On the extent of the market: a Monte Carlo study and an application to the United States egg market. Aklesso Egbendewe-Mondzozo

Transcription

1 1 On he exen of he marke: a Mone Carlo sudy and an applicaion o he Unied Saes egg marke Aklesso Egbendewe-Mondzozo Deparmen of Agriculural Economics A&M Universiy 14 TAMU College Saion, TX aklesso@amu.edu Seleced Paper prepared for presenaion a he Souhern Agriculural Economics Associaion Annual Meeing, Alana,, January 31-February 3, 009 Copyrigh 009 by Aklesso Egbendewe-Mondzozo. All righs reserved. Readers may make verbaim copies of his documen for non-commercial purposes by any means, provided ha his copyrigh noice appears on all such copies.

2 Summary This paper invesigaes he exen of he marke, using a swiching regimes model similar o hose used in sochasic froniers esimaions. We sared by performing a Mone Carlo simulaion on our model, seeking o evaluae is performance in erms of correcly esimaing he probabiliy of inegraion of wo markes. Our Mone Carlo resuls under he assumpion of half-normal and exponenial disribuion of he errors, revealed ha hese wo disribuions predic almos correcly he probabiliy of inegraion of wo markes. The half-normal error disribuion model ends o slighly underesimae he rue probabiliy of inegraion, while he exponenial error disribuion model ends o slighly overesimae he rue probabiliy of inegraion. We, finally, applied he model o he Unied Saes egg marke using daa from hree highly producive saes and one less producive sae. The model predics ha, he markes pairs considered are inegraed. Tha is, he four markes sudied belong o he same economic marke in he sense of Marshall. Furher, based on our Mone Carlo sudy, we find ha he rue probabiliy of inegraion of wo given markes lies in beween he half-normal model esimaes and he exponenial disribuion model esimaes. 1. Inroducion Since he seminal work of Sigler and Sherwin (1985) on he exen of he marke, a body of lieraure has been developed o analyze he marke mosly in erms of is geographical definiion. Mos of he works in he area have been moivaed by he definiion of anirus markes compared o he economic markes. As defined by Alfred Marshall (1961), a marke for a good is he area wihin which he price of a good ends o uniformiy, allowance being made for ransporaion. This definiion is relaed o he economic marke where differences in prices of he same commodiy observed a differen places are due o ransacion coss. Therefore, according o he definiion of a marke, in he same geographic region i is almos impossible ha prices of he same commodiy display a greaer difference han he ransacion coss over a long period of ime. The reason is ha, arbirage will always occur when he price differences in differen geographic regions are more han he ransacion coss. From his definiion, we can readily infer ha he anirus markes are jus he opposie of he economic markes in he sense ha greaer difference in prices (where price differences canno be explained by ransacion coss alone) will be observed as non ransiory beween wo geographical markes. Hence, he place ha mainains he higher price is a poenial anirus marke. We can see from hese wo marke definiions, ha an anirus marke canno be incorporaed in he economic marke. This poin of view was defended by Spiller and Huang (1986). The iniial sudies in his field, paricularly in Sigler and Sherwin (1985), he analysis were oriened owards coinegraion of prices. Tha is, geographical locaions ha happen o be in he same marke will display a parallel rend of heir prices once care has been aken o remove he common shocks. Technically, prices colleced from wo differen locaions are said o be coinegraed if hey are boh inegraed of he same order, say I(d), and ha heir linear combinaion generaes an innovaion erm ha is inegraed of order I(d-1). The es of coinegraion can usually be obained by he Augmened Dickey-Fuller (ADF) es. This es is popular bu can someimes fail o do a good job when we have a near uni roo siuaion. Tha is, a

