# Fixed Coefficients Production Function

# Fixed Coefficients Production Function

A production function associates the maximum level of output producible with given amounts of inputs. If the inputs must be combined in fixed proportions, like the ingredients of a recipe in a cookbook, the function is a fixed coefficients production function. It is also called a Leontief function, after its inventor, the economist and Nobel Prize winner, Wassily Leontif. Call centers require a one-to-one proportion between workers and telecommunication equipment. Denoting the input quantities by *L* and *K*, the isoquants are L-shaped (with the kink on the 45 degree line).

To introduce the formal definition, denote the quantities of inputs required per unit of some output by *a* _{1}, …, a_{n}, where *n* is the number of inputs. These so-called input coefficients constitute the recipe or technique for the production of the output considered. Denote the available

amounts of inputs by *x* _{1}, …, x_{n}, respectively. Then the attainable level of output is given by *y* = min {*x* _{1}/*a* _{1}, …, x_{n} */* a_{n} *}*. This is the defining formula of the fixed coefficients production function. The inputs for which the minimum value is assumed are called the bottlenecks.

The fixed coefficients production function is the cornerstone of input-output analysis, the quantitative economic tool developed in 1936 by Leontief, who traced the origin to Francois Quesnay’s *Tableau Économique* of 1758. Scholars Heinz Kurz and Neri Salvadori described the roots of input-output analysis in detail in their 2000 work, and the theory is exposited in Thijs ten Raa’s *Economics of Input-Output Analysis* (2005). The fixed coefficients function is popular, because only a single observation is needed to calculate it, making use of the input coefficients a_{i} = x_{i}/*y*. The connection between fixed coefficients and input-output analysis is as follows.

Since inputs are produced (such as electricity) or non-produced (such as labor services), we may label them 1, …, *m, m* + 1, …, *n*, where the last *n – m* inputs are the nonproduced or so-called factor inputs. Denote the input coefficients of output *j* by a_{ij}. The matrix of intermediate input coefficients is *A* = (a_{ij} *)* _{)} _{i, j}_{=} _{1,} _{…,} * _{m}* and the matrix of factor input coefficients is

*B*= (a

_{ij}

*)*

_{)}

_{i}_{=}

_{m}_{+ 1,}

_{…,}

_{n, j}_{= 1,}

_{…,}

*. The matrix of factor input coefficients gives the direct factor requirements of products. Post-multiplication of*

_{m}*B*with the so-called Leontief inverse, (1 –

*A*)

^{–1}= 1 +

*A*+

*A*

^{2}+ …, yields the matrix of total factor requirements or factor contents of products. The total requirements include the factor requirements of the produced inputs,

*BA*, the factor requirements of the produced inputs of those inputs,

*BA*

^{2}, etcetera.

An important application is the Marxian theory of labor values, in which all commodities are produced, directly or indirectly, by labor. Then the factor input coefficients matrix *B* reduces to a row vector of direct labor coefficients and the total requirements becomes a row vector of labor contents, one for each product. Another application is energy economics. Here the direct coefficients measure the energy used per unit of output and the total coefficients measure the total amount of energy embodied in products. The inclusion of the indirect effects may cause reversals in the energy intensity of products, when the production of an output requires little energy, but much intermediate input of which the production is energy intensive. The inclination of politicians to subsidize goods of which the direct energy requirements are low may therefore be ill conceived.

Input coefficients tend to be fixed at the level of the firm. Indeed, managers know how many workers are needed to operate the machines. Input coefficients vary between firms though and, therefore, the fixed coefficients production function is less appropriate for industries or economies. For example, if the wage rate increases relative to the rate of interest, labor-intensive firms may shut down and capital-intensive firms may expand to full capacity. As a result, the economy will be more capital intensive. Though derived from micro fixed coefficients production functions, the macro production function will thus feature input substitutability, much like the Cobb-Douglas function. In fact, the latter can be derived mathematically if the production capacity across firms follows a Pareto distribution, which is defined by the same formula as the Cobb-Douglas function. Most applied general equilibrium models feature production functions with a mixture of fixed and variable coefficients, but even when all the production functions are of the fixed coefficients variety, the response to price shocks may be the same as in a model with variable coefficients production functions.

**SEE ALSO** *Input-Output Matrix; Leontief, Wassily; Production Function*

## BIBLIOGRAPHY

Kurz, Heinz D., and Neri Salvadori. 2000. “Classical” Roots of Input-Output Analysis: A Short Account of Its Long Prehistory. *Economic Systems Research* 12 (2): 153–179.

Leontief, Wassily W. 1936. Quantitative Input and Output Relations in the Economic System of the United States. *The Review of Economics and Statistics* 18 (3): 105–125.

ten Raa, Thijs. 2005. *Economics of Input-Output Analysis* Cambridge, U.K.: Cambridge University Press.

*Thijs ten Raa*

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