Skip to main content

Über dieses Buch

This study is the result of an interest in the economic theory of production intermittently pursued during the past three years. Over this period I have received substantial support from the Office of Naval Research, first from a personal service consulting contract directly with the Mathematics Division of the Office of Naval Research and secondly from Project N6 onr-27009 at Princeton Univer­ sity under the direction of Professor Oskar Morgenstern. Grateful acknowledgement is made to the ·Office of Naval Research for this support and to Professor Morgenstern, in particular, for his interest in the puolication of this research. The responsibility for errors and omissions, how­ ever, rests entirely upon the author. Professor G. C. Evans has given in terms of a simple total cost function, depending solely upon output rate, a treatment of certain aspects of the economic theory of production which has inherent generality and convenience of formulation. The classical approach of expressing the technology of production by means of a production function is potentially less restrictive than the use of a simple total cost function, but it has not been applied in a more general form other than to derive the familiar conditions between marginal productivities of the factors of produc­ tion and their market prices.



Cost and Production Functions


1. The Process Production Function

Consider a single production process yielding one homogeneous output and let U denote the output per unit time at some time t. We suppose that the process involves the use of n factors of production, and write x1,x2,...,xn as amounts per unit time of these factors associated with the output U.
Ronald W. Shephard

2. Heuristic Principle of Minimum Costs

Let p1, p2,..., pn be prices per unit of the factor applications x1, x2,..., xn respectively. Then cost per unit time of producing an output U at the time t is given by
$$q = \,\sum\limits_{{i = 1}}^{n} {{p_{i}}{\kern 1pt} } $$
, ignoring any fixed charges independent of price and quantity.
Ronald W. Shephard

3. The Producer’s Minimum Cost Function

For given rates of output U and prices pi of the factors of production, the rates xi at which the factors are used satisfy equation (2) and minimize (5), if the process is organized instantaneously for minimum cost. Necessary conditions for this minimum property are
$$ \begin{gathered} {p_{i}} = \,\lambda .\frac{{\partial \psi }}{{\partial {x_{1}}}}\quad \left( {i = 1,{\kern 1pt} \,2,...,\,n} \right) \hfill \\ \psi \left( {U,{x_{1}},\,...,\,{x_{n}}} \right) = 1 \hfill \\ \end{gathered} $$
where the Lagrangian multiplier λ may depend in general upon U, p1,..., pn.
Ronald W. Shephard

4. Dual Determination of Production Function from Cost Function

Cost and production functions have sometimes been used in economic literature as separate and not necessarily equivalent specifications of production technology. But it is to be expected, in some sense, that the functions Φ or ψ and the function Q given by (8) are equivalent statements of production alternatives, assuming that the process is organized for minimum costs. The arguments of this Section are directed to the establishment of the precise correspondence between cost and production function.
Ronald W. Shephard

5. Geometric Interpretation of the Duality Between Cost and Production Function

The geometric duality between cost and production function may be described in simple terms as follows: Let the plane of Figure 1 be the plane of the factor space, considering only two factors of production. The coordinate axes X1, X2 are used Interchangeably for amounts x1, X2 and prices p1, p2 of these factors. In this coordinate system the production curve ψ(U,X1X2) =1 (U constant) and the unit cost curve Q(U,X1X2) =1 (U constant) are approximately sketched. Let p1, p2 be some arbitrarily given prices of the factors of production. These prices define a direction OR, where the coordinates p1; p2 of R are proportional to p1, p2, that is p1 = τp1 and p2 = τp2’. The proportionality factor τ is taken so that
$$Q\left( {U,\,{p_{1}},\,{p_{2}}} \right) = \,T.\,Q\left( {U,{p_{1}}',{p_{2}}'} \right) = 1 $$
which can be done since Q Is homogeneous of degree one In the prices and the point R(p1, P2) lies on the unit cost surface. The amounts x1, x2 of the factors of production which minimize cost, for given values U, P1’, p2’, define a point P(x1, x2) on the production curve ψ(U,X1X2) = 1.
Ronald W. Shephard