3 3 near uni roo siuaion makes he es less powerful for his kind of analysis. Furhermore, he coinegraion mehod falls shor when i comes o calculaing he average ransacion coss. Average ransacion coss are used o measure differences in prices of wo marke locaions. To overcome hese problems, Spiller and Huang (op. ci.) have developed a swiching regimes regression model ha performs well in erms of disinguishing economic markes from poenial anirus markes. In his paper, we will use a similar model o analyze he U.S marke for eggs. Our approach will be a bi differen from ha used by he above auhors in ha we will adap he model o possibly uilize he enire price daa observaions wihou having o spli hem in wo. Also, we propose differen error disribuions o evaluae he robusness of our mehod. The remainder of his paper is organized as follows: In secion, we will inroduce our heoreical model, in secion 3 we will carry ou a Mone Carlo sudy on he model involving Half-normal and Exponenial disribuions. In secion 4, we will briefly presen he Unied Saes egg marke, in he secion 5, we will apply our model o ha marke and finally in secion 6 we will conclude.. The model As we menioned above, he model is in he spiri of he Spiller and Huang (1986). Suppose we are looking a wo geographical locaions A and B wih observed A B prices p and p respecively. These prices move around heir means wih evenually some random shocks. For simple exposiional purpose, suppose ha we have he siuaion p > p. Le T be he ransacion coss of moving he commodiy from he locaion B o he locaion A. We have he following siuaion: If p p < T [1] i will be impossible o have arbirage beween he wo locaions because he seller in locaion B will have o pay more han he ransacion coss o sell a he locaion A and he assumpion of raionaliy of he producer is violaed. Alernaively, if we have: p p > T [] a possible arbirage can occur beween he wo locaions and his will lead o an equalizaion of prices wih he ransacion coss wedge. In oher words, if boh locaions belong o he same economic markes, we will likely see ha only ransacion coss will separae he prices in hose markes. Tha is we have: p p = T [3] Now le suppose ha he ransacion coss are disribued geomerically wih mean k such ha we have: T = k exp( ν ) [4] where ν is a random variable normally disribued wih mean zero and variance ν. From equaion [1], we can define he probabiliy of having he price differences less han he ransacion coss as: Pr ob [ p p < T ] = Pr ob[ p p < k exp( ν )] = β [5]

4 4 To depar from he Spiller and Huang model, equaion [5] can be rewrien by squaring boh erms in he bracke before we ake he logarihm in order o relax he assumpion of 0 < p p made in heir model. The poin we wan o make is ha he sign of price differences can change arbirarily o include he case where p p < T in equaion [5]. The reason is ha, he scalar β is he probabiliy ha he price difference is less han he ransacion coss. This can also be looked a as he number of imes where we have B A 0 < p p and p p < T or 0 < p p and p p < T. Hence, he parameer β is he relaive frequency wih which we canno move commodiies beween markes because of our price differences gain being less han he ransacion coss 1. Tha is, a high β implies a less inegraed marke. The advanage of squaring he erms in equaion [5] is ha in our esimaions we will no need o spli our sample in wo in order o conform o he assumpion of 0 < p p as Spiller and Huang did. Furhermore, he new formulaion implies ha he ransacion coss beween wo geographic markes mus be unique wheher we move commodiies from locaion A o B or from locaion B o A. We, herefore, have he following expression: 1 Pr ob [log( p p ) log k < v ] = Pr ob[ log( p p ) log k < v ] = β [6] We are now ready o se up our swiching regimes model. Firs, we define a posiive random variable η > 0 so ha we have he following expressions: 1 1 log( p A p B ) = D +ν η wih probabiliy β [7] log( p p ) = D + ν wih probabiliy 1 β [8] where D = log k. The equilibrium in equaion [7] corresponds o he no arbirage opporuniy where he difference beween he wo prices is usually less han he ransacion coss. The equilibrium in equaion [8] corresponds o he arbirage sae where he equaliy beween he wo prices prevails mos of he ime. Wih he assumpion ha η is half-normally disribued and runcaed above zero wih variance, we can express he likelihood funcion by he following equaion: C n = 1 1 L = [ β f + (1 β ) f ] [9] f and 1 f are he densiy funcions of ε = ν η in equaion [7] and ν in equaion η B A [8] respecively. By leing Aigner e al., 1977): Y 1 A = log( p B p ) we have he following equaions (see 1 Basically, wha is required in our model is ha he absolue value of he price differences should be less han he ransacion coss so ha arbirage does no occur in boh direcions. This will mean ha he markes are isolaed from each oher and are similar o he auarky markes.

