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Appendix C Econometric Mode! of Production and Technical Change The following material g Ives a detailed description of the econometric model of production and technical change that uncle rlies Chapter 3. * The quantity VT, here called "rate of technical change, " is called "product ivity growth rate" in the text of Chapter 3. The quant ities KT, TILT' ~ET' ANT, and ALIT, here called "biases of technical change with respect to price, " are called "biases of productivity growth with respect to price" in the text of Chapter 3. *Excerpted by permission f ram D. W. Jorgenson, The Role of Energy in Productivity Growth. EPRI EA-3482, Research Proj ect 1152-6, Final Report. Palo Alto, California: Electric Power Research Institute. May, 1984 . 143
144 TECHNICAL APPENDIX The development of our econometric model of production and technical change proceeds through two stage s. tic f irst specify a functional form for the sec- toral price factions, say {Pil, talking into account restrictions on the parameters impl fed by the theory of production. Secondly, are formulate an error structure for the econometric model and discuss procedures for estima- t ion of the Thorn parameter s. Our first step in formulating an econometric model of production and technical change is to consider specif ic forms for the sectoral price function {pil: qua 5 raO + ad In pi + AL In ~ + aE In PE + aN In ~ + aM In PM + aT · T + ~ ~(ln pE:)2 + pRI, In pi In ~ + pi In pi In pi + pi in pit in PN + HEM in ply in PM + ABE in ply · T + ~ In p];)2 LE PL ~ pLN in ~ in PN + ~ M in ~ in PM + FELT 18 pi T + ~ pEE(ln pE)2 + FAN in ~ in PN + REM in PE in Pi + BET in ~ T + ~ ~ (in p~)2 + pNM in PN in PM + ANT in ~ T + ~ ~MM(ln pM)2 + IT in pal · T ~ ~ ~ T21 , (i = 1, 2 n). For these price functions, the prices of out pot s are transcendental or, more specifically, exponential functions of the logarithms of the prices of capital (a), labor (L), electricity (E), nonelectrical energy (N) and materials (M) inputs. Ve refer to these forms as transcendental logarithmic price functions or, more simply, translog price functions, indicating the role of the vari- ables that enter into the price functions. 1. Homozeneitv and symmetry. The price functions {pi] are homogeneous of degree one in the input prices. The translog price function for an industrial sector is characterized by homogeneity of degree one if and only if the param- eters for that sector satisfy the conditions: (1) · . · . - ail + AL + aE + aN + aM = 1, ~ GIRL ORE + ION + ARM = 0, 8;1E \;L + ELF, + KILN + ELM = 0 HER ~ MEL + BEE + HEN + HEM = 0, FOR + ~NL + ONE + ON + MOM Di + pi + pi + pi + IMP = 0 GOT + 8;T + BET + ANT + MT = 0, (i = 1, 2 n). (2)
145 For each sector tho valne shares of capital, labor, electricity, nonelectrical energy, and materials inputs, say {vRl, {v~,}i 1VE}, {VN}, and {vi}, can be expressed in terms of logarithmic derivatives of the sectoral price functions with ~respect to the logarithms of. price of the corresponding input: VE = OR + 0= 1n PE + '!L 1n PL + FRE 1n PE + 0KN 1n PN + OEM 1n P1!I + ~i L L DEL 1n PE + ~LL 1n PL + ~LE 1n PE + ~LN 1n P~ + ~LM 1n PM +-§LT VE = OE + 'RE 1 n PE + ~LE 1 n PL + FEE 1 n PE + ~EN 1 n PN + ~EM 1 n P i + FET VN = aN + 0EN 1 n PE + ~LN 1 n PL + BEN 1 n PE + ~NN 1 n PN + ~NM 1 n PM + ~NT ~M = GM + ~RM 1 n PE + ~LM 1 n PL + DEM 1 n PE + PNM 1 n PN + ~i2M 1 n PM + ~MT (i = 1, 2 ... n). Finally, for each sector the rate of technical change, say {VT], can be expressed as the negative of the rate of growth of the price of sectoral out- put with respect to time, holding the prices of capital, labor, electricity, nonelectrical energy, and materials inputs constant. The negative of the rate of technical change takes the following form: T T T ~, -Vt = ~T + ~RT ln PR + I]LT ln PL + pET ln PE + ,BNT ln PN + fMT ln pi + pTT T ( i = 1, 2 . . . n) . (4) Given the sectoral price functions {pil, we can define the share elasticities with resvect to ~ricel9 as the derivatives of the value shares with respect to the logarithms of.the prices of capital, labor, electricity, nonelectrical energy, and materials inputs. For the translog price functions the share elasticities with respect to price are constant. We can also characterize these forms of constant share elasticity or CSE price functions indicating the interpretation of the fixed parameters that enter the price functions. The share elasticities with respect to price are symmetric, so that the parameters satisfy the conditions: Q1 _ Q1 ~EL - ~LI: ' . . ~IEE ~ER i _ i N - ~NE . . ~EM p ~' . . a1 _ nl ~LE ~EL ' · . ~N = ~NL ' · . 8;M 05lL · - ~EN ~NE · . Q1 _ Q1 ~EM ~ ~ME · . ~iM 03IN (i = 1, 2 n) . (5) 19. The share elasticity with res ect to price was introduced by Christensen, Jorgenson, and Lau (1971, 197~) as a fixed parameter of the translog production function. An analogous concept was employed bY Samuelson (1973). The terminology is due to Jorgenson and Lau (1983).
