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

Our approach to constructing index numbers for aggregate price risk is in the spirit of the economic analysis of index numbers under certainty. In our case, price risk for individual outputs contributes to risk regarding total revenues: variances and covariances for output prices contribute to variance of total revenue. Ignoring output (quantity) risk or uncertainty, price risk and output levels jointly contribute to revenue risk as [VR.sub.t] = [y.sup.T.sub.t][Vp.sub.t][y.sub.t], where Vp is the price covariance matrix and y is a vector of output levels. Our index number theory is invariant to the time dimension of price risk, i.e., the price covariance matrix Vp can be viewed as reflecting either (e.g.) annual, monthly, or daily price risk. Of course, the particular time dimension is important in empirical applications of the theory.

An appropriate aggregation procedure for price risk Vp will preserve the contribution of Vp to revenue risk while controlling for effects of output levels y. This is the fundamental criterion in designing index number approaches to aggregation of price risk over commodities, and it is similar in spirit to standard index number theory. Standard index number problems are best addressed in terms of value ratios rather than levels (Diewert 2004a), and we proceed in a similar manner.

The following "Laspeyres" index is the most obvious approach to aggregating price risk over commodities:

(4) [(VP.sub.1]/[VP.sub.0])L = [y.sup.T.sub.0][Vp.sub.1][y.sub.0]/ [y.sup.T.sub.0][Vp.sub.0][y.sub.0]

using base period weightings [y.sub.0] throughout the index. An analogous "Paasche" index [(VP.sub.1][VP.sub.0]).sup.P] can be defined using base period weightings [y.sub.1]. Such weightings would obviously be appropriate if outputs were in fixed proportions. However, in principle this approach misrepresents the contribution of price risk Vp to revenue risk VR under general changes in output levels y, somewhat as aggregation with standard Laspeyres indexes generally loses the economic meaning of the subaggregates (Diewert 1981). (5) An analogous "Fisher" index is

(5) [([VP.sub.1]/[VP.sub.0]).sup.F]

= [{[([VP.sub.1]/[VP.sub.0]).sup.L]([VP.sub.1]/ [VP.sub.0]).sup.P]}.sup.1/2].

Fisher indexes typically have better properties than do Laspeyres or Paasche indexes. Nevertheless, I am unaware of any applications or references even to these Laspeyres, Paasche, or Fisher indexes for aggregation of price risk.

Aggregation of Price Variances and Covariances: A Tornqvist Index Approach

Similar to the product test equation (1), a fundamental goal of index number theory for aggregating output price risk should be to decompose the change in revenue risk between two periods, [y.sup.T.sub.1][Vp.sub.1][y.sub.1]/[y.sup.T.sub.0][Vp.sub.0][y.sub.0], into a price risk change part [VP.sub.1]/[VP.sub.0] and a quantity change part [Y.sub.1]/[Y.sub.0]. So in the spirit of standard index number theory and elementary statistics, [VP.sub.1]/[VP.sub.0] and [Y.sub.1]/[Y.sub.0] should satisfy the following equation analogous to (1):

(6) [([VP.sub.1]/[VP.sub.0])([Y.sub.1]/[Y.sub.0]).sup.2] = [y.sup.T.sub.1][Vp.sub.1][y.sub.1]/ [y.sup.T.sub.0][Vp.sub.0][y.sub.0].

Then a price risk index can be correctly calculated from (6) given an appropriate aggregate output index. This connects aggregation of price risk to the formulation of aggregate index numbers for multiple outputs.

Suppose an output quantity index [Y.sub.1]/[Y.sub.0] is superlative under appropriate behavioral assumptions. Then (6) implies an aggregate price risk index [VP.sub.1]/[VP.sub.0]

(7) [VP.sub.1]/[VP.sub.0] = ([y.sup.T.sub.1][Vp.sub.1][y.sub.1]/ [y.sup.T.sub.0][Vp.sub.0][y.sub.0]/ [([Y.sub.1]/[Y.sub.0]).sup.2]

that is exact and superlative in terms of preserving the contribution of price risk for commodities to revenue risk.

