# Cobb–Douglas production function

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Wire-grid Cobb-Douglas production surface with isoquants
A two-input Cobb–Douglas production function with isoquants

In economics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs, particularly physical capital and labor, and the amount of output that can be produced by those inputs. Sometimes the term has a more restricted meaning, requiring that the function display constant returns to scale (in which case ${\displaystyle \beta =1-\alpha }$ in the formula below). The Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1927–1947.[1]

## Formulation

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)
• L = labor input (the total number of person-hours worked in a year)
• K = capital input (the real value of all machinery, equipment, and buildings)
• 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.

Further, if

α + β = 1,

the production function has constant returns to scale, meaning that doubling the usage of capital K and labor L will also double output Y. 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.

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.

## History

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 = ALβK1−β, previously used by Knut Wicksell. 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.[2]

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."[2] 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.[2] 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.[3]

## Difficulties and criticisms

### Lack of microfoundations

The Cobb–Douglas production function was not developed on the basis of any knowledge of engineering, technology, or management of the production process[citation needed]. It was instead developed because it had attractive mathematical characteristics[citation needed], 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. Crucially, there are no microfoundations for it. In the modern era, economists try to build models up from individual agents acting, rather than imposing a functional form on an entire economy[citation needed]. However, many modern authors[who?] have developed models which give Cobb–Douglas production function from the micro level; many New Keynesian models, for example.[4] It is nevertheless a mathematical mistake to assume that just because the Cobb–Douglas function applies at the micro-level, it also always applies at the macro-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).[5]

## Cobb–Douglas utilities

The Cobb–Douglas function has been applied to many other contexts besides production. It can be applied to utility[6] as follows:

${\displaystyle u(x_{1},x_{2})=x_{1}^{\alpha }x_{2}^{\beta };}$

where x1 and x2 are the quantities consumed of good #1 and good #2.

In its generalized form, where x1, ..., xL are the quantities consumed of good #1, ..., good #L, a utility function representing the Cobb–Douglas preferences may be written as:

${\displaystyle {\tilde {u}}(x)=\prod _{i=1}^{L}x_{i}^{\lambda _{i}},\qquad x=(x_{1},\cdots ,x_{L}).}$

Let λ = λ1 + ... + λL, since the function ${\displaystyle x\mapsto x^{\frac {1}{\lambda }}}$ is strictly monotone for x > 0, it follows that ${\displaystyle u(x)={\tilde {u}}(x)^{\frac {1}{\lambda }}}$ represents the same preferences. Setting αi = λi/λ it can be shown that

${\displaystyle u(x)=\prod _{i=1}^{L}x_{i}^{\alpha _{i}},\qquad \sum _{i=1}^{L}\alpha _{i}=1.}$

The utility may be maximized by looking at the logarithm of the utility

${\displaystyle \ln u(x)=\sum _{i=1}^{L}{\alpha _{i}}\ln x_{i}}$

which makes the consumer's optimization problem (where ${\displaystyle w}$ is the total wealth of the consumer):

${\displaystyle \max _{x}\sum _{i=1}^{L}{\alpha _{i}}\ln x_{i}\quad {\text{ s.t. }}\quad \sum _{i=1}^{L}p_{i}x_{i}=w}$

This has the following solution:

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

An interpretation of this solution is that the consumer uses a fraction αj of his wealth in purchasing good j.

The indirect utility function can be calculated by substituting the demand into the utility function. Ignoring a certain multiplicative constant which depends only on the ${\displaystyle \alpha _{i}}$s, we get:

${\displaystyle v(p,w)={\frac {w}{\prod _{i=1}^{L}p_{i}^{\alpha _{i}}}}}$

which is a special case of the Gorman polar form. The expenditure function is the inverse of the indirect utility function:[7]:112

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

## Various representations of the production function

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

${\displaystyle Y=(I_{1})^{a_{1}}*(I_{2})^{a_{2}}\cdots }$

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.

### Translog (transcendental logarithmic) production function

The translog production function is a generalization of the Cobb–Douglas production function.[8] The name translog stands for 'transcendental logarithmic'.

The three factor translog production function is:

{\displaystyle {\begin{aligned}\ln(Y)&=\ln(A)+a_{L}\ln(L)+a_{K}\ln(K)+a_{M}\ln(M)+b_{LL}\ln(L)\ln(L)+b_{KK}\ln(K)\ln(K)+b_{MM}\ln(M)\ln(M)\\&{}\qquad \qquad +b_{LK}\ln(L)\ln(K)+b_{LM}\ln(L)\ln(M)+b_{KM}\ln(K)\ln(M)\\&=f(L,K,M).\end{aligned}}}

where A = total factor productivity, L = labor, K = capital, M = materials and supplies, and Y = output.

## Derived from a CES function

The constant elasticity of substitution (CES) function is

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

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.[9]

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

## References

1. ^ Cobb, C. W.; Douglas, P. H. (1928). "A Theory of Production" (PDF). American Economic Review. 18 (Supplement): 139–165. Retrieved 26 September 2016.
2. ^ a b c Douglas, Paul H. (October 1976). "The Cobb-Douglas Production Function Once Again: Its History, Its Testing, and Some New Empirical Values". Journal of Political Economy. 84 (5): 903–916. doi:10.1086/260489.
3. ^ Filipe, Jesus; Adams, F. Gerard (2005). "The Estimation of the Cobb-Douglas Function: A Retrospective View". Eastern Economic Journal. 31 (3): 427–445. JSTOR 40326423.
4. ^ Walsh, Carl (2003). Monetary Theory and Policy (2nd ed.). Cambridge: MIT Press.
5. ^ Houthakker, H.S. (1955), "The Pareto Distribution and the Cobb–Douglas Production Function in Activity Analysis", The Review of Economic Studies, 23 (1): 27–31, doi:10.2307/2296148, JSTOR 2296148
6. ^ Brenes, Adrián (2011). Cobb-Douglas Utility Function.
7. ^ Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0393957357.
8. ^ Berndt, Ernst R.; Christensen, Laurits R. (1973). "The Translog Function and the Substitution of Equipment, Structures, and Labor in U.S. manufacturing 1929–68". Journal of Econometrics. 1 (1): 81–113. doi:10.1016/0304-4076(73)90007-9.
9. ^ Silberberg, Eugene; Suen, Wing (2001). "Elasticity of Substitution". The Structure of Economics: A Mathematical Analysis (Third ed.). Boston: Irwin McGraw-Hill. pp. 238–250 [pp. 246–7]. ISBN 0-07-234352-4.