Aggregation of price risk over commodities: an economic index number approach.

This article develops index number methods for aggregating price risk over commodities in production. The standard economic theory approach to construction of index numbers is extended here to price risk, and the resulting index numbers are closely related to modified Tornqvist and Fisher output quantity indexes. Given the multicommodity nature of price risk, these index number methods should have important empirical applications.

The methodology is applied to annual market price data for the six major Manitoba crops over 1950-2002 and to monthly price data for five crops over 1990-2005. Four index number formulas suggested by this article are compared with a variance of a Tornqvist price index, a common but in principle inferior measure of aggregate price risk. Commodity-level price risk is measured both from a naive expectations model and a two-step multivariate GARCH model assuming constant conditional correlations. Although we advocate modified Tornqvist and Fisher output quantity indexes in constructing our measures of aggregate price risk, data limitations required use of standard Tornqvist and Fisher output quantity indexes. Even so, results indicated that our four measures of aggregate price risk are highly correlated with each other and less highly correlated with the variance of the Tornqvist price index. These results illustrate the potential empirical importance of the index number methods for aggregating price risk over commodities.

Moreover, the economic index number approach to aggregation of price risk over commodities developed here has empirical importance in cases where the standard index number theory does not. Although the standard index number theory for aggregation of prices or quantities over commodities is in principle superior to Laspeyres or Paasche indexes, in practice correlations can be quite high (e.g., Diewert 1976). Similarly in this study, Tornqvist and Fisher output quantity indexes are very highly correlated with a Laspeyres index (table 2). Nevertheless, the aggregate indexes of price risk proposed here are not highly correlated with the standard index of price risk, i.e., a variance of an aggregate price index. In this sense, the extensions of index number theory to aggregation of price risk should be particularly important in practice.

Appendix

Proof of Proposition 1: Nonjoint CRTS technologies imply that the total cost function over all outputs can be expressed as

(A1) C(w,y) = [[summation].sub.i=1.,.m] [C.sup.i](w,[y.sub.i])

and marginal cost equals average cost

(A2) [partial derivative][C.sup.i](w, [y.sub.i])/ [partial derivative][y.sub.i] = [C.sup.i] (w, [y.sub.i])/[y.sub.i] i = 1,., m.

Define the following Tornqvist-like output quantity index [Y.sub.1]/[Y.sub.0]:

(A3)

log[Y.sub.1]/[Y.sub.0]) = [[summation].sub.i=1,.,m] {([theta].sub.i1] + [[theta].sub.i0])/2} log([y.sub.i1]/[y.sub.i0])

where [[theta].sub.it] [equivalent to] [C.sub.it]/[C.sub.t] is the share of output i in total cost at time t. This is similar to a standard Tornqvist input quantity index or output quantity index.

Proceeding similarly to Diewert (1976), we can show that this is a superlative output quantity index corresponding to a Translog cost function C(w, y) and nonjoint CRTS technologies, irrespective of risk aversion and output price risk or uncertainty. Since a Translog cost function C(w, y) is quadratic in logarithms, the quadratic lemma of Diewert (1976) implies

(A4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by Shephard's lemma (assuming all inputs are in static equilibrium) and (A1)-(A2)

= log([W.sub.1]/[W.sub.0]) + log([Y.sub.1]/[Y.sub.0])

where [S.sub.wj] [equivalent to] {([w.sub.j1][x.sub.j1]/[C.sub.1]) + ([w.sub.j0][x.sub.j0]/[C.sub.0])}/2 and [S.sub.yi] [equivalent to] ([[theta].sub.i1] + [[theta].sub.i0])/2 is defined as for (A3). The two sums in the second equality of (A4) are designated as log([W.sub.1][W.sub.0]) and log([Y.sub.1]/[Y.sub.0]), respectively. The index [W.sub.1]/[W.sub.0] can be interpreted as an index for the ratio of an aggregate average cost [AC.sub.1]/[AC.sub.0] under our assumptions, by arguments similar to the standard case (Diewert 1976, 1980a). Then (A4) implies

(A5) [Y.sub.1]/[Y.sub.0] = ([C.sub.1]/[C.sub.0])/ ([AC.sub.1]/[AC.sub.0])

i.e., [Y.sub.1]/[Y.sub.0] is equal to ratio of total cost divided by an average cost index. This implies that [Y.sub.1]/[Y.sub.0] is an exact output quantity index, and it is superlative.

Proof of Proposition 2: Assuming a Translog multioutput cost function C(w, y) and proceeding similarly to (A4),

(A6) log([C.sub.1]/[C.sub.0]) = log([W.sub.1]/[W.sub.0]) +log([Y.sub.1]/ [Y.sub.0])

by Shephard's lemma and the first order conditions [partial derivative][U(x)/[partial derivative]y = 0 for (10). Here [W.sub.1]/[W.sub.0] is defined as in (A3) and [Y.sub.1]/[Y.sub.0] as in (11). Then, proceeding as above, [Y.sub.1]/[Y.sub.0] is a superlative aggregate output quantity index.

