Retail to Farm Link! for a Complete Demand System of Food Commodities Michael K. Wohlgenant Richard C. Haidacher

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Retail to Farm Linkage for a Complete Demand System of Food Coimnodities. By Michael K. Wohlgenant and Richard C. Haidacher. Commodity Economics Division, Economic Research Service, U.S. Department of Agriculture. Technical Bulletin No. 1775.

Abstract

A new conceptual model is needed to estimate the impact of demand, supply, and marketing margins on retail food prices and farm prices. This model should be theoretically consistent with group behavior. The present study critiques past approaches to modeling farm-to-retail price spread behavior, to emphasize the need for a larger framework for the analysis of retail-to-farm linkages. This enlarged model consists of reduced-form retail and farm price equations derived from the behavior relationships which characterize aggregate output supply and input demand responses of the marketing sector. The model is then applied empirically to a set of eight food commodities. The results are^ generally consistent with theory, which indicates compatibility with competitive behavior and an aggregate constant-returns-to-scale technology of food processing. The results also show, in general, the, need to allow for variable factor proportions between farm and marketing «inputs. Farm-level derived demand elasticities (except for poultry) are more than 40 percent larger than those obtained by assuming fixed proportions.

Keywords: farm-level demand interrelationships, food commodities, input substitution, marketing margins, retail-to-farm price linkages

Acknowledgments

This research is supported through USDA cooperative agreement No. 58-3J23-4- 00278, entitled "Development of a Conceptual and Empirical Framework for the Retail-to-Farm Linkage in a Complete Demand System of Food Commodities."

Michael K. Wohlgenant is professor of economics and business, North Carolina State University. Richard C. Haidacher is an agricultural economist with the U.S. Department of Agriculture, Economic Research Service. Douglas Beach, Eric Brainich, and Jane Hopkins provided able research assistance at various phases of this project. James R. Blaylock, Jean-Paul Chavas, Bruce Gardner, and Kuo S. Huang provided useful comments on earlier drafts.

Washington, DC 20005-4788 December 1989

Contents

Summary iii

Introduction 1

Evaluation and Critique of Alternative Approaches to Modeling Marketing Margins 1

Traditional Approaches to Modeling Marketing Margins 2 Demand for Farm Output as a Factor of Production 9

A Conceptual and Empirical Framework for Retail to Farm Price Linkage 15

A Conceptual Model of Retail to Farm Price Linkage 15 Empirical Specification of Retail to Farm Price Linkage 22

Empirical Application to a Complete System of Food Commodities 24

Econometric Results 25 Flexibilities and Elasticities 30

Conclusions and Implications 32

References 35

Appendix A: Review of Alternative Derivations of Derived Demand 39

Appendix B: Data Sources and Derivations 44

11

Siiimnary

The report develops a conceptual and empirical framework for estimating the impacts of changes in demand, supply, and food marketing margins on retail food prices and farm prices. A critique is given of analytical approaches to modeling farm to retail price spread behavior, and this points up the limitations of existing approaches, and shows the advantages of viewing the farm product separately as a factor of production in food processing. Modeling quantity of the farm product as an input in food processing is here attempted, which has the advantage that a set of theoretically consistent behavioral relationships characterizing marketing group behavior can then be derived through aggregating output supply and input demand functions of individual competitive firms.

Direct measures of retail quantities for disaggregated individual food commodities are generally unavailable. For that reason, the conceptual model was formulated to consist of reduced-form behavioral equations for retail and farm prices. Under this approach, structural parameters of the equations describing aggregate marketing group behavior are underidentified. However, with given values for the retail demand elasticities, these structural parameters can be identified, and unique values for these parameters can be obtained by imposing the parametric restrictions of homogeneity and symmetry of the output supply and input demand functions. Additional restrictions are derived for a constant-returns-to-scale aggregate production function of food processing.

The methodology is illustrated by a model of retail-to-farm price linkage applied empirically to a set of eight disaggregated food commodities over the time period 1955-83. The commodities analyzed are beef and veal, pork, poultry, eggs, dairy products, fresh fruit, fresh vegetables, and processed fruits and vegetables. Separate reduced-form retail and farm price equations are estimated for each of these commodities. Homogeneity and syrmnetry restrictions are imposed on the reduced-form parameters. Extraneous estimates of retail demand elasticities are used in order to identify the structural parameters of the marketing behavior equations and to provide theoretically consistent linkages between the retail demand and derived demand parameters. With the exception of fresh fruits and processed fruits and vegetables, the econometric results are found to be consistent with theory. The statistical results generally indicate that the restrictions of symmetry and constant- returns-to-scale aggregate technology cannot be rejected. This means that food marketing behavior can be generally characterized as competitive with constant returns to scale in food processing-marketing.

Flexibilities and elasticities of demand for farm products are derived, and these take into account total effects of exogenous demand and supply shifters. These estimates, consistent with previous findings, indicate that own-price elasticities are less than 1 in absolute value, all income flexibilities are positive, and the majority of cross-price flexibilities and elasticities indicate a substitution pattern among farm outputs.

Finally, the own-price elasticities for farm outputs derived from the conceptual model developed in this study are compared with those derived using the traditional methodology based on fixed input proportions where insignificant input substitution was found. The comparison indicates that allowing for variable factor proportions results in own-price elasticities at least 40 percent larger in absolute value.

iii

Retail to Farm Linkage for a Complete Demand System of Food Commodities

Michael K. Wohlgenant Richard C. Haidacher

Introduction

This report develops a conceptual and empirical framework of the retail to farm linkage of a complete demand system of food commodities. Justification for this research is twofold. First, the U.S. Department of Agriculture's Economic Research Service has a responsibility to develop estimates of food marketing margins for public information, for policy evaluation, and for determining the impact of economic and other factors on retail food prices and farm prices. Second, although the USDA currently has empirical estimates of theoretically consistent consumer demand response and farm supply response parameters for all major food groups, missing is a consistent framework in which to measure farm to retail marketing margins and to evaluate effects on retail and farm prices of changes in retail demand, farm product supply, and costs of food manufacturing and distribution. The objective of this study is to develop a conceptual and empirical framework for evaluating impacts of changes in demand, supply, and marketing margins which has the same consistency of analysis from farm to consumer as current research has provided separately at the consumer and supply levels.

The specific objectives of the research are to:

(1) Develop a consistent theoretical framework between consumer demand for food commodities, farm level demand for those commodities, and relationships to other inputs in the marketing process.

(2) Evaluate the usefulness of the developed framework through empirical estimation of behavioral relationships explaining farm to retail linkages of a complete demand system of food commodities.

Evaluation and Critique of Alternative Approaches to Modeling Marketing Margins

This section reviews and critiques existing analytical approaches to modeling price spread and farm level demand for food commodities. In the first subsection, significant literature on traditional approaches to modeling retail to farm demand linkage is evaluated. The main focus of this subsection is on conceptual deficiencies stemming from restrictive assumptions regarding the nature of the food processing technology and problems of aggregation over products and firms. The second subsection reviews significant literature on derived demand, which views the demand for farm output more generally as demand for a factor of production in food processing. The main focus of this subsection is on analytical approaches which yield testable implications of theory for shortrun industry factor demand and output supply response behavior.

Traditional Approaches to Modeling Marketing Margins

The main conceptual framework used in the past to model marketing margins and farm level demand for food commodities is the theory of joint demand. This theory starts with the notion that demand for the final product is a joint demand for all the factors of production (Marshall, chapter 6; Friedman, chapter 7). In the extreme case of fixed proportions between the final product and inputs, a derived demand schedule for a given factor is obtained by subtracting per-unit costs of all other factors from the demand price for the final product. In the case of food, derived demand for the farm product is obtained by subtracting per-unit processing-marketing costs from the retail demand function for the product (Tomek and Robinson, chapter 6). The relationship between these two demand curves in the special case where the supply function of processing-marketing inputs is perfectly elastic is shown in figure 1. Farm output is expressed in equivalent units to retail product (for example, 2.4 pounds of beef at farm level equals 1 pound at retail, 4.8 pounds of beef at farm level equal 2 pounds at retail, etc.) so that farm price is price per equivalent unit of the retail product (for example, cents per 2.4 pounds of beef at the farm level required to equal 1 pound of beef at retail). The derived demand relation at the industry level is a partially reduced-form relation, showing how the equilibrium farm price changes as retail demand changes at various quantities. The equilibrium price of the farm product is obtained where the derived demand curve intersects the producer supply curve. And given the assumption of fixed input proportions, intersection of the equilibrium quantity with the retail demand curve determines the equilibrium price of the retail product. The marketing margin is defined as the difference between the (equilibrium) retail price and the (equilibrium) farm price, per equivalent unit. So, in the case of fixed

Price per unit of retail product

margin

Figure 1

Illustration of industry derived demand for farm product and marketing margin with fixed input proportions

s, (Farm product supply)

3 (Supply of w marketing inputs)

D,. (Retail demand)

f (Derived demand for farm product)

Quantity (Units of retail product and farm product equivalent)

factor proportions, derived demand for the farm output can be obtained directly by subtracting the marketing margin from the retail demand function.

In some cases, marketing margins will remain constant as the quantity of the commodity marketed changes, but in other cases margins will vary. Given fixed input proportions, the way in which the margin changes as the quantity marketed changes will depend on the nature of the supply function of marketing inputs. If the supply curve is perfectly elastic, then the margin will remain unchanged as the quantity marketed changes. And, the derived demand curve will be parallel to the consumer demand curve. This is the situation portrayed in figure 1.

When the supply curve of marketing inputs is upward sloping, the margin relationship rises as the quantity marketed increases. And, the derived demand curve for farm output falls more rapidly than the consumer demand curve. An equivalent way of saying this is that, with the consumer demand curve stationary, the margin would be negatively related to retail price. This situation is shown in figure 2 where the margin falls from MQ to M^^ as retail price rises from Pj.° to P^.^.

However, empirical evidence is frequently inconsistent with this predicted relationship. More often than not, margins show a negative relationship to quantity and a positive relationship to retail price (Buse and Brandow, Waugh, George and King; Tomek and Robinson, chapter 6). According to the conceptual framework of derived demand based on fixed input proportions, this behavior of marketing margins ought to result from a negatively sloped supply curve of marketing inputs, as shown in figure 3. But, such a result, while compatible with longrun industry competitive pricing with external economies to the marketing sector, is inconsistent with shortrun competitive pricing. Thus,

Figure 2

Relationship between the marketing margin and volume marketed when the supply curve of marketing inputs is upward sloping

'■;

Price "? per unit of retail product

"}

Q Quantity (Units of retail product and farm product equivalent)

Price per unit of retail product

Figure 3

Relationship between the marketing margin and volume marketed when the supply curve of marketing inputs is downward sloping

Q. Quantity (Units of retail product and farm product equivalent)

the simple conceptual framework of derived demand described above, seems deficient in its ability to explain observed relationships between marketing margins and volume marketed.

Another approach to modeling marketing margins and farm level demand for food commodities is based on the concept that price spread behavior depends on the pricing practices of market middlemen. This approach, summarized well in George and King (pp. 55-59), begins with the hypothesis that price spreads are a combination of absolute amounts and constant percentages of retail price. For a linear function,

(1) M = a + bP^,

where M = Pj. - Pf is the farm-retail price spread, Pj. is the retail price, and Pf is the farm price expressed in equivalent units to the retail product. Justification for this specification is primarily empirical. Dalrymple cites a number of studies documenting wholesale margin behavior as that of constant percentage markup, retail behavior as constant absolute behavior, and aggregate behavior as a combination of the two. Past studies by Thomsen and by Shepherd indicate margins consist of elements which are about one-half absolute amounts and one-half constant percentages. Buse and Brandow, in their analysis of marketing margins of 20 food commodities, found a positive and significant relationship between margins and retail price. Waugh (p. 20) indicates that, "Many studies of this matter in the Department of Agriculture suggest that price spreads are neither constant percentages nor constant absolute amounts, but somewhere in between the two." Finally, in their analysis of 19 food commodities, George and King find that the majority of the

commodities exhibit combinations of both constant absolute and constant percentage behavior. ^^

The effect of the type of margin behavior shown in equation (1) on derived demand at the farm level can be determined in a straightforward manner. George and King show that the elasticity of industry derived demand at the farm level can be obtained as the product of elasticity at the retail level and the elasticity of price transmission. That is,

(2) E^j = e^j . Hj,

where E^^ is the elasticity of farm-level demand for commodity i with respect to the farm price of commodity j , e^^ is the retail demand elasticity of commodity i with respect to the retail price of commodity j, and n^ is the elasticity of price transmission for commodity j, that is, the proportionate change in retail price divided by the proportionate change in farm price (Hildreth and Jarrett). From empirical estimates of margin behavior based on equation (1), farm-level derived demand elasticities can be derived directly from a knowledge of retail elasticities of demand, according to equation (2).

