Retail-to-Farm Demand Linkages, Imperfect Competition, and Short-Run Price Determination

Abstract

In this chapter, we consider both vertical and horizontal effects on derived demand for raw materials, and total effects of exogenous variables on retail and farm prices. In the last chapter we showed how retail-to-farm price linkages could be formulated and estimated consistently. In this chapter, these relationships are combined with the retail demand elasticities for meats from Chapter 5 to derive total impacts of retail demand, farm supply, and marketing costs on derived demand and retail and farm prices. We derive the properties of these generalized derived demands as well as formulas for the total elasticities. This chapter also evaluates this approach to retail-to-farm price linkages in light of role imperfect competition may play on the price relationships. Finally, causes of short-run price determination from lags between retail and farm prices are evaluated, and an empirical application to beef price spreads is presented.

Appendix: Derivation of Inventory Cost Function

The purpose of this Appendix is to derive the inventory cost function according to the method used by Holt et al. (1960, Chapter 11). The more realistic case of n products is analyzed by Holt et al. (1960), but I analyze the simpler case of a single product, which produces essentially the same result. The inventory balance condition is

$$h_{t} = h_{t - 1} + q_{t} - x_{t}$$

where $h_{t}$ is the end-of-period inventory level, $h_{t - 1}$ is the beginning-of-period inventory, $q_{t}$ is production during that period of time, and $x_{t}$ is sales during time t. Letting overbars denote expected values, we know that with production known at time t,

$$h_{t} = \overline{h}_{t} + \overline{x}_{t} - x_{t}$$

For simplicity, the firm only incurs inventory holding costs (e.g., storage rental, handling, insurance, interest, etc.) and inventory depletion costs. Let $c_{h}$ denote per unit holding costs and $c_{d}$ denote per unit depletion costs from unmet demand. Then, omitting time subscripts for notational convenience, expected inventory holding and depletion costs equals:

$$\overline{c}_{I} = c_{h} \mathop \smallint \limits_{0}^{{\overline{h} + \overline{x} }} \left( {\overline{h} + \overline{x} - x} \right)f\left( x \right)dx + c_{d} \mathop \smallint \limits_{{\overline{h} + \overline{x} }}^{\infty } \left( {x - \overline{h} - \overline{x} } \right)f\left( x \right)dx$$

where $\overline{c}_{i}$ is expected inventory costs and $f\left( x \right)$ is the probability density function (pdf) of sales. Noting that expected end-of-period inventory equals

$$\overline{h} = \mathop \smallint \limits_{0}^{{\overline{h} + \overline{x} }} \left( {\overline{h} + \overline{x} - x} \right)f\left( x \right)dx + \mathop \smallint \limits_{{\overline{h} + \overline{x} }}^{\infty } \left( {\overline{h} + \overline{x} - x} \right)f\left( x \right)dx$$

expected inventory costs can be written as:

$$\overline{c}_{I} = c_{h} \overline{h} - \left( {c_{h} + c_{d} } \right)\mathop \smallint \limits_{{h + \overline{x} }}^{\infty } \left( {x - \overline{h} - \overline{x} } \right)f\left( x \right)dx$$

Differentiating with respect to expected inventory gives the marginal cost of inventory holding:

$$mc_{I} = \frac{{\partial \overline{c}_{I} }}{{\partial \overline{h} }} = c_{h} - \left( {c_{h} + c_{d} } \right)\left[ {1 - F\left( {\overline{h} + \overline{x} } \right)} \right]$$

where $F\left( {\overline{h} + \overline{x} } \right) = \mathop \smallint \limits_{0}^{{\overline{h} + \overline{x} }} f\left( x \right)dx$ is the probability distribution function.

