A NONPARAMETRIC APPROACH TO SHORT-RUN PRODUCTION ANALYSIS IN A DYNAMIC CONTEXT By Elvra Slva * ABSTRACT A nonparametrc approach to short-run producton analyss from a cost and proft perspectves s developed n the context of an adjustment-cost model. The adjustment-cost hypothess s ncorporated n the theoretcal framework n the form of the propertes of the frm's technology wth respect to the quas-fxed factors. Nonparametrc tests for consstency of a data seres wth short-run cost mnmzaton and proft maxmzaton are developed and the emprcal mplementaton of these tests to U.S. agrculture s presented. The emprcal evdence does not refute that agrcultural producers behave collectvely as f they were a short-run cost mnmzng and proft maxmzng frm. Keywords: nonparametrc revealed-preference approach, short-run producton analyss, long-run adjustment, adjustment cost. Faculty of Economcs of Porto, Unversty of Porto, Rua Dr. Roberto Fras, 4200 Porto, * Portugal, Phone: 351-2-5571100, Fax: 351-2-5505050, e-mal:eslva@fep.up.pt. Paper to be presented at the 1999 AAEA annual meetng n Nashvlle, Tennessee. Copyrght 1999 by Elvra Slva. All rghts reserved. Readers may make verbatm copes of ths document for noncommercal purposes by any means, provded that ths copyrght notce appears on all such copes.
1. Introducton Varan (1984) proposes a unfed nonparametrc revealed-preference approach to statc producton analyss nvolvng nonparametrc tests for consstency wth specfc behavoral (e.g., cost mnmzng behavor) and technologcal (e.g., homothetcty) assumptons as well as a nonparametrc method to recover technologcal nformaton from observed frm behavor. Nonparametrc tests to check consstency of a data seres wth cost mnmzng and proft maxmzng behavor are presented n the form of a necessary and suffcent condton known as the Weak Axom of Cost Mnmzaton (WACM) and the Weak Axom of Proft Maxmzaton (WAPM), respectvely (Varan, 1984). The WACM and WAPM are derved assumng all nputs are varable. Ray and Bhadra (1993) modfy Varan's nonparametrc test for cost mnmzng behavor to allow some nputs to be quas-fxed. Usng the reverse nestedness property of the condtonal nput requrement sets, Ray and Bhadra (1993) derve the Weak Axom of Varable Cost Mnmzaton (WAVCM) from Varan's WACM. However, the WAVCM proposed by Ray and Bhadra (1993) s developed n a statc framework characterzed by nstantaneous adjustment n all factors of producton and the lack of ntertemporal lnkage of producton decsons, and whch may be the sources of observed volatons of the varable cost mnmzaton hypothess. The adjustment-cost model of the frm developed ntally by Esner and Strotz (1963) and further elaborated, among others, by Lucas (1967), and Treadway (1969, 1970) provdes a consstent dynamc theoretcal framework to analyze the frm's behavor and the underlyng producton technology. The relatve fxty of captal s ncorporated explctly n the frm's optmzaton problem n the form of an adjustment cost functon where costs per unt of nvestment ncreases wth the nvestment rate (Lucas, 1967; Treadway, 1970). The adjustment cost functon ncorporates all the forces that slow down the captal adjustment process. The
mpact of quas-fxed factor adjustment on varable nput use s extensvely dscussed n the dynamc lterature. 1 Ths paper ncorporates the sluggsh adjustment n some factors of producton and the ntertemporal lnkage of producton decsons n the theoretcal framework by specfyng the propertes of the frm's producton technology wth respect to the dynamc factors. Nonparametrc tests for consstency of a data seres wth short-run varable cost mnmzaton and proft maxmzaton are developed n a long-run adjustment context. The emprcal mplementaton of these tests to U.S. agrculture s presented whch permts nvestgatng the valdty of treatng the collecton of agrcultural producers as f they were a short-run varable cost mnmzng and proft maxmzng frm. 2. General Representaton of Dynamc Producton Possbltes n the Short-Run A well-behaved dynamc technology can be descrbed n the short-run by a famly of condtonal nput requrement sets or by a condtonal producton possbltes set satsfyng some regularty condtons. Let {V(y(t): I(t), k(t))}, t=1,...,t, be a famly of condtonal nput requrement sets descrbng the short-run technologcal lmts on the actons of the frm. Defne V(y(t): I(t), k(t)) as follows (1) V( y( t): I (t),k( t)) { x(t) :(x(t),i(t)) V(y(t):k(t))} where x(t) s the perfectly varable nput vector at tme t; I(t) s the gross nvestment vector at tme t; k(t) s the ntal captal stock vector at tme t; y(t) s the output level at tme t; V(y(t): k(t)) s the nput requrement set for y(t) gven the ntal captal stock vector k(t); and x(t) 8 8 and k(t) 8. o o 2 + + m +, I(t)
