Cobb–Douglas production function

In its most standard form for production of a single good with two factors, the function is

${\displaystyle Y=AL^{\beta }K^{\alpha }}$

where:

• Y = total production (the real value of all goods produced in a year or 365.25 days)
• L = labour input (person-hours worked in a year or 365.25 days)
• K = capital input (a measure of all machinery, equipment, and buildings; the value of capital input divided by the price of capital)
• A = total factor productivity
• α and β are the output elasticities of capital and labor, respectively. These values are constants determined by available technology.

Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus. For example, if α = 0.45, a 1% increase in capital usage would lead to approximately a 0.45% increase in output.

Sometimes the term has a more restricted meaning, requiring that the function display constant returns to scale, meaning that doubling the usage of capital K and labor L will also double output Y. This holds if

α + β = 1,

If

α + β < 1,

returns to scale are decreasing, and if

α + β > 1,

returns to scale are increasing. Assuming perfect competition and α + β = 1, α and β can be shown to be capital's and labor's shares of output.

In its generalized form, the Cobb–Douglas function models more than two goods. The Cobb–Douglas function may be written as[3]

${\displaystyle f(x)=A\prod _{i=1}^{n}x_{i}^{\lambda _{i}},\qquad x=(x_{1},\ldots ,x_{n}).}$

where

• A is an efficiency parameter
• n is the total number of input variables (goods)
• x₁, ..., x_n are the (non-negative) quantities of good consumed, produced, etc.
• ${\displaystyle \lambda _{i}}$ is an elasticity parameter for good i

Paul Douglas explained that his first formulation of the Cobb–Douglas production function was developed in 1927; when seeking a functional form to relate estimates he had calculated for workers and capital, he spoke with mathematician and colleague Charles Cobb, who suggested a function of the form Y = AL^βK^1−β, previously used by Knut Wicksell, Philip Wicksteed, and Léon Walras, although Douglas only acknowledges Wicksteed and Walras for their contributions.[4] Estimating this using least squares, he obtained a result for the exponent of labour of 0.75—which was subsequently confirmed by the National Bureau of Economic Research to be 0.741. Later work in the 1940s prompted them to allow for the exponents on K and L to vary, resulting in estimates that subsequently proved to be very close to improved measure of productivity developed at that time.[5]

A major criticism at the time was that estimates of the production function, although seemingly accurate, were based on such sparse data that it was hard to give them much credibility. Douglas remarked "I must admit I was discouraged by this criticism and thought of giving up the effort, but there was something which told me I should hold on."[5] The breakthrough came in using US census data, which was cross-sectional and provided a large number of observations. Douglas presented the results of these findings, along with those for other countries, at his 1947 address as president of the American Economic Association. Shortly afterwards, Douglas went into politics and was stricken by ill health—resulting in little further development on his side. However, two decades later, his production function was widely used, being adopted by economists such as Paul Samuelson and Robert Solow.[5] The Cobb–Douglas production function is especially notable for being the first time an aggregate or economy-wide production function had been developed, estimated, and then presented to the profession for analysis; it marked a landmark change in how economists approached macroeconomics from a microeconomics perspective.[6]

The function has been criticised for its lack of foundation. Cobb and Douglas were influenced by statistical evidence that appeared to show that labor and capital shares of total output were constant over time in developed countries; they explained this by statistical fitting least-squares regression of their production function. There is now doubt over whether constancy over time exists.

The Cobb–Douglas production function was not developed on the basis of any knowledge of engineering, technology, or management of the production process. This rationale may be true given the definition of the Capital term. Labor hours and Capital need a better definition. If capital is defined as a building, labor is already included in the development of that building. A building is composed of commodities, labor and risks and general conditions.

It was instead developed because it had attractive mathematical characteristics, such as diminishing marginal returns to either factor of production and the property that the optimal expenditure shares on any given input of a firm operating a Cobb–Douglas technology are constant. Initially, there were no utility foundations for it. In the modern era, some economists try to build models up from individual agents acting, rather than imposing a functional form on an entire economy. The Cobb–Douglas production function, if properly defined, can be applied at a micro-economic level, up to a macro- economic level.

However, many modern authors have developed models which give microeconomically based Cobb–Douglas production functions, including many New Keynesian models.[7] It is nevertheless a mathematical mistake to assume that just because the Cobb–Douglas function applies at the microeconomic level, it also always applies at the macroeconomic level. Similarly, it is not necessarily the case that a macro Cobb–Douglas applies at the disaggregated level. An early microfoundation of the aggregate Cobb–Douglas technology based on linear activities is derived in Houthakker (1955).[8]

The Cobb–Douglas function is often used as a utility function.[9][3] Utility ${\displaystyle {\tilde {u}}}$ is a function of the quantities ${\displaystyle x_{i}}$ of the ${\displaystyle L}$ goods consumed:

