AN ASSESSMENT OF DYNAMIC BEHAVIOR IN THE U.S. CATFISH MARKET: AN APPLICATION OF THE GENERALIZED DYNAMIC ROTTERDAM MODEL

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1 AN ASSESSMENT OF DYNAMIC BEHAVIOR IN THE U.S. CATFISH MARKET: AN APPLICATION OF THE GENERALIZED DYNAMIC ROTTERDAM MODEL Andrew Muhammad Department of Agrcultural Economcs, Msssspp State Unversty, PO Box 5187, Msssspp State, MS 39762; Phone: (662) ; E-mal: Kethly G. Jones U.S. Department of Agrculture, Economc Research Servce, 1800 M Street NW, Washngton DC; Phone: ; Emal: kones@ers.usda.gov Selected Paper prepared for presentaton at the Southern Agrcultural Economcs Assocaton Annual Meetng, Atlanta, Georga, January 31- February 3, 2009 Copyrght 2009 by Andrew Muhammad. All rghts reserved. Readers may make verbatm copes of ths document for non-commercal purposes by any means, provded that ths copyrght notce appears on all such copes. 1

2 ABSTRACT: Dynamc demand systems have been employed n a number of studes to account for habt formaton and nventory adustments n demand. Few studes have attempted to provde a theoretcal foundaton for the dynamc demand structures employed. Recently, Bushehr (2003) showed how a generalzed dynamc Rotterdam model could be derved from the neoclasscal ntertemporal utlty maxmzaton problem; however, no emprcal applcaton s provded n hs study. Ths paper provdes an emprcal applcaton of the generalzed dynamc Rotterdam model to the demand for processed catfsh products n the U.S. The two-perod dynamc Rotterdam model explaned a sgnfcant amount of the varaton n U.S. catfsh demand and was preferred to the one-perod and statc models. Estmates suggest that buyers adust short-run nventores such that the past sales negatvely affect current sales. Gven nventory adustment behavor, demand was relatvely more nelastc n the long-run. I. INTRODUCTION Dynamc demand systems have been employed n a number of studes to account for habt formaton and nventory adustments n demand (Sexauer, 1977; Blancfort and Green, 1983; Pollak and Wales, 1992; Arnade and Pck, 1994; Balcombe and Davs, 1996; Karaganns, Katrands and Velentzas, 2000). The conventonal approach has been to nclude lag terms as demand determnants or to employ tme seres methods (e.g. an error correcton model). Few studes have attempted to provde a theoretcal foundaton for the dynamc demand structures employed. An excepton s Brown and Lee (1992); however, they employed a method (translaton) that s typcally used to ncorporate any non-prce/ncome varable (.e. advertsng) nto a demand system. More recently, Bushehr (2003) showed how a generalzed dynamc Rotterdam model could be derved from the neoclasscal ntertemporal utlty maxmzaton problem; however, no emprcal applcaton s provded n hs 2

3 study. Ths paper provdes an emprcal applcaton of the generalzed dynamc Rotterdam model developed by Bushehr (2003) to the demand for processed catfsh products n the U.S. II. MODEL DERIVATION Bushehr (2003) shows how a generalzed dynamc Rotterdam model can be derved from the consumer s ntertemporal utlty maxmzaton problem. In ths secton we provde the mathematcal dervatons; however, readers are referred to Bushehr (2003) for the complete theory. Gven the ntertemporal utlty maxmzaton problem we can defne the optmal demand for the th good at tme t as follows: q ( t) = g ( x( t), p( t), h ( t)). (1) q () t s the quantty of good ; g denotes the demand functon; x() t s consumer expendtures; p() t s an n-vector of prces where n denotes the total number of goods wthn the consumer s choce set; and h() t s an n-vector of stock of habts. All are specfed at tme t. The above specfcaton requres an addtonal stage n the consumer budgetng process. The conventonal utlty tree approach assumes that consumers frst allocate total expendtures across product groups and then allocate group expendtures on goods wthn groups (Thel, 1980; Deaton and Muellbauer, 1980). To arrve at equaton (1), t must also be assumed that at the ntal stage of the budgetng process, consumers allocate lfetme wealth to specfc tme perods and that expendtures are allocated across goods (or groups) wthout reconsderng the ntertemporal optmzaton problem. Otherwse, demand at tme t would be a functon of lfetme wealth and the statc condtons mpled by consumer demand theory would not hold (Bushehr, 2003). 3

