer of expected utility of final wealth, and if the marginal utility of final wealth is positive, then a limit order to buy below the market price is sent to the specialist if the expected percentage gain from that order is greater than the certain percentage profit

OBST: ORGANIZED MARKETS I85

that would be obtained by purchasing at the market price. Since P is the subjective probability of success for the order at price p l , Result 1 follows directly from (6).

The second model to be presented removes the assumption that purchases can be made with certainty at the last market price. (This assump tion contradicts the fact that not all buyers can be satisfied at a fixed price if the total quantity demanded is greater than the quantity supplied at that price.)

Price po will now represent a limit buy order price above pl. Price po may be above or below the last price traded. Again, the conditions to be studied are those under which the limit order at pl will be placed. Final wealth can now be described as

(7) w,= ~ ~ ~ ~ y ~ ( ~ - p o ) + w0 if pl<+<po w,= W F ~ ~ ( ~ O + Y ~ ) & if o < @ < p ,

where W, = yo& + y1p1.4 P is the investor's subjective probability that O<t$<pl. Let P* be the probability that pl <q3 <po. Then Z(yo) E(U(W,)) is equal to

If Z'(Wo/p,)<O, the maximum of Z(yo) will occur at yo*< Wo/& or at yl *>O. Since

then the assumption that U'(WF)>O for every WF implies Z'(Wo/po)< 0 if

Result 2: Under the above assumptions, a limit order to buy will be inserted below the market price if the expected percentage gain from that order is greater than the expected percentage gain from an order placed at any higher price. Proof: The probability that the order at price & will be filled is the probability that 0 < 6 < po- That probability is P + P*. The Result follows

4. It has been implicitly assumed that the investor's order at po will always be filled at po although the fust price traded can be less than po It has been judged that the additional results obtained by revising this assumption do not justify the extra complications involved.


directly from equation ( 10). For what follows it will be helpful to know under what conditions an

increase in wealth results in an increase in the quantity of buy limit orders below the market price. The following demonstration will postulate the same assumptions as the second model, although the results would be the same under the assumptions of the first model. Price po again represents a limit buy order price above p l . The existence of an interior solution is also assumed. This implies Z'(yo*) = 0. From equation (8) it follows that

( 1 1) z'(yO*) = P*(& - pb) v ( w F ' ) + p[& - ( e & / p ] ) ] u'(wF2)

Taking the derivative of ( 1 1) with respect to Wo and solving for dyo*fdWo yields

(12) dYo*IdWo=-[P*(~- h)u"(WF') + P(&I)+)(&- ( ~ p ~ / p ~ ) ) u " ( w ~ ~ ) l f LP*(& - u"( wF') + p(& - (pF&/pl)) r( wF2)]

But Wo = &yo* + pl y1 *, since all wealth is reserved for purchases at po and pr. Differentiating with respect t o Wo and solving for dyl *ldWo gives

(13) dYI *ldWo = I 1 - pO(dy0 */dWo)I/p~

Denoting the denominator of the right side of equation (12) by D and using (1 2) and (1 3) yields

(14) dy1 *ldWo = [P*&(& - &)VYWF') + @(& - (&&/P~))U"(WF~)JIP~D

If risk aversion is assumed, then U"(W)<U for any W>U. Therefore D is negative. Using (1 4) yields the condition that

(1 5) dyl */dWo> 0 if and only if

P*&(& - ~O)UYWF'I + P&(& - ( p F p ~ / p I ) ) u " ( w F ~ ) 0

But from (1 1) and Z'(yo*) = 0 it follows that

( 16) m& - &)fpc& - (&&/PI))] = - f v ( W F 2 ) f u'(wF1)j

From (1 5) and (16), it can be concluded that

( 17) dyl */d Wo> 0 if and only if

-[un(wF1)/ u ( w F ' ) ] > - [ u"(wF2 )/ u'( wF2) l

Since an interior solution was assumed, both yo* and yl * are non-zero. Therefore WF2> WF'. Condition (17) can then be interpreted as saying the quantity of buy orders at the lower price pl will increase with wealth if and only if the investor exhibits decreasing absolute risk aversion.


Result 3: Under the above assumptions, if non-zero limit orders to buy are placed at both price po and price p I , then the quantity of orders at the lower price pI varies directly with wealth if and only if decreasing absolute risk aversion is assumed.

I I . MARKET DYNAMICS

The results of the preceding section on investor behavior together with assumptions concerning the institutional structure under which the trader operates will now be used to determine a theory of market dynamics. A mathematical representation of a market adjustment process will be derived. Theorems will be obtained, and the economic causes and implica- tions of these theorems will be explored.

It is assumed that the institutional structure is that of an organized market. Price is set by the central agent to clear the market if possible. Each trading agent sends his order to the specialist. This non-trading central agent sets a price to create zero excess demand if possible, or, if not, to create non-positive excess demand at any higher price and non- negative excess demand at any non-negative lower price. That price, chosen by the specialist according to the sum of the individual limit mes- sages, will be called a quasi-equilibrating price. Exchanges occur based on the limit orders effective at the quasi-equilibrating price and on a priority system when excess demand is not exactly zero.’

