第15章 《佃農理論》英語原著 (9)

[1].Insofarasthereismorethanoneindividualwhowantsthesamepropertyresource,petitionisimplied,andthenumberofpetitorsincludesnotonlythosewhoareactuallyusingtheresourcebutalsopotentialowners（users）.Theassumptionofzerocontractingcostsubstitutesforthesometimesdubiousassumptionof"pure"petition.Includedinthegeneralterm"contractingcost"arethecostsofnegotiatingandthecostsofenforcingthestipulationsofthecontract.Ishalldiscusstheseandotherproblemsoftransactioncostinchapter4.

[2].Dependingontheproductionfunction,thesepairsofcurvesmaynotbeidentical.Weshalldiscussthislater.

B.TheSolutioninMathematics

Forsimplicityofpresentation,assumetherearetwohomogeneousfactorsofproduction,handt,wherehistheamountoflandpertenantfarmandtistheamountoftenantlaborperfarm.Furtherassumethattheproductionfunctionsofeachtenantfarmareidentical.Withtheseassumptions,itfollowsthatthelandsize,h,andtherentalpercentage,r,contractedforeachtenantfarmwillbethesameinequilibrium.

Leteachtenantfarm'sproductionfunctionbe

q=q（h,t）

Theamountoflandperfarm,h,isequaltothetotallandholdingofthelandowner,H,dividedbythenumberoffarms,m;thatis,

Thelandowner'stotalrent,R,isthenequaltothenumberoffarmstimestherentperfarm;thatis,

R=m·r·q（h,t）

Underpetition,

Wt=（1-r）q（h,t）

whereWisthemarketwagerateofthetenantlabor,t.

TheproblemofthelandowneristhentomaximizeR,throughthechoiceofm,r,andt,subjecttotheconstraintofpetition;[1]thatis,

max.R=m·r·q（h,t）

{m,r,t}

subjecttoWt=（1-r）q{h,t）

FormingtheLagrangeanexpression,theproblemisthusthemaximizationof

Thenecessaryconditionsare:

Fromequation（2）above,wehave

λ=m

Andnotingthat

,equation（1）bees

thatis,

Thisindicatesthatrentperacreoflandequalsthemarginalproductoflandinequilibrium,aconditionidenticaltothatofafixed-rentcontract.

Fromequation（3）,wehave

orthemarginalproductoftenantlaborequalsthewagerate,aconditionidenticaltothatofawagecontract.Finally,solvingequations（1）and（4）forr,

Thatis,inequilibrium,therentalpercentagemustsimultaneouslysatisfythelasttwoterms.Inotherwords,inequilibrium,theelasticityofoutputwithrespecttoland,

,equalsthetotalyieldnetoftenantcost（rent）asaportionofthetotalproduct,

[1].Notethattandmneednotbetreatedseparately.Givent,anadjustmentofmyieldsthesameresultasadjustingtwhileholdingmconstant.Theyareseparatedhereforthepurposeofconvenientlyderivingalltheconditionsinequilibrium.

C.TheGeometricSolutionandFurtherExposition

Theresultsderivedintheprecedingsectioncanbedemonstratedgeometrically.Infigure3wehavethesamedimensionsasinfigure2.Butinfigure3weconcentrateononlyoneofthetenants,whichmeansthatthetotallandspaceownedbythelandlordmaynotbeexhausted.Thecurveq/hrepresentstheaverageproductoflandwithonetenantfamilyemployed;thatis,theaverageproductwithrespecttothelandsizewhileholdingallotherfarminginputs（ofonetenantfamily）constant.Thecurvef/h,orthefixedtotaltenantfarmingcostdividedbylandarea,revealsthecostoffarminginputs（otherthanland）whichyieldtheexpectedq/h.Assumeforthemomentthatallnonlandfarminginputsarebornebythetenant;thef/hcurveisthetotalcostotherthanlanddividedbytherespectivelandspace.Itincludesthecostoflabor,seeds,fertilizers,andfarmingequipmentassociatedwiththeperiodoftimeoftherelevantproductionrun.[1]

Thatis,f/h=（pt·t+pz·z+……）/h;wherefistheconstanttotalcostotherthanland,andpt,pz,···arethefactorpricesoftenantlabor,t;fertilizers,z;……Sinceweholdthefarminginputsconstant,thef/hcurveisarectangularhyperbola.Theverticaldifferencebetweenq/handf/hdefines（q-f）/h,therentperunitofland,takingintoaccountthealternativecostofthetenant.[2]

Thetotalamountoftenantinputswhichdefinef/harecontrac-tuallystipulated,whichisessentialbecausethetenantwouldmitlessifonlytherentalpercentagewereprescribed.Givenanyrentalpercentage,onlyaportionofeveryunitofoutputproducedwillgotothetenant.Iffarmingdecisionsweremadeentirelybythetenant,itwouldbetohisinterestthatthecostofincrementaltenantinputbelessthantheassociatedmarginalproduct.Thiswouldresultinaconditioninconsistentwithequilibrium.Afullerdiscussionofthispointwillbepresentedinthenextchapter.