V(y(t): I(t), k(t)) s a subset of V(y(t): k(t)) defned as the ntersecton of V(y(t): k(t)) and a hyperplane. Consequently, the propertes of the condtonal nput requrement set, V(y(t): I(t), k(t)), can be derved from the propertes of V(y(t): k(t)). {V(y(t): I(t), k(t))}, t=1,...,t, satsfes the followng regularty condtons: (a.1) closeness and non-emptness; (a.2) nestedness n y(t); (a.3) postve monotoncty n x(t); (a.4) convexty n x(t); (a.5) nestedness n I(t); and (a.6) reverse nestedness n k(t). Alternatvely, a condtonal producton possbltes set can be used to descrbe the technologcal possbltes of the frm n the short-run. Let T(I(t), k(t)) be the condtonal producton possbltes set at tme t. Defne the condtonal producton possbltes set as (2) T(I( t),k(t)) {(y( t), x(t)) : (y(t), x( t), I (t)) T( k( t)) } where T(k(t)) s the producton possbltes set at tme t gven the ntal captal stock vector, k(t). 3 T(I(t), k(t)) s a subset of T(k(t)) defned as the ntersecton of T(k(t)) and a hyperplane. As a result, the regularty condtons of T(I(t), k(t)) are derved from the propertes of T(k(t)). The condtonal producton possbltes set satsfes the followng propertes: (b.1) for each (I(t), k(t)), T(I(t), k(t)) s a non-empty closed subset of 8 1+m ; (b.2) negatve monotoncty n y(t); (b.3) negatve monotoncty n x(t); (b.4) for each (I(t), k(t)), the technology T(I(t), k(t)) s convex n (y(t), -x(t)); (b.5) reverse nestedness n I(t); (b.6) nestedness n k(t); (b.7) for each (I(t), k(t)), the T(I(t), k(t)) s bounded from above for each fnte x(t). Property a.5 (b.5) s derved from the property of negatve monotoncty of V(y(t): k(t)) n I(t) (postve monotoncty of T(k(t)) n I(t)), mplyng that current addtons to the captal stock vector are output decreasng. However, current addtons to the captal stock ncrease the future stock of captal leadng to potental ncreases n output n the future (property a.6 or b.6). Propertes a.5 and a.6 (or, b.5 and b.6) and convexty of V(y(t): k(t)) n I(t) (or, convexty of
T(k(t)) n I(t)) reflect the presence of sluggsh adjustment n the quas-fxed factors and the ntertemporal lnkages of producton decsons. Convexty of V(y(t): k(t)) n I(t) (or, convexty of T(k(t)) n I(t)) mples that the more rapdly the quas-fxed factors are adjusted the greater the cost leadng to sluggsh adjustment n the quas-fxed factors. 4 3. Short-Run Varable Cost Mnmzaton and Proft Maxmzaton Consder the followng data set S {(y(t),x(t),i (t),k(t),w(t), p( t) ); 1,...,n; t1,...,t } where y(t) s the output level for observaton at tme t; x(t) s the perfectly varable nput vector for observaton at tme t; I(t) s the gross nvestment vector for observaton at tme t; k(t) s the ntal captal stock vector for observaton at tme t; w(t) s the perfectly varable nput prce vector for observaton at tme t; and p(t) s the output prce for observaton at tme t. 3.1. Determnstc and Stochastc Tests for Short-Run Varable Cost Mnmzng Behavor It can be shown the followng condton (3) w ( t) x j ( t) w ( t) x ( t); y j (t) y ( t), k ( t) k j (t), I j (t) I ( t), s necessary and suffcent for the data set S to be consstent wth short-run varable cost mnmzaton n a long-run adjustment context. Condton (3) s the Weak Axom of Short-Run Varable Cost Mnmzaton (WASRVCM) and can be used to test consstency of the data set S wth short-run varable cost mnmzng behavor n a long-run adjustment context. Assumng observatons are perfect measurements, the WASRVCM depends on drectly observed varables. S s sad to be consstent wth short-run varable cost mnmzng behavor f and only f the WASRVCM holds at all data ponts.
