A Model of the Consumer s Bid Price Determination with Adjustment Costs 1

Transcription

1 A Model of the Consumer s Bid Price Determination with Adjustment Costs 1 Cheoljoon Kang Astract How do consumers make a idding decision when they arrive at the market to realize their redetermined consumtion lan, esecially when there is uncertainty on the rosect for his/her order eing matched at the market and costs are ensued from the matching failure? We develoed a simler model than the Nash equilirium models y assuming the roaility of id matching deends on the market rice. We showed, y a simulation with artificial data, that consumers can sumit ids aove the sujective value of the consumtion with the introduction of the costs from the matching failure. 1. ntroduction How do consumers make a idding decision when they arrive at the market to realize their redetermined consumtion lan? Traditionally economists resort to the Warlasian tatônement rocess to give exlanations for this. However the rocess will e different if we introduce uncertainty that some of consumer ids fail to e matched at the market and costs arise from the susequent consumtion adjustment. Vickrey1961 was the first who dealt with this issue. He formulated a non-cooerative Nash equilirium model of idding y risk neutral economic agents in single unit auctions. Later Harris and Raviv1981 extended the Vickrey model to multile unit auctions. Cox, Smith and Walker1984 reorted the results of various exeriments to test the emirical roerties of individual idding ehavior. While the game models use the interactive roaility distriutions considering all the other idders in the auction, they introduced comlicated assumtions to make the solution tractale, which restrict the extension of the auction models to the consumer s id ehavior in the market in general. For examle, the model requires information on the total numer of auction articiants, N, and the quantity offered, Q, which is hardly availale for most of the markets excet a certain form of a secified auction markets. These game models focus mainly on verifying the otimality of the market equilirium and comaring the efficiency of the alternative auction systems such as Dutch auction and English auction. n addition, all of these auction models use the assumtion that individual ids for a 1 Comments from Jin Choi, Jaejoon Woo, and Larry Morgan have imroved the earlier draft. Still all the remaining errors are the author s resonsiility. Korea Banking nstitute and DePaul niversity. Comments are welcome. cjk@ki.or.kr. 1

2 given commodity can t exceed the redetermined values in the otimization rocess. This condition, which was initially introduced y Chamerlin1948, has een consistently used in the exerimental economics researches. This aer attemts to develo a theoretical model to analyze the consumer s idding ehavior at the market when the consumer can adjust his/her consumtion lan resonding to the market rice changes. nlike the Nash game models, the roaility of the ids eing matched is assumed to deend uon the market rice which follows a certain roaility distriution. Our main interest lies secifically in the determination of the id rice when consumers are conscious of the costs when their ids are not matched in the market. We further look into the ossiility that a id higher than the reservation rice roduces more welfare to the uyer than a id lower than the reservation rice. Our ultimate goal is to develo a id-generating system, ased on the traditional consumtion theory, which can roduce the humed shae of the limit order ook oserved in most of the exchange markets. n section we revisit the traditional consumer s rolem and suggest an alternative way of modeling to introduce uncertainty aout the id matching. Section 3 introduces an alternative consumers idding model and section 4 resents the simulation results, using the artificial data to find out the roerties of the otimum ehavior derived from the model. Conclusions and suggestions for further research are given in the last section.. Consumer s rolem revisited Let s say there are n consumers who solve the following otimization rolem simultaneously, Max C 1, C : Θ i, Σ i, Ω, suject to i 1 C 1 + C, 1 where C 1 ; commodity 1 that is concerned, C ; collection of all the other commodities, 1, ; rices of the corresonding commodities, resectively, i ; income of the i th consumer, Θ i, Σ i ; reference and information sets of the i th consumer, resectively, Ω; market matching system imlied in Smith1964,198. We assume no cross effect of consumtion etween two commodities. n addition, we assume 0,C >0 for C >0; namely, even no consumtion of commodity 1 at all roduces ositive utility. This assumtion hels the consumer to refrain from aying an extreme rice for commodity 1 when the availaility of commodity 1 gets very difficult.