4 Dfferentatng Equaton 1 wth respect to tme yelds: g g g q = x + p + h. (2) x() t p () t h () t n n = 1 = 1 Note that for any varable y, y = d()/d y t t. If we dvde both sdes of equaton (2) by q () t, and f we multply the frst, second and thrd terms on the rght hand sde by x()/ t x() t, pt ()/ pt () and ht ()/ ht (), respectvely, wth some manpulaton we get the followng growth equaton: n n q x p h = η + η + φ. (3) q () t x() t p() t h () t = 1 = 1 Note thatη s the expendture elastcty and η s the uncompensated prce elastcty. φ = ( g / h )( h / g ) s the responsveness of the quantty demanded for good to changes n the stock of habt for good. The last step s to substtute the Slutsky equaton for the uncompensated prce elastcty and to multply both sdes of Equaton 3 by the th budget share w = p q / p q. 1 Ths yelds the followng demand equaton: * = n h + n + p n q x p w wφ wη w wη. (4) q() t = 1 h() t xt () = 1 pt () =1 pt () Wthout the stock of habts term φ ( h / h ( t )) equaton (4) s smlar to the w absolute prce verson of the Rotterdam model n Thel (1980) and Thel and Clements (1987) where the term n brackets s the change n real expendtures and the last term s the mpact of prces on quantty demanded. The dynamc Rotterdam model s used n estmatng the demand for U.S. catfsh products. Thel (1980) and Thel and Clements (1987) show that the statc Rotterdam model s a theoretcally separable functonal form, that s f product groups are separable (weak or strong), the Rotterdam model s suffcent for representng the demand for goods wthn a sngle product 4

5 group. We assume the same holds true for the dynamc Rotterdam model whch should be the case f lfetme wealth s pre-allocated (Bushehr, 2003). To put equaton (4) n emprcal form, we replace contnuous changes wth dscrete tme changes. Thel (1987) and Bushehr (2003) suggest the one-perod log dfference whch s used n most demand studes. Therefore, we approxmate the changes n quanttes and prces as follows: qt = log qt log qt 1 q / q( t ) and t t t 1 p = log p log p p / p( t). The term n brackets n equaton (4) s equal to the Dvsa volume ndex (Thel, 1980). We replace ths term wth a dscrete measure of the Dvsa volume ndex Q t where n n n Q = w q = x w p x / x( t) w ( p / p( t )). (5) t = 1 t t = 1 = 1 Bushehr (2003) also suggests the followng habt specfcaton for dscrete tme perods: n p n * = + k q t k = 1 h () t k= 1 = 1 h φ α α (6) where αk q k t k s a dstrbuted lag of the quanttes consumed n logdfference form. Gven equatons (5) and (6), the emprcal verson of the dynamc Rotterdam model s expressed as follows: p n n * t t = + k t k + t + t + t k= 1 = 1 = 1 w q γ γ q θ Q π p ε (7) * * where wt = 0.5( wt + wt 1) ; w = p q / p q ; γ = w α γ = w α ; θ = w t η ; t t t t t t ; k t k and π = w η. * t * γ, γ k, θ and π are parameters to be estmated and ε t s a random dsturbance term. Equaton (7) suggests that the effects of habt on consumpton s captured by past consumpton where consumpton of a 5

6 partcular good depends not only on present expendtures and prces but also on the past consumpton of that good and all other related goods. γ * Demand theory requres the followng restrctons on parameters: = 0, k γ = 0 for all and k, θ = 1, π = 0 (addng up); π = 0 (homogenety); π = π (symmetry); and Π n n= π s negatve semdefnte (negatvty). The short-run condtonal expendture and compensated prce elastctes (Hcksan) are defned as θ / and π / w respectvely. The short-run w uncompensated prce elastcty (Marshallan) s defned as π / w θ w / w (Seale, Sparks and Buxton, 1992). The long-run expendture elastcty, compensated prce elastcty, and uncompensated prce elastcty are respectvely defned as (Bushehr, 2003) L θ η = w ( γ k ) π * L η = w k k ( γ k) (8) (9) η L π = w θ ( γ ) w k ( γ k ) k k w. (10) III. EMPRICAL RESULTS Monthly dsaggregated catfsh quanttes at the processor level measured n 1,000 pounds and U.S. processor prces measured n dollars per pound were provded by the USDA, Natonal Agrcultural Statstcal Servce (NASS). The tme perod for the data was January 1996 to January Processed catfsh was dsaggregated nto sx products: fresh whole fsh, fllets and other; frozen whole fsh, fllets and other. The other category ncluded steaks, nuggets and 6

