Derived Demand, Marketing Margins, and Relationship Between Output and Raw Material Prices

Abstract

This chapter reviews the historical approach to modeling derived demand for raw materials, with a focus primarily on derived demand for agricultural inputs. This discussion is followed by the presentation of a general model of derived demand that allows one to link the system of consumer demand functions to the corresponding system of derived demand functions for agricultural inputs. The focus is on developing a framework that is theoretically consistent with consumer demand behavior and market intermediary behavior. The Gardner (1975) model develops theoretical implications for a long-run competitive behavior. The Wohlgenant (1989) model is an empirical model used to estimate and test basic tenets of competitive behavior for retail-to-farm price linkage of a complete system of demand functions. Empirical results from Chapter 5 for consumer demand for meats with linkages to other goods are used in the estimation of reduced-form equations for retail and farm prices, thereby linking retail demand to farm-level demand for each of the meat commodities. Tests for symmetry and constant returns to scale show consistency with the Gardner model.

Appendix: Input Substitution and Output Demand Uncertainty

The purpose of this Appendix is to show how continuous, convex-shaped isoquants can result from a linear programming problem of choosing a variable factor, $v$, in combination with a quasi-fixed factor, $m$, via two production techniques to minimize input use subject to given expected output, $q$, and a fixed available level of the factor, $b$. It is assumed that both factors, $v$ and $m$ are divisible and that $v$ is freely available in supply. This corresponds closely to livestock slaughter and processing where predetermined quantities of the livestock are scheduled for processing ahead of time, and where the plant adjusts its variable factors to meet expected demand. The Appendix parallels Liviathan (1971) for the simplest case of two production techniques.

The processor minimizes $v_{1} + v_{2}$ (equivalently, minimizing costs, $w\left( {v_{1} + v_{2} } \right.), {\text{where}} w {\text{is fixed}}$) subject to the production function, $q \le a_{q1} v_{1} + a_{q2} v_{2}$, and the restriction on the availability of the raw material, $b \ge a_{m1} v_{1} + a_{m2} v_{2}$, with $v_{i} \ge 0, \forall i$. The deterministic solution can be seen graphically from the figure below.

With the way the two constraints are shown, the solution to the problem is at point B where the curve intersecting point B gives the lowest possible sum of $v_{1}$ and $v_{2}$. For a given level of output, represented by the ordinate q/a_q1, there is a tradeoff between $v$ and $m$ between points A and B. At point B both constraints are satisfied, but at point A only the production constraint is met. Suppose point C is the ex ante choice of inputs with expected output given by the dotted line. This point predetermines the quasi-fixed input $m$. Suppose ex post output is given by the line segment AB. With the quasi-fixed factor fixed by point C, the input combination $v_{1}$ and $v_{2}$ will be directly below point C on the line segment AB. With the variable factor freely available, it will adjust downward to meet output $q$. In general, for the output $q,$ combinations of $v_{1}$ and $v_{2}$ would be distributed between A and B. For points to the left of the ray OC, whose slope is $\frac{{{\text{a}}_{q1} }}{{{\text{a}}_{m1} }}$, the optimum relationship between $v {\text{and}} b$ for given $q$ is $v = \gamma^{1} q - \beta^{1} b$ where $\beta^{1} = 0$. For points to the right of $\frac{{{\text{a}}_{q1} }}{{{\text{a}}_{m1} }}$, the optimum relationship between $v {\text{and}} m$ for given $q$ is $v = \gamma^{2} q - \beta^{2} b$.