George and King (p. 60) show that, according to equation (1), elasticity of demand at retail can be expected to be higher than elasticity of demand at the farm level when the intercept is nonnegative. In the extreme cases of constant absolute margin and constant percentage margin, elasticity at the farm-level is less than the retail elasticity and equal to the retail elasticity, respectively (Waugh, George and King). Gardner (pp. 404-405) points out that this rel^ationship between the retail elasticity of demand and farm-level elasticity of demand depends on the assumption of fixed proportions between the quantities of the product at the farm and retail level. If input substitution occurs, then it is possible for demand at the farm level to be more price elastic than demand at the retail level.

The main problem with modeling margin behavior as a combination of absolute amounts and constant percentage is that it lacks theoretical justification. Gardner (p. 406) shows, "...that no simple markup pricing rule--a fixed percentage margin, a fixed absolute margin, or a combination of the two--can in general accurately depict the relationship between the farm and retail price." This is because, even if such an equation perfectly fits changes generated by shifts in the farm supply function, such a model may not account simultaneously for shifts in retail demand and in farm supply. Indeed, in more recent work, Wohlgenant and Mullen show that competitive price behavior implies that the elasticity of price transmission can vary systematically with changes in the quantity of the commodity processed and marketed. In their empirical application to beef, they find that the data support this proposition and that the markup pricing rule, such as that used by George and King in equation (1), is misspecifled.

It is often suggested, usually without formal justification, that markup pricing behavior such as that exhibited in equation (1) can be ascribed to imperfect competition. While this can be true under certain conditions, it need not hold generally. Consider the extreme case of pure monopoly in which

■^George and King's econometric estimates do not control for changes in processing costs. To the extent that they are correlated with retail price, these estimates will show an upward bias in the slope of the coefficient when farm price is regressed on retail price.

the firm produces where marginal revenue (MR) equals marginal cost (MC). Since MR = Pj. • (1 + 1/e) , where e is the elasticity of demand facing the firm, it follows that price behavior of a monopolist can be characterized as

(3) P^ = __L • MC. 1+e

Assume that the farm product is expressed in equivalent units to the retail product so that MC = Pf + MCP, where Pf is the farm price per unit retail product and MCP is marginal cost of processing the product. Substituting this expression for MC in (3) and solving this equation for the farm-retail price spread, M, as a function of the retail price yields

(4) M - MCP - (1/e) • P^.

Equation (4) has exactly the same form as equation (1) where a = MCP and b = -1/e. However, equation (4) reveals that both the intercept and the slope of this relationship are, in general, not constants. The intercept depends on prices of inputs and volume of output processed. Except in the special case when demand facing the firm is a constant elasticity demand curve, the slope of equation (4) will change when retail price changes. In fact, it is not even possible to predict whether the margin will rise or fall as retail price rises without prior knowledge of the curvature of the demand curve. This can be shown by noting that in the extreme case of a linear demand curve, equation (3) reduces to

Pj. " 2 • MC + intercept,

and equation (4) becomes

M = intercept -P^.,

which implies a negative relationship between the margin and retail price.

Cowling and Waterson and Berck and Rausser consider the implications of oligopoly behavior and monopolistic competition, respectively, for markup pricing behavior. The relationships they derive, however, imply the same general relationship between the margin and retail price as the simple monopoly example presented in equation (4).^ In other words, it does not seem possible to justify the markup pricing rule in equation (1) on the basis of these particular models of imperfect competition.

Moreover, without prior knowledge of the curvature of the demand curves facing individual firms in the industry, models of imperfect competition cannot yield unambiguous predictions on the relationship between the farm-retail price spread and retail price.

Despite the deficiency of the conceptual framework of derived demand outlined earlier in explaining observed relationships between marketing margins and retail prices, it is instructive to return to this framework to examine the implications for margin and derived demand behavior when the assumption of fixed input proportions is relaxed. Figure 4 is a modification of figure 1 to show what happens when other factors are substituted for the farm product when the supply curve of the farm product shifts to the left. Two assumptions are

^Berck and Rausser (p. 119) say that raw product prices are expected to have a negative effect on margins under monopolistic competition, but their equation (36) indicates the opposite relationship.

Figure 4

Illustration of effect of variable input proportions on the relationship between retail and farm prices

Price per unit of retail product

(Price/2.4 lb. farm product)

(Prlce/1.92 lb. farm product)

(Variable proportions)

D J (Fixed proportions)

Quantity

500 1000 (Units of retail product)

1200 2400 (Corresponding units of farm product--^^ )

960 2400 (Corresponding units of farm product--^^)

made for simplification. The supply curve for the farm product is assumed to be perfectly inelastic. The supply curve of marketing inputs is assumed to be perfectly elastic so that, at the initial equilibrium point A, the slope is the same for both the retail demand curve and the fixed-proportions derived demand curve. To make the illustration concrete, consider the case of beef where it is assumed initially that 2.4 pounds of the farm product are required to equal 1 pound of retail cuts. Thus, i:"'itially at point A the farm output of 2,400 is equivalent to 1,000 at retail and the initial equilibrium farm price PfO is the price per 2,4 pounds of beef at the farm level. Now suppose the farm product supply curve shifts to the left causing the farm price to to rise. Under the assumption of fixed proportions (D^"'-) , the farm price for 2.4 pounds of farm product will rise from Pf° to P^^. However, if marketing inputs can be substituted for the farm product, this means that fewer units of the farm product will be used for each unit of the retail commodity produced. For example, if the farm product-retail product ratio declines by 20 percent to 1.92, only 960 instead of 1,200 units of the farm product will be required to equal 500 units at retail (Df^) . But now processors will be willing to pay a maximum of P^^ for this new quantity (1.92 units) rather than P^^. (The farm price for a single unit of farm product is the same at P^^ and P^^) . Marketing costs per unit of the retail product are now larger as a result of substitution of marketing inputs for the farm product. Therefore, the effect of substitution of other factors for the farm product is to cause the derived demand curve, consistent with the initial equilibrium conditions, to rotate counterclockwise through point A.

An implication of variable factor proportions, therefore, is that derived demand for the factor in question becomes more elastic. This, of course, is one of the well-known determinants of the elasticity of derived demand for a factor of production (Marshall, chapter 6). With respect to margin behavior,

variable factor proportions also imply that the farm-retail price spread increases as the quantity processed declines. With a stationary retail demand curve, this is equivalent to saying that the price spread and retail price will be positively related. Therefore, substitution possibilities between the farm product and marketing inputs provides a theoretical justification for this observed margin behavior."^

Even though input substitution provides theoretical justification for observed margin behavior, there is a view in the profession that such substitution possibilities are quite limited, and are restricted to reduction in the amount of wastage and spoilage as the price of the farm product rises (see, for example, Tomek and Robinson, chapter 6). This view of input substitution, however, is too narrow. It ignores the fact that a firm can choose among one or more production processes, or technologies, at any time. For example, fresh produce could be shipped to market by truck, by rail, or by boat, depending on the particular locations of production and consumption and the distance to the market. For each mode of transport, different combinations of marketing services and farm product are available to produce the retail product, and so opportunities for input substitution can arise from switching between transport modes.

Even if all firms employ inputs in fixed proportions, we still should expect to observe input substitutability at the industry level. The reason is that firms, because of differences in firm size, often use different input proportions (through different production processes or activities) to produce similar products. Differences in firm size could arise from differences in information about market conditions and available technologies as well as differences in entrepreneurial capacity. Thus, as relative input prices change, input substitution can occur solely because of a change in the distribution of industry output among firms. Diewert (1981) provides a rigorous justification for the proposition that input substitution at the industry level is larger than at the firm level by demonstrating that the industry production function can be viewed as the asymptotic technology to the technologies of individual firms.

Another source of input substitutability is interproduct substitution. Many retail food products are not single commodities but are composites of several individual commodities. For example, beef is a composite commodity consisting of ground beef and several types of retail cuts. This means that in response to an increase in the price of cattle (the farm product) there exists an incentive for firms to reduce production of those commodities which use relatively more of the raw material (for example, ground beef), and to increase production of those commodities which use relatively less of the raw material (for example, sirloin steak). The overall effect then, is a substitution of other inputs in processing (labor, capital) for the raw material.

Another source of input substitutability is substitution of quality for quantity as the price of the farm product rises. This theory is explained well by Barzel. In this theory, commodities are viewed as bundles of

^This of course does not rule out the possibility that the price spread and retail price are negatively related. For any commodity, the net relationship between the price spread and retail price depends on how steeply sloped the supply curve of marketing inputs is.

characteristics. Suppose there are two sets of characteristics: a set of characteristics A, which is closely linked to the factor of production, and a set B, which uses relatively less of the factor. In response to a higher factor price, the market will react by reducing the output of the commodity, and by substituting toward production of those characteristics which use relatively less of the higher priced input. With respect to food, A might be viewed as those characteristics linked directly to the raw food input such as taste and nutritional content. The set of characteristics B would then consist of characteristics linked to market-produced services purchased by consumers. In this case, an increase in the price of the farm product could lead to a substitution of marketing inputs for the purchased raw material through a decrease in production of those characteristics linked closely to the raw product, and an increase in those characteristics which are tied closely to marketing services.

More generally, consumer demand for food-marketing services can be viewed as a joint demand for home-produced as well as purchased services (Waldorf). As a producing unit, the household also is a supplier of food-marketing services. In the context of household production theory (see, for example, Deaton and Muellbauer, chapter 10), the household can be viewed as purchasing raw food products and other materials, and combining these purchased market goods with household time to produce nonmarket goods, which are consumed directly by the household. This broader view of food processing and marketing suggests that opportunities for input substitutability between the raw food product and marketing services exist in the household as well as the processing industry. Thus, even with very limited substitutability in processing plants, significant substitution between the farm product and marketing inputs can occur because of substitution between household time and purchased food-marketing services.

In sum, while a number of theories are available to explain observed relationships between retail and farm product prices, the most fundamental explanation must certainly involve opportunities for input substitution between the farm product and marketing inputs. For this reason, the next subsection reviews significant literature on derived demand, which views demand for farm output more generally as demand for an input into food processing.

Demand for Farm Output as a Factor of Production

It is important to develop a conceptual framework for retail to farm price linkage, which views the quantity of the farm output as a separate factor of production in food processing. However, few studies have actually adopted this approach. Conceptually, retail price may be viewed as determined by the intersection of retail demand and supply of the product in question. In a similar fashion, farm price would be determined by the intersection of demand and supply for the product at the farm level. Only in the case of fixed proportions between the farm and retail product can estimates of the derived demand schedule be obtained by subtracting per-unit processing-marketing costs from the retail demand schedule. With variable factor proportions, it is necessary to have, in addition to retail demand, estimates of retail supply and farm-level demand for the product.

In this subsection, alternative approaches to modeling retail to farm price linkage, are reviewed and evaluated, in the context of market equilibrium models of the industry. In the review, particular attention is given to

recent developments in the theory of the firm and the industry which allow for diversity between firms. For the most part, the discussion is limited to models based on competitive behavior. Implications of imperfectly competitive behavior for estimation of price behavior will be discussed in a following section, which concerns the relationships between the reduced form and structure. The discussion will focus on models derived from static theories of the firm and industry, which is feasible because the subsequent empirical analysis of retail to farm price linkage is based on annual data where timelags between retail and farm prices are relatively unimportant.^ Appendix A contains a review of the alternative derivations of derived demand at the firm and industry level.

Input Demand for a Firm

The neoclassical theory of the firm provides a foundation for development of models of industry behavior. It is, therefore, useful to review the basic elements of the theory. For a competitive firm, output supply and input demand functions are obtained through maximizing profit (total revenue minus total variable costs) subject to the firm's production function relating output quantities to a vector of input quantities. Let P^. denote the retail product price, Pf denote the farm product price, and W denote a vector of prices of marketing inputs (for example, labor, packaging, energy prices, etc.). Also let the corresponding quantities be denoted by Q^, Qf, and L. Then output supply and input demand behavior of an isolated competitive firm can be characterized by

(5a) Q^ = S, (P,, Pf, W),

(5b) Qf = Df (P^, Pf, W),

(5c) L = DL (P^, Pf, W),

where (5a) is the retail supply function, and (5b) and (5c) are input demand equations for the farm product and vector of marketing inputs, respectively. A number of economists (most notably, Mosak and Samuelson) have analyzed the implications of profit maximization for output supply and input demand behavior. The properties of these functions can be summarized as follows: (1) the slope of the output supply curve with respect to output price is always positive, (2) the slopes of the factor demand curves with respect to their own prices are always negative, (3) output supply and input demand functions are homogeneous of degree zero in prices, (4) the change in demand for one input resulting from a change in price of another input equals the change in demand for the other input resulting from a change in price of the first input (that is, input demand functions are symmetric with respect to input prices), and (5) the change in output supply from a change in price of an input equals the negative of the effect of a change in output price on demand for the input in question.