Each market trader decides the quantity of his income to devote to purchasing the good in question (security) per month. This quantity can be inserted as initial wealth, and it, together with all assumptions needed to derive Results 1 through 3 above for all investors, will yield limit order demand and supply functions for each day of the month. For different initial market prices, various levels of income will be devoted to purchasing and selling the good each month. The resulting schedules are flow demand and supply functions. Given the initial market price and the flow demand and supply functions, limit order demand and supply functions will be sent to the specialist. The specialist will in turn set a quasi- equilibrating price.

To study price dynamics, additional assumptions are needed to relate the flow demand and supply functions to the limit order demand and supply functions. Assumption I: “Month” is defined to be the length of time during which changes in price and in resource holdings do not affect the decisions of each investor on limit order prices and quantities. Assumption 2: If aggregate excess flow demand is zero at the prevailing

5. See Obst [ 81 for a descriptive treatment of the institutional structure of organized markets.


quasi-equilibrating price and if both of the flow curves are fixed, then the same aggregate limit order demand and supply functions are unchanged each day except for those orders initially filled. Assumption 3: If aggregate excess flow demand is positive (negative) at the prevailing quasi-equilibrating price, then the same aggregate limit order demand and supply functions are repeated the next day except the limit order demand (supply) function is increased or shifted to the right, horizontally, or the limit order supply (demand) function is shifted to the left, horizontally, in a manner related to the size of excess flow demand. (These shifts will be assumed to be parallel for mathematical convenience.) All relations between flow demand and limit order demand, and between flow supply and limit order supply are similar. Assumption 4: The market is sufficiently populated with traders having different subjective probability distributions so that there will always be some buy orders at a price lower than every sell order, and some sell orders at a price higher than every buy order.

Assumption 1 is essential if the complications of non-tiitonnement processes and of feedback effects are to be avoided. A complete theory would relate individual limit order prices to changes in the probabilities P and P* of the preceding section and would relate these probabilities to changes in market price. The effects of shifting resource holdings would also have to be considered.6 The theory to be considered here, therefore, is a short-run theory valid only so long as changing resource holdings and feedback effects either do not occur or are not sufficiently strong to alter the results to follow.

Assumption 2 would be valid if investors made their decisions at the beginning of the month and if the absence of aggregate excess flow demand or supply implied investors would merely not change their behavior. Price would remain unchanged.

Assumption 3 presents a way in which aggregate excess flow demand can change the market price. It would be valid if investors make their decisions on the first day of the month, and, when aggregate excess flow demand is non-zero at the resulting price, those investors who placed no orders subsequently decide to buy or sell at any price, or those investors who had placed orders leave them in and place new ones at all prices, or (for non-parallel shifts) those investors who had placed orders decide to raise or lower their limit order prices depending on the size and sign of aggregate excess demand.

Finally, Assumption 4 guarantees the existence of an aggregate limit

6. Several of the problems of intermonth dynamics are considered explicitly in my paper, 'Shocks, Expectations, and the Dynamics of Organized Markets," currently in progress.


order excess demand function at every point in time. Lemma I: If shifts in the flow demand and supply functions result in changes in wealth to be allocated to the security, then an increase in flow demand per month implies new orders t o buy are inserted below the market price, and an increase in flow supply per month implies new orders to sell are inserted above the market price. Proof: Shifts in the flow functions imply changes in initial wealth in the sense of Result 3 above. Since the assumptions under which that Result holds have also been assumed here, and since Assumption 3 guarantees the similarity between demand and supply relationships, the Lemma follows.

For ease of exposition, linear limit order demand and supply functions will be assumed. Let: D ( p ) = u (-up + b) be the aggregate limit order demand function; and S ( p ) = v(cp + e ) be the aggregate limit order supply function, where all parameters are strictly positive. Denote the flow demand function as QJp) , and the flow supply function as SF(p). Then, convert the above assumptions into mathematical statements involving the notation. The limit order excess demand function can be written as

(18)

The strict version of Assumption 3 above can be written as

(19) u (dbldt) - v (deldt) = X &(p) - SFCp)J

where h > 0 . This equation states that the limit order excess demand function will shift in a parallel manner according to the size and sign of excess flow demand. The right side of equation (19) is the linear form of the Walrasian adjustment mechanism. Note that the quantitative avoid- ance of accounting for completed transactions (for mathematical convenience) has no effect on the qualitative economic consequences of the model.

D ( p ) - S ( p ) = -(ua + v c ) p + (ub - vel

A mathematical formulation of Lemma 1 is

where DF2(p) >DF1(p) for every p > 0 implies u2 > u' , and SF2(p) > S ~ l ( p ) for every p > O implies v2 > v I , where ui = f {DF'(p)( p > 0) and v i = g{S~' (p ) l p > 0 ) , i = I, 2. A shift in flow demand implies a change in the quantity of limit orders below the market price as well as at other prices, and hence a change in the d o p e of the limit order demand function. If- flow demand increases, then f, an increasing function, implies u increases which, in turn, implies that the limit order demand function be- comes flatter. Thus the increase in limit orders below the market price flattens the limit order curve, which is being interpreted mathematically


as an increase in u. A similar interpretation also holds for the relation between flow supply and v .

