HUPPERT: ANALYSIS OF UNITED STATES DEMAND FOR FISH MEAL 



curve. In this case the estimated parameters are 

 said to suffer from simultaneity bias. 



The general statistical problems associated 

 with estimation of individual structural relation- 

 ships in a simultaneous equation system were 

 first examined by Haavelmo (1943). Development 

 of appropriate statistical methods for estimating 

 simultaneous equation systems has been a major 

 area of research for econometricians over the last 

 two decades (Kmenta 1971). In estimating the 

 demand curve for fish meal, however, direct re- 

 gression estimates seem appropriate, because 

 most of the observed variations in annual fish 

 meal supplies are due to exogeneous shifts rather 

 than price-induced movements along a stable 

 supply curve. Production offish meal is subject to 

 wide fluctuations due to uncontrolled variations 

 in the fish stocks exploited (Kolhonen 1974). At 

 the same time, formula feed and poultry indus- 

 tries have remained relatively stable during the 

 last 20 yr except for the secular growth accounted 

 for in the analysis. Under conditions in which the 

 random shifts in supply are much greater than 

 the corresponding shifts in demand, the ordinary 

 least squares procedure results in no significant 

 simultaneity bias (Rao and Miller 1971). 



effect of a price change may be drawn out over 

 several periods of time. A fairly simple model for 

 representing a lagged response is the "partial ad- 

 justment model" originally developed by Nerlove 

 (1958). Corresponding to any given level of the 

 independent variable, p, there is an optimal or 

 desired level of the dependent variable q. For a 

 demand function with one independent variable, 

 the level of demand fully adjusted to input prices 

 by formula manufacturers represents the desired 

 level offish meal usage: 



qf=bp^ +u, 



(4) 



where the superscript d signifies desired level. 



Because purchasers of meal cannot im- 

 mediately adjust to this desired level of usage, the 

 demand Equation (4) is not directly observable. 

 By assuming a simple structure to the adjust- 

 ment process, however, an estimable equation is 

 obtained. The partial model assumes that a fixed 

 percentage of the adjustment to desired level is 

 made each year. This introduces the difference 

 equation 



Qt -9m = yiqf-qt.i) 



(5) 



Lagged Response Mechanisms 



The use of annual price and quantity data for 

 estimating the demand function requires that the 

 response to a change in price occurs rather 

 rapidly, at least within a period of time much 

 shorter than a year. Since most domestic formula 

 feed manufacturers employ professional nutri- 

 tionists and cost-minimizing computer routines in 

 calculating formulas, the response to changes in 

 the vector of prices is probably rapid. If so, each 

 annual quantity consumed may be assumed to 

 represent at least approximately an equilibrium 

 demand response to the set of independent vari- 

 ables. The assumption of rapid response and 

 equilibrium approximation, however, has not 

 been directly verified. In the interests of rigor it is 

 useful, therefore, to consider alternative assump- 

 tions. 



A lagged response to a change in price may 

 occur due to rigidities in mixing procedures or 

 personnel, inventory management problems, or 

 time lags in renegotiating contracts for supply of 

 input or sales of products. If any of these factors 

 results in a sluggish response in the substitution 

 between fish meal and other protein meals, the 



Solving this for q^ and substituting from Equation 

 (4) yields 



q=b yp, ^ (1 - y)(7,^i + yu,. 



(6) 



The adjustment parameter, y, must be a positive 

 number <1. Larger values of y imply more rapid 

 adjustment to changes in the independent vari- 

 able. The impact of a unit change in p, is distrib- 

 uted over time in an exponentially decaying fash- 

 ion with successive annual changes in q being 

 equal to by, by (1 - y), by [1 - y)^, and so forth. 

 The ultimate change in q due to a change in p is 



Iq =blp lyil 



y)J = blp 



(7) 



where j = lag. The elements in the sequence 

 under the summation sign are all positive frac- 

 tions, and sum to one, so that the sequence can be 

 treated like a probability distribution. Each ele- 

 ment represents the percentage of the total effect 

 occuring in year t, and the mean of the distribu- 

 tion, ( 1 - y)/y, represents the mean lag in the 

 adjustment process. Distributed lag models like 

 that in Equation (4) result from other conceptual 

 models such as models of expectations formation 



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