An alternative way to characterize firm behavior is to view the problem facing the producer as twofold: (1) minimizing total variable costs subject to a given level of output and (2) maximizing profit given the total cost function for producing various levels of output. Output supply and input demand behavior in this case can be characterized by

^Studies which have developed theoretical models describing the intrinsic dynamic features of the price transmission process include Heien and Wohlgenant (1985).

10

(6a) P^ = MC (Q^, Pf, W),

(6b) Qf = Df^ (Q^, Pf, W),

(6c) L = DL" (Q^, Pf, W),

where (6a) is the condition that output price equals marginal cost, and (6b) and (6c) are the output-constant (expenditure-compensated) input demand functions for the farm output and vector of marketing inputs, respectively. Samuelson demonstrates that these equations satisfy the following conditions: (1) price effects of input demand functions are symmetric, (2) the input demand curves are downward sloping, (3) the change in any input demand with respect to an increase in quantity of output must equal the change in marginal cost with respect to a change in price of that input, (4) input demand functions are homogeneous of degree zero in input prices, (5) the marginal cost function is homogeneous of degree one in input prices, and (6) the partial derivative of the total cost function with respect to any input price equals the cost-minimizing level of that input. Clearly, equations (6a)-(6c) yield the same solutions for Q^,, Qf, and L as equations (5a)-(5c), given the same output and factor prices.

Equations (5a)-(5c) and (6a)-(6c) can be derived directly using duality theory. The supply and demand relations in (5a)-(5c) can be obtained through differentiating a profit function with respect to output price and input prices, respectively. The price equation and input demand relations in (6a)-(6c) can be obtained through differentiating a cost function with respect to output quantity and input prices, respectively (see Diewert (1974) or Varian).

Additional developments in the theory of the firm have focused on the nature of input price and output price changes with inferior factors (Bear, Ferguson 1968, Syrquin) and longrun adjustment of the competitive firm (Bassett and Borcherding 1970a, b, c; Ferguson and Saving; Silberberg). The latter case refers to the situation in which the output price adjusts in response to entry or exit from the industry until it equals minimum average cost of the firm. Other research on the theory of the firm has focused on the implications of imperfect competition in the product market for output price and input demand behavior (Ferguson, 1966), and implications of sales maximization behavior for firm output and input demand behavior (Portes). Mundlak has explored the implications of various assumptions of output effects from input price changes--output constant, total cost constant, and marginal cost constant--for the nature of the firm's input demand „response,

Input Demand for a Group of Firms

Two key problems are encountered when proceeding from relationships of the firm to those for the industry. These are (1) aggregation of functions for diverse firms into a single function for the industry and (2) taking account of changes in prices of inputs and outputs that are fixed for a single firm but not fixed for the industry as a whole (Brandow). The traditional approach to these problems has been to assume identical firms and free entry into and exit from the industry. This implies that the industry production function exhibits constant returns to scale, even though individual firms possess well defined production functions (Muth, Friedman, Diewert 1981). Since constant returns to scale implies the industry total cost function is linear in output,

11

price equals minimum average cost for each firm in equilibrium, and all output expansion paths are straight lines through the origin. This implies that (6a)-(6c) would simplify to

(7a) P^ - AC (Pf, W),

(7b) Qf - Q, . Df^ (Pf, W),

(7c) L -. Q^ . DL" (Pf, W),

where AC(») is the average cost function. Diewert (1981) has proved that these equations can be derived through duality theory by differentiating the linear total cost function, TC = c (Pf, W) • Q^., with respect to Q^., P^, and W, respectively. Thus, AC(«) is the unit cost function c(«), and Df^(*) and DL^(») are the partial derivatives of the unit cost function with respect to Pf and W, respectively.

With constant returns to scale, the retail supply curve (holding factor prices constant) is perfectly elastic so industry equilibrium output is determined where the horizontal supply curve intersects the retail demand curve. Let Dj. (Pj., Z) denote the retail demand function where Z represents exogenous retail demand shifters. Substituting this function for Qj. in (7b) and employing the comparative static methods of Diewert (1971) yields an expression for the price elasticity of derived demand for the farm product

(8) Eff = Sf (aff + e),

where Sf is the farmer's share of the retail dollar, e is the price elasticity of retail demand, and aff is the Allen own partial elasticity of substitution. This is precisely the formula obtained by Allen (pp. 503-505) using the more traditional methods of comparative statics based on the first-order conditions for profit maximization. In the two-input case, equation (8) becomes

(9) Eff - - (1 - Sf)c7 + SfC,

where a is the elasticity of substitution between the farm product and marketing inputs in producing the retail food product. Note that the second term in equation (9) has the same form as the George and King formula in equation (2) except that the elasticity of price transmission is used in place of the farmer's share of the retail dollar in the George and King formula. Also, observe that, with nonzero substitution between the farm product and marketing inputs, derived demand for the farm output will be more elastic compared with the case of zero substitution.

Hicks has derived a more general expression for the price elasticity of derived demand for a factor in the two-factor case when the supply of the other factor is less than perfectly elastic. Diewert (1971) generalized Hicks' formula to the case of N factors. This generalization is not considered in this study because of the assumption that costs of food processing and marketing are exogenous.

Muth has extended the Hicks-Allen results in the two-factor case to include (1) comparative static effects of output demand and the other factor supply changes on derived demand, and (2) comparative static effects of determinants of the industry product supply schedule. Assuming the supply curve of the farm product is perfectly inelastic (this assumption is justified in the next

12

section), and assuming all nonfarm inputs can be aggregated into one input (which are perfectly elastic in supply), approximate total relative changes in derived demand for the farm product and total relative changes in supply of the retail product can be characterized as

(10a) dlnQf = (- (l-Sf)a + S^e) • dlnPf + Sf(a+e) • dlnW + e^ • dlnZ,

(10b) dlnQ^ = ((1 - Sf)/Sf) o • dlnP,. + dlnQf - ((1 - Sf)/Sf)a • dlnW,

where e^ is the partial elasticity of retail demand with respect to a 1- percent change in Z. These expressions are general equilibrium functions of demand for the farm output and supply of the retail product. They will prove useful in the empirical analysis in testing and assessing the implications of constant returns to scale in food processing. Notice that the industry derived demand schedule for the farm product is negatively sloped and that the retail supply schedule is positively sloped, even with supply of the farm product fixed. This latter condition obtains with variable proportions because, as the retail price rises, the price of the farm product rises, and other inputs are substituted for the farm product, causing the marginal product of these inputs to fall and therefore marginal cost to rise as retail output increases. As shown in equation (10b), only when there is no substitution between the farm product and other inputs in food processing will the relative change in retail supply equal the relative change in farm product supply. Finally, notice that (1) an increase in retail demand will always lead to an increase in derived demand for the farm product, (2) the relationship between derived demand for the farm product and price of other inputs is ambiguous depending upon the relationship between o and e, and (3) the relationship between retail supply and W is always nonpositive.

Brandow was the first agricultural economist to use the above framework to analyze the implications of market equilibrium for farm-level demand and retail supply. Hov/ever, his analysis has limited usefulness because he assumed a Cobb-Douglas production function which implies a unitary elasticity of substitution.

Building on previous work by Brandow and Floyd, Gardner used a two-input, single product model of the food processing industry to analyze the implications of market equilibrium for the relationship between retail and farm prices. His main conclusions, which can be verified using equations (10a) and (10b) and specifications for retail demand and input supply functions, can be summarized as follows: (1) an increase in retail demand will decrease the retail-farm price ratio when marketing inputs are more elastic in supply than the supply curve for the farm product; (2) an increase in supply of the farm product will increase the price ratio; (3) an increase in the supply of marketing inputs (decrease in the price of marketing inputs) will decrease the price ratio; (4) a price ceiling at retail will decrease farm price; (5) farm price supports will reduce the retail-farm price ratio; (6) derived demand for the farm product will be more elastic than retail demand when o is larger than the absolute value of e; (7) the farmer's share of the retail dollar and retail-farm price ratio are analytically distinct concepts, and the two measures of price spread behavior will behave differently under changing market conditions except when the elasticity of substitution is zero; and (8) the elasticity of substitution can be estimated by dividing changes in the farmer's share of the retail dollar by changes in ratio of farm to retail price changes.

13

Wohlgenant (1982) has extended Gardner's analysis to the case of several marketing inputs with additional insight into the effect of substitutability between marketing inputs. Fisher has applied Gardner's model to the incidence of marketing charges on farm price. And Miedema, following Muth, has extended Gardner's results to include the effects of technical change on the relationship between retail and farm prices.

While the market equilibrium approach based on constant returns to scale has provided considerable insight into the relationship between retail and farm prices, it is not without limitations. The main problem, as Panzar and Willig point out, is in the assumption that all firms in the industry are marginal. Allowing for inframarginal firms can cause the product supply curve to be less than perfectly elastic when input prices are held constant, thereby invalidating the assumption of an industry constant-returns-to-scale production function. Panzar and Willig also show that Ferguson and Saving's proposition that longrun equilibrium output price and factor prices are positively related regardless of factor classification holds generally only for normal factors of production.

One longstanding problem in the theory of the firm and industry has been to show rigorously that industry factor demand curves are unambiguously negatively sloped. One reason for attachment to the assumption of a constant returns-to-scale longrun production function is that, prior to recent work by Heiner, industry factor demands could be shown to be unambiguously negatively sloped only if (1) all firms are identical or (2) the input-output ratio of the marginal firm is representative of the industry (Bassett and Borcherding, 1970a).

In a recent paper, Heiner proves that in the short run, without imposing any limitations on firm diversity, industry derived demand curves are unambiguously negatively sloped. He proves this by demonstrating that the shortrun industry responses are bounded between the industry output constant effects and the industry output-price constant responses. Moreover, he proves that these input responses are symmetric and that the industry supply curve is always steeper (more inelastic) than the curve constructed as the horizontal summation of individual firms' supply curves. Braulke (1987) has recently extended Heiner's results to show that longrun industry factor demand curves, where entry and exit from the industry occur, are also negatively sloped, even with very dissimilar firms.

While Heiner's results provide theoretical justification for using either equations (5a)-(5b) or (6a)-(6b) as a basis for formulating industry-level behavioral equations, the question remains as to which set of firm-level behavioral equations to use. In aggregating across firms, it seems reasonable to assume prices are the same, or at least price differences resulting from transport costs, quality differences, etc., remain the same over time. On the other hand, quantities of outputs can vary widely across firms and across time so the existence of an aggregate cost function which underlies equations (6a)-(6b) becomes dubious.^ Thus, it would seem that using the output supply and input demand functions in equations (5a) and (5b), in which output price is constant, for aggregation to industry response functions is the more theoretically sound approach.

^A sufficient condition for the existence of an aggregate cost function is that firms have homothetic cost functions (Lopez 1984b).

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Empirical applications of the market equilibrium approaches described in this subsection to derived demand for farm products and industry supply of the retail product are quite limited. Early work by Fox, Waugh, and others used specifications that could be roughly viewed as industry derived demand functions for the farm product. Indeed, by comparing their reduced-form derived demand estimates with those estimates obtained from the corresponding retail demand specifications, they frequently found that margins were neither strictly constant percentage nor constant absolute. Their estimates of retail to farm price linkage were incomplete, though, because they did not take account of industry supply behavior.

More recent approaches have utilized duality theory in estimating both farm-level demand and retail price behavioral equations. Ball and Chambers and Dunn and Heien apply the translog cost function to the meat products group and the major food groups (meat, dairy, poultry and eggs, and fruits and vegetables), respectively. Using a profit function approach, Lopez (1985) and Hopkins estimate behavioral equations for Canadian food processing and U.S. food processing, respectively. In all cases, the empirical results were somewhat mixed with a number of inconsistencies including incorrect signs, implausible elasticities, and anomalous economies of scale when estimated with cost functions. The shortcomings of these studies probably should not be taken too seriously, given the degree of product and input aggregation in these studies as well as the generally poor quality of the data used. The studies provide motivation for focusing on more dissaggregated food commodities.

A Conceptual and Empirical Framework for Retail to Farm Price Linkage

The previous section provides background information for formulation of a model to estimate retail to farm price linkage of a complete system of food commodities. The model developed should be consistent with the theoretical framework of industry output supply and input demand behavior derived from either equations (5a)-(5c) or (6a)-(6c). If time-series data on retail and farm prices and quantities were available, estimation of retail supply and farm product demand parameters would be relatively straightforward. These parameter estimates, together with estimates of consumer demand for food, could be used to estimate effects of changes in retail demand, farm supply, and marketing costs on retail and farm prices. The problem, though, is that published retail quantity data for individual food commodities are derived by applying fixed input-output coefficients to the USDA's disappearance data, so no new information would be provided by estimating retail supply functions with these data, and no new information would be provided on the degree of input substitutability. For this reason, an alternative approach is taken for estimating retail to farm price linkage, based on reduced-form specifications for retail and farm prices. This is the subject of the first subsection. The second subsection is concerned with empirical specification of retail and farm product price equations.

