How to read this
With constant returns, . The translog is a second-order Taylor approximation of in the log prices, so it fits any unit cost function near . By Shephard's lemma its cost shares are linear in log prices (&) and can be estimated by OLS.
- Approximate a CES: move the prices away from and see where the approximation drifts.
- Your own parameters: the restrictions from economic theory are checked one by one.
- The elasticities (%), (#), ($) and the results of Arnberg and Bjørner (2007) have panels of their own.
Parameters
Current prices
A second-order approximation
Cost shares are linear in log prices
Elasticities at the current prices
Restrictions from economic theory
Results: Arnberg and Bjørner (2007)
Danish industrial companies, firm-level panel, . Inputs: 1 electricity, 2 other energy, 3 labour, 4 machine capital. Reported numbers only; nothing is re-estimated here.
Price elasticities (part of their Table 5)
Row: input whose demand responds; column: price that changes. At sample-mean cost shares; standard errors in parentheses; * 10 %, ** 5 %, *** 1 %. Own-price elasticities are small; substitution between electricity and other energy is small; energy and machines are complements (negative cross-price elasticities).
Translog parameters (part of their Table 3)
Why firm-level panel data?
Two firms with Leontief technologies, so no substitution inside either firm: energy intensive and capital intensive . Energy gets dearer and output moves from the first firm to the second.
More in “Substitution or Composition?” → (prices, consumers' choice and the econometrician's view)