aeb 6184 lau s normalized restricted profit function n.
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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.
conjugate duality

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

Conjugate Duality
economic problem

Consider the economic problem where p is a vector of output prices and y is a vector of outputs (subject to a technology set)

    • Consider
    • So the profit as a function of price is the supremum of the profit function choosing the feasible combination of inputs and outputs.
Economic Problem

From the other side of the expression

    • 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 sub- problems

It is elementary that

  • And consequently
saddle point condition

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

  • Compare this approach with that from Hotelling-Samuelson
Saddle-Point Condition
definitions and conventions

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, … Kmare 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.
Definitions and Conventions

Just a little clarity –

    • The set of activities can be defined as
    • This vector is contained in a technology set T
    • This technological mapping is used to define the normalized production function
properties of the technology

Origin: 0  T

  • Closure: T is closed
  • Convexity: Tis convex
  • Montonticity: If


  • Nonproducibility: If
Properties of the Technology
properties of the production function

Production Function

    • 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
Properties of the Production Function
conjugate duality of profit

Given that yn+1 = -F(y,K), the profit function can be rewritten as

  • The unrestricted profit function can be written as
  • The conjugate duality proof results from
Conjugate Duality of Profit