Econ Agri Production
Econ Agri Production AEB 6184
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This 4 page Class Notes was uploaded by Weldon Rau I on Friday September 18, 2015. The Class Notes belongs to AEB 6184 at University of Florida taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/206594/aeb-6184-university-of-florida in Agricultural Economics And Business at University of Florida.
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Date Created: 09/18/15
Shephard s Duality Proof Part II Lecture XVII I Basic Functions A Following Shephard s development from the last lecture we have two basic groups of functions 1 The distance function production function and associated level set a The level set La uis de ned as the set of possible combinations of inputs that can be used to produce the output level u b Given the level set we can de ne a distance function as x Pux 3l0 min lllxeL quot2xquot P c The production function can then be de ned as xmaxu PuxZlxeD d In addition we can de ne the set of ef cient input vectors by the distance function Eu xl Fuax 1 2 The function and the cost structure can be de ned based on the cost minimization problem a The cost function is de ned as Qup minp39xxeL b The cost structure is then de ned for the set of all possible input prices in a similar way as the level sets of inputs are defined in output space Speci cally AQ PQupZlpzo With the equality QuPl being established by the normalization of fp for the po which we will discuss 0 below c By symmetry we can de ne a set of ef cient prices B The claim of Shephard duality is then Qup minp39x l ux Z lu Z 0p Z 0 I uxirfp39xQupZ lu ZOxZ 0 AEB 6184 7 Production Economics Lecture XVII Professor Charles Moss Fall 2005 C Geometric Relationship between Dual Cost and Production Structures There is a rather simple and elegant geometric relationship between the isoquants of the sets La of the production structure and the isoquants of the sets AQ of the cost structure for any positive output rate u 2 The proof of the correspondence involves showing the relationship between the efficient sets E defined from the distance function and defined from the cost function x 0 P p39x Pu x 7 EWCQW P 5 ForxQup0 g EuC39Pux a Starting from a positive output rate u and an arbitrary price vector p0 the efficient points of bounded set of may be generated from the contacts points of the hyperplanes 1 Given the basic price ratio p0 we can define the entire ray is defined by Bpol Z 0 II The hyperplane p 39x Qup Is the support plane for the production possibility set La uwith contact at some point 9E0 AThat is that a price vector p0 and a level of output u determine the level of input use fee on the cost surface Qup AEB 6184 7 Production Economics Lecture XVII Professor Charles Moss Fall 2005 B Put differently 0 Qu p0 C This point is not necessarily unique b Based on this relationship we define the ray A P P Qup I Working backwards 0 Qltua 0Q Hap 0 1 Qup II e EAQ since the cost function is the distance function for the set AQ Further 0 P H Qup c Letting 770 be the intersection between the cost function at u p0 and the ray Bpol Z 0 30 A Qup p 6p llQuap 6quotT 2 Quap I 3quot770quot quotpol ll oll The distance of o from the origin is the reciprocal of the norm of 770 from the origin which is the distance of po39x Qu p0 from the origin 3 1 Working from the other side we define support of Ag as x p I ux In this direction e is the contact point between the ray defined by x0 and the level set I ux I Defining point 9E0 as A0 x I u x0 II Therefore xo39 o I ux03 Aco39po 1 because po39fco Qup 3 I ufc l III Further AEB 6184 7 Production Economics Lecture XVII Professor Charles Moss Fall 2005 A x0 nx n lial IV Back to the distance function representations we let go be the intersection of the ray AxollZO with the hyperplane xo39p Pu x0 Hence I a 0 x yyyioxouxpux 3lo 1 ux 1 2n5 nvux u w 3 Thus o and 9E0 are boundaries of the sets AQu and Lqu respectively These are reciprocally related to each other that is the inverse of the supporting hyperplane