Note that Assumption 2 implies the limit order excess demand function is unchanged if excess flow demand is zero and the flow demand and supply curves are unchanged. Equations (19) and (20) will yield the same conclusion :

(21) The parameters Q and c are invariant with respect to flow demand, flow supply, excess flow demand, or time.

If (19),(20) and (21) are assumed, then ub - ue and UQ +vc will not vary if excess flow demand is zero and the flow functions are unchanged. Hence the limit order excess demand function is unchanged. This result is equivalent to Assumption 2 in terms of its effect on the price mechanism.

Finally, it is necessary to include the function of the specialist in an organized market:

(22) D(p) = S(p)

The specialist sets a price so that limit order excess demand is zero. Thus, the conditions shown in (1 8) through (22) represent a dynamic

price adjustment model for organized markets that explicitly considers individual behavior under uncertainty and the environment under which agents are exposed. Results will now be derived from this model and compared to the usual results under a simple Walrasian mechanism.

Differentiating (18) with respect to time, assuming that DF(p) and SF(p) are fixed schedules, and using (19), (20), (21) and (22) yields

(23) dpldt = [h/(Ua + VC)][DF(P) - SF(P)]

where [h/(ua + vc)] is the speed of adjustment of the process. Result 4: For this market process representing investor behavior under uncertainty in organized markets, the speed of adjustment of the price mechanism varies directly with the negative of the slope of the limit order excess demand function (where price is on the vertical axis). pt.oof: The slope of (1 8) is -Z/(ua + vc). Result 4 then follows from (23). The major implication of this Result is that the flatter the limit order excess demand function, the slower will price converge (process less damped). The economic reason for this is that the flatter limit order function means that there are buy orders representing larger quantities below the market price and sell orders representing larger quantities above the market price. This implies that the same excess flow demand (supply) will raise (lower) price by a smaller amount, since there are more orders to fill at nearby prices.


Result 5: For the above market process, the speed of adjustment of the price mechanism is slower (faster) if flow demand and flow supply increase (decrease) even though excess flow demand at every price may be unchanged. Proof: Suppose DF1(p) increases to DF2(p) and SF1(p) increases to SF2(p). Then by (20) u' increases to u2 and v' increases to v2. The positiveness of the parameters a and c implies u2a + v2c >u'a + v'c. The result follows from equation (23). Result 6: For the above market process, the speed of adjustment of the price mechanism will be more rapid from a decrease in flow demand than from an equivalent increase in flow supply. PLoof: Suppose DF'(p) decreases to DF2(p). Equation (20) implies u1>u2. Therefore

(24) u'a + v'c > u2a + v'c

It is necessary to compare this with what occurs when SF*(p) increases to SF2(pl. Equation (20) implies v2> v'. Therefore

(25) u'a + v2c > u'a + v'c

Inequalities (24) and (25) yield

(26) The required result follows from equations (23) and (26).

Results 5 and 6 are related. Result 5 suggests that the larger the flow demand and supply curves, the more limit orders below and above the market price. Therefore, the limit order excess demand curve will be flatter, and the process will be less damped. Result 6 demonstrates the asymmetry between shifts in flow demand and flow supply. Although excess flow demand may remain unchanged, the decrease in flow demand leads to more rapid convergence than does the increase in flow supply. The economic reason for this is that the decrease in flow demand results in fewer buy orders below the market price, and hence more rapid convergence even though excess flow demand is unchanged.

u'a + v2c > u2a + v'c

I l l . CONCLUSION

The Walrasian adjustment mechanism, together with assumptions concerning both investor behavior under uncertainty and the environment under which the investor trades, have led to conclusions unobtainable without explicit consideration of these behavioral and environmental


effects. The simple dynamic price mechanism provides no theory of the speed of adjustment coefficient. The simple mechanism considers excess flow demand as the only variable. Result 5 suggests that the absolute size of flow demand and flow supply are also crucial in the dynamic process. More important, the asymmetry Result 6 declares that changes in flow demand have effects different from changes in flow supply, although excess flow demand may be unchanged. Clearly, the simple dynamic adjustment mechanism is an inadequate representation of price behavior in organized markets under uncertainty.

Two attributes of organized markets are liquidity and efficiency. The market is perfectly liquid if an investor is able to buy and sell any quantity without influencing the price. The market is efficient if the allocation of funds is, in some sense, optimal. The equilibrium set of prices under rather broad assumptions are efficient.' The results of this paper suggest that as the economy grows the market will become more liquid and less efficient. As the economy grows, flow supply and demand increase. Result 5 states that convergence to equilibrium will then be slower. The market is more liquid because of the existence of a larger quantity of limit orders both below and above the market price. The market is less efficient because prices are further away from their equilibrium values for longer periods of time. The techniques developed in this paper together with a complete specification of the environment as the economy grows should aid in constructing measures designed to provide an optimal mix of liquidity and efficiency.

7. See Arrow and Hahn [4] for an abbreviated treatment of efficiency, and for references to other work in the optimality of competitive equilibrium.

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on organized markets under uncertainty

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