5 5 1 Y ( Y D) D η f = φ 1 Φ ν + η ν + η ν ν + η [10] 1 Y D f = φ [11] ν ν Under an alernaive assumpion ha η follows exponenial disribuion, we will have an 1 η exponenial densiy funcion g ( η ) = *exp( ). The mean of he exponenial θ θ disribuion is θ and he variance isθ. Wih a lile algebra (see Greene, 003) one can show ha we will have he following expression: 1 1 Y D ν ( Y D) 1 ν f = Φ + 1 exp + [1] θ ν θ θ θ f being he same as in equaion[11]. Φ( ) is a sandard normal CDF and φ( ) is a sandard normal PDF. The maximum likelihood esimaion of he logarihm of [9] will lead o he esimaes of he parameers of ineres ( ν, η, D, β ) for he half-normal specificaion and he parameers ( θ, η, D, β ) for he exponenial disribuion. Therefore, he probabiliy of inegraion of any wo markes is equivalen o1 β. Tha is a lower β corresponds o higher inegraion beween wo markes and a higher β corresponds o lower inegraion. 3. Mone Carlo experimens 3.1 Daa generaion process To es our model, we se up an experimen where we have generaed price differences daa consisenly wih equaions [ 7] and [ 8]. The average ransacion coss is aken o be k = such ha log k = D = 3.. We generae one housand observaions of he error erms v which are normally disribued wih mean zero and variance ν = 0.0 so ha we have v ~ N(0,0.0). We also generae one housand observaions of he error ermsη. In he case where η follows half-normal disribuion, he variance is aken o be η = 1.16 so ha we have η ~ HN(0,1.16) and in he case where η follows an exponenial disribuion, he variance is aken o be θ = so ha we have η ~ Exp(0, 0.36). The relaive frequency of no arbirage (he probabiliy of no inegraion) beween markes is aken o be β = 0. 3; which means ha he rue probabiliy of inegraion in he experimenal marke pairs is 1 β = Once we generaed he random samples for he composie error erms v and η, we calculaed he variable Y in such a way ha i represens he expeced value of he equaions [7] and

6 6 [8]. Tha is, we have generaed Y = β ( D + ν η ) + (1 β )( D + ν ). To analyze he resuls as a funcion of he sample size, we generae esimaion sample sizes of 00, 50 and 500 observaions. The resuls of he experimens for boh half-normal and exponenial disribuion models are given in he able below. 3. Mone Carlo experimens resuls Table1: Simulaion resuls for half-normal and exponenial disribuion models β =0.30 v =0.5 Half-Normal η =1.16 D =-3. β =0.30 Exponenial v =0.5 θ =0.36 D = Mean Bias MSE Mean Bias MSE Mean Bias MSE The resuls from able1 above show ha boh error specificaions (half-normal and Exponenial) esimae closely he relaive frequency of no arbirage β for he wo experimenal markes. The half-normal specificaion overesimaes he frequency of no arbirage by 0.07 and he Exponenial specificaion underesimaes he frequency of no arbirage by These biases of he esimaes are almos zero in he boh specificaions. In fac, he probabiliies of inegraion ( 1 β ) of he wo markes in he experimen are esimaed as 0.63 by he half-normal model and as by he Exponenial model. Tha is, boh models perform well in he esimaion of he parameers of he models wih minor biases. However, we acknowledge ha he rue probabiliy of inegraion is bounded below by he half-normal disribuion model esimaes and bounded above by he exponenial disribuion model esimaes as he rue probabiliy of inegraion is 0.7. In he nex secion we will apply our model o he U.S egg marke. 4. The U.S egg marke Eggs are usually classified as a poulry produc by he Unied Sae Deparmen of Agriculure (USDA). Their producion is concenraed in he easern and souhern of he Unied Saes. The egg firms locaed in he res of he counry are scaered and smaller. This is probably because of he climaic condiions of he cener and he norhern areas of he Unied Sae (see map in he appendix). From his picure, our conjecure is ha he prices will end o be lower in he souhern and easern pars han he res of he counry, as he souh and he eas have higher egg producion. Furhermore, for example, if he ransporaion of eggs canno be easily susained from he souh and he eas o he norhwes (mainly because of he disance), he markes locaed in he norhwes will end