${\displaystyle {\tilde {u}}(x)=\prod _{i=1}^{n}x_{i}^{\lambda _{i}}}$

Utility functions represent ordinal preferences and do not have natural units, unlike production functions. As a result, a monotonic transformation of a utility function represents the same preferences. Unlike with a Cobb-Douglas production function, where the sum of the exponents determines the degree of economies of scale, the sum can be normalized to one for a utility function because normalization is a monotonic transformation of the original utility function. Thus, let us define ${\displaystyle \lambda =\sum _{i=1}^{n}\lambda _{i}}$ and ${\displaystyle \alpha _{i}={\frac {\lambda _{i}}{\lambda }}}$, so ${\displaystyle \sum _{i=1}^{n}\alpha _{i}=1}$, and write the utility function as:

${\displaystyle u(x)=\prod _{i=1}^{n}x_{i}^{\alpha _{i}}}$

The consumer maximizes utility subject to the budget constraint that the cost of the goods is less than his wealth ${\displaystyle w}$. Letting ${\displaystyle p_{i}}$ denote the goods' prices, he solves:

${\displaystyle \max _{x_{i}}\prod _{i=1}^{n}x_{i}^{\alpha _{i}}\quad {\text{ subject to the constraint }}\quad \sum _{i=1}^{n}p_{i}x_{i}=w}$

It turns out that the solution for Cobb-Douglas demand is

${\displaystyle \forall j:\qquad x_{j}^{\star }={\frac {w\alpha _{j}}{p_{j}}}}$

Since ${\displaystyle \alpha _{j}={\frac {p_{j}x_{j}^{*}}{w}}}$, the consumer spends fraction ${\displaystyle \alpha _{j}}$ of his wealth on good j. Note that this is the solution for either ${\displaystyle u(x)}$ or ${\displaystyle {\tilde {u}}(x),}$ since the same preferences generate the same demand.

The indirect utility function can be calculated by substituting the demands ${\displaystyle x_{j}}$ into the utility function. Define the constant ${\displaystyle K=\Pi _{i=1}^{n}\alpha _{i}^{\alpha _{i}}}$) and we get:

${\displaystyle v(p,w)=\prod _{i=1}^{n}\left({\frac {w\alpha _{i}}{p_{i}}}\right)^{\alpha _{i}}={\frac {\Pi _{i=1}^{n}w^{\alpha _{i}}\cdot \Pi _{i=1}^{n}\alpha _{i}^{\alpha _{i}}}{\Pi _{i=1}^{n}p_{i}^{\alpha _{i}}}}=K\left({\frac {w}{\Pi _{i=1}^{n}p_{i}^{\alpha _{i}}}}\right)}$

which is a special case of the Gorman polar form. The expenditure function is the inverse of the indirect utility function:[10]^:112

${\displaystyle e(p,u)=(1/K)\prod _{i=1}^{n}p_{i}^{\alpha _{i}}u}$

The Cobb–Douglas function form can be estimated as a linear relationship using the following expression:

${\displaystyle \ln(Y)=a_{0}+\sum _{i}a_{i}\ln(I_{i})}$

where

• ${\displaystyle Y={\text{output}}}$
• ${\displaystyle I_{i}={\text{inputs}}}$
• ${\displaystyle a_{i}={\text{model coefficients}}}$

The model can also be written as

${\displaystyle Y=e^{a_{0}}(I_{1})^{a_{1}}\cdot (I_{2})^{a_{2}}\cdots }$

As noted, the common Cobb–Douglas function used in macroeconomic modeling is

${\displaystyle Y=K^{\alpha }L^{\beta }}$

where K is capital and L is labor. When the model exponents sum to one, the production function is first-order homogeneous, which implies constant returns to scale—that is, if all inputs are scaled by a common factor greater than zero, output will be scaled by the same factor.

The constant elasticity of substitution (CES) production function (in the two-factor case) is

${\displaystyle Y=A\left(\alpha K^{\gamma }+(1-\alpha )L^{\gamma }\right)^{1/\gamma },}$

in which the limiting case γ = 0 corresponds to a Cobb–Douglas function, ${\displaystyle Y=AK^{\alpha }L^{1-\alpha },}$ with constant returns to scale.[11]

To see this, the log of the CES function,

${\displaystyle \ln(Y)=\ln(A)+{\frac {1}{\gamma }}\ln \left(\alpha K^{\gamma }+(1-\alpha )L^{\gamma }\right)}$

can be taken to the limit by applying l'Hôpital's rule:

${\displaystyle \lim _{\gamma \to 0}\ln(Y)=\ln(A)+\alpha \ln(K)+(1-\alpha )\ln(L).}$

Therefore, ${\displaystyle Y=AK^{\alpha }L^{1-\alpha }}$.

• Leontief production function
• Production–possibility frontier
• Production theory

• Renshaw, Geoff (2005). Maths for Economics. New York: Oxford University Press. pp. 516–526. ISBN 0-19-926746-4.

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