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where PQ is the price of output. This equation can be rewritten in discrete time using a second-
order Tornqvist approximation of this expression. This is done by replacing differentials with
difference operators and replacing shares with averages of their beginning-of-period and end-of-
period values, sj .6

ΔInC-Δln(QPq) = 11 - 1∣∆InQ + sl (ΔInPI -ΔlnPQ) + sL(ΔInPL -ΔlnPQ)
In )

+ sK(ΔlnPκ-ΔlnPo) + I AlnC ∣ΔlnA

K( k Q) UlnA )

where Δ is the difference operator for period t versus t-1 and the overbars denote cross-period
averages of the associated variabIe.7 The interpretation of each term remains the same when
discrete changes in Iogarithms are used.8

6 This equation hoIds if the (unobserved) cost function is a restricted (homothetic, constant returns)
transIog. Even if these assumptions do not hoId, equation 6 provides a second order approximation.

7 To deaI with muItipIe outputs and inputs, we aggregate across the individuaI commodities using
Tornqvist indices. For exampIe, at a given firm, the rate of growth in the output price index is
Δ ln PQ = ∑θj Δ ln (Pj1Q), where Pj1Q is the price in period t of the jth product the firm sells, and

(suppressing Q superscripts on prices) θ j = .5
.

I              λ

Pjt-1 q jt-1

'

I           λ

Pjt Qjt

is the average share

J„  ^

∑ Pit-1 Qit-1

< i=1                         ■■

J~ ^
ΣPit Qit
< i=1                 ■■

of the jth product in totaI revenues in periods t and t-1. AnaIogous methods Iead to Tornqvist indices of
logarithmic changes in the price of inputs and the price of labor.

8 The approximation is nearly exact for small rates of growth.

12



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