Most economic models are expressed in terms of prices and quantities. In practice, however, economic data is often available only in terms of revenues or expenditures (value data). Value data is the product of prices and quantities, but does not provide explicit price and quantity information. Cobb and Douglas [1] and Farrell [2] are two canonical examples using such data. Shephard [3] developed duality theory to utilize value data¹. However, his results depend on price information.

Bowley [4] introduced a ratio-based price index, calculated as the geometric mean of the Paasche and Laspeyres indexes². Fisher [5] identified it as “ideal” for satisfying a set of mathematical properties, collectively the Test Approach (see Fisher [6] ), and emphasized its use as a quantity index. Konüs [7] showed Fisher’s ideal index to be an exact solution for a quadratic production function. Diewert [8] demonstrated its exactness for a flexible second-order quadratic mean aggregator (utility or production) function, showing it to be superlative and completing the Economic Approach. Finally, Diewert [9] extended its use to Malmquist [10] output, input, and productivity indexes.

In a separate line of inquiry, Afriat [11] exploited properties of the Weak Axiom³ (Cyclical Consistency) to recover unobserved utility from consumer-level price and quantity data and later unobserved technology from production prices and quantities [12] . Cross and Färe [13] extended Afriat’s approach, using nonlinear programing to recover unobserved price relatives from firm-level revenues and expenditures.

Balk [14] uses dimensional invariance to construct Fisher ideal price and quantity indexes with value data and price relatives, instead of prices and quantities. Statistical agencies utilize this method, augmenting value data with secondary price surveys and constructing elementary price indexes [14] .

In this article, we show how to construct Fisher ideal price indexes directly from value data, without price relatives or the secondary price survey. In the next section, we review the value-based Fisher index, expressed in terms of value data and price relatives. We then recover the Fisher index price relatives from value data, without prices. In Section 3, we evaluate the accuracy of our value-based index in a simple Monte Carlo experiment in which firms respond rationally, but not optimally, to price changes. We then conclude.

Let and be vectors of (input) quantities and their prices, and denote output/utility and the production/utility function

³We use the term Weak Axiom of Profit Maximization (Weak Axiom) discussed by Varian [15] , though many alternatives exist, notably Afriat’s [11] Cyclical Consistency.

⁴For the existence of a minimum, see Färe and Primont [16] . Note that is the inner product of prices and quantities.

⁵Constant price relative across firms is a weaker restriction than constant prices, the so called “Law of One Price” [17] .

The cost function⁴ is defined

Konüs [7] defines a price index for situations 0 and 1 as the ratio of cost functions,

Here, the price index is defined for a fixed output level.

Consider a unit cost function which is homogeneous, quadratic mean of order. Here is the price vector for goods. According to Diewert [9] , this cost function is flexible, since it can provide a second order differential approximation to any linearly homogeneous, twice continuously differentiable, positive function. When a price function is exact for a flexible function, like the quadratic mean, it is said to be superlative.

The Fisher index is exact for a second-order Quadratic cost function. It is the geometric mean of two component indexes. The Laspeyres index is a ratio of prices, weighted by past period quantities, whereas, the Paasche index is weighted by current period quantities:

The Fisher is then.

Following Balk [14] , we can write the value-based Fisher index in terms of value data and price relatives for situations 0 and 1, constant across firms⁵,

The expenditure share is given by, and is the input quantity vector.

We wish to recover price information from the inner product of price and quantity. It is well known that prices and price relatives cannot be recovered from the inner product alone, since it is an under- identified system of 2N equations and 4N unknowns:

Here, all inputs and outputs are included for situations 0 and 1.

Identification is trivial when firms are strictly cost minimizing and the technology is known a priori. To see why, suppose the cost function is differentiable in. By Shephard’s lemma, the solution vector is a function of prices, and the value vector is an exactly identified system of 2N (possibly nonlinear) equations and 2N unknowns,

Price relatives may still be recovered from this system when firms are not strictly cost minimizing, but are instead weakly rational. To illustrate how this is possible, consider first an economy in which there is no quantity response to price change, due to some regulation, capital constraint, or information asymmetry. The response vector is exactly zero for each firm in the economy. Exact price relatives are recoverable directly from the value ratios of any perfectly constrained firm:

This economy is not cost minimizing, but satisfies Afriat’s [12] Weak Axiom of Profit Maximization,

In fact, the perfectly constrained firm defines the null vector 0 of Weak Axiom’s convex response-cone, and the profit maximizing firm defines its upper bound, illustrated in Figure 1.

Under dimensional invariance (see Balk [14] ), the inequality (9) can be rewritten in value and price relative terms:

The vector of shadow price relatives then solves the minimization problem, given values for firms,

To explore the accuracy of the proposed value-based Fisher index, we simulate a two-period, two-input, unit output economy and recovered price relatives from problem (11). Firms respond to input price changes rationally, but incompletely in both magnitude and direction, as illustrated in Figure 1.

Both response magnitude and direction are random variables, following two independent 2-parameter Beta distributions, with identical shape parameter sets. Response magnitude ranges between zero and the cost minimizing response to problem (2). Response direction has compact support from the rationalizing price cone formed by price line and shifted price line. Such responses lead to Farrell [2] technical and allocative inefficiency bounded by the Weak Axiom.

Define accuracy of price index as the ratio of the value-based price index with recovered price relatives and the full information price index:. Then, is accurate. Value-based indexes overstate price changes when and understate when. All statistics are reported for 300 iterations.

Economic data are frequently available in value terms, rather than price and quantity terms. We extended the superlative Fisher index to value data and recovered price relatives by exploiting the Weak Axiom. Our value-

based Fisher model eliminates the need for supplementary price information. The value-based model’s accuracy is encouraging, when firms respond to price changes rationally, but with error. We did not explore econometric approaches to price recovery. We also did not explore the accuracy of the value-based model when firms respond irrationally to price changes, violating the Weak Axiom. Such would be the case when the response error term is two-sided. Both extensions would be of practical interest.

Authors thankfully acknowledge Bert Balk’s extensive criticisms and suggestions for this project.