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# 625 Class Note for ECON 65200 at Purdue

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This 18 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 18 views.

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Date Created: 02/06/15

Nonparametric Efficiency Testing I A linear programming based approach to efficiency assessment Cross section or time series inputoutput data viewed as observations from the technically feasible set Assumptions for what follows includes that each observation comes from the same technology set Modifications exist to account for technical progress Purdue University Ag Econ 652 Lecture 11 Nonparametric Efficiency Testing cont d The easiest way to think of efficiency is by looking at a single inputsingle output system Which ofthe following observations are ef cient Observation Output Input 1 1 3 2 3 5 3 2 6 4 6 6 5 3 9 6 7 11 7 8 15 Purdue University Ag Econ 652 Lecture 11 2 Nonparametric Efficiency Testing cont d Purdue University Ag Econ 652 Lecture 11 Nonparametric Efficiency Testing cont d I What can you say about the efficiency ofthese observations I Are some observations more efficient than others Purdue University Ag Econ 652 Lecture 11 Nonparametric Efficiency Testing cont d Purdue University Ag Econ 652 Lecture 11 Nonparametric Efficiency Testing cont d Purdue University Ag Econ 652 Lecture 11 Nonparametric Efficiency Testing cont d I What are you assuming about returns to scale I What happens if you change your returns to scale assumption Purdue University Ag Econ 652 Lecture 11 Nonparametric Efficiency Testing cont d Purdue University Ag Econ 652 Lecture 11 Nonparametric Efficiency Testing cont d I What else are you assuming I Are you assuming anything about prices Can you generalize these ideasto higher dimensions ie multiple inputs andor multiple outputs Purdue University Ag Econ 652 Lecture 11 9 Nonparametric Efficiency Testing cont d I Consider a two inputone output case Which ofthe following observations are ef cient Observation Output Input 1 Input 2 1 5 1 1 2 5 2 1 3 6 1 2 4 8 1 3 5 8 2 2 6 9 3 3 7 10 4 2 Purdue University Ag Econ 652 Lecture 11 10 Nonparametric Efficiency Testing cont d Which ofthe following observations are ef cient Observation Output Input 1 Input 2 1 5 1 1 2 5 2 1 Note that 2 uses more ofinput 1 the same amount of input 2 and produces no more output than 1 So 2 is inefficient Purdue University Ag Econ 652 Lecture 11 11 Nonparametric Efficiency Testing cont d Which ofthe following observations are ef cient Observation Output Input 1 Input 2 1 5 1 1 3 6 1 2 4 8 1 3 Note that input use for observation 3 is the average of input use for observations 1 and 4 However the average output level for observations 1 and 4 is 65 strictly greater than observed output for 3 Hence we conclude 3 appears to be inefficient Purdue University Ag Econ 652 Lecture 11 12 Nonparametric Efficiency Testing cont d Which ofthe following observations are ef cient Observation Output Input 1 Input 2 1 5 1 1 5 8 2 2 6 9 3 3 Note that the input usage for 5 is the average ofthe input usage for 1 and 6 but the output production 8 is greater than the average output 7 592 What are we assuming about returns to scale Purdue University Ag Econ 652 Lecture 11 13 Nonparametric Efficiency Testing cont d I If we assume constant returns to scale CRTS what happens I Observation Output Input 1 Input 2 1 5 1 1 5 8 2 2 6 9 3 3 I With CRTS we can double inputs and outputs for observation 1 So with 2 units of each input we can get 10 units ofoutput Similarly 3 units of each input we get 15 units of output 80 5 and 6 are inefficient Purdue University Ag Econ 652 Lecture 11 14 Nonparametric Efficiency Testing cont d Let us formalize our assumptions Free disposability of inputs and outputs Convexity of input requirements and production possibilities I No errors in the data Purdue University Ag Econ 652 Lecture 11 15 Nonparametric Efficiency Testing cont d Assume that you have a bunch of data for K different firms indexed by IF1K The output of firm k is denoted by ukand inputs are denoted by Xnk where n indexes inputs Let s not assume constant returns to scale I Considerthe set of technically feasible input combinations that produce a given amount ofoutput 0 Purdue University Ag Econ 652 Lecture 11 16 Nonparametric Efficiency Testing cont d Associate a weight with each observation the single weight applies to all inputs and outputs The technically feasible set of inputs producing u0 is K uo x clxN l Zuklquot 2M0 161 K k k xnl an n1N 1 21k 1 1quot 2 0 k 1K k1 Purdue University Ag Econ 652 Lecture 11 17 w ll Nonparametric Efficiency Testing cont d The role ofthe M s is as weights in forming a convex combination of the inputoutput vectors The final line makes the weights nonnegative and sum to unity The first inequality defining the set ensures that At least u0 will be produced by the linear combination ofobserved inputoutput vectors Use of the inequality implies free disposal of output Purdue University Ag Econ 652 Lecture 11 18 Nonparametric Efficiency Testing cont