7 other products not elsewhere specfed. Varable statstcs are presented n Table 1. Gven that a dynamc model of lag-length k s nested wthn a model of lag-length k + 1, a lkelhood rato (LR) tests can be used to test for the approprate lag (Brown and Lee, 1992). Tests results are presented n Table 2. LR tests ndcated that the statc Rotterdam model was reected n favor of the one-perod lag model, and that the one-perod lag model was reected n favor of the two-perod lag model. However, there was lttle dfference between the log-lkelhood values for the three-perod and two-perod lag models. Therefore, we assume a maxmum lag length of two months. Addtonally, homogenety and symmetry faled to be reected n the 2-perod lag model. All results that follow assume a two-month lag, homogenety and symmetry. Estmaton of the dynamc Rotterdam model was accomplshed usng the LSQ procedures n TSP (verson 5.0) (Hall and Cummns, 2005). Prelmnary results ndcated that the constant term should be excluded from the model. Thus, equaton (7) was estmated wthout a constant term whch suggests that there were no trendng varables (n levels) that determned catfsh demand (Seale, Marchant and Basso, 2003). Overall, the dynamc Rotterdam model performed well. All expendture effects θ (margnal shares) were postve and sgnfcant at the 1% level. These estmates reflect how a dollar ncrease n real expendtures s allocated across the sx products. Gven that fllets (fresh and frozen) are the more popular products, ther margnal share estmates were relatvely larger. All own-prce effects π were negatve and sgnfcant at the 1% level (except frozen whole fsh). A number of cross-prce effects reflected a compettve relatonshp between products, partcularly between fresh fllets and frozen fllets (0.385). 7

8 The lag effects are presented n Table 4. Postve own-lag effects reflect habt formaton and negatve effects reflect short-run nventory adustments (Sexauer, 1977). Note that all own-lag effects are sgnfcant and negatve for all products suggestng nventory adustment behavor on the part of buyers. Gven the relatve durablty of frozen products, there own-lag effects were negatve n both perods whereas the own-lag effects for the fresh products were mostly sgnfcant n the frst perod only. The sgns and magntudes of the cross-lag effects depend on the relatonshp between products (substtutes versus complements) and the adustment behavor of buyers (habts versus nventores). For example, f any two products are substtutes (complements) and unrelated to all other goods, we would expect ther cross-lag effect to be postve (negatve) f buyers adust consumpton behavor based on short-run nventores. Lastly, the short-run and long-run expendture and prce elastctes are presented n Table 5. Gven nventory adustments, demand n the short-run was relatvely more elastc. The short-run expendture elastctes were close to unty for all products. In the long-run, the effect of expendtures on demand was sgnfcantly smaller. The short-run own-prce elastctes (Hcksan and Marshallan) ndcated that the demand for the fresh products was elastc and the demand for frozen fllets was also elastc. In the log-run, the demand for all products except fresh fllets was nelastc. Gven that fresh fllets are relatvely more pershable the mpact of short-run nventory adustments was relatve small resultng n smlar short-run and long-run own-prce elastctes. IV. SUMMARY & CONCLUSION Ths paper provdes an emprcal applcaton of the generalzed dynamc Rotterdam model presented by Bushehr (2003) whch was used n estmatng 8

9 dsaggregated catfsh demand. Test results showed sgnfcant nformaton when the statc model or 1-perod lag model was assumed. The 2-perod dynamc model explaned a sgnfcant amount of the varaton n U.S. catfsh demand. The lag estmates suggest that buyers adust short-run nventores such that sales n the prevous two perods have a negatve effect on current perod sales. Gven nventory adustment behavor, the demand was relatvely more nelastc n the long-run. 9

10 NOTES 1 * * The Slutsky equaton s defned as η = η η w, where η s the compensated prce elastcty and w = p q / p q s the budget share for good. REFERENCES Arnade, C., Pck, D. and Vasavada, U. (1994) Testng dynamc specfcaton for mport demand models: the case of cotton, Appled Economcs, 26, Balcombe, K. G. and Davs, J. R. (1996) An applcaton of contegraton theory n the estmaton of the almost deal demand system for food consumpton n Bulgara, Journal of Agrcultural Economcs, 15, Blancfort, L. and Green, R. (1983) An almost deal demand system ncorporatng habts: an analyss of expendtures on food and aggregate commodty groups, The Revew of Economcs and Statstcs, 65, Brown, M. G. and Lee, J. Y. (1992) A dynamc dfferental demand system: an applcaton of translaton, Southern Journal of Agrcultural Economcs, 24(2), Bushehr, M. A. M. (2003) Dynamc generalzaton of the Rotterdam model, Appled Economcs Letters, 10, Deaton, A. and Muellbauer, J. (1980) Economcs and Consumer Behavor, Cambrdge Unversty Press, Cambrdge. Hall, B. H. and Cummns, C. (2005) TSP Internatonal Reference Manual Verson 5.0, TSP Internatonal, Palo Alto, Calforna. Karaganns, G., Katrands, S. and Velentzas, K. (2000) An error correcton almost deal demand system for Greece, Agrcultural Economcs, 22, Pollak, R. A. and Wales, T. J. (1992) Demand System Specfcaton & Estmaton, Oxford Unversty Press, New York. Seale, J. L., Marchant, M. A. and Basso, A. (2003) Imports versus domestc producton: a demand system analyss of the U.S. red wne market, Revew of Agrcultural Economcs, 25, Seale, J. L., Sparks, A. L., and Buxton, B. M. (1992) A Rotterdam applcaton to nternatonal trade n fresh apples: a dfferental approach, Journal of Agrcultural and Resource Economcs, 17, Sexauer, B. (1977) The role of habts and stocks n consumer expendture, Quarterly Journal of Economcs, 91, Thel, H. (1980) The System-Wde Approach to Mcroeconomcs, The Unversty of Chcago Press, Chcago. Thel, H. and Clements, K. W. (1986) Appled Demand Analyss: Results from System-Wde Approaches, Ballnger Publshng Company, Cambrdge. 10