3 Once we are given 1,, i, Θ i, Σ i, Ω, we can solve 1 in the straight forward way to get the otimum values for C 1 and C. When we change 1, with other variales fixed, the units of C 1 demanded and the units of C demanded will change interactively. When there is no cost incurred in shifting around etween C 1 and C, the value of one dollar sent either for C 1 or C will e the same at the otimum. Now assume the quantity of commodity 1 chosen y these n consumers is fixed at one unit for each consumer. That is, the marginal utility from the second unit of consumtion of commodity 1 dros suddenly to zero. For n consumers, there will e n sujective values, v 1, v, - -, v n, derived from the unit consumtion of commodity 1. We can derive the market demand for commodity 1 y adding u the units at these values horizontally and the equilirium rice will e set once the market suly is given in the erfect information environment. As in <Figure1>, individuals who id elow v will e denied the allocation of commodity 1 and have to adjust their consumtion lans. The information on the level of v is not known a riori. Only consumers whose id exceeds v will e ale to carry out the initial consumtion lans. <Figure 1> Market demand for C 1 and the allocation of individual ids. Sujective value v 1 v v v n C 11 C 11+ C 1 C 11 +C 1 + C 11 +C 1 + +C 1n C 1i ; i consumer s quantity demanded for commodity 1. Now what if our consumers are concerned aout the matching roaility of their ids? We can find a clue from Vickrey1961. He formulated a non-cooerative Nash equilirium model of idding y risk neutral economic agents in single unit auctions. Later Harris and Raviv1981 extended the Vickrey model to multile unit auctions. Cox, Smith and Walker1984 reorted the results of various exeriments to test the emirical roerties of individual idding ehavior. The Nash gaming model summarized in Cox, Smith and Walker1988 is as follows. 3

4 When there are N idders, cometing for Q units of a homogeneous good offered in erfectly inelastic suly, each idder sumits a id for a single unit y maximizing the following ojective function. Max v i - i,θ i G i where i ; individual s id rice v i ; monetary value for i, θ i ; arameters reresenting the individual s characteristics, G i ; roaility that all N-1 rivals of idder i will id rices less than or equal to i, i.e., roaility that i s id will e acceted. When idder i raises his/her id, the roaility that it will eat the rival s ids increases while the welfare value of the id=v i - i declines. Each idder must thus alance these two factors in determining his/her id, considering the roale ids of the others. Since this is a multi-agent non-cooerative game, we need comlicated assumtions to make the solution tractale, which restrict the extension of the auction models to the consumer s rolem in the market in general. For examle, the model requires information on the total numer of auction articiants, N, and the quantity offered, Q, which is hardly availale for most of the markets excet a certain form of secified auction markets. Also these game models used simler distriutions like uniform distriution for v i and G to get the solution for real market alication. n addition, all of these auction models use the assumtion that individual ids for a given commodity can t exceed the redetermined values in the otimization rocess. This assumtion, which was initially introduced y Chamerlin1948, has een consistently used in the exerimental economics researches. Chamerlin1948 takes these values as the Marshallian demand rice which works as a limit rice in the market 3. Smith196 defined this rice as the reservation rice, the maximum rice that a uyer is willing to ay for one unit of the fictitious commodity, and the articiants in his exeriments are not allowed to lace ids eyond the reservation rices. ndividual uyers are assumed to determine their ids to maximize the difference etween these reservation rices and the actual ids 4. However, when we introduce uncertainty on the rosect individual s order eing matched at the market and costs are ensued from the matching failure, there might e a ossiility that a id 3 Chamerlin1948, Smith196,.11. 4

