Production Function Estimation

Transcription

1 Producton Functon Estmaton Producton functon L: labor nput K: captal nput m: other nput Q = f (L, K, m ) Example, Cobb-Douglas Producton functon Q = AL α K β exp(ɛ ) ln(q ) = ln(a) + αln(l ) + βln(k ) + ɛ

2 ln(q ) = ln(a) + αln(l ) + βln(k ) + ɛ Both L and K are endogenous,.e. correlated wth the error term ɛ because frms decde ther nputs so that ther profts are maxmzed,.e. cost s mnmzed. Both labor and captal nput wll change dependng on the productvty shock. If the productvty s hgh, frms wll use less nput gven the same output. Frms perhaps wll use more nputs and produce more output.

3 Frm Cost Mnmzaton Problem Gven output Y MnC = wl + rk s.t. Y = AL α K β exp(ɛ) Lagrangan: wl + rk + λ [Y AL α K β e ɛ] F.O.C wl = λαy rk = λβy L = αr βw K

4 By substtuton, and usng Y = AL α K β e ɛ we get cost mnmzng K, L gven Y { [ ( ) αr α ] } 1 1 α+β K = A e ɛ Y βw [ ( βw L = A αr ) ] β 1 e ɛ Y 1 α+β Both K and L depend on the error term ɛ. Hence, OLS of the producton functon estmaton results n bas.

5 Instrumental Varables Estmaton Input prces can be consdered as nstruments. That s, ln(q ) = ln(a) + αln(l ) + βln(k ) + ɛ wth w, r as nstruments for ln(l ), ln(k ). One can assume that the market level nput prces are not correlated wth the productvty shock of the ndvdual frm ɛ. The two stage least squares would look lke ln(q ) = ln(a) + α [ˆγ wl w + ˆγ rl r ] + β [ˆγ wk w + ˆγ rk r ] + ɛ (1) where ˆγ s are the frst stage regresson coeffcents.e. L = γ wl w + γ rl r + v L K = γ wk w + γ rk r + v K

6 However, equaton 1 s not a relatonshp between the wages and rental rate of captal to output mpled by the frm s cost mnmzaton and proft maxmzaton problem. Proft maxmzaton does not say that f wage changes by w, frms should change output by [αγ wl + βγ wk ] w It s better to estmate the relatonshp between nput prces and ouput derved drectly from the cost mnmzaton problem.

7 Cost Functon C(w, r, p E, Q) = Mn {L,K} wl + rk + p E E s.t. Q = f (L, K, E) E: energy nput. Notce that gven output, the varables n the cost functons are all nput prces, whch are exogenous. From Shephard s Lemma or C w = L, C r = K, lnc lnw = wl C, lnc lnr = rk C, C p E = E lnc = p E E lnp E C

8 Functonal Form C(w, r, Q) = exp [β 0 + β w lnw + β r lnr + β E lnp E +β w2 (lnw) 2 + β r2 (lnr) 2 + β E2 (lnp E ) 2 or +β wr (lnw)(lnr) + β we (lnw)(lne) + β re (lnr)(lnp E ) +β wq (lnw)(lnq) + β rq (lnr)(lnq) + β EQ (lnp E )(lnq) +β Q lnq + β Q2 (lnq) 2 + ɛ ] lnc(w, r, Q) = β 0 + β w lnw + β r lnr + β E lnp E +β w2 (lnw) 2 + β r2 (lnr) 2 + β E2 (lnp E ) 2 +β wr (lnw)(lnr) + β we (lnw)(lne) + β re (lnr)(lnp E ) +β wq (lnw)(lnq) + β rq (lnr)(lnq) + β EQ (lnp E )(lnq) +β Q2 (lnq) 2 + ɛ

9 Express all the nput prces as p, = 1, 3. Then, lnc = β 0 + β lnp + β Q lnq + 1 γ j lnp lnp j + 2 j γ QQ (lnq) 2 γ Q lnp lnq

10 Restrcton on the Cost Functon Coeffcents An advantage of the cost functon approach s that one can mpose theoretcal restrctons mpled by the cost mnmzaton on the cost functon. Symmetry That s, lnc p p j = γ j = γ j lnc p j p

11 Homogenety of degree one If all the nput prces ncrease by s, cost also ncreases by s as well. That s, ln(sc) = lns + lnc = β (lns + lnp ) + 1 γ j (lns + lnp )(lns + lnp j ) 2 j + γ Q (lns + lnp )lnq + terms wthout (lns + lnp ) Hence, RHS equals lns + lnc for any p, Q f and only f β = 1 γ j = 0 j γ Q = 0

12 Estmaton Cost functon One can estmate the cost equaton by OLS, gven one has data on cost, nput prce and output. lnc = β 0 + β lnp + β Q lnq + 1 γ j lnp lnp j + 2 j γ Q lnp lnq γ QQ (lnq) 2 + ɛ 1 (1)

13 But there are other equatons that can be ncluded n the estmaton. Share Equatons 3 Share functons: {L, K, E} p C = β + j γ j lnp j + γ Q lnq + ɛ 1+ (2) Notce that snce the cost share sum up to one,.e. p C = 1 only 2 of the 3 cost share equatons are ndependent. Therefore, we use the cost functon and the 2 of the 3 cost share functons to estmate the parameters of the cost functon, wth the restrctons of the symmetry and homogenety of degree 1.

14 Maxmum Lkelhood Estmaton Assume that the error terms of the cost functon and the share functons are dstrbuted jontly normal. Then, the log lkelhood ncrement of frm k s l k = 1 2 ln(π) 1 2 ln( Ω ) 1 2 ɛ kω 1 ɛ k [ ɛ k1 = lnc k β 0 + β lnp k + β Q lnq k... ɛ k,+1 = p C β + γ j lnp j + γ Q lnq j where Ω s the varance covarance matrx. log lkelhood s ] l = k l k Choose parameters of the model subject to symmetry and homogenety of degree 1 restrctons to maxmze the lkelhood.

15 GMM Estmaton GMM objectve functon s [ ] [ ] F = ɛ k W ɛ k k Choose parameters to mnmze the objectve functon, subject to the restrcton of symmetry and homogenety of degree 1. The weghtng functon s set to be the estmate of the varance covarance matrx of k ɛ k. k