Transcript PowerPoint
Shephard’s Duality Proof:
Part I
Lecture XVI
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: D1 x x 0
The boundary points of D excluding the origin:
n
D2 x x 0, xi 0
i 1
D2 is further subdivided into two regions
D2 x x D2 , x L u for some u 0, 0
D2 x x D2 x L u for some u 0, 0
The last segregation segregates D2 into those
points that are on a level set (LΦ(u) ) and those
points not on a level set
Looking at the definition if
n
x D2 x 0, xi 0
i 1
at least one xi is equal to zero, but production is
still possible (i.e., the input is not strongly
necessary). Thus,
x L u :u 0
The second possibility is that inputs are strictly
necessary.
However,
D2 D2 D2
Thus, the possible set of inputs is defined
as the union of all of these disjoint sets:
D 0 D1 D2 D2
The distance function (Ψ(u,x) ) is then
defined on D for the production possibility
sets
x
for x D1 D2 , u 0
u, x 0 for x 0 D2, u 0
for x D, u 0
0 x
0 min x L u
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
0 x L u
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.
If the input bundle gives a valid output
level the distance function is defined as
x
x
u, x
0 x
Defining the norm of a vector
x
n
2
x
i
i 1
Thus
u, x
x
0 x
1
0
x
0 x
If the original input vector does not yield a valid
level set then
x 0 D2, u 0 u, x 0
Otherwise
x D, u 0 u, x
Proposition 14: For any u=[0,)
L u x u , x 1, x D
So the level set is defined as those input
vectors x that have a distance function value
Ψ(u,x) ≥ 1.
x L u , u 0
x
1
x L u , x
x
1
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:
x max u u, x 1 , x D
Thus, the distance function defines the
production function
Shephard’s Cost Function
Following the general framework above for
the distance function, the cost function is
defined as
Q u, p min px x L u , p D, u 0,
x
From this definition
L u x px Q u , p p 0
Duality
Looking ahead, the two functions Q(u,p)
and Ψ(u,x) are dualistically determined if
Q u, p min px u, x 1 , u 0, p 0
x
u, x inf px Q u, p 1 , u 0, x 0
p