5 higher than the reservation rice roduces more welfare to the consumer than a id lower than the reservation rice, where the enefit of raising the id matching roaility dominates. n this situation restricting the articiants in the exeriments not to make ids eyond their reservation rices may revent them from revealing their economic motivations to the full extent. n the real life we are often forced to accet the rice higher than the rice that we have set efore we go on shoing. Other exenditures may have to e foregone thereuon. Also in the commodities and securities markets long hedgers often sumit ids higher than the forward sold rice to sto the loss of their oen osition in the volatile market. 3. Model of consumer s id rice determination with adjustment costs n the following we develo a model of consumer s id rice determination where consumers face the uncertainty aout their ids eing matched and can adjust their consumtion lan to accommodate this 5. We assume the market to which our consumers sumit their ids is a discrete call auction market so that they have to wait for a while etween the sessions 6. nlike the Nash game models, the roaility of the ids eing matched is assumed to deend only uon the market rice which follows a certain roaility distriution. This assumtion enales us to avoid the comlexity of defining all the other articiants idding strategy distriutions. Esecially in a dynamic market environment where it is difficult to otain the information on the numer of uyers and the total quantity availale, it would e more economical and thus realistic for uyers to calculate their id success roaility simly from the current market rice distriution, not from all the other uyers id distriutions. n fact the information on the latter distriution can e said to e a suset of the information on the former distriution since all the other uyers ehavior will e reflected in the market rice eventually. With the introduction of the uncertainty aout the id matching, the reresentative consumer s rolem 1 changes to determining the id rice,, to maximize the following welfare function. Max W= 0 1, C1 ' d + 0, C0 ' d 3 where ' ; roaility density function of, ; market rice of commodity 1, 0,, C 1 ; consumtion of commodity when the id is acceted, C 0 ; consumtion of commodity when the id is not acceted, 5 Here consumers include traders who have long hedging needs, i.e., ona fide hedgers with forward sold ositions in cash commodities. 6 According to Smith1993, idding ehavior is affected y the matching system in the market. 5

6 6 : id rice for commodity 1 and 0,. From the udget constraint, C 1 =-/ and C 0 =/. Sustitute these into 3 and introduce v, the reservation rice of commodity 1, then 3 ecomes; W + + = d d v C ' 0, ' 1, 0, 4 where / v C =, the amount of consumtion when 1 =v. v can e defined as the maximum rice to otain one unit of consumtion 1 without consideration of the id matching roaility, given and. We assume the individual consumer s choice of doesn t affect. Proosition : At otimum =, ' / 0, 1, =. 5 where / 1, C v =. Proof: Since the consumer s choice of id doesn t affect, 4 can e rewritten as follows. 1 0, 1, W + = = W 0 ' 0, 1 1, ' 1, = + ' ] 0, 1, [ 1, 1 = The second order condition is; 0. ] 0, 1, [ ' 1, 1, < + <QED> Condition 5 means that, at the otimum, the ratio of the marginal utilities of consumtion 1 and consumtion equals to the contriution of the marginal density to the total roaility. For the normal distriution, the RHS of 5 continues to increase as the id rices are raised 7. As the value of consumtion 1 ecomes more recious comared to consumtion, consumers seek to raise the roaility of id matching. 7 See that / rises as ids increase in <Figure->.

7 v Corollary: From 5, 1, C + 0, / > 0. This is readily otained from 5 since > 0, / >0. That is, allocating money for consumtion 1 is etter than eing enforced to send all the money on consumtion. This difference can e defined as an adjustment cost of the consumtion lan since the first term reresents the welfare achieved when the id is acceted in the market and the second term is the welfare otained when the id is not acceted. The value of one dollar sending is different etween those two states 8. We denote this adjustment cost as Δ. Proosition : When >v, W>Wv if, Δ > [ 1, C 1, C + v ] / Γ. 6 v And when =v, W =Wv. Here, Γ =. v Proof: From 4, W - Wv = 1, + 0, 1 v [ 1, v + 0, v = 1, C + 1, C v + 0, v v = [ 1, C + 0, ] [ 1, C 0, ] v > 0, if 1, C 0, >. v v 1, C + 0, Sutract 1 from oth sides to get 6. When =v, W -Wv=0. QED. 1 v] Proosition imlies that consumers may determine the id level higher than the reservation rice if the adjustment cost incurred with the id failure exceeds the welfare loss from aying higher rice for the consumtion, discounted y the roaility imrovement from the higher id sumission. n 6, Γ reresents the relative roaility imrovement in the id accetance and the numerator is the welfare loss from choosing higher than v. Next, we are interested in the roerties of the otimum id rice,, of N consumers with 8 The lum sum onus aid when the ids are matched in the exeriments of Smith1993 can e included in these costs. 7