A Conceptual Model of Retail to Farm Price Linkage

Based on previous theoretical considerations, it seems desirable to derive the industry behavioral equations from the behavioral equations of the firm specified in equations (5a) and (5b) in which the output price is constant. Conceptually, the complete structural model for a particular commodity.

15

assuming perfect competition in the output and input markets, takes the following form:

(11a) Q/ = D, (P,, Z), (retail demand)

(lib) Q/ = 2S/ (P^, Pf, W), (retail supply)

(lie) Q/ = SD/ (P^, Pf, W), (farm-level demand)

(lid) Qf^ predetermined,

(lie) Q, Qr,

(llf) q/ = Qf^ = Q,,

(farm-level supply)

(retail market clearing)

(farm-level market clearing)

where all variables are defined as before. Note that the retail supply and farm-level demand functions are explicitly obtained as horizontal summations of the supply and demand functions of individual firms, where i denotes an individual firm. It is assumed that the number of firms in the industry is constant for any given time period. Also note that if some inputs are held fixed, the arguments of the functions could be expanded to include these fixed inputs as parameters. For the sake of convenience, these fixed inputs are subsumed in the supply and demand functions. (Time trends are included in the empirical model to account for changes over time in these inputs.) Finally, observe that in (lid) the quantity of the farm output is assumed to be predetermined with respect to the current period farm price. The condition that supply of the farm product is a function of lagged rather than current year prices is a consequence of biological lags in agricultural production processes.

Using equations (11a) and (lib) to eliminate Q^., the system (11a)-(llf) may be written as the two-equation system:

(12a) ES/ (P,, Pf, W) - D^ (P^, Z) = 0,

(12b) Qf - SD/ (P^, Pf, W) = 0.

The total differentials of equations (12a) and (12b) may be written

"âp^~ dP^ + ^" dPf + ^- • dW ^ • dP^

^ aw aPj- ^ aD^ az dZ

ciQ 1- . dP^ i- . dPf ap. d?4

EaPf^ "aw " • dW = 0.

Combining like terms and expressing these differentials in the form of partial elasticities of supply and demand yields

(13a) (^,

(13b) - e

e) • dlnP^ + ^rf • dlnPf e^ • dlnZ - ^i dlnW,

fr dlnP. Çff . dlnPf = ^f^ . dlnW - dlnQf,

where dlnP^. = dPj,/Pj., etc., and the ^'s and e's denote the partial elasticities of the supply and demand functions, respectively. Note that the elasticities of aggregate retail supply and aggregate farm-level demand are appropriately defined as quantity-share weighted sums of the respective elasticities of supply and demand for individual firms. For example.

16

^rr - Sir/ (QrVQr) » ^here ^^^i = (as//aP^) (P^/Q^) ÍS the price elasticity of retail supply of the ith firm.

What are the implications of the restrictions among elasticities at the firm level for the industry-level behavioral relations? First, it should be clear that if the zero homogeneity condition of output supply and input demand functions holds at the firm level, it will hold at the industry level as well. Since retail demand functions are also zero homogeneous in prices and income, this implies that equations (13a) and (13b) are invariant to proportional changes in P^., Pf, W, and those elements of Z which are retail prices of other consumer goods and consumer income. Second, the symmetry relationship between the effect of the farm price on retail supply and the effect of retail price on farm-level demand holds at the industry level as well. To see this, recall that for an individual firm (see appendix A)

(14) OS

5P. f r

Summing over all firms and converting to elasticities obtains

or after multiplying the left-hand side of equation (15) by Q^/Qr» ^^^ right-hand side of (15) by Qf/Qf, and rearranging terms yields

(16) Crf = - SfC fr'

which is precisely the aggregate counterpart to equation (14) expressed in elasticity form.

Comparative Statics of Reduced Form

The comparative statics of the reduced-form equations

(17a) P^ - P^ (Z, W, Qf),

(17b) Pf - Pf (Z, W, Qf),

can be characterized by solving the system of equations (13a) and (13b) for dlnPj. and dlnPf. These solutions, through application of Cramer's rule, have the form

A^^ • dlnZ + A^ (18a) dlnP^

(18b) dlnPf = Af^ • dlnZ + A^^

dlnW + Aj.f • dlnQf,

dlnW + A ff dlnQf,

where

(19a) A„ = -iffe,/D,

(19b) A,, = (^,,?,, - erfifw)/D,

(19c) A^f = e,f/D,

17

(19d) Af, = Cfrez/D.

(19e) Af„ = (-?f,^r„ + (^„ - e)Çf„)/D,

(19f) A„ = - (^„ - e)/D,

(19g) D = - (?„ - e)?« + ?,f^f,.

By the results of the previous section, it is possible to sign the partial derivatives of equations (17a) and (17b). First, observe from equation (18b) that the reciprocal of Aff is the industry derived demand elasticity for the farm product, holding prices of other inputs constant.^ Heiner proved that, in the short run, this elasticity is unambiguously negative. Moreover, since the retail supply elasticity (Crr) ^^ positive and the retail demand elasticity (e) can be expected to be negative in all normal cases, by equation (19f) this implies that D > 0. This immediately implies, since ^^^ < 0, that Aj.2 has the same sign as e^, which we would expect to be positive in all normal cases. Second, if the farm product is a normal input--which seems plausible for the commodity aggregates analyzed in this study--then ^^.^ is negative (see Ferguson 1968) , and ^¿j. is positive in view of equation (16) . This implies by equation (19c) that A^^ < 0, and by equation (19d) that Af^ takes the same sign as e^. Finally, the signs Aj.„ and Af^ are generally indeterminate because the sign of ^^^ is ambiguous. Note that from equations (18a) and (18b) that if Z depends on retail prices of other consumer products which depend on Qf and W, then the elasticities in equations (19a)-(19f) give only the partial effects of changes on Pj. and Pf, respectively. More will be said in the next subsection on how to compute the total effects of changes in Qf and W on retail price and derived demand for the farm product, holding prices of other farm products constant.

Relationships Between the Reduced Form and Structure

A question of interest in the present study is whether or not it is possible to obtain unique values for the ^'s and e's from the reduced-form parameters in equations (19a)-(19f). The answer is generally no, because the system is under-identified; that is, there are six reduced-form parameters and eight elasticities of the structural equations. However, if values of the retail demand elasticities (e and e^) are known, then unique estimates of the ^'s can be obtained. The structural and reduced-form parameters are related to one another as follows

<^rr - ^) ^rf

-Î fr -Î ff

A ^ A A ^ rf rw rf

^fz ^fw ^ff

Î 0 rw

^fw- 1

^At the industry level, derived demand is conceptually equivalent to a partially reduced-form relation showing the total effect of a change in price of the input on quantity demanded of the input. In the case examined in the text (as well as in Heiner's paper), equilibrium adjustment in the output market is taken into account. When all firms reduce input purchases in response to a given increase in price, aggregate output supply decreases causing output prices to rise. Therefore, input demand falls less than indicated by the demand curve constructed as the horizontal summation of the individual firm's demand curves.

18

The solutions to these six equations for the structural retail supply and farm-level demand elasticities, given values for the retail demand elasticities, are

(20a) Ç„ = e + Affe^/B,

(20b) Ç,f = -A,f/B,

(20c) Ç^ - (A,fAf„ - AffA^„)/B,

(20d) Cfr = -Af,/B,

(20e) if„ = A„/B,

(20f) Cfw = (Af,A^ - Af^„)/B,

where

(20g) B » A„Aff - A,fAf,.

It is interesting to observe from these expressions that, even if the retail demand elasticities are not known, values for the supply and demand elasticities except ^^.j. can still be determined from the reduced-form estimates. Also, observe that when e^ = 1, which occurs under the conditions specified in the next subsection, then by the syirmietry restriction (16)

(21) A^f = -SfAf,,

which is a linear cross-equation restriction on the reduced-form retail and farm price equations for given values of the farmer's share of the retail dollar (Sf). This restriction will be useful in the following econometric analysis.

In the previous section, it was argued that with diversity among firms, the existence of an aggregate cost function seemed doubtful. However, if the existence of such a cost function is posited, then it is possible to obtain an estimate of the output compensated input demand elasticity of the farm product from estimates of the output-price constant demand and supply functions. Using the result of Lopez (1984a), the output compensated demand elasticity for the farm output can be estimated as

(22) ^,,= = |„ - Cfr^rf/^rr-

By the theory of input demand behavior, this elasticity is unambiguously negative. Also, in the two-input case, dividing this parameter estimate by -(1 - Sf) yields an estimate of the elasticity of substitution between the farm product and marketing inputs in food processing (see, for example, Allen, pp. 503-505).

In this study, there is also interest in testing for a constant returns-- to-scale aggregate production function in light of the extensive use of this model. The implications of constant returns to scale for the relationship among elasticities in the reduced-form equations (18a) and (18b) can be derived through use of equations (10a) and (10b) in conjunction with the total

19

differential of the retail demand function in equation (11a), which can be written as

dlnQ^ = e • dlnP^ + e^ • dlnZ.

Substituting this expression for the left-hand side of equation (10b), and solving both equations (10a) and (10b) for dlnPf and dlnPj. obtains the following restrictions among the reduced-form elasticities in equations (18a) and (18b) when e^ = 1

(23a) A^, = -A^f,

(23b) Af^ = -Aff.

Thus, we have the rather intuitive result that constant returns to scale implies that equal proportionate changes in retail demand and quantity of the farm output will leave both retail and farm prices unchanged. Also note that this restriction is equivalent to assuming the retail supply curve is perfectly elastic, since equations (23a) and (23b) imply by (20g) that B = 0, and by (20a) that the limit of ^^.^ goes to infinity as B approaches zero.

Elasticity of Price Transmission and Derived Demand Elasticity

It was noted earlier that in previous work on price spreads the concept of elasticity of price transmission played a crucial role in obtaining estimates of the elasticity of derived demand for the farm product. Two conceptually different measures of the elasticity of substitution have been developed by Hildreth and Jarrett and George and King. Hildreth and Jarrett (p. 110) define elasticity of price transmission as "...the relative change in retail price to the relative change in producers' price when other factors affecting processor behavior are held constant." On the other hand, George and King (p. 61) define the elasticity of price transmission as "...the ratio of relative change in retail price to the relative change in the farm-level price."

Using George and King's definition in the context of the present model, an expression for the elasticity of price transmission (n) can be obtained by solving equation (18b) for dlnQf, substituting into equation (18a), setting dlnZ = dlnW = 0, and solving for dlnPj./dlnPf to obtain

(24) n = A^f/Aff.

Conceptually, this definition of elasticity of price transmission gives the total percentage change in equilibrium retail price resulting from a 1-percent change in farm price. In this definition, the change in farm price, causing the change in equilibrium retail price, can be thought of as caused by a given change in farm product supply that produces a 1-percent change in farm price.

To see how the elasticity of price transmission and retail demand elasticity are related to the industry derived demand elasticity, E^^ = l/A^f, it is useful first to define Eff in terms of the derived demand elasticity obtained when retail demand is perfectly inelastic. When the retail demand curve is perfectly inelastic, the retail demand elasticity, e, is zero and the retail

20

quantity is fixed. From equation (19f) this implies the output-constant derived demand elasticity, E^ ^ ^ , equals

which is precisely the form given in equation (22). Using the definitions for Ef£ and n in terms of the structural parameters in equations (19c) and (19f), this allows us to relate Eff to E^ ^ ^, n, and e as follows:

Eff = E,£^ + n . e (^f,/^„) .

Thus, the elasticity of price transmission, n, is a crucial parameter in relating the retail demand elasticity, e, to the industry derived demand elasticity. However, in contrast to George and King's formula (2), elasticity of derived demand at the farm level cannot in general be obtained as the product of the elasticity of price transmission and elasticity of demand at the retail level. The above formula shows that this approach to obtaining derived demand elasticities is valid if and only if Eff^ = 0 and ^^^ = ^j.^.. These conditions will obtain when the retail product is produced in fixed proportions with the farm product, but would not be expected to hold otherwise.

Implications for Monopoly Power

The final issue to consider in this subsection is the implication of monopoly power in the retail product market for the elasticities in equations (18a) and (18b) . To show that monopoly power can lead to ambiguous signs for A^.^ and Af2, and to violation of the symmetry restriction equation (21) when e^ = 1, it suffices to rework the comparative statics of equations (17a) and (17b) in the extreme case of pure monopoly. Following Diewert (1974), the first thing to note is that, with monopoly power, the same retail and farm quantities would result from profit maximization when the average retail price, P^, in equations (5a) and (5b) is replaced by the marginal retail price

P/ = P, + D,(P,, Z)/D,,(P,, Z),

where Dj.j. (•) is the partial derivative of the retail demand function with respect to the retail product price. Substituting this expression for Pj. in equations (5a) and (5b) and retracing the steps leading up to equations (18)-(19) yields the following comparative static results:

(25a) A„* = -^ff*(e, -e„*)/D*,

(25b) A,„* - Utt* ^rw* - Crf* e£„*)/D*.

