Mscellaneous Notes on roducton Economcs ompled by eter F Orazem September 9, 00 I Implcatons of conve soquants Two nput case, along an soquant 0 along an soquant Slope of the soquant,, MRTS s the margnal rate of techncal substtuton A conve shape requres that nputs have dmnshng margnal productvty, the necessary assumpton for downward slopng derved demand for factor nputs Ths requres that as we move down an soquant toward more and less, f falls and f ncreases, and so 0 Slope Note that,, ; and by Young s theorem,? 0 (I) If term n brackets < 0, then of techncal substtuton 0 mplyng conve soquants and dmnshng margnal rate Dmnshng MRTS s consstent wth stage of producton n nput case
II Elastctes of substtuton s a measure of substtutablty, but t s senstve to unts of the nputs Eample: obb-douglas ; measured n horsepower or dollars changes slope, as does L measured n hours, weeks, Senstvty to unts makes nterpretaton mpossble Need unt free measure of substtuton Followng RGD Allen (938), John Hcks (93), %, % %, %,? 0 (II) Ths s symmetrc and unt free If 0, then, 0 whch mples that and are substtutes f ndfferences are conve Wth two nputs, dmnshng returns, and ether cost mnmzaton or proft mamzaton, the two nputs have to be substtutes Wth more than two nputs, complements are possble but at least one par of nputs has to be substtutes as shown below
3 III Generalzaton of the elastcty of substtuton The general form of the elastcty of substtuton (III) Where F s the, cofactor of F, the Hessan matr of the producton functon Estmatng the elastcty of substtuton usng the prmal s complcated because t requres both frst and second order dervatves of the unknown producton functon to estmate the elements of the matr F 0 f f f n f f f f n F f f f f n f n f n f n f nn Specal case of constant returns to scale (RS) The general form of the elastcty of substtuton s unweldy, and so we often mpose restrctons that smplfy the form Output elastctes sum to whch means that (III) Wth constant returns, each factor s pad ts margnal contrbuton to producton To see ths, multply both sdes by output prce p Total revenue pq s equal to each factor s margnal revenue product (equal to nput prce at optmum) tmes the amount of the nput Usng (III), we can defne factor shares wthout knowng nput or output prces by / / (III3) And so wth constant returns to scale, factor shares equal the rato of the output elastcty to the scale elastcty
4 The mposton of constant returns to scale smplfes the estmaton of the elements of F somewhat, although t s dffcult to demonstrate that wthout usng a partcular specfcaton whch we wll demonstrate wth the translog producton functon below From the general form above, we can wrte the RS form of the elastcty of substtuton as Whch does not smplfy much However, n the two nput form, we can show some substantal smplfcatons (III4) Ths can be further smplfed by mposng the restrctons mpled by whch mples 0 0 0 0 Insertng these nto the denomnator of the second term n (III4), we have lacng ths nto the RS form of the elastcty of substtuton, we have (III5) IV Translog roducton Functon To estmate the elements of matr F, we need frst and second partals of the producton functon The translog s a second-order appromaton to an unknown producton functon q f(,,, n ) ln (IV) Frst dervatves of (IV) wll yeld output elastctes (IV) So
5 Second dervatves of (IV) are of the form Rearrangng: And so Smlarly If you wsh, you can populate the Hessan matr of the producton functon, F, usng the estmated frst and second partal dervatves of f( ) and compute the elastctes of substtuton The RS form of the translog producton functon mposes the restrcton that the output elastctes must add up to Inspecton of (IV) reveals that the restrctons mpled by RS are ; 0 We can also mpose symmetry so that Often the restrcton mprove the precson of the estmates by reducng collnearty among the frst- and second-order terms and reducng the number of parameters to be estmated
6 V Generalzed ost Mnmzaton Assume the frm wants to produce a level of output the frm s a prce taker n nput markets The frms obectve s to choose nputs through n so as to mnmze the cost of producng,,, Frst order condtons: 0 0,,, 0 For nputs for whch equalty holds, long-run optmum for any two nputs and Frm wll set margnal cost of all nputs equal so that, Assume all frst order condtons hold wth equalty Totally dfferentatng, we get 0 0 0 Rearrangng and applyng Young s theorem 0 -f -f f n dλ - -f -λf -λf -λf n d -dw -f -λf -λf λf n d -dw -f n -λf n -λf n λf nn d n -dw n H