8 different levels of v when all the other variales are given. Prolem 4 can e simlified as a function of v- and when, P, Θ i, Σ i, and Ω are given as in 7. Max H [ v, :,, Θ, Σ, Ω] 7 i i A. Tye consumer Here we develo the model only for risk neutral consumers to simlify the simulation work later. Following the game models, we first derive the otimum ehavior of the consumerscalled as idders in those game models who try to maximize only the exected welfare surlus or return when their ids are successful. We name them as Tye consumers. nder this condition, the integrated utility functionh in 7 will e increasing monotonically in the exected value of surlus or returnev-=er. This can e exressed as the roduct of the osition rofit, v-, and the roaility of the id for commodity 1 eing matched at the market, t, where the market rice of commodity 1, t, is assumed to e uncertain until the end of the current eriod, t, as in 8. ER = t v-. 8 and t = ' t d, where is the density function of the market rice. We assume 0 t t 0=0, t =1. We also assume t has the first and the second moments and t is differentiale with resect to. The shae of t will deend uon each individual s exerience or information on the characteristics of t changes and the order matching system of a given market. But we assume it is identical for all consumers in the market. Equation 8 imlies that a consumer laces his/her id considering the return or welfare surlus realized from the current osition and the roaility of his/her id eing hit at the market. As the consumer raises his/her id, the return from consumtion 1 will deteriorate while the roaility of the order eing hit will increase. 9 That is, / = ' > 0. The otimal condition will e; E R = ' v = 0, 9 ' v =, 10 v = / ', As the consumer raises her id aove the reservation rice, she still derives a ositive utility since we assume 0, C >0. Therefore she may choose to stay out of market if she has to ay an unexectedly high rice which exceeds the reservation rice y more than a certain amount. 8

9 The second order condition will e v. Condition 10 imlies that, at the otimum, the degree of the marginal return decrease or loss increase LHS should not exceed the cumulative roaility of the id eing hit RHS as the id rice goes u. From 10-1, v->0, and hence <v, since / >0. Here, / measures the contriution of the marginal roaility to the cumulative roaility of a given id rice change. This otimum condition means that, if a consumer considers only to maximize the exected return, his/her id rice will e always less than the reservation rice given y the initial consumtion lan. Therefore, the condition resumed y Chamerlin1948 and Smith196 can e justified. n stock markets we can find traders who stick to the rule given y 10, that is, who never sumit ids at a loss. One of the major risks accomanying this rule is the ossiility that losses accumulate fast in a short eriod of time. For examle, assume a trader s forward sold rice is 90 and the market is fluctuating around 100. The trader can lace his/her id at 89 and kee waiting for a very thin oortunity of realizing a rofit. A more sensile trader would consider equally this risk of fast loss accumulation when his/her id is left unmatched in a volatile market. The otimization rolem 8 needs to e changed for this tye of consumers since they are concerned aout their ids not acceted at the market and costs arise in adjusting their consumtion lans according to changing market conditions. B. Tye consumer To incororate the cost of adjusting the consumtion lan when the id fails to e matched at the market, we make a change to the rolem 8 as follows. EAR = 0 v ' t d t - Δ' t d t = t v--1- t. 11 Here reresents the cost arising from the id failure and is assumed to e non-negative 10. t is assumed to e indeendent of the distriution function, t, and the choice of. The first term in the RHS of 11 is the exected rofit when his/her id is successfully matched and the second term is the exected cost of the id failure. Accordingly, the otimum condition 10 changes to 1. ' v + Δ = 1 10 To secify the relationshi etween Δ in 6 and Δ in 11, we need to introduce more comlicated assumtions on the welfare function structure and the rice distriution. n order to simlify the discussion, here we just assume that a fixed cost arises when the id fails to e matched in the market. 9