146 Similarly, given the sectoral price factions {PiI, Ire can define the biaso' of ~c" cha,8eo trite .rospect to Price as; derivatives of the value shares with respect to tiee.20 Alternatively, Ire can define the biases of technical change with respect to price in terms of the torivatives of the rate of technical change with respect to the logarithms of the price of capital, labor, electricity, nonelectrical energy, and materials inputs. Those two definitions of biases of technical change are equivalent. For the tranelog price factions the biases of techni- cal change troth respect to price are constant; these parameters are symmetric and satisfy the conditions: Q1 _ o! it= - i~T~ J · - FAT Is PAL ~ · ~ FAT ~ WE I · - ~ ~, AT RAM ( i 3 1, 2 ~ ~ ~ n) ~(6) Finally, we can define the rate of chaneo of the negative of the rate of tochnic41 chance FAT ( i = 1, 2 . . . n) as the derivative of the negative of the rate of technical change with respect to time. For the translog price func- tions these rates of change are constant. 2. Concavitv. Our nest step in considering specif ic forms of the sectoral price functions {Pal is to derive restrictions on the parameters impl fed by the fact that the price functions are increasing in all five input prices and the concave in the five input prices. First, since the price functions are increasing in each of the five input prices, the value shares are nonnegative: vat 2 0 ~ . VL 2 0 . . VE 2 0 , . VN ~ O VM 2 0 , Under homogene i ty these value shares son to unity: · . · . - vie + vL ~ vE + VN + VII = 1 J ( i = 1J 2 e e ~ n) . (7 ) (i = t. 2 eve n). (8) 20. The bias of productivity growth was introduced by flicks (1932). An alternative definition of the bias of productivity growth eras introduced by Bins~ranger (1974a, 1974b). The definition of bias of productivity growth to be employed in our econometric model is due to Jorgenson and Lau (1983) .
147 Concavity of the sectoral price functions [Pi} implies that eLc matrices of sec.ond-ordor partial derivatives {Bi] are negative semi-definite,21 so that the matrices Phi + mini' _ via arc negative semi-definite, whero: Qi Hi · Qi = pi + viv! - Vi , (i - 1, 2 ... n). where: Qua ~ ply O O O O O pj! O O O O O p[O O O' O O Pi) 0 O O O O PM Hi 5 vE O O O O O VL O O O O O VE O O O O O VN O O O O O VM_ 'vie' VL | , vi = YE AN VM (i = 1, 2 ... n) , , and {Ui] are matrices of constant share elasticities, defined above. Without violating the nonnegativity restrictions on value shares we can set the matrices {vivi - Vi} equal to zero, for example, by choosing the value shares: . . . VK = 1 VL = 0 VE = 0 VN = 0 vie = 0 , Necessary conditions for the matrices {Ui + vivi - Vi} to be negative semi- definite are that the matrices of share elasticities {Ui] must be negative semi-definite. These conditions are also sufficient, since the matrices · · . - {v~vt - Vt] are negative semi-definite for all nonnegative value shares s~m- ming to unity and the sum of two negative semi-definite matrices is negative semi-definite. To impose concavity on the translog price functions the matrices {011 of con- stant share elasticities can be represented in terms of the their Cholesky factorizations: 21. The rate of change was introduced by Jorgenson and Lan (1983). . 22. The following discussion of share elasticities with respect to price and concavity follows that of Jorgenson and Lan (1983). Representation of conditions for concavity in terms of the Cholesty factorization is due to Lau (1978).
