Long Run Production Function
The long run production function is the relationship between the maximum output that a firm can produce and the quantities of all inputs, when all inputs are variable. The long run production function is distinguished from the short run production function, in which at least one input is fixed.
Concept
The long run production function describes the technological possibilities available to the firm when it can choose the quantities of all inputs. It is the fundamental constraint on production, and it determines the cost function and the supply behaviour of the firm.
The function exhibits the property of constant, increasing, or decreasing returns to scale according to how output responds to a proportional increase in all inputs. If output rises more than proportionally, the function exhibits increasing returns to scale; if less, decreasing returns.
Returns to Scale
Returns to scale are a property of the technology, determined by the physical relationship between inputs and output. They are distinguished from returns to a factor, which concern the response of output to an increase in one input with others held constant.
Increasing returns to scale arise from the specialisation of labour and of capital, from the use of more efficient technologies at larger scale, and from the spreading of fixed costs. Decreasing returns to scale arise from the difficulties of managing a large organisation and from the limits of the available inputs.
The returns to scale determine the shape of the long run average cost curve. Increasing returns imply a downward-sloping curve, decreasing returns an upward-sloping curve, and constant returns a horizontal curve.
Isoquants and the Choice of Technique
The long run production function can be represented by isoquants, which show the combinations of inputs that produce a given level of output. The slope of the isoquant is the marginal rate of technical substitution, which measures the rate at which one input can be substituted for another.
The firm chooses the combination of inputs that minimises cost for a given level of output. The choice depends on the relative prices of the inputs and on the shape of the isoquants. The cost-minimising combination is where the marginal rate of technical substitution equals the ratio of input prices.
The analysis of the choice of technique is central to the analysis of the effects of changes in input prices. A change in the relative price of labour and capital induces firms to substitute the cheaper input for the more expensive one, a process known as factor substitution.
Applications
The long run production function is used in the analysis of growth, where the accumulation of capital and the growth of the labour force shift the function and raise output. It is used in the analysis of technical progress, which changes the relationship between inputs and output.
The function is also used in empirical work, where it is estimated from data on firms and used to assess the returns to scale, the efficiency of firms, and the effects of policy. The estimation requires attention to the functional form and to the potential for endogeneity.
Technical Progress
Technical progress changes the long run production function, allowing more output to be produced from the same inputs. The progress can be embodied in new capital goods, or it can be disembodied, in the form of improved knowledge and methods.
The analysis of technical progress distinguishes between neutral progress, which raises the productivity of all inputs proportionally, and biased progress, which raises the productivity of some inputs more than others. The bias of technical progress affects the demand for the inputs and the distribution of income.
Technical progress is a central determinant of long-run growth, since it raises the productivity of the economy and allows output to grow even when the inputs are fixed. The sources of technical progress, including research and development and the diffusion of knowledge, are analysed in the theory of growth.
The Cobb-Douglas Function
The Cobb-Douglas production function is the most widely used functional form in the analysis of production. It expresses output as a function of the inputs raised to constant exponents, and it exhibits constant returns to scale when the exponents sum to one.
The function has convenient properties. The exponents represent the shares of the inputs in output, and the elasticities of substitution between the inputs are equal to one. The function is therefore easy to estimate and to interpret.
The function has been extended to include human capital and other inputs, and it is used in the analysis of growth and of the sources of growth. The estimates of the exponents are used to decompose the growth of output into the contributions of the inputs and of the residual, which is often interpreted as the contribution of technical progress.
Empirical Estimation
The estimation of production functions is a central task of empirical economics. The estimation requires data on the inputs and the outputs of firms or of industries, and it must address the problems of simultaneity and of measurement.
The simultaneity problem arises because the inputs are chosen by the firm, and the choice may be correlated with unobserved factors that affect output. The estimation must therefore use instruments or other methods to identify the parameters.
The measurement problem arises because the inputs and the outputs must be measured in consistent units. The measurement of capital is particularly difficult, since the capital stock is the result of past investment and its value depends on the prices of the assets.
Summary
The long run production function describes the technological possibilities available to the firm when all inputs are variable, and it is central to the analysis of returns to scale, of technical progress, and of growth. Its estimation requires attention to the functional form and to the problems of simultaneity and of measurement.