7 The second order condton for cost mnmum s that H < 0 Usng the mplct functon rule, H 0 mples that there are well defned reduced form equatons rulng nput demand of the form,,,,,,,,,,,, If there are multple constrants, the condton for a mnmum s sgn H (-) m where m s the number of constrants In two nput case, H s 0 -f -f -f -λf -λf <0 for cost mnmum -f -λf -λf But ths s the same condton n (I) that assures conve soquants and dmnshng MRTS: stage To get the Hamermesh form (page 35), dvde all n equatons by λ across both sdes of the equaton and multply both sdes by (-) The equatons wll have the form: The matr form wll be 0 f f f n dλ/λ d f f f f n d dw /λ f f f f n d dw /λ f n f n f n f nn d n dw n /λ 0 F Multplyng a row by (-) reverses the sgn of the prevous determnant For n+ equatons, F -[/(λ n )] H f n s even F [/(λ n )] H f n s odd
8 Note the F s the bordered hessan matr of the producton functon A concave producton functon requres that sgn( F ) (-) n VI Output constant elastcty of factor demand The output constant elastcty s the nput demand along an soquant We can relate the elastcty of substtuton to the output constant labor demand elastcty as follows: Applyng ramer s rule to the Hamermesh form of the second-order equatons (page 35), we have the cross-prce effect defned by The frst order condtons mply We can plug these nto the general form of the elastcty of substtuton (III), gettng ; where k s the cost share for the th nput, and s the output constant elastcty of the th nput wth respect to the th nput prce Rearrangng, we can wrte the output-constant demand elastcty as (VI) If we plug back for σ and k, we get We can sum all of these output constant demand elastctes as 0 (VI) The numerators are an eample of epanson by alen cofactors shown below We have shown that the sum of output constant demand elastctes s zero The cost shares must be postve The own demand elastcty σ < 0, and so at least one 0 (e substtutes) That
9 means that n every producton process, there must be at least some substtutes Not all nputs can be complements for whom 0 because that would volate the condton 0 Sdebar: Eample of epanson by alen cofactors
0 VII Demonstratng the dual relatonshp between the cost functon and the producton functon From before, we mnmzed cost subect to a desred output level Wth equalty of the frst order condtons and a nonzero determnant of the Hessan matr, we get a system of nput demands,,, If we multply these by ther respectve nput prces,,,,, we get the cost of producng output If we generalze to all output levels, we have a cost functon of the form,,,, Shepherd s Lemma,,,, whch s the nput demand equaton 0 s the slope of the nput demand equaton s the cross-prce effect whose sgn s uncertan 0 means nputs and are substtutes 0 means that nputs and are complements Under the dual, the elastcty of substtuton can be wrtten (VII) Whch s much easer to estmate than the elastcty of substtuton usng the prmal (III) It requres only nput prces, nput quanttes, and dervatves of the nput demand equaton roof: Applyng ramer s rule to the Hamermesh form of the second-order equatons (page 35), we have the cross-prce effect defned by We wll use that and the result from the frst order condton that The defnton of the elastcty of substtuton under the prmal (III) s _//
We can derve the output constant elastcty of demand by multplyng and dvdng the rghthand-sde by That means we have the dual verson of the prmal result (VI) derved before It s much easer to show condton (VI), 0 usng the dual Subtractng from both sdes, we have 0 Multplyng both sdes by w and dvdng by, multply each term by we have 0 0 0 Usng that and, 0 _// 0
VII Estmatng the Dual The Translog ost Functon ln V 0 + V y ln Y + n V ln + n n ln ln + n y ln ln Y Where Y s output, s prce of nput Notce: producton functon) are obb-douglas 0, y that the cost functon (and To guarantee homogenety of degree one n the cost functon, mpose that V, 0, y 0 One could estmate ln drectly but there s a problem wth multcollnearty gven all the nd order terms n the regresson But t s possble to estmate the parameters through the nput demand equatons Shepherd's Lemma ln ln / so the dervatve of ln wrt ln yelds an equaton for factor shares n V + ln + y ln Y The same can be done for the other nput prces whch yelds a system of n factor share equatons n factor prces and output ) σ + (substtuton effect)
3 roof: / ln ln ln so Solvng for and multplyng both sdes of the resultng equaton by, + rearrangng, σ + σ + ) η + (cross demand elastcty) 3) ( ) σ + 4) η + (own prce elastcty)