*
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
K : X Y  , 
where X and Y are arbitrary sets, and define
f  x   sup K  x, y 
yY
g  y   sup K  x, y 
xX
*
* Consider the economic problem where p is a vector of output
prices and y is a vector of outputs (subject to a technology set)
 : py  ,  where y T
* Consider
f  p   sup   p, y 
yY T
* 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
g  y   inf   p, y 
pP
* 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 f  x  over all x  X
maximize g  y  over all y  Y
* It is elementary that
f  x   K  x, y   g  y 
* And consequently
inf sup K  x, y   inf f  x   sup  supinf K  x, y 
xX yY
xX
yY
yY xX
f  x
g  x
*
f  p
g  y
*
* 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 f  x   sup g  y 
xX
yY
* Compare this approach with that from Hotelling-Samuelson
  p, w   py  c  w, y 
 L  p, w, y     p, w   py  c  w, y 
*
  p, w
py  c  w, y 
*
* It is assumed that there are (n + 1 + m) commodities.
* The commodities y1, y2, … yn+1 are assumed to be variable, and the
quantities of the last m commodities K1, K2, … Km are assumed to
be fixed.
* 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
y1*, y2*, … yn*; the normalized prices for the fixed commodities are
denoted K1*, K2*, … Km*.
* 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
yn 1
 y1 
 K1 
K  : y    and K   
 yn 
 K m 
* This vector is contained in a technology set T
 y
yn1 K  T
* This technological mapping is used to define the normalized
production function
F  y, K    sup  yn 1  y
*
yn 1
K   T 
* Origin: 0  T
* Closure: T
* Convexity:
is closed
T is convex
* Montonticity: If  y
yn1 K  T
*
*




y
y
K

T

y
then 
n 1
n 1  yn 1

* Nonproducibility: If  y
*
yn1 K  T , yn1  0
* Production Function
F  y, K    sup  yn 1  y
yn 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  yn 1  y* y  yn 1
*
* Given that yn+1 = -F(y,K), the profit function can be rewritten
as


P  y* y  F  y, K   H  y * , K   sup y * y  F  y , K 
y
* The unrestricted profit function can be written as


F *  y* , K *   sup y* y  K * K  F  y, K 
y,K
* The conjugate duality proof results from


H *  y, K   sup y* y  H  y* , K   F  y, K 
y
*