To proceed further, we need to specify a specific pdf for $f\left( x \right)$. Following Holt et al. (1960), I assume a normal distribution so that

$$\left[ {1 - F\left( {\overline{h} + x} \right)} \right] = \left[ {1 - N\left( {\overline{h} /\sigma } \right)} \right] = N\left( { - \overline{h} /\sigma } \right)$$

where $N\left( t \right) = \mathop \smallint \limits_{ - \infty }^{t} \frac{1}{\surd 2\pi }e^{{ - z^{2} /2}} dz$. Totally differentiating the above marginal cost function yields:

$$dmc_{I} = \left[ {N^{\prime}\left( { - \overline{h} /\sigma } \right)\left( {c_{h} + c_{d} } \right)/\sigma } \right]\left[ {d\overline{h} - \left( {\overline{h} /\sigma } \right)d\sigma } \right]$$

If we also assume as in Holt et al. (1960) that the standard deviation of demand is proportional to expected demand or sales, i.e., $\sigma = \nu \overline{x}$, then the total differential of marginal costs equals:

$$dmc_{I} = \left[ {N^{\prime}\left( {\frac{{\overline{h} }}{{\nu \overline{x} }}} \right)\frac{{\left( {c_{h} + c_{d} } \right)}}{{\nu \overline{x} }}} \right]\left[ {d\overline{h} - \left( {\overline{h} /\overline{x} } \right)d\overline{x} } \right]$$

The linear approximation at $\overline{h}^{0} , \overline{x}^{0}$, and $mc_{i}^{0}$ is:

$$mc_{I} = mc_{I}^{0} + \left[ {N^{\prime}\left( {\frac{{\overline{h} }}{{\nu \overline{x} }}} \right)\frac{{\left( {c_{h} + c_{d} } \right)}}{{\nu \overline{x} }}} \right]\left[ {\left( {\overline{h} - \overline{h}^{0} } \right) - \left( {\frac{{\overline{h} }}{{\overline{x} }}} \right)\left( {\overline{x} - \overline{x}^{0} } \right)} \right]$$

(A.1)

Consider now the quadratic approximation for inventory costs presented in Eq. (8.32):

$$c^{*} = \frac{1}{2}c_{I} \left[ {h - g_{0} - gx} \right]^{2}$$

(A.2)

The parameters of (A.1) can then be matched with those of the derivative of (A.2):

$$\frac{{\partial c^{*} }}{\partial h} = c_{I} \left[ {h - g_{0} - gx} \right]$$

This will be the same formula for marginal costs in (A.1) if:

$$c_{I} = \left[ {N^{\prime}\left( {\frac{{\overline{h} }}{{\nu \overline{x} }}} \right)\frac{{\left( {c_{h} + c_{d} } \right)}}{{\nu \overline{x} }}} \right]$$

$$g_{0} = \frac{{ - mc_{I}^{0} }}{{c_{I} }}$$

$$g = \frac{{\overline{h} }}{{\overline{x} }}$$

These results therefore give us the desired cost coefficients.

Problems

• 8.1 Show that the submatrix of price effects, $\varvec{m}_{r}^{*} \left( {\varvec{r},\varvec{w},\varvec{I}} \right)$, from Table 8.1 is negative semi-definite at the sample means. Use data from Table 5.2 and Table 7.1 in your calculations.

• 8.2 Re-calculate the incidence of market charges using the formula shown in problem 7.2 using the results from Table 8.1. Compare the results with those obtained in problem 7.2. If there are differences, explain why you think they are different.

• 8.3 The food stamp program, SNAP (Supplemental Nutrition Assistance Program), is said to have significant effects on commodity prices. The maximum amount spent by consumers through SNAP was $76 billion in 2011. PCE was $ 10,641.1 billion in 2011. SNAP benefits can only be used for food consumed at home. Proportions of meats consumed at home are estimated to be approximately 0.47 for beef and veal, 0.79 for pork, and 0.78 for poultry (Reed et al. 2002). Estimate the effects of SNAP on retail and farm prices of beef and veal, pork, and poultry using the elasticities reported in Table 8.1.

• 8.4 Suppose that market intermediaries form quasi-rational expectations on raw material prices such that $E_{t} r_{t + 1} = b_{0} + b_{1} r_{t} + b_{2} r_{t - 1}$ Substitute into (8.40) to show the price spread can be expressed as a distributed lag in current and past raw material prices. Under what conditions would this specification result in a negative relationship between the price spread and the current period raw material price? Is this purely coincidental or does it have a theoretical basis? (Hint: see Wohlgenant [1985]).