(25c) A,,* = ^,f*/D*,

(25d) Af/ = (Cfr*(e, - ?„*) + (?rr* " e)Çf,*)/D*,

(25e) Aj„* - (-Cfr*Cr„* + (?rr* " e)Ç£„*)/D*,

(25f) A„* - - (e„* - e)/D*,

where ^j,^* ^^^ Cfz* ^^^ partial elasticities of retail supply and farm-level demand with respect to a change in Z. These partial derivatives exist

21

because, with imperfect competition, output response and input demand generally depend on both retail price and exogenous retail demand shifters. An asterisk is used to indicate that elasticities are evaluated at the marginal price P/ instead of the average price Pj.. D* has precisely the same form as equation (19g) with the retail elasticity evaluated at P^* instead of Pj.. Comparing equations (25a)-(25f) with (19a)-(19f) reveals that the impact of Z on retail price and farm price is ambiguous, and that the symmetry restriction equation (21) generally no longer holds when monopoly power is present. It can be shown that symmetry holds in the presence of monopoly behavior only when the retail demand elasticity is constant. The significance of the comparative static results equations (25a)-(25f) for econometric analysis is that negative signs for A^.^ and Af^, and violation of the symmetry restriction equation (21) could indicate imperfectly competitive behavior. These results do not necessarily imply nonprofit-maximizing behavior.

Empirical Specification of Retail to Farm Price Linkage

Three separate problems must be resolved as we go from the conceptual framework in the previous subsection to an empirical specification of retail and farm price behavior in this subsection. These problems are: (1) selection of appropriate functional forms for equations (17a) and (17b), (2) specification of retail demand shifters subsumed in the variable Z, and (3) development of formulas to calculate total effects of exogenous retail demand shifters, marketing costs, and farm output quantities on retail and farm prices.

Econometric Specification

The question of functional form specification for econometric analysis of (17a) and (17b) is generally an open one. For primarily pragmatic reasons, the approach taken in this study is to assume the elasticities in equations (18a) and (18b) are approximately constant and to replace instantaneous relative changes in the variables by first-differences in the logarithms; that is

(26a) AlnP^^ « A„ • AlnZ^ + A^^ • AlnW^ + A^^ • AlnQ^^ + A^^ + U^^,

(26b) AlnPft -= Af, . AlnZt + Af^ • AlnW^ + A^^ • AlnQf, + Af^ + U^^,

where t denotes the time period, 11^.^ and U^^ denote random disturbance terms, and Aj.Q and A^^ are intercept values which reflect changes in prices due solely to trend. Trend effects can occur because of the presence of technical change in marketing and shifts in consumer demand not accounted for by relative prices and real income. The approximate nature of this form is apparent in light of the symmetry restriction equation (21), which holds only for a given value for the farmer's share of the retail dollar, S^. By the results of the previous subsection, the homogeneity restriction can be imposed by deflating all nominal values in equations (26a) and (26b) by a general price deflator such as the consumer price index (CPI). Imposing the homogeneity restrictions and symmetry restriction equation (21) makes the parameter estimates of equations (26a) and (26b) consistent with the theories of industry behavior specified previously. Also, given the homogeneity and symmetry restrictions, one can impose and test for the implications of a constant-returns-to-scale industry production function according to equations (24a) and (24b). Structural parameter estimates of equations (10a) and (10b) can then be

22

obtained from the reduced-form parameters given retail demand elasticity estimates of equation (11a) expressed in first-differences in logarithms.

Specification of the form of Z is particularly important in order to make the parameter estimates of equations (26a) and (26b) internally consistent with the consumer demand estimates. Theoretically, in the context of a system of consumer demand functions, the total differential of retail demand for the ith commodity in elasticity form can be written

(27) dlnQ^i = e^i • dlnP^^ + ^ ^ij * dlnP^j + e^y • dlnY + dlnPOP,

where e¿¿ is the own-price elasticity of demand for commodity i, e^j is the cross-elasticity of demand for good i with respect to the price of good j, e^y is the income elasticity of good i, Y is per capita consumer total expenditures (income), and POP is the total consuming population. In order to have internally consistent estimates of retail demand shifters in equations (26a) and (26b) , the variable AlnZ^ is defined precisely as the sum of the last three groups of terms in equation (27) expressed in first-differences in logarithms; that is, for the ith good

(28) Aln Zit = 2 e^j . AlnP^^^ + ^^y • Aln Y^. + AlnPOP^.

When each Z variable is defined in this way, the value for the partial elasticity e^ in equation (13a) is identically equal to 1 because equation (28) shows the impact on demand of a 1-percent change in demand. However, in order to construct such a variable, values for the cross-elasticities of retail demand and income elasticity for each commodity are required. What is proposed here is to use internally consistent demand elasticities estimated from previous research.

At this point, the reader may wonder if use of extraneous information to construct values for equation (28) involves circular reasoning, since values for these parameters are typically estimated from retail quantity data which are derived as fixed proportions of farm output disappearance data. This is somewhat problematical, but the procedures in obtaining equation (28) for arbitrary values of the retail elasticities is theoretically consistent with the theory of consumer behavior and specification of retail and farm price behavior. Moreover, the first part of equation (28), which depends on related retail prices, may be viewed as a price index in itself. Indeed, this price index may be viewed quite generally as a Divisia price index, with weights chosen as cross-elasticities of demand instead of commodity consumer expenditure shares. The index equation (28) ought to be relatively insensitive to the particular weights chosen for the price and income variables; because, in a time-series context, prices and income are typically highly collinear. Therefore, using extraneous demand elasticity parameters for equation (28) estimated from retail disappearance quantity data is not viewed as a serious source of error in empirical implementation of equations (26a) and (26b).^

^Wohlgenant (1989) uses the Hausman specification test to test for the exogeneity of AlnZ in the reduced-form retail and farm price equations. In every case, the null hypothesis that AlnZ is exogenous was not rejected.

23

Total Effects of Exogenous Changes on Retail and Farm Prices

The final issue to consider at this time, prior to empirical implementation of equations (26a) and (26b), is development of formulas to calculate total effects of exogenous changes on retail and farm prices. The problem with using equations (26a) and (26b) directly is that retail and farm prices for a given commodity depend upon retail prices of other commodities, which also depend on retail price of the commodity in question. Having formulas which express changes in retail and farm prices solely in terms of the exogenous variables is essential for quantifying the effects of food policy and farm program changes on retail and farm prices. For example, it is not possible to use an equation like equation (26b) directly to compute the impact of a change in income on farm price (caused, for example, by a change in the food stamp program) because a change in income could also lead to changes in equilibrium values of other retail prices subsumed in Z.

Substituting equation (28) into (26a) and (26b) and rearranging terms obtains the following system of retail and farm price equations:

(29a) Aln P^ = A^^ • (e" • Aln P^ + e° • Aln P^" + e^ • Aln Y + 1 • AlnPOP) + A^^ . Aln W + A^f • Aln Qf,

(29b) Aln Pf = Af2 • (e" • Aln Pj. + e° • Aln P^" + e^ • Aln Y + 1 • AlnPOP) + Af^ • Aln W + Aff • Aln Qf,

where underbars denote vectors and matrices of appropriate dimensions. The matrix e" is the n x n matrix of price elasticities of retail demand with zero diagonal elements; the matrix e° is the matrix of cross-price elasticities of retail commodities whose prices are taken as exogenous (for example, nonfood goods); ey is the column vector of income elasticities; and 1 is a column vector of I's. To obtain total effects of the exogenous variables on retail and farm prices, solve equations (29a) and (29b) by matrix methods to obtain

(30a) Aln P^ = (I^ - A^^ • e")"^ • (A^^ • e° » Aln P^° + A^ % Aln Y + A^2 • 1 • AlnPOP + A^^ • Aln W -I- A^f • Aln Qf ) ,

(30b) Aln Pf = Af^ . (e- • (I^ - A^, • e")"^ • A^, + I^) • (e° • Aln Pj.° + ey . Aln Y + 1 • AlnPOP) + (Af, • e- • (I^ - A^, • e")"^ • A^^ + A^J • Aln W + (Af^ • e" . (I^ - A^^ • e")"^ • A^f + A^f) • Aln Qf,

where 1^ is the n-th order identity matrix. Estimates of derived demand elasticities for farm outputs are obtained by inverting the matrix that premultiplies Aln Qf in equation (30b).

Empirical Application to a Complete System of Food Commodities

In this section, the analytical model of retail to farm price linkage developed in the previous section is applied to estimation of a complete system of food commodities. There is particular interest in the influence of the degree of processing on estimated price relationships, so a range of commodities is included in the analysis. The commodities included in the analysis are: (1) beef and veal, (2) pork, (3) poultry, (4) eggs, (5) dairy products, (6) processed fruits and vegetables, (7) fresh fruits, and (8) fresh vegetables. Particular attention will be given to the meat items (1-3), which

24

are viewed as the relatively intensive processed items, in contrast to fresh fruits and vegetables, which involve little or no processing. For each commodity, the parameters of the reduced-form retail equation (26a) and reduced-form farm price equation (26b) are estimated. Data required for estimation include time series on retail and farm prices, time series on the retail demand shift variables Z (which are functions of all retail food and nonfood prices, income, and population in view of equation 28), time series on the index of marketing costs W, and time series on the quantities of the various farm products produced (denoted by Qf) .

Farm output and price data for these commodities were obtained from time-series data and conversion factors published by the USDA. Farm price data are producer price indexes for crude foodstuffs. Retail price data are consumer price indexes for the corresponding farm categories and income and population data are total personal consumption expenditures per capita and midyear U.S. civilian population, respectively. These data, which are also published by USDA, were furnished by Kuo S. Huang of the USDA, who also provided a matrix of consumer demand elasticities for construction of the retail demand shift variables. The commodities included in his m.odel were: (1) beef and veal, (2) pork, (3) poultry, (4) eggs, (5) dairy products, (6) processed fruits and vegetables, (7) fresh fruit, (8) fresh vegetables, (9) fish, (10) sugar, (11) fats and oil, (12) cereals, (13) beverages, and (14) nonfood. Farm price linkages for items 9-13 were not included either because appropriate data were lacking or because it was impossible to identify a corresponding raw product for the corresponding retail product. The marketing cost variable used is the index of food marketing costs (Harp). The time period for estimation is 1956-83, with 1955 used to generate the initial values for the first-difference variables. Data sources, together with details on transformations and data imputations, are reported in appendix B.

Econometric Results

Equations (26a) and (26b) were estimated for the eight different food commodities (beef and veal, pork, poultry, eggs, dairy products, processed fruits and vegetables, fresh fruit, fresh vegetables) by the joint generalized least squares technique. These equations were estimated with and without the symmetry restriction equation (21) and the constant-returns-to scale restrictions equations (23a and 23b).

Unrestricted estimates of equations (26a) and (26b) for the eight sets of price equations are displayed in table 1. With a few exceptions, the parameter estimates are consistent with prior expectations. Retail demand shifters, measured by Z, and farm output variables are generally highly statistically significant determinants of retail and farm prices. With the exception of fresh fruit, increases in retail demand are positively related both to retail and to farm prices. With the exception of processed fruits and vegetables and fresh fruit, all farm price flexibilities are larger than 1 in absolute value. This pattern of change is broadly consistent with previous empirical work (see, for example, Fox, Waugh, Dunn and Heien). The marketing cost index is statistically insignificant in virtually every case, and often displays a sign contrary to prior expectations in the retail price equations. However, the marketing cost variable is usually insignificant.