7 7 o be differen from hose locaed in he souh and he eas. In oher words, hese wo locaions (he eas and he souh on one hand, and he norhwes on oher hand) will be less likely o be in he same marke. In our empirical sudy, we will use he price daa of and o represen souhern saes and he prices daa of and Washingon Sae o represen, respecively, he easern and norhwesern saes. The choice of our daa is dicaed by heir availabiliy and our desire o look a he markes in he souh and he eas versus he markes in he norhwes. 5. Daa and esimaion resuls 5.1 Daa The daa are 7 monhly egg prices of 4 saes obained form he Unied Saes Deparmen of Agriculure - Naional Agriculural Saisics Service (USDA-NASS). We have daa for he period of January 00 o December 007. The markes of he saes analyzed in his sudy are, and for he souhern and easern markes and Washingon Sae for he norhwesern marke. As menioned in he secion above, our choice of hese markes is moivaed by he fac ha he egg indusry is more producive in he souhern and easern saes han in he res of he counry. For he norhwes saes, only he Washingon Sae s daa is considered because we do no have any daa for saes such as Monana and Oregon. We, also, believe ha he disance of Washingon Sae from he Souh and he Eas is big enough for he possibiliy of isolaed markes. Obviously, he ransporaion issues are very imporan in deermining wheher he markes in wo differen geographic locaions are inegraed. The summary saisics of he four markes are represened in he able below. Table : Summary saisics of price daa Saes Mean Sandard deviaion Minimum Maximum Washingon From hese summary saisics we can see ha he average egg prices of he easern and souhern saes (, and ) are lower han he average prices of he norhwesern sae (Washingon Sae). A look a he map in he appendix shows ha among all he Saes, had he highes producion in 007 (abou 6,39 millions of eggs) and herefore, had he lowes price. The nex highes producive sae in our sudy was (abou 4,994 millions of eggs) followed by (abou 4,79 millions of eggs) and lasly we had Washingon Sae (abou 1,50 millions of eggs). Also among he four Saes, Washingon Sae has achieved he highes minimum price and he highes maximum price. The graphs showing he rend of he pairs of marke prices used in he sudy are presened below.

8 exas georgia Washingon Washingon Washingon -Washingon Washingon Washingon Figure1: Plo of he level of prices for he six marke-pairs.

9 9 5. Esimaion resuls The maximum likelihood esimaion resuls of he model described in equaion [9] is given in he ables below. We firs presen he resuls under he half-normal disribuion assumpion, and second we presen he resuls under he exponenial disribuion assumpion. Table 3: Half-Normal Disribuion Model resuls Washingon Washingon Washingon β (0.1) (1.30) (.0)* (4.950)* (1.960)* (1.850)** ν (5.74)* (3.030)* (3.170)* (3.10)* (4.40)* (3.030)* η (0.0) (1.590) (.00)* (3.070)* (.110)* (1.600) D (-6.10)* (-1.650)* ( )* (-1.880)* (-7.10)* (-5.640)* EXP (D) (.088)* (6.711)* (8.333)* (6.39)* (10.989)* (.695)* LogL Table 4: Exponenial Disribuion Model resuls Washingon Washingon Washingon β (0.895) (1.848)** (1.837)** (.090)* (.140)* (.080)* ν (3.371)* (3.55)* (3.777)* (7.989)* (5.333)* (5.488)* θ (1.644)** (.061)* (1.507) (18.6)* (0.815) (7.700)* D (-1.10)* (-7.10)* (-9.730)* (-9.615)* (-3.530)* (10.11)* EXP (D) (7.46)* (6.369)* (9.091)* (4.348)* (1.81)* (3.717)* LogL The asympoic z-saisics are in he parenheses. * Significan a 5% level, ** Significan a 10% Level.