6. Constraints on the Factors of Production

The equations of the preceding sections have been calculated upon an assumption of independent factor allocations xi (i = 1, 2,..., n) in the production functions Φ and ψ, and cost per unit time given by (5) was defined without fixed charges per unit time independent of the prices and quantities. These simplifications were made for initial convenience and they may be substantially relaxed without essentially changing the previous analysis.
Ronald W. Shephard

7. Homothetic Production Functions

A Production function of the Independent factor variables x1, x2,..., xn will be called Homothetlc, if It can be written
$$ \Phi (\sigma ({x_{{1,}}}\,{x_{2}}), \ldots ,\,{x_{n}})$$
where σ is a. homogeneous function of degree one and Φ is a continuous positive monotone increasing function of Φ. The properties assumed In Section 1 for the function Φ of equation (l) are taken for the function Φ, and the production surfaces related to (31) are given by
$$ (U) = \Phi (\sigma ({x_{{1,}}}\,{x_{2}}), \ldots ,\,{x_{n}})$$
$$ f(U) = (\sigma ({x_{{1,}}}\,{x_{2}}), \ldots ,\,{x_{n}})$$
where f(U; is the inverse function of Φ(σ) in (32). Since Φ(σ) and f(U) are increasing functions of their arguments we have
$$ \frac{{d\Phi (\sigma )}}{{d\sigma }} > 0,\frac{{d\Phi (U)}}{{dU}} > 0$$
for all positive σ and U, but no special assumptions of algebraic sign are made for the second derivatives of Φ(σ) and f (U).
Ronald W. Shephard

8. The Cobb-Douglas Production Function

Consider now a special representation of the index function σ appearing in the definition of the nomothetic production surfaces (32) or (32.1). Let
$$ \sigma = {\sigma _{0}}\left[ {\mathop{\Pi }\limits_{{i = 1}}^{N} {{(\frac{{{x_{1}}}}{{{x_{1}}0}})}^{{{a_{{i\quad }}}}}}\mathop{\Pi }\limits_{{k = 1}}^{L} {{(\frac{{{x_{1}}}}{{{x_{1}}0}})}^{{{b_{k}}}}}} \right] $$
$$ \mathop{\Sigma }\limits_{{i = l}}^{N} {a_{1}} + \mathop{\Sigma }\limits_{{k = 1}}^{L} {b_{k}} = 1$$
where xi O, zk O are values at some base time (t) of the factor applications xi, zk and σO is the value of σ at (tO). We shall think of the variables zk as denoting current rates of application of primary factors of production, e.g. the services of labor and land, and distinguish the rates xi as services of non-primary factors such as fixed capital. It is assumed that all of these rates of factor applications are independent arguments of the function σ.
Ronald W. Shephard

9. The Problem of Aggregation

Economic theories are frequently expressed in aggregate terms, with propositions related to such aggregates as capital, labor, producer’s goods and consumer’s goods, particularly if these theories are to find quantitative expression of their structure or serve as qualitative guides to economic policy. It is Inconvenient to think in terms of a very large number of components of an economic system. For this reason a Robinson Crusoe type of economic theory was invented, but these contructions have not been entirely convincing and, until recently, economists have contented themselves with tacitly assuming that theories can justifiably be constructed in terms of aggregates of economic quantities by reasoning in terms of single quantity prototypes of these aggregates. The quantitative expression of such theories of aggregates has been made in terms of index number measurement of the aggregates, defining the prototype variable as some average of the micro-economic components of the aggregate which it represents.
Ronald W. Shephard

10. Dynamics of Monopoly Under Homothetic Production Function

An Interesting application of the propositions related to homothetlc production functions may be made to the dynamics of monopoly as formulated by G. C. Evans.21 Similar constructions are possible for the other economic variational problems of Evans and to the analogous dynamic theory of competition as formulated by Roos,22 but we shall restrict ourselves here to the monopoly problem for the purpose of illustrating the way in which our theory of cost and production functions may serve to broaden the basis of these mathematical economic formulations.
Ronald W. Shephard


Weitere Informationen