*
Elluminate - 15

* The purpose of this paper is to provide a complete characterization of the normalized restricted profit function via the conjugate duality approach and under conditions more general than those customarily assumed in the literature.
*
A normalized restricted profit function is defined as the maximized value of the normalized prices of the variable commodities and the quantities of the fixed commodities.
* None of the above treatments [Samuelson, Gorman, McFadden,
Diewert, Jacobsen, and Shepard] is based on the conjugacy correspondence of closed, proper, convex functions.
*

* In addition, technological assumptions are relaxed in various directions.
* First, all commodities (except one) are treated symmetrically.
* There is no artificial distinction between commodities that are net outputs and commodities that are net inputs.
* The theory of the normalized restricted profit function thus developed will not only accommodate normalized cost functions (all net outputs are fixed) and normalized revenue functions (all inputs are fixed) as special cases, but also all the mixed cases in between
(some net outputs and some net inputs are fixed; for example, when a firm has a long-term fixed output delivery as well as material supply contracts).
*
Second, there is no assumptions of free disposal except for the numeriare commodity.
*

*
Third, the assumption that the production possibilities set is a convex cone is relaxed.
* It does not need to be a convex cone, or in fact, even convex.
* Instead, a new assumption of biconvexity, which is weaker than convexity is introduced.
* Biconvexity allows for the existence of overall increasing returns while preserving the properties of diminishing marginal rates of transformation (substitution) amongst certain subsets of commodities.
* Fourth, reversibility in production is allowed in a limited sense, not all production plans are reversible, but some production plans may be reversible.
* Finally, the normalized prices of commodities are allowed to be either positive, zero, or negative. Only the price of the numeriare is required to be positive.

*
Rockafellar, R.T. 1974. Conjugate Duality and Optimization
Society for Industrial and Applied Mathematics.
* Let us start with a function of the form
  where X and Y are arbitrary sets, and define
 sup
 
 sup
 
*

*
Consider the economic problem where p is a vector of output prices and y is a vector of outputs (subject to a technology set)

:
  where y

T
* Consider
 sup
  
* So the profit as a function of price is the supremum of the profit function choosing the feasible combination of inputs and outputs.
*

* From the other side of the expression
 inf
 
,

* The infemum is the minimum set of prices that will yield a given level profit subject to a specific set of netputs (y).
*
The work of the conjugate duality is then based on two subproblems minimize
 
over all x

X maximize
 
over all y

Y

*
It is elementary that
       
*
And consequently inf sup
   inf
 sup
 sup inf
 

*

*

*
Theorem 2: A pair (x,y) satisfies the saddle-point condition if and only if x solves the minimization problem and y solves the maximization problem, and the saddle value of K exists, i.e., one has inf
 sup
* Compare this approach with that from Hotelling-Samuelson



,
    
,

    
,
    
,

*

*
  
 

* It is assumed that there are (n + 1 + m) commodities.
*
The commodities y quantities of the last m commodities K be fixed.
1
, y
2
, … y
n+1 are assumed to be variable, and the
1
, K
2
, … K m are assumed to
* All prices are measured relative to the (n + 1)st commodity. The price of the (n + 1)st commodity is always one.
* The normalized prices of the variable commodities are denoted y
1
* , y
2
* , … y denoted K
1
* n
* ; the normalized prices for the fixed commodities are
, K
2
* , … K m
* .
*
The production possibilities set is the set of all feasible production plans.
* The production function is defined by the supremum of the quantity of the (n + 1)st commodity given the quantities of all the other commodities, such that the plan is contained in the production possibilities set.
*

*
Just a little clarity –
*
The set of activities can be defined as
 y
 y n

1
K
 y
1  
 
  and K
* This vector is contained in a technology set T


K
1

 
 
 K m

 y
 y n

1
K

T
* This technological mapping is used to define the normalized production function
    sup
 y n

1
 y
 y n

1
K
 

T

*

*
*
Origin : 0  T
*
Closure : T is closed
*
Convexity : T is convex
*
Montonticity : If then
 y
 y n

1
K

T y
 y
* n

1
K
    y
* n

1
 y n

1
*
Nonproducibility : If
 y
 y n

1
K

, n

1

0

*
*
Production Function
    sup
 y n

1
 y
 y n

1
K
 
T

*
Domain: The effective domain of F(y,K) is a convex set containing the origin F(0,0) = 0.
* Closure: F(y,K) is closed in {y,K}.
* Convexity: F(y,K) is convex in {y,K}.
* Nonnegativity: F(y,K) is nonnegative.
* The profit function maximized by the producer then becomes
P
 * y y
 y n

1
 *
  y n

1

*
Given that y
n+1 as
= -F(y,K), the profit function can be rewritten
P
 *
    

*
,

 sup
 y
*
  
,


* The unrestricted profit function can be written as
F
*

*
,
*

 sup

*
  *
  
,


* The conjugate duality proof results from
H
*

,
  sup y

*
 

*
,
 
 
,

*