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The '''Leontief production function''' describes a production technology in which inputs are used in fixed proportions, so that there is no substitution between them. Named after Wassily Leontief, who used such functions in input-output analysis, it represents the extreme case of production with perfectly complementary inputs.

== Specification ==
The Leontief production function takes the form in which output is the minimum of the quantities of each input divided by its required coefficient. For two inputs, capital and labour, output equals the minimum of capital divided by its coefficient and labour divided by its coefficient. This means that output is limited by the scarcest input relative to its requirement, and additional quantities of other inputs do not increase output.

The function exhibits constant returns to scale and has isoquants that are right-angled, with the corner at the point where inputs are in the required proportions. In contrast to the smooth isoquants of the Cobb-Douglas function, Leontief isoquants permit no substitution: the ratio of inputs is fixed by technology.

== Input-Output Analysis ==
Leontief's principal contribution was to develop input-output analysis, a framework in which the production of each sector requires inputs from other sectors in fixed proportions. The input-output table records the flows of goods and services among sectors, and the Leontief inverse translates final demand into the total output required from each sector.

Input-output analysis is used to study the interdependence of industries, the effects of changes in final demand, and the economy-wide implications of shocks. It is also used in environmental analysis to compute the emissions and resource use embodied in production, and in the study of global value chains to trace the origins and destinations of value added.

== Relation to Substitution ==
The Leontief function is the opposite extreme from the linear production function, in which inputs are perfect substitutes and the marginal rate of substitution is constant. Real technologies typically lie between these extremes, allowing some substitution but not unlimited substitution.

The assumption of fixed proportions is a simplification that is useful in some contexts, particularly where technology genuinely constrains the input mix, but it can be misleading where substitution is possible. The elasticity of substitution, which is zero for the Leontief function and infinite for the linear function, summarises the degree of substitutability.

== Applications and Limitations ==
Leontief functions are used in input-output models, in the analysis of capacity constraints where production is limited by the scarcest input, and in models of complementarity in consumption as well as production. The assumption of fixed coefficients is convenient for accounting and for linear programming, and it makes the analysis tractable.

The limitation is that substitution is generally possible in the long run, as firms adjust their technology and input mix in response to relative prices. For this reason, models of long-run growth and of factor demand typically use functional forms that permit substitution, while input-output models retain the Leontief specification for its accounting clarity and empirical feasibility.

== Fixed Coefficients in Input-Output Tables ==
Input-output tables record the value of transactions among sectors and between sectors and final demand. The fixed coefficients assumption implies that the input requirements per unit of output are constant, so that the table can be used to compute the total output required to satisfy a given pattern of final demand.

The Leontief inverse, derived from the input-output table, gives the total output of each sector required per unit of final demand for a given product. It captures both the direct requirements and the indirect requirements through the supply chain, making it possible to trace the full effects of a change in demand.

== Applications in Economics ==
Input-output analysis is used to estimate the employment, output, and emissions embodied in production, to analyse the effects of a sector's expansion on the rest of the economy, and to study the structure of global value chains. Multi-region input-output models trace value added and emissions across countries, informing analysis of trade and climate policy.

The framework is also used in linear programming models of production, where fixed coefficients define the resource requirements of activities and the optimal combination of activities is found by linear optimisation.

== Elasticity of Substitution ==
The elasticity of substitution measures the responsiveness of the input ratio to changes in relative input prices. For the Leontief function it is zero, reflecting the impossibility of substitution. For the Cobb-Douglas function it is one, and for the linear function it is infinite.

Empirical estimates of the elasticity of substitution between capital and labour typically fall between zero and one, suggesting that substitution is possible but limited. The Leontief function therefore represents a limiting case that is convenient for some purposes but not generally appropriate for the analysis of long-run factor demand.

== Critique and Modern Practice ==
The assumption of fixed coefficients is criticised for ignoring the possibility of substitution, which is important in the long run when firms can adjust technology and input mix in response to prices. It is also criticised for treating technology as fixed, whereas technology changes over time, altering input requirements.

Modern practice uses Leontief functions where they are appropriate, as in accounting frameworks and short-run analyses of capacity constraints, and uses more flexible forms where substitution matters. The choice reflects a trade-off between tractability and realism, and the appropriate specification depends on the question at hand.

== See also ==
* [[Production economics]]
* [[Cost Function]]
* [[Factor Endowment]]

== References ==

<ref name="r1">Leontief, W. The Structure of American Economy.</ref>
<ref name="r2">Research on input-output analysis and its applications.</ref>
<ref name="r3">Studies of production functions and substitution.</ref>

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[[Category:Production economics]]
[[Category:Microeconomics]]
[[Category:Econometrics]]