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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  px x  L  u  , p  D, u  0,  
x
From this definition
L  u    x px  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  px   u, x   1 , u  0, p  0
x
  u, x   inf  px Q  u, p   1 , u  0, x  0
p