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Table 1--Unrestricted reduced-form econcxnetric estimates of retail and farm product prices, 1955-83

Elast icitv of price with respect to--

Index of Retail Product Farm marketing demand

Commodity price quantity (Qf) costs (W) shifter (Z) Trend (T) R2 DU

Beef and Retail -0.921 -0.513 0.664 -0.001 0.54 1.78 veal (-4.642) (-0.742) (2.153) (-0.098)

Farm -1.365 -0.879 1.354 -0.018 0.55 1.92 (-4.333) (-0.800) (2.766) (-0.811)

Pork Retail -0.966 -0.027 1.474 -0.021 0.87 2.15 (-10.964) (-0.052) (4.938) (-2.010)

Farm -2.070 -0.464 2.046 -0.019 0.89 2.03 (-13.314) (-0.502) (3.884) (-1.027)

Poultry Retail -1.280 0.217 1.185 0.002 0.81 2.30 (-6.441) (0.384) (8.068) (0.160)

Farm -2.946 -0.067 1.907 0.073 0.76 2.15 (-6.698) (-0.053) (5.866) (2.909)

Eggs Retail -4.412 -1.801 5.152 -0.064 0.53 2.28 (-4.639) (-1.597) (2.438) (-2.287)

Farm -5Jff -4.661 6.246 -0.046 0.48 2.39 (-3.681) (-2.505) (1.791) (-0.987)

Dairy Retail -0.901 0.241 1.348 -0.023 0.44 2.62 (-3.467) (0.832) (2.253) (-2.416)

Farm -1.458 0.272 2.200 -0.028 0.37 2.05 (-3.040) (0.508) (1.992) (-1.590)

Processed Retail -0.490 1.065 0.016 -0.001 0.23 2.07 fruits and (-2.192) (1.741) (0.038) (-0.040) vegetables

Farm -0.141 0.428 0.177 -0.012 0.05 2.51 (-0.705) (0.782) (0.469) (-0.928)

Fresh fruit Retail -0.779 -0.782 -0.193 0.014 0.24 2.05 (-2.477) (-0.828) (-0.360) (0.942)

Farm -0.124 -0.806 -0.350 -0.006 0.04 2.31 (-0.333) (-0.717) (-0.550) (-0.311)

Fresh Retail -0.097 -0.575 0.475 0.005 0.03 2.25 vegetables (-0.172) (-0.646) (0.495) (0.255)

Farm -2.205 -1.058 1.919 0.002 0.51 2.40 (-4.164) (-1.260) (2.119) (0.118)

Note: Values in parentheses are t-values. Farm prices for beef and veal and pork are adjusted for by-product values according to the procedure discussed in appendix B.

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Table 2 displays the regression results when the synimetry restriction equation (21) is imposed. In all cases except fresh fruit, this restriction is not rejected. For processed fruits and vegetables and fresh fruit, signs on quantity of the farm output in the farm price equation are reversed from the unrestricted estimates. However, these variables are insignificant and, even for the unrestricted estimates, take on implausibly low values. This suggests aggregation and/or data problems with these two commodities. Processed fruits and vegetables is a very heterogeneous category, and fresh fruits includes significant retail products for which there are no U.S. counterparts (for example, bananas). Overall, the results in table 2 show a general consistency with industry competitive price behavior.

Given the symmetry restriction, equations (26a) and (26b) were subjected to the constant-returns-to-scale restriction implied by equations (23a) and (23b), table 3. Consistent with the symmetry test results, all commodities except fresh fruit appear to be compatible with this restriction. This suggests that the assumption of an industry production function exhibiting constant returns to scale may be a reasonable specification for analysis of food marketing behavior.

Given the maintained hypothesis of constant returns to scale, structural parameters of the industry supply and demand functions are derived for all commodities except processed fruits and vegetables and fresh fruits. (These commodities are excluded because of the wrong signs on the farm output variables.) The structural parameters of interest in this connection are the elasticity of substitution (a) and elasticity of price transmission (n). Estimates of o are obtained through use of equation (9), estimates of Eff from the reciprocal of the elasticities of farm price with respect to farm output in table 3, and extraneous estimates of e provided by Kuo S. Huang of the USDA (see appendix B). The elasticity of price transmission in this case is equal to the farmer's share of the retail dollar, which can be verified by using equations (21) and (23b) in (24). Estimates of these structural parameters are presented in table 4. In all cases, the elasticities of substitution are positive as expected. In some cases (for example, beef and veal and dairy products), the estimates are quite large, suggesting substantial opportunities for input substitution. Whether the magnitudes of these estimates are related to the degree of processing is unclear because the elasticity of substitution estimates for fresh vegetables, which is likely to involve the least processing of any of the commodities, is the third largest. What the pattern of substitution elasticities across commodities may suggest is that substitutability is more closely associated with interproduct substitution since the three commodities with the highest values for o are all quite complex and rather heterogeneous in nature.

An important question to ask is: Are the elasticity of substitution estimates in table 4 significantly different from zero? While this question is difficult to answer directly because the estimates are derived from extraneous information about retail demand price elasticities, it is possible to test this hypothesis for given values of e and Sf. Recall that when a = 0, the elasticity of derived demand reduces to E^f = e • Sf. Since E^f = l/A^^, an approximate test for a = 0 is the t-statistic

t - (Aff - l/(e . Sf))/SE(Aff),

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Table 2--Symm€try restricted reduced-form econometric estimates of retail and farm product prices, 1955-83

Elasticity of price with respect to- _ Index of Retail F-value

Product Farm marketing demand for symmetry Commodity price quantity (Qf) costs (W) shifter (Z) Trend (T) restriction

Beef and veal Retail -0.909 -0.513 0.690 -0.002 (-5.467) (-0.742) (3.521) (-0.192) 0.012

Farm -1.348 -0.878 1.399 -0.019 (-4.931) (-0.799) (5.467) (-1.094)

Pork Retail -0.981 -0.040 1.409 -0.020 (-11.753) (-0.076) (5.197) (-1.944) 0.272

Farm -2.086 -0.518 1.783 -0.013 (-13.689) (-0.564) (11.753) (-0.900)

Poultry Retail -1.110 0.134 1.218 -0.006 (-8.226) (0.239) (8.441) (-0.609) 1.360

Farm -2.805 0.050 2.135 0.062 (-6.633) (0.040) (8.226) (2.666)

Eggs Retail -4.357 -1.769 5.485 -0.068 (-4.710) (-1.579) (3.357) (-2.832) 0.062

Farm -5.754 -4.590 7.028 -0.054 (-3.673) (-2.497) (4.710) (-1.711)

Dairy Retail -0.927 0.244 1.252 -0.022 (-3.918) (0.840) (2.898) (-3.093) 0.054

Farm -1.493 0.281 1.971 -0.025 (-3.271) (0.527) (3.918) (-2.572)

Processed Retail -0.067 0.943 0.108 -0.012 fruits and (-0.912) (1.549) (0.258) (-0.916) 4.015 vegetables

Farm 0.150 0.326 0.336 -0.022 (1.090) (0.599) (0.912) (-1.885)

Fresh fruit Retail -0.129 -0.814 0.119 0.011 (-0.719) (0.862) (0.229) (0.726) 6.343

Farm 0.344 -0.735 0.404 -0.004 (1.061) (-0.654) (0.719) (-0.246)

Fresh Retail -0.058 -0.617 0.193 0.013 vegetables (-1.973) (-0.694) (0.215) (0.864) 0.681

Farm -2.451 -1.090 1.540 0.009 (-5.600 (-1.300) (1.973) (0.606)

Note: Equations were estimated by the joint generalized least squares method with cross-equation restriction, A^^ = '^f^fz* îf^posed at 1967-69 average values for $£. Average share values for this time period by commodity are as follows: beef and veal (0.65), pork (0.55), poultry (0.52), eggs (0.62), dairy (0.47), processed fruits and vegetables (0.20), fresh fruits (0.32), and fresh vegetables (0.33). Also see table 1 for further explanation of the results.

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Table 3--Reduced-form econometric estimates of retail and farm product prices with symmetry and constant returns to scale restrictions imposed, 1955-83

Elasticity of price with respect to --

Commodity Product price

Farm quantity (Qj

Index of marketing

^) costs (W)

Retail demand shifter (Z) Trend (T)

F-value for constant returns to scale

Beef and veal

Retail -0.858 (-5.275)

-0.506 (-0.731)

0.858 (5.275)

-0.007 (-0.571) 1.;06

Farm -1.320 (-5.275)

-0.896 (-0.816)

1.320 (5.275)

-0.017 (-0.999)

Pork Retail -1.080 (-15,814)

-0.104 (-0.200)

1.080 (15.814)

-0.011 (-1.399) 2.440

Farm -1.963 (-15.814)

-0.486 (-0.530)

1.963 (15.814)

-0.019 (-1.341)

Poultry Retail -1.197 (-10.095)

0.173 (0.309)

1.197 (10.095)

-0.002 (-0.211) 1.881

Farm -2.302 (-10.095)

-0.129 (-0.106)

2.302 (10.095)

0.038 (2.090)

Eggs Retail -4.081 (-4.604)

-1.886 (-1.694)

4.081 (4.604)

-0.053 (-2.805) 0.940

Farm -6.582 (-4.604)

-4.658 (-2.535)

6.582 (4.604)

-0.048 (-1.547)

Dairy Retail -0.776 (-3.712)

0.298 (1.033)

0.776 (3.712)

-0.016 (-3.416) 1.418

Farm -1.652 (-3.712)

0.277 (0.518)

1.652 (3.712)

-0.019 (-2.188)

Processed fruits and vegetables

Retail

Farm

0.025 (0.958)

0.127 (0.958)

0.945 (1.561)

0.418 (0.772)

-0.025 (-O.0'>o;

-G.127 (Û-953)

-0.011 (-1.138)

-0.010 (-1.209)

1.762

Fresh fruit

Retail 0.093 (0.902)

-0.881 (-0.939)

-0.093 (-0.902)

0.008 (0.532) 1.321

Farm 0.289 (0.902)

-0.850 (-0.759)

-0.289 (-0.902)

-0.009 (-0.486)

Fresh vegetables

Retail -0.765 (-5.637)

-0.619 (-0.697)

0.765 (5.637)

0.012 (0.846)

0.694

Farm -2.319 (-5.637)

-1.055 (-1.258)

2.319 (5.637)

-0.000 (-0.013)

Note: Equations were estimated by the joint generalized least squares niethod with symmetry and constant returns to scale restrictions, equations (25a) and (23b), imposed. See tables 1 and 2 for further explanation of the results.

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Table 4--Estimates of elasticity of substitution and price transmission from reduced-form econometric estimates in table 3

Parameter values

Commodity Elasticity of retail demand (e)

Elasticity of price

transmission (n) Elasticity of substitution (a)

Beef and veal -0.78 0.65 0.72

Pork -0.64 0.55 0.35

Poultry -0.73 0.52 0.11

Eggs -0.09 0.62 0.25

Dairy -0.21 0.47 0.96

Fresh vegetables

-0.22 0.33 0.54

Note: Elasticities of retail demand are extraneous estimates furnished by Dr. Kuo S. Huang of the USDA (appendix B). With constant returns to scale, elasticities of price transmission are farmers' shares of the retail dollar for 1967-69 average values. Elasticity of substitution estimates obtained through use of formula in equation (9) and price flexibilities in table 3.

where SE(Aff) is the estimated standard error for A^f derived from table 3. The estimated t-values for the six commodities shown in table 4 are:

Commodity beef and veal pork poultry eggs dairy fresh vegetables

t-value for a 0 2.60 7.09 1.45 7.93

19.05 11.45

In only one case, poultry, do we unambiguously fail to reject the null hypothesis of fixed input proportions.

Flexibilities and Elasticities

Using the equations in table 3 and extraneous information on the consumer demand elasticities for the food products, the set of equations shown in (30b) is used to calculate a matrix of total effects of exogenous demand and supply shifters on farm-level prices. This estimated matrix for beef and veal, pork, poultry, eggs, dairy products, and fresh vegetables is shown in table 5. While total effects of demand and supply changes on prices for processed fruits and vegetables and fresh fruits are not included here, the unrestricted retail price equation estimates from table 1 were included in the model when

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Table 5--Matrix of flexibilities for reduced-form farm-level prices

Quant ity Marketing

Beef cost Farm price and veal Pork Poultry Eggs Dairy Vegetables index Income

Beef and veal -1.37 -0.17 -0.10 -0.14 0.04 0.00 -1.06 1.42

Pork -0.27 -2.05 -0.22 0.10 0.00 -0.02 -0.57 1.64

Poultry -0.50 -0.66 -2.42 -0.34 0.06 -0.18 -1.73 1.68

Eggs -0.70 0.07 -0.33 -6.71 -0.10 0.07 -5.00 0.29

Dairy 0.00 -0.03 0.01 -0,02 -1.65 -0.02 0.39 0.24

Vegetables -0.07 -0.12 -0.22 0.11 -0.09 -2.34 -0.98 0.20

Note: Flexibilities show percentage changes in farm prices to 1-percent changes in quantities, marketing cost index, and income. These are total flexibilities calculated from equation (30b).

deriving the flexibilities and elasticities for the other commodities. Also, in order to conserve space, only total effects of farm quantities, marketing costs, and income are presented. Effects of retail prices of other food and nonfood commodities and population are reported in appendix B. Overall, the results in table 5 are consistent with previous results indicating (1) all own-price flexibilities are larger than 1 in absolute value, (2) the majority of cross-price flexibilities display negative terms indicating substitutability among farm outputs, (3) all marketing cost variables except dairy show negative signs, and (4) all income flexibilities are positive. By solving the system of equations (30b) for percentage changes in farm quantities, we obtain total elasticities of derived demand for the farm outputs. These are shown in table 6. Again, the results are broadly consistent with previous findings. All own-price elasticities are less than 1 in absolute values and the majority of the cross-price elasticities are positive, indicating substitutability among farm outputs. The main difference from previous results is that, with the sole exception of poultry, farm-level demands are nearly as large as the corresponding retail elasticities or larger than the corresponding retail elasticities (see table 4). The reason these elasticities are so large is that there is substantial substitutability between farm outputs and marketing inputs in the corresponding food marketing industries. This is in agreement with the analysis of the previous section showing that derived demand for the farm product can be more elastic than retail demand.

