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# 544 Class Note for ECON 65200 at Purdue

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This 17 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 27 views.

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Date Created: 02/06/15

Nonparametric Efficiency Testing cont d More often than not there will be a mixture of fixed and variable inputs then we solve the following I minimize waxn x1xtlllK quot1 K subject to Zuklk 2 u0 k1 K k k anI 5xquot forlSnSt k1 K K 23631quot S xquot for n gtt 21quot 1 11 2 0 k1 k1 Purdue University Ag Econ 652 Lecture 12 Nonparametric Efficiency Testing cont d If we apply this test for firm 1 where input 1 istreated as xed we get minimize 2x2 7 7X14i14a427 1 2 3 4 4 11 81 81 s4 1 2 3 4 3 5 51 61 3x2 1 2 3 47 k Purdue University Ag Econ 652 Lecture 12 Nonparametric Efficiency Testing cont d The optimum for this problem has an objective value of 6 Does rm 1 appearto be minimizing variable costs Remember that firm 1 was using 3 units of input 2 Recall that when both inputs were variable firm 1 appeared to be inefficient Purdue University Ag Econ 652 Lecture 12 Nonparametric Efficiency Testing cont d Thus far we have seen tests for Technical efficiency in the production of one output I Allocative efficiency in the minimization of costs There are three other types of efficiency tests Multiple output technical efficiency Profit maximization Revenue maximization given input levels Purdue University Ag Econ 652 Lecture 12 Nonparametric Efficiency Testing cont d Multiple output technical ef ciency test maximize I all K subject to Zuilk 2 11317 K 2x32 Ex k1 K 21quot 1 2quot 20 w l l 1 Purdue University Ag Econ 652 Lecture 12 5 Nonparametric Efficiency Testing cont d Here we are asking ifthe entire vector of outputs can be scaled up by a single factor I We are not asking whether more of any output could be produced I What will the optimal objective value be for An ef cient producer I An inef cient producer Purdue University Ag Econ 652 Lecture 12 6 Nonparametric Efficiency Testing cont d Note that we are testing each output at a time for efficiency OB0A in this case k 2 Purdue University Ag Econ 652 Lecture 12 7 Nonparametric Efficiency Testing cont d Here is a case where this test produces an unintuitive result t A u2 B Purdue University Ag Econ 652 Lecture 12 8 Nonparametric Efficiency Testing cont d For each m we could solve the following maximize Tm rmll1lK K subject to Zumklk 2 uia m kk 0 ulZum m M M K xflk S x2 2 1 23 2 O k1 l l 1 Purdue University Ag Econ 652 Lecture 12 Nonparametric Efficiency Testing cont d 02 1 Purdue University Ag Econ 652 Lecture 12 10 Nonparametric Efficiency Testing cont d Now consider our unintuitive example A uzA B Purdue University Ag Econ 652 Lecture 12 11 Nonparametric Efficiency Testing cont d Profit maximization looks much like cost min test M 0 0 x11 l 41 ef Emu EWXquot K k k subject to Eu 7L 2 um K K k k k k Y an ltxn v1 211 gt0 Purdue University Ag Econ 652 Lecture 12 12 Nonparametric Efficiency Testing cont d Notice that this is the multiple output case Notice the similarity with the cost min test How would you identify inef ciency Where is the data in the problem that is relevant to the individual firm I How does this change with fixed inputs Purdue University Ag Econ 652 Lecture 12 13 Nonparametric Efficiency Testing cont d If we apply this test for firm 2 where the output price minimize 414 4x1 3x2 711 712 145 1414 2 u 411112 8 3 8 4 3 x1 3111 5amp2 5 3 614 s x2 2 3 4 k Purdue University Ag Econ 652 Lecture 12 14 Nonparametric Efficiency Testing cont d The optimal solution for this problem has an objective value of 9 Is rm 2 ef cient I Yes observed pro t was 9 How would this change if input 1 was fixed Purdue University Ag Econ 652 Lecture 12 15 Nonparametric Efficiency Testing cont d Test of pro t max with input 1 fixed Firm 2 minimize 414 2x2 711 712 145 1414 2 M 4211412 8112 s1 3111 5amp2 5 3 614 s x2 2 3 4 k New optimal objective is 18 and firm 2 is efficient Purdue University Ag Econ 652 Lecture 12 16 Nonparametric Efficiency Testing cont d Revenue maximization test M maximize 2 pium u1 uM ll lK quot11 K subject to zumklquot 2 um k1 K k k 0 K k k ExtL an 22L 211 20 k1 k1 Purdue University Ag Econ 652 Lecture 12 17 Nonparametric Efficiency Testing cont d Again this is similar to cost minprofit max problems and looks as follows for rm 2 minimize 414 711 712 145 1414 2 M 4211412 8112 s1 3111 512 511614 g 5 2 3 4 k Purdue University Ag Econ 652 Lecture 12 18 Nonparametric Efficiency Testing cont d In the single output case is the revenue test strange Note that technical inef