FISHERY BULLETIN: VOL. 78, NO. 2 



3) Annual domestic price of corn and annual 

 domestic price of soybean meal are introduced 

 as complementary and substitute product 

 prices. 



4) The trend in aggregate demand over time is 

 accounted for by the aggregate poultry and 

 egg production in the United States. 



All of the variables expressed in dollars are 

 deflated by the Wholesale Price Index to elimi- 

 nate spurious correlations caused by the inflation- 

 ary trend. 



Functional Form 



Demand studies typically utilize least squares 

 regression methodology with either a linear or a 

 log-linear equation. As noted by Chang (1977), 

 however, there is no a priori reason to choose one 

 of these forms. Each form imposes some fairly 

 strict conditions upon the characteristics of the 

 demand function which may contradict theoreti- 

 cal considerations or actual experience. Linear 

 equations imply that the elasticity of demand 

 with respect to any independent variable is a de- 

 creasing function of that variable; a log-linear 

 equation implies constant elasticities. Chang 

 suggests that the income elasticity of demand for 

 meat should fall with rising income. A similar 

 consideration applies to fish meal demand. At low 

 prices, feed manufacturers would use near 

 maximum amounts of fish meal allowable and 

 could easily substitute soybean meal for fish 

 meal. With relatively high fish meal prices, feed 

 manufacturers would use a smaller proportion of 

 fish meal, but as price rises further it would be 

 increasingly difficult to maintain desired quan- 

 tities of lysine and methionine by substitution of 

 soybean meal. Thus it is clearly unwarranted to 

 rule out increasing price elasticity through a 

 priori choice of functional form. 



The function to be fitted by regression analysis 

 can be chosen by determining the appropriate 

 transformation of variables for the linear least 

 squares procedure. The log-linear transformation 

 is a special case of a parametric family of trans- 

 formations introduced by Box and Cox (1964). 

 The parameter defines the transformation 



function is expressed as 



r* = 



= ix' - l)/\. 



(1) 



Equation (1) is linear for A = 1, and becomes 

 logarithmic as \ approaches zero. The demand 



270 



,* — 



6o + ^1 •''^1* + . . . + bf^Xk* + u 



(2) 



where q is the quantity demanded, thex's are the 

 independent variables affecting demand, u, is a 

 stochastic error term, and the 6, and k are 

 parameters to be determined. The superscript * 

 indicates that the variable has been transformed 

 as in Equation (1). 



Price elasticity of demand is defined as the ab- 

 solute value of the ratio of percentage change in 

 quantity demanded to percentage change in 

 price. Assuming that the first independent vari- 



able is the price, E 

 tion (2) we get 



dq 



(t) 



E = \bi\(q/x^: 



From Equa- 



ls) 



The elasticity defined in Equation (3) is an in- 

 creasing function of x^ when A>0, and is a de- 

 creasing function of x^ when \<0. Thus the esti- 

 mate of the transformation parameter X provides 

 a test of whether the price elasticity increases, 

 decreases, or remains fixed along the demand 

 curve. 



Simultaneity Bias 



In economic theory, the supply and demand 

 curves interact to determine the market price. 

 Over a period of time, shifts in both supply and 

 demand factors cause the market price and ob- 

 served quantities of products to vary. Without 

 these shifts, only one price and quantity would be 

 observed, making it impossible to estimate a de- 

 mand or supply curve. When the demand curve 

 remains stable, the observed price-quantity pairs 

 "trace out" the demand curve with, of course, 

 some stochastic error, and a regression analysis 

 will result in a demand curve estimate. When the 

 supply curve remains stable, the observed data 

 will fall along the supply curve, and a regression 

 analysis of the price-quantity relationship results 

 in a supply curve estimate. If shifts in both de- 

 mand and supply occur, the resulting data will 

 not unambiguously identify either of these two 

 curves, and an ordinary least squares regression 

 will generally result in a set of parameters re- 

 flecting neither the supply curve nor the demand 



