1、李永波The Firm: ProductionAdvanced MicroEconomicsIn this lecture we set out some of the elements needed for an analysis of the firmTechnical efficiencyReturns to scaleConvexitySubstitutabilityMarginal products.and (for next time) assuming a competitive environment.We do it within the context of a singl

2、e-output firm.But first we need the building blocks of a model.The basics of production.ziamount of input iQamount of outputThe basics of production.input vector z := (z1 ,z2 ,.,zm)wiprice of input iw := (w1 ,w2 ,.,wm)input price vector Pprice of outputNotation: PricesQ G(z1, z2, ., zm )The single-o

3、utput production functionWritten more compactlyQ G(z)technologyoutputYes, but why not = sign here?inputsThe meaning of the functionthe maximum amount of output that can be producedfrom this list of inputsUse this relation to distinguish two cases.Components of the relationshipQ G(z)Q = G(z)21The cas

4、e where production is technically efficientThe case where production is (technically) inefficientTechnical efficiencyG(z) zi_iG (z) := where differentiableSome handy notation.z2Qz10 G(z , z ) 1 2outputinput 2input 1Q G(z , z )1 2infeasiblepointsQ = G(z , z )1 2technically efficient pointsNow lets sl

5、ice this set up to get some useful toolsThe full production functionPick a particular output level QZ(Q) := z: G(z)QFind a feasible input vector z G(z) QRepeat to find all such input vectorsInput requirement setsZ(Q) := z: G(z)Qthe set of input vectors that meet the technical feasibility condition f

6、or output level Q.but what would Z look like?This depends on the assumptions we make about production.First, a “standard” case.?What is this thing.?z1z2G(z , z ) =Q1 2 infeasible points _ Z(Q)technically efficient points feasible, but inefficient pointsG(z , z ) Q1 2The input requirement set:z1z2 z

7、z Pick two boundary pointsDraw the line between themIntermediate points must lie in the interior of Zmeaning: a combination of two techniques may produce more outputG(z) =QG(z) =QG(z) QBut what if we changed some of the assumptions here? _ Z(Q)Case 1: Z is smooth and strictly convexz1z2 _ Z(Q) z z P

8、ick any two points in ZDraw the line between themIntermediate points must lie in Zmeaning: a combination of feasible techniques is also feasibleCase 2: Z convex but not strictly convexz1z2 this point isnot feasible _ Z(Q)This region causes a problemmeaning: in this region there is an indivisibilityC

9、ase 3: Z is smooth but not convexAn Example.LondonNew York3131z1z2slope undefinedat this point the only efficient point for Q =Q _ Z(Q)Case 4: Z is convex but not smoothz1z2z1z2z1z2z1z2Standard case, but strong assumptions about divisibility and smoothnessalmost conventional case: mixtures may be ju

10、st as good as single techniquesPresents problems: the dent represents an indivisibilityunusual case : only one efficient point and not smooth. But not perverse.Summary: 4 possibilities for the input requirement set Z z : G(z) = Q This is the isoquant. Lets look at its shape.Pick an output level QFin

11、d the input requirement set Z(Q)Draw the boundary of this setIsoquants z : G(z) = Q Az1inputs requiredto produce at Az2Isoquant at QSlope = z2 / z1The input ratio describes the particular techniqueThe isoquant is the boundary of ZThe isoquant through A (Q)Marginal rate of technical substitutionThe s

12、lope of the isoquant is the marginal rate of substitution at A.It measures the implicit “price” of input 1 in terms of input 2.The higher is this “price”, the smaller is the relative usage of input 1z1z2 A AG1(z)/G2(z) The responsiveness of the input ratio to the MRTS is given by the elasticity of s

13、ubstitution- log(z1/z2) log(G1/G2) Can be seen as the isoquants “curvature”input ratioNow for aspecial case.A constant elasticity of substitution:Increase the elasticity of substitution.z1z2Now look at the structure of the contour map.z1z2Homothetic contoursz1z2Qtz2tztQr1G(t z) = t G(z)rContours of

14、a homogeneous functionThe isoquants form a contour map. If we looked at the “parent” diagram, what would we see? Lets do this for 2 inputs, one output.Lets rebuild from the isoquantsz2Qz1expansionray 0G(t z) = t G(z)constant returns to scaleProportional increase in all inputsz2Qz10t1 G(t z) t G(z)In

15、creasing returns to scaleProportional increase in all inputsz2Qz10t1 G(t z) t G(z)Decreasing returns to scaleProportional increase in all inputsz2Qz1isoquantQ =Q0Take a horizontalsection.to get the isoquant again z2Qz10this gives us our next conceptNow take a vertical section. Pick a technically eff

16、icient input vectorKeep all but one input constant Q= G(z)Measure change in output w.r.t. this inputG(z) zi_MPi = Gi(z) = Marginal productsz1QG(z)z1QG(z)z1QG(z)possible relationships between output and one inputz1QG(z)Lets take the conventional case feasible setG(z)Qz1input 1 is essentialSet of technically efficient pointsTake the relationship between output