ciency implies pro t and revenue maximization inefficiency Cost minimization inef ciency implies pro t maximization inefficiency Revenue maximization inef ciency implies pro t maximization inef ciency Purdue University Ag Econ 652 Lecture 12 19 Nonparametric Efficiency Testing cont d How do we add the assumption of constant returns to scale I There is a problem with the profit maximization test when all inputs are variable and we assume constant returns to scale What is it Purdue University Ag Econ 652 Lecture 12 20 10 Nonparametric Efficiency Testing cont d Want to learn more Then read Farrell Md The Measurement of Productive Efficiency Journal of the Royal Statistical Society 1203253 290 Varian HR 1984 The Nonparametric Approach to Production Analysis Econometrica 523579597 Fare R S Grosskopf and H Lee 1990 A Nonparametric Approach to Expenditureconstrained Profit Maximization American Journal of Agricultural Economics 72574581 Purdue University Ag Econ 652 Lecture 12 21 Producer Problems Complications I Incorporating additional activitiesconstraints in producer problems a question of scope What makes producer problems interesting is their interactions with the world Market interactions purchases and sales of inputs or outputs Resource transactions factor rental service hiring borrowing Purdue University Ag Econ 652 Lecture 12 22 11 Producer Problems Complications l Regulations Prohibitions or limits on use of some inputs Prohibitions or limits on the production of some outputs Contracts with suppliers of input or demanders of outputs l Formulation of constraints and objective must re ect Producerincentives Tradeoffs available to producers Purdue University Ag Econ 652 Lecture 12 23 Endogenous Price Models I In some cases price taking behavior is not the appropriate perspective eg Where producers have market power Models ofentire sectors wherein we know there is a relationship between supply and demand quantities and market prices Purdue University Ag Econ 652 Lecture 12 24 12 Endogenous Price Models cont d The key is to know what the relationship between price and quantity is Demand QP is typically known but I What we need is inverse demand PQ so the relationship must be invertible ng QPocPs and PQQoc s Purdue University Ag Econ 652 Lecture 12 25 Endogenous Price Models cont d In optimization modeling endogenous price models are often used to model partial equilibrium problems Simple example Let s say we know that demand is DP an and supply is SP cPd Purdue University Ag Econ 652 Lecture 12 26 13 Endogenous Price Models cont d Now considerthe problem b Dlbba71b d s1ddcild 1 b 1 d subject to D S S 0 maximize S D The rstorder conditions for this problem are Da 1 P 0 where P is the shadow price on the constraint Sc1 d P 0 and DSgO Purdue University Ag Econ 652 Lecture 12 27 Endogenous Price Models cont d Notice that we can use algebra to reexpress the first order condition for D as D an And likewise for S S cPd So the shadow price on the constraint is acting like the market price and the relationships between Supply Demand and Market Price have been modeled Purdue University Ag Econ 652 Lecture 12 28 14 Endogenous Price Models cont d l How did we do this maximize Ida1bdd e Jscwds 50 subject to Dis SO l 80 our objective is the integral underthe inverse demand curve less the integral underthe inverse supply curve producer s plus consumer s surplus Purdue universityAg Eeerr 652 Lecture 12 2a Endogenous Price Models cont d Graphically here is what is going on 4 5 Demand 4 CS 3 5 339 2 5 2 l 5 Supply an EU 7n 0 an an inn Purdue universityAg Eeerr 652 Lecture 12 an 15 Endogenous Price Models cont d I It is straightforward to adapt this model to analysis of spatial equilibrium problems The base is a transportation problem transhipment problem or transhipment problem with processing Demand and or supply are no longertreated as fixed but rather integrals ofthe inverse relationships are incorporated in the objective Purdue University Ag Econ 652 Lecture 12 31 Endogenous Price Models cont d For example the transportation model with variable supplies and demands becomes mD quotS maggigj zc j Pjdjddj Pisz39dsz39 m n EECU39XU n subject to inj ng Si vxv 300391 20 m Purdue University Ag Econ 652 Lecture 12 32 16 Endogenous Price Models cont d This formulation has the same interpretation as before producers plus consumers surplus There is an adjustment for interindustry costs Graphical illustration is no longer feasible This type of model is frequently used to study markets for a single good or a sector partial equilibrium Purdue University Ag Econ 652 Lecture 12 33 17

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