Translog Cost Function E. Berndt and D. Wood, "Technology, Prices, and the Derived Demand for Energy," Review of Economics and Statistics, 57, 1975, pp E. Berndt and D. Wood, "Technology, Prices, and the Derived Demand for Energy," Review of Economics and Statistics, 57, 1975, pp

Production and Cost Functions Production function: Q = f(x) Cost minimizing factor demands: x i = x i (Q,p) Cost function: C =  i=1,…M p i x i (Q,p) = C(Q,p)

Theory of Cost Function Shephard’s Lemma: x i = x i (Q,p) =  C(Q,p)/  p i p i x i /C = (p i /C)  C(Q,p)/  p i Factor Shares: s i =  lnC(Q,p)/  lnp i Elasticity of Factor Substitution: (Own and Cross) Price Elasticity:

Theory of Cost Function Constant returns to scale: C = Qc(p) Average cost function: c(p) = C/Q Marginal cost function:  C/  Q = c(p) Linear homogeneity in prices: c(p)=c( p) 2 nd order Taylor approximation of lnc(p) at lnp = 0:

Berndt-Wood Model U.S. Manufacturing, Output and Four Factors: Q, K, L, E, M Prices: P K, P L, P E, P M The constant return to scale translog cost function: ln(C) = b 0 + ln(Q) + b K ln(P K ) + b L ln(P L ) + b E ln(P E ) + b M ln(P M ) + ½ b KK ln(P K ) 2 + ½ b LL ln(P L ) 2 + ½ b EE ln(P E ) 2 + ½ b MM ln(P M ) 2 + b KL ln(P K )ln(P L ) + b KE ln(P K )ln(P E ) + b KM ln(P K )ln(P M ) + b LE ln(P L )ln(P E ) + b LM ln(P L )ln(P M ) + b EM ln(P E )ln(P M ) Symmetric conditions:  ij =  ji, i,j = K,L,E,M

Berndt-Wood Model Factor shares: S K = P K K/C, S L = P L L/C, S E = P E E/C, S M = P M M/C S K +S L +S E +S M = 1 (because P K K+P L L+P E E+P M M = C) Factor share equations: S K = b K + b KK ln(P K ) + b KL ln(P L ) + b KE ln(P E ) + b KM ln(P M ) S L = b L + b KL ln(P K ) + b LL ln(P L ) + b LE ln(P E ) + b LM ln(P M ) S E = b E + b KE ln(P K ) + b LE ln(P L ) + b EE ln(P E ) + b EM ln(P M ) S M = b M + b KM ln(P K ) + b LM ln(P L ) + b EM ln(P E ) + b MM ln(P M ) Elasticities:  ij = b ij /(S i S j ) + 1 if i≠j;  ij = b ij /(S i S i ) /S i,  ij = S j  ij, i,j=K,L,E,M

Berndt-Wood Model Linear restrictions: b K + b L + b E + b M = 1 b KK + b KL + b KE + b KM = 0 b KL + b LL + b LE + b LM = 0 b KE + b LE + b EE + b EM = 0 b KM + b LM + b EM + b MM = 0 Stata programs and datasets: –bwp.dta, bwq.dtabwp.dtabwq.dta –bw1.do, bw2.do, bw3.dobw1.dobw2.dobw3.do