d The second set of inequalities Ensure that at least as much input as the convex combination is used I Embody free input disposal because it is an inequality Purdue University Ag Econ 652 Lecture 11 19 Nonparametric Efficiency Testing cont d For the data listed above this set is 0 71 xlx2l 1 2 3 4 5 6 7 0 52L 5 L 6 L 8 L 8 L 9 L 10 L 2M 1 2 3 4 5 6 7 12L 22 1 L 1 L 22 3 L 4 L 2161 1 2 3 4 5 6 7 12L 1 L 22 3 L 22 3 L 22 2162 1 2 3 4 5 6 7 k Purdue University Ag Econ 652 Lecture 11 20 10 Nonparametric Efficiency Testing cont d Note that this set is a function of the output level u0 Ef ciency has something to do with being on the boundary of this set The technical ef ciency test asksthe question given the input vector could higher output have been produced Purdue University Ag Econ 652 Lecture 11 21 Nonparametric Efficiency Testing cont d To answer this question we could solve the following problem mamm H K k k subject to u 1 1 IV u W a 0 kk x171 xi k1 K k k V1 211 20 Purdue University Ag Econ 652 Lecture 11 22 11 Nonparametric Efficiency Testing cont d Note that X0 are the fixed input levels for firm 0 whose efficiency is being tested and whose data is included in IF1K Can you argue that there is a lower bound forthe optimal u What is that bound How should we decide if firm 0 is ef cient I What are we assuming about returns to scale Purdue University Ag Econ 652 Lecture 11 23 Nonparametric Efficiency Testing cont d If we apply this test to the two input data and test for efficiency of rm 2 we solve the following problem maximizeu 1 2 3 4 5 6 7 52 52 62 82 82 92 102 2M 1 2 3 4 5 6 7 12 22 12 12 22 32 42 2 1 2 3 4 5 6 7 12 12 22 32 22 32 22 1 1 2 3 4 5 6 7 k Purdue University Ag Econ 652 Lecture 11 24 12 Nonparametric Efficiency Testing cont d The optimal solution forthis problem is u5 What do we conclude Does this agree with your intuition Note that this was a test oftechnical ef ciency Under what could this really be optimal Purdue University Ag Econ 652 Lecture 11 25 Nonparametric Efficiency Testing cont d I What are you assuming about prices Technical efficiency Allocative efficiency Purdue University Ag Econ 652 Lecture 11 26 13 Nonparametric Efficiency Testing cont d I Consider the following price and quantity data Output nput1 nput2 PM PM PM 5 7 3 4 2 3 4 7 4 1 3 5 4 14 2 8 3 5 3 14 2 8 16 Purdue University Ag Econ 652 Lecture 11 27 Nonparametric Efficiency Testing cont d Output nput1 nput2 q p61 m 7 34 23 l 7 41 3 5 Note that for the first and second observations the input mix is different but output is the same The cost of inputs for firm 1 is 3X42X318 The cost if firm 1 used rm 2 s inputs3x12x513 Hence firm 1 is inefficient Firm 2 observed cost is 4X13X519 achievable 4X43X325 So rm 2 appears to be efficient Purdue University Ag Econ 652 Lecture 11 28 14 Nonparametric Efficiency Testing cont d Output lnput1 lnput2 Q m m 14 2 8 3 5 14 2 8 16 Now compare firms 3 and 4 Note that firm 4 is inefficient in the way that made us uncomfortable when we looked closely at technical efficiency more input same output Firm 4 observed costs2x81x622 achievable costs2x81x521 Purdue University Ag Econ 652 Lecture 11 29 Nonparametric Efficiency Testing cont d Clearly we need a test that takes into account the value of inputs Now our question becomes how low could we drive costs and still produce as much as firm 0 Input levels now become variables in our testing problem and the output level is xed I When testing firm 0 we always use firm 0 prices in our measurement of costs Purdue University Ag Econ 652 Lecture 11 30 Nonparametric Efficiency Testing cont d To test for cost minimization efficiency we solve N 0 mlnlmlze Ewnxn a XNJI lK m K subject to 214W 2 u0 Ir1 K 2x5 S x k1 K 2 1 2V 2 0 k1 Purdue University Ag Econ 652 Lecture 1 1 31 Nonparametric Efficiency Testing cont d If we apply this test to the two input data and test for efficiency of rm 1 we solve the following problem minimize 3x1 2x2 711712 145 1414 2 7 4ll112 8 8A4 3 x1 31 1 512 5 3 614 s x2 2 3 4 k Purdue University Ag Econ 652 Lecture 1 1 32 16 Nonparametric Efficiency Testing cont d The optimal solution for this problem has an objective value of 13 with inputs at levels 1 and 5 Does it appearthat firm 1 is efficient I What should the objective be if rm 1 is ef cient I Can we attribute some degree of inef ciency How about noting 1OOX1813138 and saying that costs for rm 1 are 38 too high Purdue University Ag Econ 652 Lecture 11 33 Nonparametric Efficiency Testing cont d If we apply this test for ef ciency to firm 2 we solve the following problem minimize 4x1 3x2 711 712 145 1414 27 411112 8 3 814 3 x1 3 1 512 5 3 614 s x2 1 2 3 47 k Purdue University Ag Econ 652 Lecture 11 34 17 Nonparametric Efficiency Testing cont d The optimal solution for this problem has an objective value of 19 with inputs at levels 1 and 5 Does it appear that firm 2 is efficient I What should the objective be if rm 2 is ef cient I Will we ever find hyperefficiency ie minimum feasible costs higher than observed Purdue University Ag Econ 652 Lecture 11 35 18

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