10 Thus, v-+ >0, and for some large, >v. This means that it may e in the uyer s interest to lace the id,, aove the reservation rice, v, when the cost of his/her order not matched is considered. Even though the consumer has to ay the rice higher than his/her lanned reservation rice, he/she can e haier with the success of her id than losing the consumtion oortunity at all. 4. Simulation Result Rather than seeking analytical solutions to examine the roerties of the otimum conditions 10 and 1, we develo a simulation method using the artificial data. The simulation is focused on how Tye and Tye consumers resond to changing market exectations, resectively, for a given range of the reference rice. The artificial data for the market rice and the reference rice are constructed as follows. The market rice of commodity 1 is set at 100 at the eginning of the current eriod. However, the rice at the end of the eriod is assumed to follow the normal distriution with mean μ and volatility σ. For the ase case, μ=100 and σ=% are considered. We assume consumers are concerned aout the range of the market rice change on a secific day. So σ reresents the daily volatility. The daily volatility of % is equivalent to 3% of the annual volatility, which is twice larger than the level oserved during the normal days of the stock markets in most of the develoed countries 11. The reservation rices are given as integers etween 90 and 110. The consumers are assumed to know the mean value and the volatility of the market rice efore they sumit the ids. However, they don t exect their ids to change the market rice distriution. We look at the effect of the changes in the mean and the volatility of the market rice uon their otimum id level. <Figure > shows the otimum id determination of Tye consumers whose reservation rices are given as 98, 100, 10, resectively. The otimum id of the reak-even consumer whose reservation rice is 100 is calculated at The welfare surlus of this consumer would e 1.80 if this consumer s id is acceted at the market. However, the roaility of the id accetance is only 18%. Consumers with higher reservation rice who are exected to make rofits on the average sumit ids higher than consumers with lower reservation rice. But the marginal increase of the id rice is less than the change in the reservation rice and decreasing. For examle, the marginal increase in the otimum ids declines from 1.65 = to 1.39 = One reason to use a large volatility is to identify the roerties of the otimum ids y magnifying the change in rices. We tried 0.% of volatility and the major conclusions of the simulations were not significantly affected excet generating long decimal oints. 1 This numer is calculated y the interolation method, using the values of v- and / at =98 and

11 98.0 as the reservation rice rises from 98 to 100 and to 10, consecutively. <Tale-1> summarizes the otimum ids and their matching roailities of consumers with reservation rices from 9 to As exlained already, consumers with higher reservation rices tend to id higher ut at a decreasing rate. That is, the willingness to ay more to secure the commodity falls short of the reservation rice increase. This is reflected in the decline of the id ratio, which is defined as the ratio of the id to the reservation rice. We can attriute this result to the assumtions of the risk neutral consumer and the normal distriution of the market rice. For the normal distriution, the contriution of the marginal roaility to the total roaility, /, continues to rise as the id rice increases, and consumers ecome less willing to sacrifice the rofit to raise the id matching roaility as in <Figure->. That is, consumers are getting more interested in raising rofit =v- than enhancing the id matching roaility as the reservation rice goes u. Therefore, ids aove the mean of the rice distriution 100 tend to contract down toward the mean, and the id function curves away from the reservation rice when the reservation rice gets higher. This non-linearity of the id function is similar with Vickrey 1961 who assumes risk neutral idders and the uniform distriution of the idding rice. On the other hand, Cox, Smith and Walker1988, using the uniform rice distriution ut the constant relative risk averse utility, show the linear relationshi etween the idder s valueequivalent to the reservation rice and their ids in the first-rice auction 14. However, as Potters, Marc, Jean-Philie Bouchaud006 reort, the limit order ooks of most of the securities markets have a humed shae near the current rice and thus it would e more realistic to use a distriution whose density has its eak at the mean such as normal distriution. Nonetheless, for the risk-averse consumer, id rices may go u further with the higher reservation rice to rotect the rofit on hand. n <Tale-1> consumers with the reservation rice far elow the current market rice stick to the low ids even with a very low chance of matching. They don t have any real interest in immediate id matching. Potters, Marc, Jean-Philie Bouchaud 006 reort the emirical evidence that there are market articiants who elieve that large jums in the rice are always ossile. However, this result seems to show the limited rationality of tye consumer. <Figure-> Otimum id determination of Tye consumers 13 We confined the reservation rices range within 4 standard deviations from the mean to manage the size of the simulation. 14 The first-rice id auction is the market in which the auctioned oject is awarded to the idder who sumits the highest id at a rice that is equal to his or her id. Cox, Smith and Walker1988,.97 11