Proof of Proposition 3: Assume nonjoint CRTS technologies, so that average cost for output i is [AC.sub.i](w). Define a new Fisher-like index by substituting [AC.sub.i]([w.sub.t]) for [p.sub.t] in (3):

(A7) [([Y.sub.1]/[Y.sub.0]).sup.F*] = [[[AC.sub.0][y.sub.1]/ [AC.sub.0][y.sub.0]].sup.1/2][[[AC.sub.1][y.sub.1]/ [AC.sub.1][y.sub.0].sup.1/2]

where [AC.sub.t] [equivalent to] ([AC.sub.1]([w.sub.t]),., [AC.sub.m]([w.sub.t])). Here, e.g., [AC.sub.0][y.sub.1] [equivalent to] [[summation].sub.i=1,.,m][AC.sub.i] ([w.sub.0])[y.sub.il] = C([w.sub.0], [y.sub.i]). So (A6) can be expressed as

(A8) [([Y.sub.1]/[Y.sub.0]).sup.F*] = C[([w.sub.0], [y.sub.1]).sup.l/2]C[([w.sub.0], [y.sub.0]).sup.-1/2] x C[([w.sub.1], [y.sub.1]).sup.1/2]

Adopt the standard assumption that C(w, y) = c(w)h(y) where c(w) is a unit cost function and h(y) is an aggregator for outputs y (e.g., c(w)= [[[summation].sub.i=l],.,m] [[summation].sub.j=1.,.m] [b.sub.ij][w.sub.i][w.sub.j]].sup.1/2] implies that the standard Fisher input price index is equal to c([w.sub.1])/ C([w.sub.0])). Multiplying (A8) by C[([w.sub.0]).sup.-l/2]c [([w.sub.0]).sup.1/2] C[([w.sub.1]).sup.-1/2]C[([w.sub.1]).sup.1/2],

(A9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here the ratio between total cost and unit cost defines a corresponding output quantity aggregator, e.g., C(w, y)/c(w) = Y(w, y). So (A9) implies

(A10) [([Y.sub.1]/[Y.sub.0]).sup.F*] = [Y([w.sub.0], [y.sub.1])/Y[([w.sub.0], [y.sub.0])].sup.1/2] x [Y([w.sub.1], [y.sub.1])/Y[([w.sub.1], [y.sub.0])].sup.1/2].

This implies that [([Y.sub.1]/[Y.sub.0]).sup.F*] is an output aggregator, independently of profit maximization or risk preferences (assuming cost minimization and nonjoint CRTS technologies). Since c(w) can be a flexible functional form as discussed above, [([Y.sub.1]/[Y.sub.0]).sup.F*] is a superlative output quantity index.

[Received June 2005; accepted December 2006.]

References

Alexander, C., and E. Lazar. 2006. "Normal Mixture GARCH(1,1): Applications to Exchange Rate Modelling." Journal of Applied Econometrics 21:307-36.

Bohrnstedt, G.W., and A.S. Goldberger. 1969. "On the Exact Covariance of Products of Random Variables." Journal of the American Statistical Association 64:1439-42.

Bollerslev, T. 1990. "Modelling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model." Review of Economics and Statistics 72:498-505.

Bollerslev, T., R.Y. Chou, and K.F. Kroner. 1992. "ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence." Journal of Econometrics 52:5-59.

Bollerslev, T., R.F. Engel, and J. Woolridge. 1988. "A Capital Asset Pricing Model with Time Varying Covariance." Journal of Political Economy 96:116-31.

Chambers, R.G. 1983. "Scale and Productivity Measurement under Risk." American Economic Review 73:802-05.

Chambers, R.G., and R.D. Pope. 1991. "Testing for Consistent Aggregation." American Journal of Agricultural Economics 73:808-18.

--.1994. "A Virtually Ideal Production System: Specifying and Estimating the VIPS Model." American Journal of Agricultural Economics 76:105-13.

Chavas, J.-P., and M.T. Holt. 1990. "Acreage Decisions under Risk: The Case of Corn and Soybeans." American Journal of Agricultural Economics 72:529-38.

Clements, K.W., and H.Y. Izan. 1987. "The Measurement of Inflation: A Stochastic Approach." Journal of Business and Economic Statistics 5:339-50.

Coyle, B.T. 1992. "Risk Aversion and Price Risk in Duality Models of Production: A Linear Mean-Variance Approach." American Journal of Agricultural Economics 74:849-59.

--. 1999. "Risk Aversion and Yield Uncertainty in Duality Models of Production: A Mean-Variance Approach." American Journal of Agricultural Economics 81:553-67.

--. 2005. "Dynamic Econometric Models of Crop Investment in Manitoba under Risk Aversion and Uncertainty." Organization for Economic Co-operation and Development, Working Party on Agricultural Policies and Markets, Paris.

Davis, G.C. 2003. "The Generalized Composite Commodity Theorem: Stronger Support in the Presence of Data Limitations." Review of Economics and Statistics 85:476-80.

Davis, G.C., N. Lin, and C.R. Shumway. 2000. "Aggregation without Separability: Tests of the United States and Mexican Agricultural Production Data." American Journal of Agricultural Economics 82:214-30.

Diebold, F.X., and M. Nerlove. 1989. "The Dynamics of Exchange Rate Volatility: A Multivariate Latent Factor ARCH Model." Journal of Applied Econometrics 4:1-21.

Diewert, W.E. 1976. "Exact and Superlative Index Numbers." Journal of Econometrics 4:114-45.