Ths test s determnstc snce no probablty assessments are mpled and s a dagnostc procedure to check whether the data are fully consstent wth the short-run varable cost mnmzaton hypothess. A stochastc test of the type proposed by Varan (1985) s conducted assumng observatons are not perfect measurements. Defne the null hypothess as the "true" data s consstent wth short-run varable cost mnmzng behavor. Assumng only varable nput demand data s measured wth error, the observed quantty varable x(t) can be related to the "true" varable as follows (4) l (t) x l ( t)(1 J vl (t)), l1,...,m; =1,...,n; t=1,..,t. l(t) s the "true" nput demand of the varable nput l for observaton at tme t, and J vl (t) s a random error assumed to be ndependently and dentcally dstrbuted as 2 N(0,) (t)). Gven the assumpton n (4), the WASRVCM depends on the observed varables (w(t), y(t), I (t), k(t)) and the unobservable varable (t). The WASRVCM can be checked by runnng the followng quadratc programmng problem (5) S(t) mn l (t) { n 1 m ( l (t)/x l (t) 1)2 : ( t) 0; w (t) j (t) w (t) (t), l1 y j ( t) y (t); k ( t) k j (t); I j ( t) I (t) } 2 2 t=1,...,t. Rejecton of the null hypothess occurs when S(t)/) (t) > C, or ) (t) < S(t)/C, where 2 C s the % crtcal value from the 3 table for (nm) degrees of freedom. Defne ) 2 ( t) S(t)/C 2 as the crtcal value of ) (t), whose value s obtaned after solvng (5). If the error varance of the varable nput demand data s known and less than ) 2 ( t), the null hypothess s rejected. Thus, the stochastc test n (5) provdes a range for the error varance over whch the data set S s consstent wth short-run varable cost mnmzng behavor.
3.2. Determnstc and Stochastc Tests for Short-Run Varable Proft Maxmzng Behavor It can be shown the followng condton (6) p ( t) y (t) w ( t) x (t) p (t)y(t) w (t) x(t); k ( t) k(t); I (t) I (t) s necessary and suffcent for consstency of the data set S wth short-run varable proft maxmzng behavor n a long-run adjustment framework. Condton (6) s called the Weak Axom of Short-Run Varable Proft Maxmzaton (WASRVPM). Assumng observatons are perfect measurements, the WASRVPM depends on drectly observed varables. S s sad to be consstent wth short-run varable proft maxmzng behavor f and only f the WASRVPM holds at all data ponts. A stochastc test can be conducted to account for the possblty of measurement errors n the data. Defne the null hypothess as the "true" data consstent wth short-run varable proft maxmzng behavor. Assumng only varable nput demand data s measured wth error, defne the observed demand for each varable nput as n (4). Gven (4), the WASRVPM depends on the observed varables (p (t), w (t), y(t), k (t), I (t)) and on the unobservable varable (t). The WASRVPM can be checked by runnng the followng quadratc programmng problem (7) S( t) mn{ n l (t) 1 m l1 ( l (t)/x l ( t) 1)2 : p (t)y (t) w (t) (t) p ( t) y j (t) w (t) j (t); k ( t) k j (t); I j ( t) I (t); (t) 0}, t=1,...,t. The stochastc test n (7) s smlar to the stochastc test n (5) provdng a range for the error varance over whch the data set S s consstent wth short-run varable proft maxmzaton.