12 50 v-,/' id /'100,4 /'100, <Tale-1> Otimum ids and matching roaility for Tye consumer μ=100, σ=,cost=0 μ=100, σ=4,cost=0 Reservation id id id match id match rice id id distanc ratio distance roaility ratio roaility e

13 Numers at the ottom row are the averages. id ratio%=id/reservation rice, id distance=id-current rice/σ. t would e interesting to see how far the consumers lace their ids away from the current market rice. This is reresented y the id distance, which measures the distance from the current rice, standardized y the volatility. For examle, the reak-even consumers lace ids at aout 90% of the volatility elow the current rice. Consumers who anticiate the rofit y one volatility v=10 ut their ids at 0% of the volatility elow the current rice. However, these figures vary with the daily volatility of the market. The last column of <Tale-1> shows the simulation result when the market volatility is raised to 4%, an extreme market day of a sudden anic or jum. nteresting to note is that consumers determine ids lower than efore and the dro in the ids is most consicuous with the consumer whose reservation rice equals to the current rice 100. We can visually confirm this from <Figure-3>. The curvature of the id function ecomes more linear. The reason is that, as the market ecomes more volatile, the contriution of the marginal roaility to the total roaility of id success, /, jums u over the range of ids until the curve of / with 4% volatility crosses the curve with % volatility. This time consumers have more room to raise rofit v- at a given roaility than when the volatility was %. Further oservation from <Tale-1> is that consumers with reservation rices elow 100 are lessed with higher matching roailities even though they are lowering their ids as the market ecomes more volatile. <Figure-3> Comarison of the otimum ids with different volatilities with no cost of id failure 13

14 Reservation rice μ=100, σ=,cost=0 μ=100, σ=4,cost=0 Next we examine the case when consumers exect the market rice change 15. <Tale-> summarizes the simulation result when the mean of the market rice is exected to move either u or down in arallel y one volatility. <Tale-> Otimum ids with the exectation of rice change for Tye consumers Reserv μ=10, σ=,cost=0 μ=98, σ=,cost=0 ation id id roail id id roail id change id change rice ratio distance ity ratio distance ity We assume the consumers exectation on market rice change doesn t cause direct rice change at the end of the eriod. 14

15 Numers at the ottom row are the averages. id ratio%=id/reservation rice, id distance=id-reservation rice/σ. 3 Changes are calculated as the difference from the ase case. 4 Proailities are calculated from the normal distriution with mean=10 and 98 for each corresonding case, and volatility=%. When consumers exect the market rice is rising, they would raise the ids to kee u with the matching roaility. However, the magnitude of the id increase is less than the exected rice change ecause the desired change in the roaility weight / when the mean increases to 10 is smaller than the change in the exected rice. Thus the roailities of id matching in the sixth column are smaller than those in the ase case where the rice is exected to remain unchanged. Note that the matching roailities calculated now are lower than the revious case with mean 100. The roailities will e different deending uon whether the rice rise is exected only for an individual consumer under consideration or for all the market articiants. Here we assume the rice change is ulic information. Also <Tale-> shows the id resonse across different reservation rices. Consumers with higher reservation rice raise ids relatively more comared with those with lower reservation rice. This is ecause the roaility of the id matching is deteriorating fast as the mean of the rice distriution aroaches their ids from the left. <Figure-4> exlains the otimum id resonse to the exected rice rise of Tye consumers when the reservation rices are given at 98, 100, and 10, resectively Symmetrically, when the rice is exected to decline, consumers cut down their ids to secure more rofit as the matching roaility is imroving, ut at a decreasing rate. 15

16 n all the magnitude of the id rice adjustment is less than the exected market rice change for Tye consumers. <Figure-5> comares id resonses to rice changes across the different reference rices. <Figure-4> Otimum id resonse to the exected rice rise of Tye consumer v-,/' v-98, v-100, v-10, /'100, /'10, <Figure-5> Bid distriutions of Tye consumers when the market rice is exected to move y one volatility 16