148 Bi BiL BiE 81CN Bi BKL BLL BLE BLN BLM Bi Bi Bi Bi Bi 8KN BLN ~EN B." B~ Bi Bi Bi B1 Bt, 1 \216i O O ~ \21 1 li Ai 1 0 41 4 2 4 3 A51 Ai52 A53 A54 1 O O 61 0 O O O O O 0 43t o O O 0 64 0 O O 62 o o o o o 65J 21 >31 1 >32 0 1 O O O O A41 A51 A42 AS2 4 3 5 3 1 A1 0 1 A316l 41 1 A5l~i Ai Ai li i i A 1 +AA 1 l+xi i 1 i+ 1 21 1 21 21 1 2 21 31 1 32 241 21 1 42 2 51 21 1 A52 1 A ~t A1 1 dl+Ai i i i i 1 i i i Ai i i 1 i i i >1 Ai i+ i Ai i+A i 31 1 31A21 1 3262 >31A3161+A32~3262+63 41 316l+A42A32A2+A4363 51 31 1 52 32 2 53 3 i i Ai + i i i i i+i Ai i+Ai i i Ai i+Ai i i+i Ai i+i i i i i i i i i i i i 1 1 41 21 1 42 2 41 31 ~ 42 32 2 43 3 41 41 1 42 42 2 43 43 3 4 A41lslAl+142A52A2+~43A5363+A5654 A i ~ i A i 1 t ~ i+A i ~ i A i ~ i ~ i+) i A i ~ i+) i ~ t ~ t ~ i ~ i+} i ~ i ~ i+A i A i ~ i+A i ~ i A i A i ~ i+) i A i ~ i+> 1 ~ ~ ~ i ~ i ~ ~+6 i Under constant retorns to scale tho constant shero elasticitios satiefy ~ye motry rostrictions and restrictions implied by homogeneity of degree one of the price f~nction. Thoso restrictions imply that the parameters of the i i i i i A1 , A , Ai , 6 , Choleety factorizations {A21, A31, A41, A51, X32, A42, AS2, 43 53 54 B:, 63i, 64, Si} must satiefy the follo~ring contitions: 1 + A21 + A31 + A41 + AS1 1 + A32 + A42 + AS2 1 + A`3 + As3 s 0 , 1 ~ AS4 ~ O Sc ~O. O. O. Under those conditions shore is a onc--to-one transfor~ation betwoen the con- stant share elasticities {0KK, ~,. ~E, ~N, ~M, ~LL, ~LE, ~LN' ~LM' ~EE ~EN' ~EH' ~NN, ~M, ~MM'} and the p~r.~eters of the Choles~ factorizations. Tho matricos of sharo elasticitios aro negative semi-definite lf and only if the diagonal elements {6i, 5i, 6~, F'4} of the matrices {Di} are non~positive. This complete s tho specif ication of onr modol of prodoction and tochnical chango . 3. Indcx n~bo`~. Tho nogativo of tho sverage ratos of technical chango in an~r two points of time, say T and T-1, can bo expressed as the differonce
149 between successive logarithm of the price of output, less i weighted average of the differences between successive logarithms of the prices of capital, labor, electricity, nonelectrical energy and materials inputs, with freights given by the average value shares: -vT=loqi(T) - log, (T-l) - v~Elopi(T) - lnp':(T-l) where: -vLEloptfT) - lapi;(T-l) ~ - vE[lopEtT) - lnpF(T-l) -vNElnpN(T) - lnpNtT-l) ~ - vMElnpM(T) - lapM(T-l) ~ , (i = 1, 2 ... n), VT -~ ~ [vT(T) + vTfT-l) ~ , and the average value shares in the two periods are given by: vR = ~ Ev~(T) + v~(T-l)l, · ~ ~ vL = ~ EVL(T) + vLfT-l)l, · ~ ~ VE = ~ EvE(T) + vE(T-l) ~ , vN = ~ EVN(T) ~ vN(T-l)l, VM = ~ EvM(T) ~ vM(T-l) ~ , (i = 1, 2 ... n), We refer to the expressions for the average rates of technical change {VT} as the translo~ price index of the sectoral rates of technical change. Similarly, we can consider specific forms for prices of capital, labor, elec- tricity, nonelectrical energy, and materials inputs as functions of prices of individual capital, labor, electricity, nonelectrical energy, and matcriale inputs into each industrial sector. Ve assume that the price of each input can be expressed as a translog function of the price of its components. Accordingly, the difference between successive logarithms of the price of the input is a weighted average of differences between successive logarithms of prices of its components. The weights are given by the average value shares of the components. He refer to these expressions of the input prices as translo~ indexes of the price of sectoral inputs. 23 23. The price indexes were introduced by Fisher (1922). These indexes were first described from the translog price function by Diewert (1976). The corresponding index of technical change was introduced by Christensen and Jorgenson (1970). The translog index of technical chance was first derived from the translog price funct ion by Diciest ( 1980) and by Jor Benson and Lan ( 19 83 ) .