Table 7 shows derived demand elasticities calculated using the traditional methodology of multiplying elasticities of price transmission (which equal the farmer's share of the retail dollar in this case) times retail demand elasticities, equation (2). The own-price elasticities of derived demand by this procedure are considerably smaller than those shown in table 6. Indeed, with the exception of poultry, the own-elasticities in table 6 are at least 40 percent larger in absolute value. The reason for this difference in estimated elasticities, of course, is that this formula assumes fixed input proportions. Given the magnitudes of the errors from using the traditional formulas, special care should be taken in using this methodology for calculating derived demand elasticities.

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Table 6-"Matn"x of elasticities for reduced-form farm-level input demands

Price Marketing

Beef cost Farm quanti ity and veal Pork Poultry Eggs Dairy Vegetables index Income

Beef and veal -0.76 0.06 0.02 0.02 -0.02 0.00 -0.76 1.02

Pork 0.09 -0.51 0.04 -0.01 0.00 0.00 -0.29 0.64

Poultry 0.10 O.U -0.42 0.02 0.01 0.05 0.74 0.13

Eggs 0.08 -0.02 0.02 -0.15 0.01 -0.01 -0.73 -0.05

Dairy 0.00 0.01 0.00 0.00 -0.61 0.01 0.19 0.08

Vegetables 0.01 0.01 0-06 -0.01 0.03 -0.43 -0.16 -0.21

Note: Elasticities are total derived demand elasticities calculated through inversion of the matrix of flexibilities in table 5.

Table 7--Matrix of elasticities of farm-level derived demands under fixed proportions assumption

Price

Farm Beef and quant i ty veal Pork Poultry Eggs Dairy Vegetables Income

Beef and veal -0.50 0.06 0.02 0.02 -0.02 0.00 0.95

Pork 0.09 -0.36 0.04 -0.01 0.01 0.00 0.64

Poultry 0.12 0.13 -0.38 0.02 -0.02 0.03 0.32

Eggs 0.07 -0.02 0.02 -0.05 0.01 0.00 -0.06

Dairy 0.00 0.01 0.00 0.00 -0.10 0.00 0.14

Vegetables 0.01 0.01 0.04 -0.01 0.03 -0.07 -0.03

Note: Derived demand elasticities are derived as product of elasticities of price transmission and retail demand elasticities according to equation (2).

Conclusions and Implications

The purpose of this report was to develop a conceptual and empirical framework for estimation of retail to farm linkage of a complete demand system of food commodities. The main findings of this study can be summarized as follows:

(1) The traditional approach to modeling derived demand for food at the farm level, based on fixed input proportions, is generally deficient in its ability to explain observed relationships between marketing margins and retail prices. While imperfect competition can account for margin behavior resembling markup behavior, the most fundamental explanation for this behavior is input substitution between the farm product and marketing inputs. In addition to the traditional view of input substitution (i.e., reduction in the amount of wastage and spoilage from relative price changes), it is argued that input substitution also can arise from interfirm diversity, interproduct

32

Substitution in commodity aggregates, and quantity/quality substitution in food processing-marketing.

(2) Viewing quantity of the farm product as a factor of production in food processing has the advantage that a set of theoretically consistent behavioral relationships characterizing marketing group behavior can be derived through aggregating output supply and input demand functions of individual competitive firms. Behavioral equations derived in this manner, with parametric restrictions of firm behavior and market clearing conditions imposed, yield a theoretically consistent framework for evaluating impacts of changes in consumer demand, farm supply, and marketing input costs on retail and farm prices.

(3) Because direct estimates of retail quantities for dissaggregated individual food commodities are generally unavailable, the conceptual model of retail to farm price linkage was specified to consist of reduced-form behavioral equations for retail and farm prices. In general, the structural parameters of the behavioral equations characterizing industry marketing group behavior are underidentified. However, given values for the retail demand elasticities, and upon imposing the parametric restrictions of homogeneity and symmetry on the reduced-form parameters, unique estimates of the structural parameters can be obtained. Additional restrictions are developed for imposing and testing for the existence of a constant-returns-to-scale aggregate production function.

(4) In order to implement empirically the conceptual model of retail to farm linkage, extraneous estimates of retail demand elasticities are required. These estimates were obtained from a previously estimated system of consumer demand functions which satisfy the homogeneity, symmetry, and Engel aggregation conditions of consumer theory. Thus, with homogeneity and symmetry conditions imposed on the reduced-form retail and farm price equations, the resulting parameter estimates linking retail demand to derived demand for farm outputs automatically satisfy the theoretical restrictions of consumer behavior and marketing group behavior.

(5) The conceptual model of retail to farm price linkage was applied to a set of eight food commodities: beef and veal, pork, poultry, eggs, dairy products, fresh fruits, fresh vegetables, and processed fruits and vegetables. With the exception of fresh fruits and processed fruits and vegetables, the econometric results are found to be consistent with theoretical specifications and a priori expectations. With the exception of fresh fruits, the statistical results indicate that the parametric restrictions of symmetry and constant returns to scale cannot be rejected. This means that, for the most part, food marketing behavior can be characterized as competitive with constant returns to scale in food processing-marketing. Given these restrictions, estimates were derived of the elasticity of substitution between the farm product and marketing inputs. With the exception of poultry, these elasticities are all significantly different from zero. This suggests that the assumption of fixed input proportions is generally inappropriate.

(6) Flexibilities and elasticities of demand for farm products were derived which take into account total effects of exogenous demand and supply shifters. These estimates are consistent with previous findings indicating that all own-price flexibilities are larger than 1 in absolute value, all own-price elasticities are less than 1 in absolute value, all income flexibilities are

33

positive, and the majority of cross-price flexibilities and elasticities among farm outputs display a substitution pattern. The pattern of impacts of marketing costs (represented by the index of food marketing costs) and income elasticities are less consistent, but generally conform to a priori expectations.

(7) A comparison of the own-price elasticities for farm outputs derived from the conceptual model developed in this study with those derived using the traditional methodology based on fixed input proportions indicates substantial differences in the elasticities. With the exception of poultry, this comparison indicates that relaxing the assumption of fixed input proportions results in price elasticities at least 40 percent larger in absolute value. This suggests analysts should take extreme caution in using the traditional methodology to estimate retail to farm demand linkages. At the very least, reduced-form farm price equations should be estimated directly when attempting to quantify the impacts of retail demand and/or farm supply shifts on farm price.

There are a number of areas of future research deserving attention. First, the system needs to be extended to encompass additional commodities through dissaggregation of a number of the composite food commodities. Possibilities include dairy products, fresh fruits, fresh vegetables, and processed fruits and vegetables. Second, the forecasting performance of the derived farm price and retail price equations needs to be assessed. Among other things, this would provide useful information regarding areas for improvement in econometric specification. Third, the conceptual and empirical framework needs to be broadened to encompass commodities which yield joint products at the retail level (for example, fresh and processed dairy products). Fourth, additional work needs to be undertaken on ways to quantify the impact of marketing costs on individual food commodities. This could provide useful information for construction of cost components that would enhance their usefulness for econometric analysis of retail to farm linkages.

34

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Huang, K., and R.C. Haidacher. "Estimation of a Composite Food Demand System for the United States." Journal of Business and Economic Statistics 1(4): 285-291, 1983.

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Appendix A: Review of Alternative Derivations of Derived Demand

This appendix reviews mathematical derivations and properties of alternative derived demand functions obtained at the firm and industry level.

Firm Level

At the firm level, demand functions for factors of production can be derived through cost minimization or profit maximization. The demand functions obtained are commonly referred to as output constant and output-price constant factor demand functions, respectively. In order to facilitate the analysis, the properties of these demand functions are examined through use of duality theory.

Let C(Qj,, Pf, W) be the total cost function of producing output Q^. given factor prices Pf and W. This cost function has the following properties (Varian, chapter 1): (1) nondecreasing in Pf and W, (2) homogeneous of degree 1 in Pf and W, (3) concave in Pf and W, and (4) continuous in Pf and W. Application of Shephard's lemma yields the output constant input demand (that is, demand functions where output is parametric) functions

(A.l) Qf = aC (Q,, Pf, W) = Df^ (Q,, Pf, W), aPf

(A. 2) L » aC (Q^, Pf, W) = DL^ (Q^, Pf, W), dh

where Qf is the quantity of the farm product and L is the quantity of marketing inputs. By the derivative property of homogeneous functions, the first-order partial derivatives of the total cost function with respect to factor prices are homogenous of degree zero in factor prices. The property that the total cost function is concave in factor prices implies that the matrix of second-order partial derivatives

a^c d^C apT" aPfSw

d^C d^c dvd?f aw^

is negative semidefinite. This property implies that the diagonal elements of this matrix are nonpositive and that the off-diagonal elements are symmetric. In view of equations (A.l) and (A.2) we find that

aPf ap^ - ' d\j ' d^ - '

aw aPfaw awaPf aPf*

That is, output constant input demand functions are negatively sloped and symmetric with respect to price changes.

The derivative of the total cost function with respect to output is marginal cost

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(A.3) dc aQr

- MC(Qr, Pf, W).

Because the total function is homogeneous of degree 1 in Pf and W, the marginal cost function also is homogeneous of degree 1 in Pf and W since differentiation is only with respect to Q^. From equations (A.1)-(A.3) we find that

a^c a^c aPfaq, aq.aPf

aMC aPf

dL d^c a^c aMC aQr awaq^ aq^aw d\j-

That is, the change in any factor demand with respect to a change in quantity of output equals the change in marginal cost with respect to a change in the price of that factor.

The profit function, 7r(Pj., Pf, W) , is defined as the maximum profit for prices Pj., Pf, and W. This function has the following properties (Varian, chapter 1) : (1) nondecreasing in P^., nonincreasing in Pf and W; (2) homogeneous of degree 1 in P^., Pf, and W; (3) convex in P^,, Pf, and W; and (4) continuous in Pj., Pf, and W. Application of Hotelling's lemma yields the output-price constant supply and factor demand functions

(A.4) Qr

(A.5) Qf

(A.6) L

_dn__ (P^, Pf, W) = S^ (P^, Pf, W), ÔP,

-dn (P^, Pf, W) = Df (P^, Pf, W), aPf

-djL (Pr, Pf, W) = DL (P^, Pf, W). aw

Because 7r(») is homogeneous of degree 1 in all prices, the supply and factor demand functions (which are obtained as first-order partial derivatives of the profit function) are homogeneous of degree zero in prices. The property that the profit function is convex in prices implies that the matrix of second- order partial derivatives

d^n a^TT a^TT ap/ ap^aPf ap^aw

a^TT

aPfap, IT^ aPfaw

d^i^ d^n awap^ awaPf aw^

is positive semidefinite, matrices, we find that

Hence, by the properties of positive semidefinite

3Qr = aPr aPr^

dL -d^n aw dW

> 0,

< 0,

aPf

a^TT d^ir ■dQi aPf aPj.aPf aPfap^ ap^ '

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aw a^TT

aPj,aw a^TT

awaPr -aL ap.

aQf aw

a^^r aPfaw

a^TT awaPf

aL aPf-

That is, output supply functions are upward sloping, output-price constant factor demand functions are downward sloping, and the supply and demand functions are symmetric with respect to all price changes.

Industry Level

Conceptually, industry demand for a given factor of production cannot be obtained as the horizontal summation of individual firms' demand functions. The reason can be seen in the following example. If, say, all firms should reduce input purchases in response to an input price increase, the output supply curve would shift to the left, causing output prices to rise. In turn, the firms' input demand schedules would then shift somewhat to the right. The expected net effect is for the industry derived demand schedule to be steeper than the schedule constructed as the horizontal summation of individual firms' demand functions. In a similar manner, the industry product supply curve will be a general equilibrium relationship, showing the total change in quantity supplied to a given change in output price. This schedule is also expected to be steeper than the curve constructed as the horizontal summation of the individual firms' supply curves. This is because of the effect the collective output response of firms has on factor prices whose supply schedules are not perfectly elastic.

