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In this paper we show how to use value data (price times quantity) to construct Fisher price and quantity indexes. In particular, we think of revenue and expenditure data. This model extends the work of Cross and F?re, who showed how to recover relative prices from value data with no explicit price or quantity information. We examine the accuracy of our model over a range of price changes, firm sample sizes, and response variation, in a Monte Carlo experiment in which firms respond to price changes with error. The model outperforms it component indexes with accuracy levels that increase with response variation.

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}. However, his results depend on price information.

Bowley [^{2}. Fisher [

In a separate line of inquiry, Afriat [^{3} (Cyclical Consistency) to recover unobserved utility from consumer-level price and quantity data and later unobserved technology from production prices and quantities [

Balk [

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

^{3}We use the term Weak Axiom of Profit Maximization (Weak Axiom) discussed by Varian [

^{4}For the existence of a minimum, see Färe and Primont [

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

The cost function^{4} is defined

Konüs [

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

Consider a unit cost function

The Fisher index

The Fisher is then

Following Balk [^{5},

The expenditure share is given by

We wish to recover price information from the inner product of price and quantity

Here, all inputs and outputs are included

Identification is trivial when firms are strictly cost minimizing and the technology

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

This economy is not cost minimizing, but satisfies Afriat’s [

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

Under dimensional invariance (see Balk [

The vector of shadow price relatives

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

Both response magnitude and direction are random variables, following two independent 2-parameter Beta distributions, with identical shape parameter sets

Define accuracy

We consider a Translog unit cost function and impose parameter restrictions

The value-based Fisher index is understated,

Value-based Paasche and Laspeyres component indexes are under and overstated, respectively, at all levels of volatility and price deviations. This is reasonable, because they reference only a single-period market basket, whereas the Fisher index averages over the two-period market basket.

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-

Paasche | Fisher | Laspeyres | |||||||
---|---|---|---|---|---|---|---|---|---|

15 | 20 | 30 | 15 | 20 | 30 | 15 | 20 | 30 | |

0.14 | 0.94 | 0.94 | 0.93 | 1.04 | 1.03 | 1.03 | 1.14 | 1.14 | 1.14 |

0.29 | 0.91 | 0.91 | 0.90 | 1.00 | 1.00 | 0.99 | 1.10 | 1.10 | 1.09 |

0.41 | 0.91 | 0.90 | 0.90 | 0.99 | 0.99 | 0.98 | 1.08 | 1.08 | 1.07 |

Paasche | Fisher | Laspeyres | |||||||
---|---|---|---|---|---|---|---|---|---|

50% | 100% | 200% | 50% | 100% | 200% | 50% | 100% | 200% | |

0.14 | 0.99 | 0.97 | 0.94 | 1.00 | 1.01 | 1.04 | 1.01 | 1.04 | 1.14 |

0.29 | 0.99 | 0.96 | 0.91 | 1.00 | 1.00 | 1.00 | 1.01 | 1.03 | 1.10 |

0.41 | 0.99 | 0.96 | 0.91 | 1.00 | 0.99 | 0.99 | 1.01 | 1.03 | 1.08 |

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.