Define $\alpha \left( b \right) = \frac{{{\text{a}}_{q1} }}{{{\text{a}}_{m1} }}$ and let $f(q,m) = \gamma q - \beta b$ and let the probability density function for demand be $h\left( q \right)$ such that $\mathop \smallint \limits_{L}^{U} h\left( q \right)dq = 1$. The relationship between the expected value of $v$ and output is $E\left( v \right) = \mathop \smallint \limits_{L}^{\alpha \left( b \right)} f\left( {q,b} \right) h\left( q \right)dq + \mathop \smallint \limits_{\alpha \left( b \right)}^{U} f\left( {q,b} \right)h\left( q \right)dq$. Partially differentiating with respect to $b$ we obtain

$$\frac{\partial E\left( v \right)}{\partial b} = \mathop \smallint \limits_{L}^{\alpha \left( b \right)} \frac{{\partial f\left( {q,b} \right)}}{\partial b}h\left( q \right)dq - \mathop \smallint \limits_{\alpha \left( b \right)}^{U} \frac{{\partial f\left( {q,b} \right)}}{\partial b}h\left( q \right)dq$$

$$= \mathop \smallint \limits_{L}^{\alpha \left( b \right)} - \beta^{1} h\left( q \right)dq - \mathop \smallint \limits_{\alpha \left( b \right)}^{U} - \beta^{2} h\left( q \right)dq\quad \text{or}$$

$$\frac{\partial E\left( v \right)}{\partial b} = \beta^{2} \mathop \smallint \limits_{\alpha \left( b \right)}^{U} h\left( q \right)dq$$

because $\beta^{1} = 0.$ The marginal rate of substitution between b and v is ${\text{MRS }} = - \frac{\partial E\left( v \right)}{\partial b}$. Differentiating with respect to $b$:

$$\frac{\partial MRS}{\partial b} = - \beta^{2} h\left( {\alpha \left( b \right)} \right) \alpha \left( b \right) < 0$$

Therefore, the MRS is continuously differentiable and convex to the origin as postulated.

Problems

• 7.1 It is frequently observed that income and the farm value share are negatively correlated. Using the Gardner long-run competitive model of the food industry, explain two ways in how this could occur. Do these conditions seem reasonable?

• 7.2 For pork where we find the elasticity of substitution is not significantly different from zero, calculate the new reduced-form parameters using Eqs. (7.78)–(7.80) using data from Table 7.2.

• 7.3 Define incidence of marketing charges on farmers as $I_{r,w} = \frac{{\left| {\frac{dlog r}{dlog w}} \right|}}{{\left( {\left| {\frac{dlog r}{dlog w}} \right| + \left| {\frac{dlog p}{dlog w}} \right|} \right)}}$. Calculate values of $I_{r,w}$ for beef and veal, pork, and poultry from the empirical results in Tables 7.6 and 7.8 for beef and veal and poultry, and from problem 7.2 for pork. Explain why there are differences in the incidence of marketing charges across commodities.

• 7.4 The Wohlgenant (1989) model takes farm quantities as predetermined in the short run. Suppose over a period of time, producers are able to adjust supplies in response to the change in farm price. In particular, suppose that the supply elasticities for beef and veal, pork, and poultry are 0.5, 1.0, and 3.0. Using the results from Tables 7.6–7.8 estimate the effects of exogenous shifts in demand, processing costs, and farm supplies on retail and farm prices for each commodity. Assume that for each commodity, supply changes can be written as $Er_{i} = \left( {1/\varepsilon_{i} } \right)Em_{i} + Es_{i}$, where $\varepsilon_{i}$ is the supply elasticity of commodity and $Es_{i}$ is the (vertical) relative change in supply. Discuss the new results incorporating the supply functions, and compare with the results in Tables 7.6–7.8.

• 7.5 The COVID-19 pandemic caused disruptions in red meat and poultry processing. In some instances, there was more than a 40 percent reduction in the workforce. Using the comparative static results from Gardner’s model with the parameter estimates from this study, calculate the effect of a 10 percent reduction in employment on retail-to-farm price ratios for the three meat commodities. For determinate solutions, assume that the short-run supply elasticity for each farm input is 0.1. Discuss the results and explain why meat processors would not necessarily profit from the increase in the price spread, in this instance.