Assuming risk aversion, index numbers for multiple outputs should not be calculated from profit maximization, and in general index numbers depend on knowledge of risk preferences or the corresponding dual utility function (Chambers 1983). Nevertheless, we can show that a Tornqvist-like aggregate output quantity index is appropriate assuming a (static) Translog cost function and constant returns to scale (CRTS) in nonjoint technologies, which are common assumptions in index number theory. Later, we will relax these assumptions. This result is stated as the following Proposition (see Appendix for proof).

PROPOSITION 1. Define the output quantity index [Y.sub.1]/[Y.sub.0]:

(8) 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.i0] [equivalent to] [C.sub.it]/[C.sub.t] is the share of output i in total cost at time t. Assume CRTS non joint technologies, (static) cost minimization, and a Translog cost function C(w, y). Then

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

where [C.sub.1]/[C.sub.0] is ratio of total cost and log ([AC.sub.1]/[AC.sub.0]) = [[summation].sub.i=1,.,n][S.sub.wj] log([w.sub.j1]/[w.sub.j0]) [S.sub.wj] [equivalent to] {[w.sub.j1][x.sub.j1]/[C.sub.1] + ([w.sub.j0][x.sub.j0]/[C.sub.0])}/2. [AC.sub.1]/[AC.sub.0] is an index of aggregate average cost. Thus [Y.sub.1]/[Y.sub.0] is an exact output quantity index, and it is superlative.

In contrast, if technology is joint or not CRTS, then the above result does not apply. Then an aggregate output index generally depends on the properties of risk preferences as well as technology. For example, assume constant absolute risk aversion (CARA) and the firm solves the utility maximization problem

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [alpha] is the constant coefficient of absolute risk aversion. This leads to the following Proposition.

PROPOSITION 2. Define the output quantity index [Y.sub.1]/[Y.sub.0]:

(11) log([Y.sub.1]/[Y.sub.0])

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

where [[gamma].sub.it] [equivalent to] ([P.sub.it][y.sub.it] - [alpha][VR.sub.t])/[C.sub.t] Assume CARA utility maximization (10) and a Translog cost function C(w, y). Then [Y.sub.1]/[Y.sub.0] satisfies (9).

However, it is important to note that this index requires an estimate of the coefficient of absolute risk aversion [alpha]. (6)

Extensions to a Fisher Index

A Fisher ideal index [([Y.sub.1]/[Y.sub.0]).sup.F] (3) is often advocated for aggregation. The economic interpretation of this index is typically based on revenue maximization: then an aggregator function of the form f(y)= [[[summation].sub.i=1,.,m] [[summation].sub.i=1,.,m][a.sub.ij][y.sub.i][y.sub.j]].sup.1/2] implies that [([Y.sub.1]/[Y.sub.0]).sup.F] = f(y.sub.1])/f([y.sub.0]). Thus, this index is superlative assuming revenue or profit maximization (Diewert 1976, 2004a). However for our purposes, a Fisher output quantity index should be defined from cost minimization and a cost function, independently of profit maximization or risk preferences. (7)

A Fisher-like output quantity index can be developed from a cost minimization model as follows. 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](wt) for [p.sub.t] in (3):

(12) [([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.i1] = C([w.sub.0], [y.sub.1]). The following Proposition shows that (under standard assumptions in the literature) this is a superlative output quantity index.

PROPOSITION 3. Assume nonjoint CRTS technologies and a cost function C(w, y) = c(w)h(y). Then

(13)

[([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]

where Y(w, y) is an output aggregator function.

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, [([Y.sub.1]/[Y.sub.0]).sup.F*] is a superlative output quantity index.

Thus, we have developed two economic index numbers for aggregating outputs based on cost minimization, the Tornqvist index [([Y.sub.1]/[Y.sub.0]).sup.T] (8) and the Fisher index [([Y.sub.1]/[Y.sub.0]).sup.F*] (12). In either case, the output quantity index can be substituted into (7) to calculate a corresponding index of price risk over commodities.

Extensions to Higher Moments, Subindexes, and Output Risk