17 Reservation rice μ=10, σ=,cost=0 μ=100, σ=,cost=0 μ=98, σ=,cost=0 Now we look into the ehavior of Tye consumers who are conscious of the costs arising from the id failure. The difficulty here is to define the costs of the id failure in the simulation. When a consumer s id is not matched, he/she has two choices: one is to give u the urchase of commodity 1 and change the initial consumtion lan, and the other is to wait for another match in the next eriod. n oth cases costs tend to increase as the market rice ecomes more volatile. For examle, traders maintain cash reserves in their transaction accounts to wait for the next trading oortunity, which tend to e linked with the market volatility. Following this market convention, we assume the cost of id failure is roortional to the daily volatility. We aly two different levels of costs, σ=100% and σ=00% to the otimum condition The simulation results for each case are shown in <Tale-3> and <Tale-4>, resectively. <Tale-3> Otimum ids for Tye consumers when the market volatility=% Reservation rice id change μ =100, σ=,cost=σ μ =100, σ=,cost=σ id id id id roaility id change ratio distance ratio distance roaility 16 To exactly quantify the costs, we need to select the secific form for the utility function. 17

18 Numers at the ottom row are the averages. id ratio%=id/reservation rice, id distance=id-reservation rice/σ. 3 'change' is the difference from the ase case for Tye consumers. When the id failure costs are considered, consumers have to raise the id to increase the matching roaility aove the level given for Tye consumer and thus avoid the costs incurred when their ids are not acceted. <Tale-3> shows the result when the market volatility is % with the unmatched cost equal to 100% and 00% of σ, resectively. The id rices of Tye consumers are higher than those of Tye consumers y 1.9 oints on the average. This time consumers with lower reservation rice are more eager to raise the ids. Furthermore, consumers with the reservation rice elow 98 even choose for their ids to exceed the reservation rice, so are willing to accet the trading loss. However, the id increase doesn t exceed the size of the unmatched cost. <Figure-6> comares the otimum ids for the three different levels of the unmatched cost at the same time; with the unmatched cost=0, 100%, and 00% of σ, resectively. <Figure-6> Comarison of ids with the unmatched cost when the volatility=% 18

19 Reservation rice μ=100, σ=, cost=σ μ=100, σ=,cost=0 μ=100, σ=, cost=σ This result is in contrast to the condition in the exeriments y Chamerlin 1948 and Smith 196, 1964, Esecially in the exeriments carried out y Smith articiants were strictly foridden to sumit ids aove the reservation rice to otain the lum sum onus given to them when their orders are matched. The reservation rices in those exeriments were given y the random selection of cards carrying the aritrary numers 17. When the market volatility is given at 4%, an extreme market of a sudden anic or jum, id rices rise steely with the consideration of the unmatched cost, as shown in <Tale-4>. This time the sloe of the idding function gets steeer with higher cost, meaning that consumers with higher reservation rices ecome more sensitive to the unmatched cost than when the volatility is given at %. We can see this from <Figure-7>. n a market anic, consumers who have large oen osition rofits on hand are forced to resond to the higher unmatched costs. n conclusion we can ursue numerous simulations y introducing various shaes of unmatched costs and rice distriutions, such as making the unmatched cost as a function of the 17 This condition in those exeriments might have een inevitale ecause they couldn t charge uon the articiants, mostly college students, the loss amount when their ids exceeded the reference rice. 19

20 market rice as in 13. To find out the solution for 13, we may need to introduce more comlicated assumtions. EAR = 0 - Δ v ' t d t ' d 13 t t t <Tale-4> Otimum ids for Tye consumers when the market volatility=4% μ =100, σ=4,cost=σ μ =100, σ=4,cost=σ Reservation rice id change id id id id roaility id change ratio distance ratio distance roaility Numers at the ottom row are the averages. id ratio%=id/reservation rice, id distance=id-reservation rice/σ. 3 'change' is the difference from the case when the volatility=4% and no unmatched cost. 4 Proailities are calculated with the normal distriution with mean=100 and volatility=4%. <Figure-7> Comarison of ids with the unmatched cost when volatility=4% 0