10 10 The resuls from he half-normal disribuion assumpion in Table 3 sugges ha all egg markes are highly inegraed excep - markes and -Washingon markes. Indeed he probabiliy of inegraion is for - markes, for - markes, for - Washingon markes, for - markes, for - Washingon markes and for -Washingon markes. The variances ( ν ) of he price random componens are saisically significan suggesing ha here are many sochasic shocks ha affec he prices. Also, he average ransacion coss beween he markes are saisically differen from zero. This is evidence ha a leas he ransporaion coss will be non zero beween hese markes pairs. The resuls from he assumpion of exponenial disribuion in Table 4 sugges ha all he β s are low implying ha he markes pairs are highly inegraed wih his model as well. These findings are similar o he resuls of he assumpion of half-normal disribuion. However, he degrees of inegraion of markes are higher in he exponenial model han in he half normal model consisenly wih our previous Mone Carlo sudy. Typically, our empirical resuls and he Mone Carlo resuls suppor he fac ha he halfnormal model, ends o underesimae he degree of inegraion while he exponenial model ends o overesimae he degree of inegraion. In our model, we also assume ha if hese markes are inegraed in he sense ha hey are in he same economic markes, he average difference in prices of he marke will be closely equivalen o he ransacion cos found in our model. Table 5 below displays his comparison.. Table 5: Model performance in he esimaion of he ransacion coss Washingon Washingon Washingon Acual Daa Half-Normal Exponenial Overall, boh models performed well in erms of predicing he ransacion coss. Some of he differences beween he acual ransacion coss and he models predicions migh be relaed o he relaively small sample size of our daa and o he exogenous shocks on he prices ha we found very significan. 6. Conclusion In his paper, we applied a modified version of he 1986 Spiller and Huang paper o analyze he degree of inegraion of he Unied Saes egg marke. We found ha he markes considered in he model are highly inegraed excep he - markes and he -Washingon markes ha are less inegraed. The wo differen error disribuion assumpions made did no yield significanly differen esimaes. Also, he models reveal he presence of significan sochasic shocks on he prices. When we compare he price differences of he marke o he ransacion cos calculaed from he model, we find ha he model performs well in erms of he ransacion coss esimaion. A furher invesigaion of his model will be o apply i o

11 11 commodiy markes wih large sample observaions. These fuure works may validae our Mone Carlo resuls and give us a beer esimae of he ransacion coss. References Aigner, D. Lovell, C. A. K. and Schmid, P. (1977), Formulaion and esimaion of sochasic fronier producion models, Journal of Economerics, Vol.6, No.1 (July), pp Greene, W. (003), Economeric analysis, 5 h ed., Prenice Hall, New York. Marshall, A., (1961), Principles of economics 35, Variorum ediion. Spiller, P.T. and Huang, C. J. (1986), On he exen of he Marke: Wholesale gasoline in he norhern Unied Sae. Journal of indusrial economics, Vol. XXXV, No.. (December). R Developmen Core Team (008), R: A language and environmen for saisical compuing. R Foundaion for Saisical Compuing, Vienna, Ausria. ISBN , URL hp:// Sigler, G.J and Sherwin, R.A. (1985), The Exen of he marke. Journal of Law and Economics, Vol.8, No.3. (Oc.), pp Appendix