In this subsection, the properties of industry factor demand are examined alternatively in the short run and the long run. Throughout, input prices are taken as parametric so that total input response to a given change in input price can be evaluated. Intuitively, the comparative static results can be thought of as general equilibrium responses produced by a shift in input supply that produces a 1-percent change in the input price. The terms "output constant" and "output-price constant" are standard terminology for output and output price being parametric, respectively.

Following Heiner, it is useful to relate firm-level input demand behavior corresponding to the two cases of output constant and output-price constant demand functions. These two cases can be related by substituting the profit-maximizing output level, equation (A.4), into the output variable of the cost-minimizing input choice, equations (A.l) and (A.2), to obtain

A

(A.7) Df(P^, Pf, W) = Df<=[S^(P^, Pf, W), Pf, W], A

(A.8) DL(P^, Pf, W) - DL^[S^(P^, Pf, W), Pf, W].

To obtain the shortrun factor demand responses, we must specify the manner in which output price adjusts to clear the market in response to a factor price change. Let i index firms. Then, from equation (A.4) the aggregate output response of all firms holding output and factor prices constant is SSj,^(Pj., Pf, W) . Let D(Pj.) denote the output demand function so that market clearing in the output market is defined as

(A.9) D(P,) - SS/(P^, Pf, W) » 0.

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Applying the implicit function theorem to equation (A.9), we find that the output price response to a change, for example, in the price of the farm input is

(A.10) - - S aPf ap. - 2

as. a?.

Since profit-maximizing behavior implies that supply curves of individual firms are upward sloping, and assuming the output demand curve is downward sloping, the denominator of equation (A.10) is negative so the sign of (A.10) depends on the sign of the numerator. In all normal cases (for example, normal factors) the numerator of equation (A.10) is negative, so output price and factor price changes can be expected to move in the same direction.

Now by substituting the reduced form solution of equation (A.9), Pr(Pf, W) , into equations (A.7) and (A.8) we obtain a firm's shortrun factor demand functions which take into account the effect of output price always adjusting to equate demand with industry output supply. That is,

A

(A.11) D£(Pj, W) - Df[P^(P£, W), Pf, W], A

(A. 12) DL(Pf, W) - DL[Pr(Pf, W), Pf, W].

By differentiating equation (A.11) with respect to Pf we obtain

(A.13) 3Df ÔP.

dDf 3P, 3Df aPr ôPf aPf

Differentiating (A.7) with respect to Pf obtains

(A.14) dDf dU¿ âSj. ^ dPf" as. dP, d?f

Thus, upon substituting (A.14) into (A.13), and indexing this response for the ith firm, we obtain

(A.15) £5x1 ap.

-âBill as. ap.

aOf^ ap. dPf

Therefore, shortrun response of the ith firm to a change in factor price consists of three effects: (1) output constant effect, (2) output response effect, and (3) output-price response effect. The first two effects together represent the profit-maximizing response holding output price constant. The sum of these two effects is unambiguously negative. However, the third effect may be either positive or negative, and it is normally expected to be positive. Thus, for an individual firm, the law of demand can be violated.

However, as Heiner proves, the ambiguity of the sign of the shortrun response of a firm to changing factor price becomes unambiguously negative when the responses are aggregated over all firms. That is, the sign of E aÖ^^/aPf is unambiguously negative. While the proof of this result is tedious (and is therefore not given here), the intuition is straightforward. In particular, Heiner shows that the aggregate factor demand response is bounded by the two extreme cases when demand for the output is perfectly inelastic and perfectly elastic. That is, the industry factor demand curve lies between two demand curves: one constructed as the horizontal summation of the constant output

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demand responses and the other constructed as the horizontal summation of the output-price constant demand responses, which are both unambiguously negatively sloped. An intuitive discussion of this result can be found in Friedman (p. 184). Braulke (1984) shows that this result still holds if the industry faces less than infinitely elastic supply curves in some of its input markets.

The traditional approach to analysis of longrun industry behavior is to assume free entry into and exit from the industry and to assume identical firms, since this will avoid aggregation problems. Under these assumptions, there exists an aggregate production function which exhibits constant returns to scale. Constant returns exist because, with given technology, fixed factor prices, and determinate firm size (that is, each firm operating at minimum longrun average cost), a doubling of all inputs could be achieved by doubling the number of producing firms, which in turn would lead to a doubling of industry output. Under these conditions, the industry total cost function is linear in output, output price equals minimum average cost of the representative firm, and the relative input demands are independent of the level of output (Silberberg, Diewert 1981). That is,

(A.16) P^ - AC(Pf, W) ,

(A. 17) Qf = 11- (Pf, W) . Q^ = ACf(P£, W) . Q^,

rIAC (A. 18) L - — (Pf, W) . Q, = AC,(Pf, W) . Q,,

where AC(») is the average cost (unit cost function), ACf(«) is the relative factor demand function for the farm input, and AC^ (•) is the relative factor demand function for other inputs.

The comparative statics of the industry demand for a factor in the long run with identical firms can be conducted upon substituting the output demand function D(Pj.) for Qj, in equations (A. 17) and (A. 18) and then differentiating equations (A.16)-(A.18) with respect to the appropriate factor price. For the farm input, the slope of the industry factor demand curve can be characterized as

aQf aACf o + Ar ^D ^AC aPf " aPf ^^ ^ aPf ap^ '

or, since the Allen own-elasticity of substitution equals

AC * aACf/aPf

""'' '^ (Qf/Qr)^ '

(Diewert 1974, p. 114), we have that

9^L Qi! . „ . Qi„ + Ar 2 30 dt ^^ '' ~Q? ' "äF,

or

E ff ^ JÈfls . _ZiQf .a + Q¿ £t I^ £H

aPf Qf p,Q, " Q^ p^ q, dF,'

or

(A. 19) E„ = Sf (aff + e).

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since P = AC and ACf = (Qf/Qj.) by equations (A.16) and (A.17), respectively. Here Sf is the farmer's share of the retail dollar and e is the elasticity of demand for the retail product. Equation (A.19) indicates that the industry elasticity of demand for a factor may be decomposed into a substitution effect (Sf'Off) and an output effect (S^e) . The first term in equation (A. 19) is precisely the output constant own-price elasticity of demand for the farm input. By the zero homogeneity property of output constant factor demand functions, the sum of the own and cross price elasticities must equal zero. This implies in the two-factor case that

Sfaff + SLafL = 0,

or Sfaff = -SL^fL = -SL^,

where a = afL = Allen partial elasticity of substitution between the farm input and marketing inputs, and SL = 1 - Sf is the marketing group's share of the retail dollar. This result implies equation (A.19) can be written as

(A.20) Eff = -(1 - Sf) a + S^e.

Equation (A.20) clearly shows that, under the conditions specified (identical firms in longrun competitive equilibrium), industry demand for the factor in question is downward sloping (assuming demand for the retail product is downward sloping).

Diewert (1981) has shown longrun equilibrium behavior can be characterized by equations (A.16)-(A.18) under less stringent conditions than identical firms. Necessary conditions that the industry production function exists and exhibits constant returns to scale, are that optimal plant size (for a typical firm) be small relative to industry output, and that there are no barriers to entry into the industry. Braulke (1987), pointing out that entry and exit ensure that the net addition to industry supply must always be nonnegatively correlated with changes in longrun equilibrium prices, proves that the law of demand holds for a competitive industry in the long run regardless of whether firms are similar or different from one another in any way.

Appendix B: Data Sources and Derivations

The basic data required for estimating reduced-form retail and farm price equations include prices for each commodity at the retail and farm levels, farm-level quantities for each commodity, an index of food marketing costs, and data for construction of the retail demand shift variables.

Price and quantity data were generally taken from the USDA's Food Consumption, Prices, and Expenditures (FCPE). With the exception of beef and veal, and pork, values for per capita farm output were obtained directly from FCPE. Values for total farm output were derived by multiplying per capita quantities by the July 1 U.S. civilian population. Retail price data are consumer price indexes, and farm price data are producer price indexes of crude foodstuffs published in FCPE. The income variable used is total personal consumption expenditures. All price data (including the index of food marketing cost index) and income were deflated by the consumer price index for all items.

Farm output values for beef and veal, and pork, were obtained by dividing per capita disappearance levels (carcass-weight equivalent) reported in FCPE by

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the average dressing yields for animals slaughtered under Federal inspection published in USDA's Livestock and Meat Statistics.

Farm prices for beef and veal, and pork, were adjusted for by-product values using the formula

Aln P, - ^ .in P,* - i^ M„ P,,.

where GFV is the gross farm value, NFV is the net farm value, BPV is the farm by-product value, Pf* is the unadjusted producer price index, and P^p is the price of by-products. Producer price indexes are for cattle and calves and hogs. Values for GFV, NFV, and BPV (GFV - NFV + BPV) are fixed at their average 1967-69 values reported in Livestock and Meat Statistics. Values for the price of by-products for each meat category were derived by converting farm by-product values reported in Livestock and Meat Statistics to farm- weight equivalents. Adjustment factors to convert to farm-weight equivalents are unpublished data provided by Lawrence A. Duewer of the Economic Research Service, USDA.

The marketing cost variable is the index of food marketing costs developed by Harry Harp, of USDA's Economic Research Service, with unpublished values for this index from 1960 to 1968 also provided by Harp. Values for 1955 to 1960 were estimated by regressing the food marketing cost index on overlapping values of the discontinued index of intermediate goods and services for 1960- 68 published in the USDA's Marketing and Transportation Situation. The regression equation was then used to predict values of the index of food marketing costs for 1955-59 using values of the index of intermediate goods and services for this period.

Consumer demand elasticities used as weights for construction of the retail demand shift variables (the Z's in equation 28) are shown in appendix table 1. These elasticity estimates globally satisfy the homogeneity condition; they locally satisfy the symmetry and Engel aggregation conditions at the 1967-69 average values for the commodity consumer expenditure shares. These elasticity estimates are unpublished values provided by Kuo S. Huang of the Economic Research Service using the composite demand system specified in Huang and Haidacher. The same consumer price indexes used in estimating the matrix of elasticities in appendix table 1 are used in estimating the reduced-form price equations in this study.

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Appendix Table 1--Matrix of price and income elasticities used in empirical analysis

Elasticity with respect to price of--

Beef and Conmodity veal Pork Poultry Fish Eggs Dairy

Processed Fats Fresh Fresh fruit and Non- Income and oil fruit vegetables vegetables Cereals Sugar Beverages food elasticity

Beef and veal

Pork

Poultry

Eggs

Dairy

Fresh fruit

Fresh vegetables

Processed fruit and vegetables

-0.7765 0.1019 0.0479

0.1387 -0.6464 0.0833

0.1850 0.2349 -0.7274

0.1143 -0.0323 0.0377

-0.0054 0.0190 -0.0063

0.0032 0.0251 -0.0354 0.0069 -0.0366

0.0239 -0.0186 0.0109 -0.0110 -0.0069

0.0202 -0.0358 -0.0326 -0.0353 0.1452

-0.0296 -0.0933 0.0243 -0.0376 -0.0096

-0.0002 0.0043 -0.2108 -0.0075 0.0159

-0.0617 0.0185 0.1432 -0.0575 -0.0018 0.0966 -0.0062 -0.1177

0.0140 0.0210 0.0785

-0.0465 0.0528 -0.1142

-0.0095 -0.0144 0.0558 0.0037 0.0784

-0.0083 -0.0016 0.1374 -0.0165 0.0465

0.0060 -0.0387

0.0027 0.0458

0.1036 -0.2663

0.0187 0.0104

0.0165 0.0890

0.1085 0.1325

0.2226 0.1017

0.0508 -0.3115

0.0132 -0.0445 0.0022 -0.2072 0.9445

-0.0353 -0.0208 -0.0264 -0.1789 0.6390

0.1686 -0.2034 0.0232 0.0308 0.3179

0.0621 -0.0064 0.0403 0.0025 -0.0642

-0.0084 -0.0014 0.0030 -0.0435 0.1358

-0.0433 0.1108 -0.0486 0.6055 -0.8787

-0.0838 0.0820 -0.0139 -0.0623 -0.-288

0.0768 -0.0984 0.0144 -0.4007 0.6190

Appendix Table 2--Flexibilities of farm prices with respect to other retail prices and population

Retail price

Farm Fats Non- price Fish and oils Cereals Sugar Beverages food Population

Beef and veal 0.01 0.00 0.04 -0.08 0.01 -0.30 0.96

Pork 0.06 -0.02 -0.04 -0.10 -0.05 -0.42 0.96

Poultry 0.06 -0.10 0.40 -0.51 0.05 -0.19 1.44

Eggs -0.19 -0.26 0.49 -0.14 0.28 -0.10 0.97

Dairy 0.00 -0.01 -0.02 0.00 0.00 -0.08 0.97

Fresh vege- tables -0.01 0.00 -0.17 0.13 -0.03 -0.21 1.08

ii>U.S. Government Printing Office : 1989 - 261-499/20020

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