1. Translog Cost Function E. Berndt and D. Wood, "Technology, Prices, and the Derived Demand for Energy," Review of Economics and Statistics, 57, 1975, pp 376-384E. Berndt and D. Wood, "Technology, Prices, and the Derived Demand for Energy," Review of Economics and Statistics, 57, 1975, pp 376-384.

2. 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)

3. 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:

4. 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:

5. Berndt-Wood Model U.S. Manufacturing, 1947-1971 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

6. 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 ) + 1 - 1/S i,  ij = S j  ij, i,j=K,L,E,M

7. 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