The name is absent

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, s_j .6

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

+ sK(ΔlnP_κ-ΔlnP_o) + 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 j^th product the firm sells, and

of the j^th 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.

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