生产者理论知识(-)1课件

The Firm:Comparative StaticsMicroEconomicsProductionOptimisationThe Firm and the MarketComparative StaticsComparative StaticsOptimisationThe Firm and the MarketProductionThe FirmOverview.m.we derive the firms reactions to changes in its environment.mThese are the response functions.mWe will examine three types of them,treating the firm as a.m.Black Boxthe firmMoving on from the optimum.the firmoutput level;input demandsmarketpricesHow it works.lUse the fact that the firm is an optimiserlBehaviour can be predicted by necessary and sufficient conditions for optimumlThe FOC can be solved to yield behavioural response functions.lTheir properties derive from the solution function The firm as a“black box”The FirmProductionOutput SupplyOrdinaryInput DemandOptimisationComparative StaticsThe Firm and the MarketOutput SupplyConditionalInput DemandOrdinaryInput Demandblack box problemsChoose z to minimiseS S wi zi m i=1Q G(z).subject to the production constraint.z 0.and the obvious non-negativity conditionsThe solution to the first-stage problem.C(w,Q):=min wi zi vector ofinput pricesSpecified output leveland,presumably.Yields minimised cost as a function of exogenous variables.one for each of the m inputsz1*=H1(w,Q)z2*=H2(w,Q).zm*=Hm(w,Q).optimal input demands as a function of exogenous variablesdemand for input i,conditional on output Q We need to examine the first stage of the optimisation processzi*=Hi(w,Q)A function of input prices.and output levelconditional input demand function(our first response function)Result depends on shape of Zz1z2z1z2z1z2z1z2z2z1Take the conventional case.Map the optimum into(z1,w1)-spacez2z1w1z1Start with an arbitrary value of w1.Do it again for a lower value of w1.and again to get.H1(w,Q)the conditional demand curvehIn the conventional case.h.the constraint set is convex,with a smooth boundaryhWe find the solution is a continuous map.h.that is single valued.Points to noteResult depends on shape of Zz1z2z1z2z1z2z1z2Z(Q)_ 2z2z1What about the non-convex case.?again map the optimum into(z1,w1)-spaceZ(Q)_ 2w1z2z1z1.now try a very low value of w1But what happens in between?a demandcorrespondenceNonconvex Z:jumps in z*w1z1no price yields a solution heremultiple inputsat this priceThe demand correspondencePoints to notehIn this case.h.the constraint set is nonconvex hWe find the solution is a discontinuous maphThe map is multivalued at the discontinuity.Let us set this difficulty aside.mLets take it for granted that single-valued input-demand functions exist.mHow are they related to the cost function?mWhat are their properties?mHow are they related to properties of the cost function?Do you remember these.?Assume the existence of a conditional input demand functionC(w,Q)wi=zi*_Remember this.?slope of thecost functionoptimal demand for input iCi(w,Q)=zi*.yes,its Shephards lemmaAnd so.Ci(w,Q)=Hi(w,Q)i _wj Now lets differentiate this.conditional inputdemand functionWhich gives us.Cij(w,Q)=Hji(w,Q)secondderivativeCji(w,Q)=Cij(w,Q)Why.?And now for a simple result:=2_wj wi 2_wi wj Second derivatives commute.The effect of the price of input j on conditional demand for input iHij(w,Q)=Hji(w,Q)The effect of the price of input i on conditional demand for input jThe economic meaning.Now for an even simpler result:Cii(w,Q)=Weve put j=i.Special caseHii(w,Q)this must be negative.so this must be negative too.and so:Because the cost function is concave in prices:Consider the demand for input 1conditional demand curve ziwiHi(w,Q)Hii(w,Q)0The conditional demand curve slopes downwards 3Nonconvex Z yields discontinuous H3Cross-price effects are symmetric3Own-price demand slopes downward.For the conditional demand function.The FirmProductionOutput SupplyOrdinaryInput DemandOptimisationComparative StaticsThe Firm and the MarketConditionalInput DemandConditionalInput DemandOrdinaryInput Demandblack box problemsmax PQ-C(w,Q)s.t.Q 0Yields optimaloutputThe second-stage problemQ*=S(w,P)supply of outputWe need to examine the second stage of the optimisation process(our second response function)PQFor a given P read off optimal QQCQC/QPPQPQQPQPPPNow let P fall.Note what happens below Average Cost.PQ_ _Q=S(w,P)no price gives solution hereSupply curvez2Q0IRTS hereProduction function with local IRTSz1lSupply curve slopes upwardlNonconcave G yields discontinuous SlIRTS means G is nonconcave and so S is discontinuousFor the supply function.OptimisationThe FirmProductionOutput SupplyOrdinaryInput DemandComparative StaticsThe Firm and the MarketConditionalInput DemandConditionalInput DemandOutput Supplyblack box problemsdemand for input i,conditional on output Q zi*=Hi(w,Q)Q*=S(w,P)supply of outputzi*=Hi(w,Q)Q*=S(w,P)Now put together the two stages of the optimisation processBy substitution:Hi(w,S(w,P)Di(w,P)demand for input i(unconditional)=:(our third response function)Differentiate for the uncompensated demandTotal=Substitution effect+Output effectUnconditional Demand can be determined from the cost functionFrom Shepherds lemmaAndSinceCan solve Sj(w,P)in terms of C(w,P)Hotellings LemmaProve Assuming one input zThe Profit Function and its derivatives areBut from cost minimisation we knowConsider the demand for input 1Change in costconditional demand curveprice fall z1*z1w1H1(w,Q)initial pricelevel.and allow the price of input 1 to fallinput-price fall:substitution effectNotional increase in factor input if output target is held constantprice fallz1*z1w1Original conditional demand curveordinary demand curvez1*Conditional demand curve at new output levelinput-price fall:total effectpNonconvex Z may yield a discontinuous D pCross-price effects are symmetricpOwn-price demand slopes downwardSame basic properties as for H functionThe ordinary demand function.ProductionOptimisationThe FirmComparative StaticsThe Firm and the MarketComparative StaticsThe Firm and the MarketProductionOverview.The FirmProductionThe Short RunOptimisationComparative StaticsThe Firm and the MarketA Special CaselT his is not a moment in time butl.is defined by additional constraints within your modelThe short run.Short-run problem builds on the standard approachChoose Q and z to maximiseP:=P Q -S S wi zi m i=1Q G(z).subject to the production constraint.Q 0z 0.and some pretty obvious conditions:zm =zmWith side condition(s)zm=zm C(w,Q,zm):=min S S wi zi The solution now involves the short run cost functionSide constraint in short runCompare this to the solution without the side constraint.C(w,Q):=min S S wi zi C(w,Q,zm)By definition.So therefore.C(w,Q)C(w,Q,zm)_ _ Q QLets look at the graphic.PQC/QQ_C/QAverage cost in the short and the long runPQCQQ_CQMarginal cost in the short and the long runRelationships between the short and the long runQ_PQC/QCQC/QCQShort runLong runThe supply curve is steeper in the short run H1(w,Q,zm)z1 w1 H1(w,Q)Short runLong runSo too is the conditional demand curvethe firmprice signalsYou must remember this.functional relationsoutput responsesBasic functional relationsHi(w,S(w,P)=Di(w,P)demand for input i,conditional on output supply of outputdemand for input i(unconditional)Di(w,P)Hi(w,Q)S(w,P)And they all hook together like this