4. Emprcal Results Annual data for aggregate U.S. agrculture over the tme perod 1948-1994 are taken from Ball et al. (1997). A detaled and exhaustve descrpton of the data seres and ts constructon s found n Ball et al. (1997). The WASRVCM n (3) and WASRVPM n (6) are slghtly modfed to accommodate 5 monotonc nonregressve techncal change n tme seres data. Tests for WAPM and WACM (Varan, 1984, 1985), and for the WAVCM (Ray and Bhadra, 1993) are also performed consderng the possblty of nonregressve techncal change. In addton, the Weak Axom of Varable Proft Maxmzaton (WAVPM) s also tested as a jont hypothess wth nonregressve 6 techncal change. The nonparametrc tests for WASRVPM, WASRVCM, WAVPM, WAVCM, WAPM and WACM are conducted consderng several combnatons of output and nput categores to nvestgate whether volatons of the behavoral axoms and ts degree of serousness are senstve 7 to the number of output and nput categores. The combnatons of outputs and nputs consdered n the emprcal analyss are: () twelve outputs, fve varable nputs (agrcultural chemcals, fuels and electrcty, feed, seed and lvestock purchases, other purchased nputs, hred labor) and four quas-fxed factors (self-employed labor, durable equpment, real estate and farm nventores); () twelve outputs, two varable nputs (ntermedate nputs and hred labor) and four quas-fxed factors; () two outputs, two varable nputs and four quas-fxed factors; (v) two outputs, fve varable nputs and four quas-fxed factors. The emprcal results for all nonparametrc tests are presented n Tables 1 and 2. The WASRVCM s fully satsfed n all cases. The nonparametrc tests for WACM and WAVCM reveal the U.S. agrcultural data do not volate these behavoral axoms n cases () and () but volatons are detected n cases () and (v). The estmates generated by the determnstc and
goodness-of-ft tests for WACM and WAVCM are qute small, ndcatng the observed departures from the jont hypothess of (total and varable) cost mnmzaton and nonregressve techncal change n the aggregate agrcultural sector data are economcally nsgnfcant. Usng the rejecton crteron of 10% measurement error, t can be nferred from the stochastc test that volatons of 8 WACM and WAVCM are not statstcally sgnfcant. Although the test results for the WACM and WAVCM ndcate mnor departures from the mantaned hypothess, the emprcal results n Table 1 suggest the number of output categores are mportant to detect volatons of the WACM and WAVCM. Volatons of the WACM and WAVCM are detected n cases where two output categores are consdered (()- (v)), and no volatons are dentfed n cases wth twelve output categores (()-()). The WASRVPM s fully satsfed n all cases. Volatons of the WAPM are detected n all cases. The determnstc test for WAPM ndcates that approxmately 87% of the observaton comparsons volate the jont hypothess of total proft maxmzaton and nonregressve techncal change n all cases. Though the percentage of volatons are smlar n these cases, volatons are economcally more serous n cases () and (v) as ndcated by the magntude of average percentage error. The goodness-of-ft test ndcates economcally sgnfcant departures from WAPM. However, the stochastc test dentfes standard errors n the varable nput quantty data due to measurement error rangng between 2.08% and 3.45% across cases. Gven the rejecton 9 crteron of 10% measurement error, the jont hypothess s not rejected n all cases. Volatons of the WAVPM are detected n all cases. The estmates generated by the determnstc and goodness-of-ft tests are qute small ndcatng the U.S. agrcultural data do not volate serously the WAVPM. The lower bound for the standard error of the nput quantty data ranges between 1.10% and 1.68%. Employng the rejecton crteron of 10% measurement error, the observed departures from WAVPM are not statstcally sgnfcant.
The test results n Table 2 reveal the observed volatons of WAPM and WAVPM and ts degree of serousness vary wth the number of output categores. The percentage of volatons of the WAPM and WAVPM s hgher n cases wth twelve outputs ((), ()) than n cases wth two output categores. However, volatons of the WAPM are economcally more serous n cases wth two output categores (() and (v)) as ndcated by the relatve magntude of the average percentage error. Volatons of the WAVPM are economcally more serous when twelve outputs are consdered (() and ()). Departures from WAPM and WAVPM are statstcally more serous n case (). 5. Concludng Comments The frm's short-run producton decsons are clearly affected by the presence of sluggsh adjustment n some factors of producton. Nonparametrc tests for consstency wth short-run varable cost mnmzng and proft maxmzng behavor are developed n a long-run adjustment context. An emprcal mplementaton of the nonparametrc tests for WASRVCM and WASRVPM under the hypothess of monotonc nonregressve techncal change s presented for a tme seres of annual U.S. agrcultural data over the tme perod 1948-1994. Nonparametrc tests for WACM, WAPM, WAVCM and WAVPM subject to nonregressve techncal change are also conducted. The emprcal results ndcate that WASRVCM and WASRVPM are fully satsfed for each of the four aggregaton levels consdered, ndcatng that agrcultural producers behave collectvely as f they were a short-run cost mnmzng and proft maxmzng frm. The WACM and WAVCM are very senstve to the number of output categores, snce no volatons are detected n cases wth twelve outputs, regardless of the number of nput categores. The WAPM and WAVPM are not fully satsfed for all aggregaton levels.