lSO 4. Stochastic soocificatloo. To formulate an cconoactric model of production and technical change we add a stochastic component to the equations for the valoc shares and the rate of technical change. Vc ass~o that each of these equations has two additlvo component e. The first is a nonrandom function of capital, labor, electricity, nonelectrical energy, and materials inputs and time; the second is an unobservable ransom distorbanec that is functionally independent of these variables. 's obtain an econometric model of production and technical change corresponding to the translog price function by adding random disturbances to all six cq~atione: Vet + B=1n pi+ p~Lln PL + ,~Eln PE + ,~Nln PN + liMln pi + B" · at+ L ~ BRL Pi ,l;Lln Pl; + Bj:;Eln PI + Bj:;Nln PN + ,l;Mln PM + ~ T+si VE=aE + 0~Eln PI + pLEln PL + ,pEln PI + BENln Pit ~ 0EMln pi + BiT T+8E V~a' + B1N18 Pi + ~LN1n PL + B N1D PE + BNNln PN + ~Mln PM BNT N V<OM + BRM1n PR + ~M1n PL + IEM1n PE + ~M1n PA + BM]~1n P1! + B" · T+SM vT~a~ + pRTln PK + 8;Tln p~, + RETln PE + gTln P' + fMTln pi + ~ · T+aT (i - 1, 2 ... n), where {aK, o];, aE. a~j, QM, 4, ~, ~L, 01iE, ~N, ~H, ~T, ~;L' 8;E' ~N' ~M, ILT. BEE' ~, IEM, ~ET, ~, ~M, ~, ~MM, lit. ~ ~ are unknown parameters and {~, eL, sE. eN, eM, sT] arc unobservable random disturbances. Since the value shares sne to unity, the onkno~rn para~neters satisfy the same restrictions as beforo and tho random disturbances corrosponding to tho fonr valne shares s=~ to zero: £K + BL + sE + Bi ~ eM s 0, (i - 1, 2 ... n), so that these random disturbances are not distributed indopendently. Vc a~s~me that thc random disturbances for all six equations have expected valne oqual to zero for all obsorvations: i' BR BL E £E1 SN Si M ~T - = 0 . (i = 1, 2 ... n) . Ve also ass~e that the random disturbances have a covariance matris that is the same for all observations; since the random disturbances corresponding to '
151 the five value shares sad to zero, this matris is positive somi-definite with rank at most equal to five. Ve assume that the covariance matrix of the random disturbances corresponding to the first four value shares and the rate of technical change, say Al, has rank five, where: V ATE I = Hi EN IT (i = 1, 2 ... n) , so the Hi is a positive definite matrix. Finally, we assume that the random disturbances corresponding to distinct observations in the same or distinct equations are uncorrelated. Under this assumption that the matrix of random disturbances for the first four value shares and the rate of technical change for all observations has the Eronecker product form: V 'sx(1) e~(2) : £~(N) £L(1) e £T(N) Since the rates of technical change {vT] are not directly observable, the equation for the rate of technical change can be written: T KT PE ~LTin PA + pETln PE + pNTI---i + pi ~ i ~ (i = 1, 2 ... n) , where £T is the average disturbance in the two periods: · · - -~ = ~ r [£T(T) + eT(T-1)] , Similarly, the equations for the value shares of capital, labor, electricity,
152 acneleetrical onorgy, ant eateriale inputs can bo ~rritten: -i ,~ ~ + ~r~lE + ~L~I, + ~E~E + ~N~N + ~M~H + ,5ZT~ + eX ; + piL~i + ~-+ 4;E~E + ~LN~N + 4;M~H + Bj;T ~ + eL i i ~i + 4; 1;~ + ~EE~E + ~EN~N + ~EM~H ~ETE i ~ ~i + ~ ~i + lEN~E + ~NN~N + ~NM~H ~ ,14 ~M + ~M~Z + 4;M~E + ~EH~E + ~NH~N + ~iM~H + ~Mr F + -aH, (i s 1, 2 n) . ~r~cr e: c' ~ ~ reK(T) + s':(T-1)] cL ~ [eL(T) + eL(T-l) -si ~ ~ [sE(T) + gE(T-l)] eN ~ ~ [eN(T) + sitT-l)] -si ~ ~ [si(T) ~ 8i(T-l)] (i ~ 1, 2 ... n) . As before, the averago veluo sheres rq. v~ . vE, ~N, vH1 sn~n to unity. so that the averago disturbances {eg, 8L, sE, si, eM} s~ to zero: £r ~ B~ + BE + eN + eM = 0, (l s 1, 2 ... n) . Tho covarianco matris of the average disturbances corresponding to the equa- tlon for tho rate of tochnical change for all observations, say Q, is a Laurent eatris: sT(2) sT(3) V sT(N) ~rhor e: ~- = 0, Il0... 0 ... O 017... 0 · e · e · ~ O O O ee. .