11 Table 1. Descrptve Statstcs for U.S. Processed Catfsh: January 1996 January 2007 Fresh Frozen Prce ($/lb) Whole Fllet Other Whole Fllet Other Mean Standard Devaton Mnmum Maxmum Monthly Quantty (1000 lbs) Mean 3,297 4,836 1,300 1,138 9,705 3,917 Standard Devaton , Mnmum 2,426 3, ,296 2,384 Maxmum 4,467 6,815 2,156 1,500 12,362 5,364 Expendture Share Mean Standard Devaton Mnmum Maxmum Table 2. Lkelhood Rato Tests Results Models Lag structure Log-lkelhood Value 3-perod 2, LR Statstc P-value 2-perod 2, (30) 1-perod 2, (30) Statc 2, (30) Economc constrants (2-perod model) Unrestrcted 2, (18) Homogenety 2, (5) Symmetry 2, (10) All models have homogenety and symmetry model. The number of restrctons s n parenthess. 11

12 Table 3. Condtonal Demand Estmates for Processed Catfsh Prce Coeffcents Products Fresh Frozen Fresh Whole Fllet Other Whole Fllet Other Whole (.017) *** (.028) * (.009) *** (.012) (.036) *** (.008) Fllet (.090) *** (.019) (.027) Other (.008) *** (.007) π (.095) *** (.018) (.023) (.005) Frozen Whole (.018) * (.029) * (.006) Fllet (.125) *** (.025) Margnal Share θ (.004) *** (.009) *** (.003) *** (.003) *** (.012) *** Other (.016) *** (.008) *** Equaton R System R 2 =.965 a Asymptotc standard errors are n parentheses. Homogenety and symmetry are mposed. *** Sgnfcance level =.01; ** Sgnfcance level =.05; * Sgnfcance level =.10 12

13 Table 4. Dynamc Adustment Estmates Lag Coeffcents γk Products Fresh Frozen Whole Fllet Other Whole Fllet Other Fresh One-perod Lag ( q 1 ) Effects Whole (.074) *** (.028) Fllet (.164) ** (.061) * (.264) Other (.049) Frozen Whole (.052) *** (.019) Fllet (.216) *** (.081) Other (.139) (.119) * (.109) * (.015) t (.241) (.018) *** (.080) *** (.072) (.052) (.038) ** (.034) *** (.084) (.010) (.025) (.084) * (.076) *** (.011) *** (.027) (.352) (.224) (.321) (.045) *** (.112) *** (.207) * (.029) *** (.073) *** Two-perod Lag ( q t 2 ) Effects Fresh Whole Fllet Other Whole Fllet Other Whole (.073) *** (.028) (.118) (.102) (.016) * (.037) *** Fllet (.162) Other (.048) (.062) (.018) Frozen Whole (.051) *** (.020) Fllet (.215) ** (.082) Other (.139) (.053) (.261) (.078) (.226) (.067) (.034) ** (.082) (.010) (.025) (.083) *** (.072) *** (.011) ** (.026) (.346) (.223) (.301) * (.046) *** (.109) (.195) ** (.030) Asymptotc standard errors are n parentheses. *** Sgnfcance level =.01; ** Sgnfcance level =.05; * Sgnfcance level = (.070) *** 13

14 Table 5. Short-run and Long-run Demand Elastctes Short-run Elastctes Long-run Elastctes Hcksan Marshallan Hcksan Marshallan Expendture Expendture Fresh own-prce own-prce own-prce own-prce Whole Fllet Other Frozen Whole a b a a Fllet Other a Sgnfcance level =.10; b Sgnfcance level =.05. All others are sgnfcant at the.01 level or lower. 14