21 Reservation rice μ=100, σ=4, cost=σ μ=100, σ=4, cost=σ μ=100, σ=4,cost=0 4. Conclusion and further research This aer attemts to develo a theoretical model to analyze the consumer s idding ehavior at the market when the consumer can adjust his/her consumtion lan resonding to the market rice changes. Our interest lies secifically in the determination of the id rice when consumers are conscious of the costs when their ids are not matched in the market. Our ultimate goal is to seek the theoretical foundation for the id generation in the market and develo a system simler than the Nash equilirium game models. While the game models use the interactive roaility distriutions considering all the other idders in the market, our model emloys only the roaility distriution of the market rice. n order to look into the economic outcomes derived from the model we resort to a simulation method with the artificial data rather than seeking analytical solutions and/or human exeriments. n the simulation we showed that id rices can exceed the sujective value of the commodity when consumers are allowed to adjust the initial consumtion lan and a certain amount of costs are incurred when their ids are not matched in the market. This is in contrast to the condition given in those exeriments y Chamerlin 1948 and Smith 196, 1964, n addition, 1

22 we relicated the result of Vickrey 1961 that, with risk neutrality, the id function curves away from the reservation rices and the ids aove the mean of the rice distriution contract down toward the mean, which can eventually roduce the humed shae of the limit order ook in the real market. Even though our analysis rovides the case where costs arise from the failure of id matching from the consumer s ersective, no exact methodologies are rovided to measure the size of the id failure costs. Also we need to introduce various reference functions and market interactions into the simulation. n order to do this, we elieve two further develoments are necessary. One is a real market survey that investigates how the hedgers with ona fide interests determine the level of their ids, given the market rice distriution and the oen osition rofit and loss. The other is develoing a simulation system equied with the market matching oeration, i.e., a system which can generate ids and offers from artificial traders and comlete the order matching and clearing. n the end the artificial market simulation method may e ale to relace human exeriments y introducing more conditions of the real economy into the model. For a theory testing urose, human exeriments are costly and constrained y the conditions of given resources. t is also difficult to relicate the exeriment results y other researchers. The artificial simulation method has rolems too. The most significant one is the difficulty in introducing market interactions. But this is ecoming a matter of technical rolem considering the current raid technology imrovement. Reference Chamerlin, Edward H.1948, An Exerimental merfect Market, Journal of Political Economy, Vol. LV, #, Aril Cox, James C., Bruce Roerson, and Vernon L. Smith198, Theory and Behavior of Single Oject Auctions, Research in Exerimental Economics, Vol.,. 1-43, JA Press nc Cox, James C., Vernon L. Smith, and James M. Walker1984 Theory and Behavior of Multile nit Discriminative Auctions, Journal of Finance Vol. XXXX, No. 4, Se Cox, James C., Vernon L. Smith, and James M. Walker1988 Theory and ndividual Behavior of First-Price Auctions, Journal of Risk and ncertainty,, Harris, Milton and Arthur Raviv1981 Allocation Mechanisms and the Design of Auctions, Econometrica 49Nov. 1981, Potters, Marc, Jean-Philie Bouchaud006, More Statistical Proerties of Order Books and Price mact, arxiv:cond-mat/ v1 31 Oct. 00. Smith, Vernon L.196, An Exerimental Study of Cometitive Market Behavior, Journal of

23 Political Economy, Vol. LXX, #, Aril, 196. Smith, Vernon L.1964, Effect of Market Organization on Cometitive Equilirium, Quarterly Journal of Economics, Vol. LXXV, #, May, Smith, Vernon L.198, Microeconomic Systems as an Exerimental Science, American Economic Review, Vol. 7, No. 5, Dec., 198, Smith, Vernon L. and James M. Walker1993, Monetary Rewards and Decision Cost in Exerimental Economics, Economic nquiry, Vol. XXX, Aril 1993, Vickrey, William1961, Counterseculation, Auctions, and Cometitive Sealed Tenders, Journal of Finance 16March 1961,