Shephard’s Duality Proof: Part I

1. Shephard’s Duality Proof: Part I Lecture XVI

2. Definition of the Distance Function • As a starting point of the discussion of the distance function, Shephard defines the space of possible inputs over the nonnegative domain D. • Then the space of all possible input bundles is segmented into a sequence of regions: • The origin: {0} • Interior points of D: • The boundary points of D excluding the origin:

3. D2 is further subdivided into two regions • The last segregation segregates D2 into those points that are on a level set (LΦ(u) ) and those points not on a level set

4. Looking at the definition if at least one xi is equal to zero, but production is still possible (i.e., the input is not strongly necessary). Thus,

5. The second possibility is that inputs are strictly necessary. • However, • Thus, the possible set of inputs is defined as the union of all of these disjoint sets:

6. The distance function (Ψ(u,x) ) is then defined on D for the production possibility sets

8. Breaking this definition down by parts, Ψ(u,x) is a function of the output level u and an input vector x. • LΦ(u) is the level-set of inputs that produce at least output • λ0 is the minimum length along that array that will generate an output on that level set

9. Under the first scenario the input bundle gives a valid output level: • If x  D1 then the input bundle produces a valid output because it is an interior point in output space. • If x  D2’ then the input bundle is on one of the axis (i.e., xi = 0 for some i), or one of the inputs is not strictly necessary.

10. If the input bundle gives a valid output level the distance function is defined as • Defining the norm of a vector

11. Thus

13. If the original input vector does not yield a valid level set then • Otherwise

14. Proposition 14: For any u=[0,) So the level set is defined as those input vectors x that have a distance function value Ψ(u,x)≥ 1.

16. Proposition 15: For any u(0,+) , the isoquant of a production set LΦ(u) consists those input vectors x ≥ 0 such that Ψ(u,x) = 1. • So the distance function can be used to define the isoquant. • Proposition 17: Thus, the distance function defines the production function

17. Shephard’s Cost Function • Following the general framework above for the distance function, the cost function is defined as

19. From this definition

20. Duality • Looking ahead, the two functions Q(u,p) and Ψ(u,x) are dualistically determined if