Table 1. Nonparametrc Test Results for Cost Mnmzaton Behavoral Objectve Determnstc Test Stochastc Test Percentage of Average Percent Crtcal Value of Volatons Error Error () WACM 0.00 0.00 0.00 WAVCM 0.00 0.00 0.00 WASRVCM 0.00 0.00 0.00 () WACM 0.00 0.00 0.00 WAVCM 0.00 0.00 0.00 WASRVCM 0.00 0.00 0.00 () WACM 0.10 0.002 0.29 WAVCM 0.096 0.004 0.29 WASRVCM 0.00 0.00 0.00 (v) WACM 0.10 0.002 0.28 WAVCM 0.096 0.004 0.49 WASRVCM 0.00 0.00 0.00 (a) Crtcal value of the standard error of the data s calculated at the 5% sgnfcance level. Table 2. Nonparametrc Test Results for Proft Maxmzaton Behavoral Objectve Determnstc Test Stochastc Test Percentage of Average Percent Crtcal Value of Volatons Error Error () WAPM 87.4 305.5 3.45 WAVPM 1.44 0.082 1.68 WASRVPM 0.00 0.00 0.00 () WAPM 87.4 306.6 2.18 WAVPM 1.44 0.081 1.10 WASRVPM 0.00 0.00 0.00 () WAPM 87.3 486.7 2.08 WAVPM 1.06 0.073 1.10 WASRVPM 0.00 0.00 0.00 (v) WAPM 87.2 485.3 3.30 WAVPM 1.15 0.072 1.65 WASRVPM 0.00 0.00 0.00 (a) Crtcal value of the standard error of the data s calculated at the 5% sgnfcance level.
FOOTNOTES 1. There are several studes emphaszng the theoretcal lnkages between adjustment costs and varable nput use (e.g., Treadway (1970), McLaren and Cooper (1980), Epsten (1981), Caputo (1990)). Adjustment costs and the relaton between quas-fxed factors and varable nputs are the focus of several emprcal studes (e.g., Vasavada and Chambers (1986), Howard and Shumway (1988), Luh and Stefanou (1991, 1996)). 2. The concept of V(y(t): k(t)) and ts regularty condtons are extensvely dscussed n Slva (1996) and Slva and Stefanou (1996a). 3. The concept of T(k(t)) and ts regularty condtons are explored n Slva (1996) and Slva and Stefanou (1996b). 4. Convexty of V(y(t): k(t)) n I(t), or, alternatvely, convexty of T(k(t)) n I(t) leads to sluggsh adjustment n the quas-fxed factors snce t mples an ncreasng margnal cost of adjustment. For a detaled dscusson of the regularty condtons of V(y(t): k(t)) and T(k(t)), see Slva (1996) and Slva and Stefanou (1996a, 1996b). 5. The procedure adopted to ncorporate nonregressve techncal change s the one proposed by Fawson and Shumway (1988). As a result, the nonparametrc test for WASRVCM (WASRVPM) determnes the extent to whch the aggregate sectoral data are consstent wth the jont hypothess of cost mnmzaton (proft maxmzaton) and nonregressve techncal change. Furthermore, testng the consstency of the U.S. agrcultural data seres wth short-run varable cost mnmzaton and proft maxmzaton requres nformaton on the nvestment levels. For each quas-fxed factor, nvestment s approxmated as the dfference between the current and lagged captal stock.
6. Modfyng Varan s nonparametrc test for proft maxmzng behavor to allow some nputs to be quas-fxed permts dervaton of the Weak Axom of Varable Proft Maxmzaton (WAVPM). The WAVPM s derved from Varan s WAPM usng the nestedness property of the condtonal producton possbltes set. The WAVPM s gven as p (t)y j (t) w (t) x j (t) p (t)y (t) w (t) x (t), k ( t) k j (t). 7. The jont hypothess for each nonparametrc test nvolves proft maxmzaton or cost mnmzaton and nonregressve techncal change. As a matter of smplfcaton, each test s denoted by the name of each behavoral axom (e.g., WAPM). 8. No measurement error nformaton s avalable for the specfc data used n ths study. In ths case, t s common practce to adopt a standard error based on evdence of measurement error n other data seres (e.g., Wllams and Shumway (1998) adopt a 10% standard error). 9. There s an nconsstency between the results of the goodness-of-ft test and the stochastc test for WAPM. The goodness-of-ft test results ndcate volatons of WAPM are economcally sgnfcant where the standard error generated by the stochastc test s small from a statstcal vewpont. Ths nconsstency can be attrbuted to the dfferent nature of the goodness-of-ft test and the stochastc test. The goodness-of-ft test attrbutes volatons to errors n optmzaton, and n the stochastc test volatons are attrbuted to errors n measurng the quantty data.
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