AtypicalMesillaValleyirrigatorgrowscottonforwhichoutputispredictedwellbythisCobbDouglasproductionfunction:1Q=0.193L0.12W0.88WhereL=haofland;W=waterin1000cubicmetersappliedperseason;Q=tonsofcottonprodu...

AtypicalMesillaValleyirrigatorgrowscottonforwhichoutputispredictedwellbythisCobbDouglasproductionfunction:1Q=0.193L0.12W0.88WhereL=haofland;W=waterin1000cubicmetersappliedperseason;Q=tonsofcottonproducedperseason.1.1Writetheproductionfunctionforagrowerwhosefarmislimitedto1haofland.Forthatlittlefarm,howmuchwaterisneededtoproduce1.2tonsoftotalcotton?1.2Howmuchwaterappliedtocottonisneededtoproduce1.2tonscottonifthegrowerallocatesonequarterofthefarm(0.25haofland)tootheruses?Whatifthegrowerassigns0.50haoflandtootheruses?Whydoesittakeasmallerincreaseinwaterusetomaintaincottonproductionwhenlandfallsfrom1.00to0.75hathanwhenlandfallfrom0.75to0.50ha?1.3Noticethatahigherwater‐to‐landratio(higherwaterappliedperha)canmaintainthebaselevelofcottonproduction(1.20).Productionismaintainedbysubstitutingwaterforland.Thisiscalledthe“lawofvariableproportions,”animportantinsightofmicroeconomictheory.Describeforfarmers,researchers,andpolicymakerswhythisisanimportantinsight.1.4Whatadditionalinformationdoestheirrigatorneedtofindhowmuchwaterandlandminimizesthecostofproducing1.20tonsofcotton?1.5Writetheequationsforthemarginalphysicalproductivityofwater.Showthatwater’smarginalproductivityisdiminishingwithincreasedwater.Showthatthisproductionfunctionhasconstantreturnstoscale.1.6Whatisthemarginalrateoftechnicalsubstitutionofwaterforlandwhenlandisfixedat1haandwaterappliedis8.04?WhatusefulinformationdoesthatMRTStellyou?ShowthatinputusemustoccurinstageIIofproduction.1.7Showthatthemarginalphysicalproductofwaterdependsononlytheratioofwaterappliedtolandandnotontheabsolutelevelofeither.1.8Canyouusethisproductionfunctiontofindtheprofitmaximizinglevelofirrigation(appliedwater)fora1hafarm?Whattwopiecesofextrainformationdoyouneed?