Cobb-Douglas - AbsoluteAstronomy.com

Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, the Cobb–Douglas functional form of production function

Production function

In microeconomics and macroeconomics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs...

s is widely used to represent the relationship of an output to inputs. Similar functions were originally used by Knut Wicksell

Knut Wicksell

Johan Gustaf Knut Wicksell was a leading Swedish economist of the Stockholm school. His economic contributions would influence both the Keynesian and Austrian schools of economic thought....

(1851–1926), while the Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb

Charles Cobb (economist)

Charles Wiggins Cobb was an American mathematician and economist. He published many works on both subjects, however he is most famous for developing the Cobb–Douglas formula in economics. He worked on this project with the economist Paul Douglas while lecturing at the Amherst College in...

and Paul Douglas

Paul Douglas

Paul Howard Douglas was an liberal American politician and University of Chicago economist. A war hero, he was elected as a Democratic U.S. Senator from Illinois from in the 1948 landslide, serving until his defeat in 1966...

during 1900–1947.

Formulation

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

where:

Y = total production (the monetary value of all goods produced in a year)
L = labor input
K = capital
Capital (economics)
In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...

input
A = total factor productivity
Total factor productivity
In economics, total-factor productivity is a variable which accounts for effects in total output not caused by inputs. If all inputs are accounted for, then total factor productivity can be taken as a measure of an economy’s long-term technological change or technological dynamism.If all inputs...
α and β are the output elasticities
Output elasticity
In economics, output elasticity is the percentage change of output divided by the percentage change of an input. It is sometimes called partial output elasticity to clarify that it refers to the change of only one input....

of labor and capital, 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

Ceteris paribus

or is a Latin phrase, literally translated as "with other things the same," or "all other things being equal or held constant." It is an example of an ablative absolute and is commonly rendered in English as "all other things being equal." A prediction, or a statement about causal or logical...

. For example if α = 0.15, a 1% increase in labor would lead to approximately a 0.15% increase in output.

Further, if:

α + β = 1,

the production function has constant returns to scale: Doubling capital K and labour 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

Perfect competition

In economic theory, perfect competition describes markets such that no participants are large enough to have the market power to set the price of a homogeneous product. Because the conditions for perfect competition are strict, there are few if any perfectly competitive markets...

and α + β = 1, α and β can be shown to be labor and capital's share 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

Least squares

The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...

of their production function. There is now doubt over whether constancy over time exists.

History

Paul Douglas

explains 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

Charles Cobb (economist)

who suggested a function of the form

previously used by Knut Wicksell

Knut Wicksell

Johan Gustaf Knut Wicksell was a leading Swedish economist of the Stockholm school. His economic contributions would influence both the Keynesian and Austrian schools of economic thought....

. Estimating this using least squares

Ordinary least squares

In statistics, ordinary least squares or linear least squares is a method for estimating the unknown parameters in a linear regression model. This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear...

, he obtained a result for the marginal productivity of labour (k) to 0.75$ --- which was subsequently confirmed by the National Bureau of Economic Research

National Bureau of Economic Research

The National Bureau of Economic Research is an American private nonprofit research organization "committed to undertaking and disseminating unbiased economic research among public policymakers, business professionals, and the academic community." The NBER is well known for providing start and end...

to be %74.1. Later work in the 1940s prompted them to allow for the exponents on C and L vary, resulting in estimates that subsequently proved to be very close to improved measure of productivity developed at that time.

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." 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

American Economic Association

The American Economic Association, or AEA, is a learned society in the field of economics, headquartered in Nashville, Tennessee. It publishes one of the most prestigious academic journals in economics: the American Economic Review...

. 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

Paul Samuelson

Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...

and Solow. 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 the presented to the profession for analysis; it marked a landmark change in how economists approached macroeconomics

Macroeconomics

Macroeconomics is a branch of economics dealing with the performance, structure, behavior, and decision-making of the whole economy. This includes a national, regional, or global economy...

Dimensional analysis

The Cobb–Douglas model is criticized by some Austrian

Austrian School

The Austrian School of economics is a heterodox school of economic thought. It advocates methodological individualism in interpreting economic developments , the theory that money is non-neutral, the theory that the capital structure of economies consists of heterogeneous goods that have...

economists such as William Barnett II on the basis of dimensional analysis

Dimensional analysis

In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...

. They argue it does not having meaningful or economically reasonable units of measurement

Units of measurement

A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...

unless

. However, other economists in reply to Barnett have argued that the units used are not fundamentally more unnatural than other units commonly used in physics such a log

Logarithm

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

temperature or distance squared.

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. It was instead developed because it had attractive mathematical characteristics, such as diminishing marginal returns

Diminishing returns

In economics, diminishing returns is the decrease in the marginal output of a production process as the amount of a single factor of production is increased, while the amounts of all other factors of production stay constant.The law of diminishing returns In economics, diminishing returns (also...

to either factor of production and the property that expenditure on any given input is a constant fraction of total cost. Crucially, there are no microfoundations

Microfoundations

In economics, the term microfoundations refers to the microeconomic analysis of the behavior of individual agents such as households or firms that underpins a macroeconomic theory....

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. However, many modern authors have developed models which give Cobb–Douglas production function from the micro level; many New Keynesian models, for example. 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).

Some applications

Nonetheless, the Cobb–Douglas function has been applied to many other contexts besides production. It can be applied to utility

Utility

In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

as follows:
U(x₁,x₂)=x₁^αx₂^β;
where x₁ and x₂ are the quantities consumed of good #1 and good #2.

In its generalized form, where x₁, x₂, ... ,x_L are the quantities consumed of good #1, good #2, ..., good #L, a utility function representing the Cobb–Douglas preferences may be written as:

with x = (x₁, x₂, ... ,x_L). Setting

and because the function

is strictly monotone for x > 0, it follows that

represents the same preferences. Setting

it can be shown that

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

which makes the consumer's optimization problem:

This has the solution that:

which has the interpretation that the per-unit fraction of the consumers incomes used in purchasing good j is exactly the marginal term

Various representations of the production function

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

Where:

Y = Output
I_i = Inputs
a_i = model coefficients

The model can also be written as

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

where K is capital and L is labor. When the model coefficients sum to one, as in this example, the production function is first-order homogeneous

Homogeneous function

In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...

, 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. The name translog stands for 'transcendental logarithmic'.

The three factor translog production function is:

where L = labor, K = capital, M = materials and supplies, and q = product.

Derived from a CES function

Constant elasticity of substitution

Constant Elasticity of Substitution

In economics, Constant elasticity of substitution is a property of some production functions and utility functions.More precisely, it refers to a particular type of aggregator function which combines two or more types of consumption, or two or more types of productive inputs into an aggregate...

(CES) function:

corresponds to a Cobb–Douglas function,

Proof:

Apply l'Hôpital's rule

L'Hôpital's rule

In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit...

Therefore,

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Formulation

History

Dimensional analysis

Lack of microfoundations

Some applications

Various representations of the production function

Translog (transcendental logarithmic) production function

Derived from a CES function

See also

External links