153 The covariance matrix of the average dlstorbancc corresponding to each cqua- tion for the four value shares is the same, so that the covariance matrix of the average disturbances for the first four value shares and the rate of technical change for all observations has the Eronecker product form: V -(2) -8~3) s~(M) eL(2) £T(N) = :l ~ Q , (i = 1, 2 ... n) . S. Estimation. Although disturbances in equations for the,average rate of technical change and the average value shares are antocorrelated, the data can be transformed to eliminate the antocorrelation. The matrix Q is positive definite, so that there is a matrix T such that: Tar' = I ,, T'T = o-1 .. . . To construct the matrix T we can first invert the matrix Q to obtain the inverse matrix O-1, a positive definite matrix. We then calculate the' Chole- sty factorization of the inverse matrix Q 1, Q-1 = LDL' , where L is a unit lower triangular matrix and D is a diagonal matrix with positive elements along the main diagonal. Finally, we can write the matrix T in the form:' .~ T = D1/2L, where D1/2 is a diagonal matrix with elements along the main diagonal equal to the square roots of the corresponding elements of D. He can transform the equations for the average rates of technical change by the matrix T = D1/2L' to obtain equations with uncorrelated random distur
154 b ance s: since: D1/2L, -vT(2) VT(3) VT(N) 1 In 4~2) ... 2 /2L. 1 On 4~3) ... 3 1 ln~ 4tN) ... N TOT' = (D1/2L,) n (D1/2L, ) ~ = I ,= si (2, + D1/2L' sT(3) sT(N) (i 2 1J 2 ... n) J The transformation T = D1/2L' is spelled to data on the average rates of technical change {hi} and data on the average values of the variables that appear on the right hand side of the corresponding equations. He can apply the transformation T = D1/2 L' to the first four equations for average value sharos to obtain equations with oncorrelated disturbances. As before, the transformation is applied to data on the average values shares and the average values of variables that appear in the corresponding equations. The covariance matrix of the transformed disturbances from the first four equations for the average value shares and the equation for the average rate of technical change has the Eronocker product form: HI ~ ~1/2L,)(~i ~ p)(I ~ D1/2L,). _ Pi e I (i 1 2 To estimate the unknown parameters of the translog price function we combine the first four equations for the average value shares with the equation for the average rate of technical change to obtain a complete econometric model of production and technical change. Ve estimate the parameters of the equations for the remaining average value shares, using the restrictions on these param- etcrs given above. The complete model involves twenty unknown parameters. total of twenty-two additional parameters can be estimated as functions of these parameters, given the restrictions. restrictions. Our estimates of the unknown parameters of thee econometric model of production and technical change wi11 he based on the nonlinear three-stage least squares estimator introduced by Jorgenson and Laffont.24 ~~~~ie Jorgenson and Laffont (1974).
