Cointegration relationships and hedonic pricing of differentiated commodities: an application to price dynamics in the US dairy sector

Abstract

This study investigates the implications of hedonic pricing for price dynamics of differentiated commodities. A conceptual model of hedonic pricing is developed under a Leontief technology, showing how commodity prices reflect the underlying value of their components. Implications for the existence of cointegration relationships among commodity prices are derived. An application to the pricing and dynamics of selected US dairy commodities is presented. It provides evidence on the role of component valuation in the dynamics of dairy commodity prices in the short run as well as in the long run. Distinguishing between market regime and government regime (when the government price support is active), the analysis finds significant differences in dairy price dynamics between the two regimes.

Acknowledgements

This research was supported by a grant from the Wisconsin Milk Marketing Board, and by a USDA grant to the Food System Research Group, University of Wisconsin, Madison.

Notes

 This argument applies in the absence of transaction costs. The presence of transaction costs (e.g. transportation costs) can be important. It can account for spatial price differences, and it can affect the long-run relationships between prices (McNew and Fackler, 1997).

 Differentiated products are products exhibiting different characteristics. In this study, we focus on the case of butter, American cheese and non-fat dry milk. Although these dairy products obtain all their components from milk, their nutritional characteristics differ across products.

 For example, milk can be processed into a variety of dairy products, including fluid milk, cheese, butter, yogurt, ice cream, non-fat dry milk, etc.

 There are many other dairy products (e.g. dry whey, whey protein concentrate). Our commodity choice is motivated in large part by data availability over the last few decades.

 See Luenberger (1995) for a discussion of the benefit function B(·) and its use in efficiency analysis.

 Note that Equation 7 can be alternatively written as B(L) p _t  = β₀ + e _t , where B(L) = [I − β₁ L − β₂ L ² − ··· − β _h L ^h ], and L is the lag operator satisfying L ⁱ p _t  = p _t−i . The VAR process Equation 7 is covariance-stationary if the roots of |B(L)| = 0 are all outside the unit circle (Hamilton, 1994, p. 259). It has at least one unit root if |B(1)| = 0, in which case the VAR is non-stationary.

 Equation 6a considers that hedonic component prices λ _t are the same across all commodities. This implicitly assumes that the reallocation of components across commodities is costless. Note that the analysis could be extended to allow for costly component reallocation. To see that, assume that Equation 6a takes the form

where λ _ikt  = λ _kt  + b _ik . This allows the shadow component prices λ _ikt to vary across commodities. However, the evolution of component prices is ‘parallel over time’ in the sense that there exists values λ _kt such that [λ _ikt  − λ _kt ] is constant over time for each commodity i and for each component k. If b _jk  = 0 for a given j and k, then λ _jk is the shadow price of the kth component in the jth commodity. Then, if positive, b _ik can be interpreted (for i ≠ j) as the marginal cost of transferring the kth component from the jth commodity to the ith commodity. This implies that Equation 6b becomes p _t  = c _t  + Z λ _t  + 

δ _ik b _ik . Given that KZ = 0, this yields

where α = K [

δ _ik b _ik ]. This is Equation 9, with an intercept α added. Thus, a modified Equation 9 can still hold when component prices vary across commodities.

 The price data for selected dairy products are obtained from Dairy Market News (1970–1999), Agricultural Marketing Service, USDA. Wisconsin assembly point prices are used for American cheese measured in 40-pound blocks. For butter, the grade A butter price in Chicago was used. This price series was discontinued in November 1998 and the (adjusted) grade AA butter price in Chicago was used for the latest months. For non-fat dry milk, we use wholesale price of non-fat dry milk for human food. All prices are measured in cents/lb.

 In contrast, the fixed proportion assumption appears less applicable to milk. Indeed, milk composition can vary with season, region, feed management and breed (St-Pierre and Scobie, 1987). For this reason, we did not include the price of milk in our analysis.

 Note that government purchases of dairy products take place regularly for reasons unrelated to the dairy price support programme (e.g. military purchases, government food programmes). An examination of the price data and government purchases data lead us to choose ‘10% of consumption’ as a threshold used to identify when government purchases can be considered as actively supporting dairy prices. Comparing the market price and support price data confirms that the threshold value of 10% reasonably identifies each regime. For example, for American cheese, the year 1977 and most of the 1980–86 period are identified to be in the ‘government regime’ as they exhibit large government cheese purchases and relative price stability. The 1990s fall into the ‘market regime’ with minimal government cheese purchase. A sensitivity analysis with different threshold values (5% and 15%) was performed. We found that these alternative thresholds did not affect our main qualitative results.

 Thus the Q _i estimates in Table 2 are to be interpreted as deviations from the fourth quarter.

 In our analysis, we treat government purchases as exogenous. Hence the dummy variables (Ds) representing regime switching are also treated as exogenous. There is a large literature dealing with unknown or endogenous regime switching models (Perron, 1980; Andrews, 1993; Andrews and Ploberger, 1994; Hamilton, 1994; Vogelsang, 1997). Analysing dairy price dynamics with endogenous government purchases is beyond the scope of this study. This is a good topic for future research.

 This neglects possible differences between the market regime and the government regime. This raises a difficult issue: the dates of regime change differ for each price. With three prices, this would imply the existence of eight regimes. In the context of an AR(7) applied to single equations, handling eight regimes becomes cumbersome (Maddala and Kim, 1998, p. 410). While we neglect such complexities, this suggests a need to interpret our ADF results with caution.

 These results are consistent with the VAR results presented (Figs. 2a–2c). Since the VAR analysis incorporates the effects of regime switching, this suggests that our ADF test results may not be adversely affected by their neglect of regime switching.

 Milk protein includes casein and milk serum protein. The protein in milk, butter and non-fat dry milk contains about 82% casein. However, the protein in cheese is made entirely of casein. As the suggestion of a review, we focus on casein as casein is typically seen as being the most valuable protein in milk and dairy products.

 One pound of butter (non-fat dry milk) has 0.0085 lb (0.3563 lb) of ‘total protein’. Total protein includes ‘true protein’: casein (77.1%) and milk serum protein (16.9%); and non-protein nitrogen (6%).

 The order in the AR(3) is consistent with the VAR(3) reported below. By not including the dummy variables D, the AR(3) for [Kp _t ] reported in Table 1 implicitly assumes that hedonic pricing holds under both regimes. Empirical support for this assumption is presented below.

 The Schwarz criterion consists in choosing the model for which [ln(maximum likelihood) − K/ln(T)/2] is largest, where K is the number of parameters and T is the number of observations.

 Note the consistency of the parameter estimates in the VAR model holds whether or not the prices are cointegrated. However, having I(1) variables and/or cointegration relationships affects the asymptotic distribution of the parameter estimates (Hamilton, 1994). The implications of a VAR model with unbalanced equations (where some of the variables are I(1) and others I(0)) are discussed in Banerjee et al. (1993) and Maddala and Kim (1998, pp. 251–2).

 Note that the parameters β _ijk and β _Dijk in Equation 10 are asymptotically normally distributed, thus justifying the use of an F-test. However, the other parameters in Equation 10 typically have a non-standard asymptotic distribution under cointegration (Hamilton, 1994, pp. 571–656).

 Under the government regime, the estimated eigenvalues are: 0.1410, 0.0011 and 0.0004.

 Under the market regime, the estimated eigenvalues are 0.2066, 0.1224 and 0.0620.