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# Cobb-Douglas

In economics, the Cobb-Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. It was proposed by Knut Wicksell (1851-1926), and tested against statistical evidence by Charles Cobb and Paul Douglas in 1928.

For production, the function is

Y = ALαKβ,

where:

• Y = total production (the monetary value of all goods produced in a year)
• L = labor input
• K = capital input
• A = total factor productivity
• α and β are the output elasticities 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. 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. That is, if L and K are each increased by 20%, Y increases by 20%. If

α + β < 1,

returns to scale are decreasing, and if

α + β > 1

returns to scale are increasing. Assuming perfect competition, 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 of their production function. There is now doubt over whether constancy over time exists.

## Difficulties

Neither Cobb nor Douglas provided any theoretical reason why the coefficients α and β should be constant over time or be the same between sectors of the economy. Remember that the nature of the machinery and other capital goods (the K) differs between time-periods and according to what is being produced. So do the skills of labor (the L).

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 to either factor of production.

Crucially, there are no microfoundations for it. In the modern era, economists have insisted that the micro-logic of any larger-scale process should be explained. The C-D production function fails this test.

For example, consider the example of two sectors which have the exactly same Cobb-Douglas technologies:

if, for sector 1,

Y1 = AL1αK1β

and, for sector 2,

Y2 = AL2αK2β,

that, in general, does not imply that

Y1 + Y2 = A(L1 + L2)α(K1 + K2)β
This holds only if L1 / L2 = K1 / K2 and α+β = 1, i.e. for constant returns to scale technology.

It is thus a mathematical mistake to assume that just because the Cobb-Douglas function applies at the micro-level, it also applies at the macro-level. Similarly, there is no reason that a macro Cobb-Douglas applies at the disaggregated level.

## Some applications

Nonetheless, the Cobb-Douglas function has been applied to a lot of other contexts besides production. It can be applied to utility as follows: U(x1,x2)=x1αx2β; where x1 and x2 are the quantities consumed of good #1 and good #2.

In its generalized form, the Cobb-Douglas utility function is written as:

$prod_\left\{i=1\right\}^N x_i^\left\{alpha_\left\{i\right\}\right\}$
where $x_i$ are the quantities consumed of each good i and $alpha_\left\{i\right\}$ are the demand elasticities of utility.

## Various representations of the production function

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

$log_e\left(Y\right) = a_0 + sum_i\left\{a_i log_e\left(I_i\right)\right\}$
Where:

• Y = Output
• Ii = Inputs
• ai = model coefficients

The model can also be written as

$Y = \left(I_1\right)^\left\{a_1\right\} * \left(I_2\right)^\left\{a_2\right\} cdots$

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

$Y = K^alpha L^\left\{1-alpha\right\}$

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, which implies constant returns to scale, that is, if all inputs are doubled that output will double.

### 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:

$ln\left(q\right)=ln\left(A\right)+aL*ln\left(L\right)+aK*ln\left(K\right)+aM*ln\left(M\right)+bLL*ln\left(L\right)*ln\left(L\right)+bKK*ln\left(K\right)*ln\left(K\right)+$
$bMM*ln\left(M\right)*ln\left(M\right)+bLK*ln\left(L\right)*ln\left(K\right)+bLM*ln\left(L\right)*ln\left(M\right)+bKM*ln\left(K\right)*ln\left(M\right)=f\left(L,K,M\right)$.

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

## Derived from a CES function

Constant elasticity of substitution (CES) function: $Y = A\left[alpha K^gamma + \left(1-alpha\right) L^gamma\right]^\left\{frac\left\{1\right\}\left\{gamma\right\}\right\}$

When $gamma = 0$, this CES function will reduce to a Cobb-Douglas function, $Y=AK^alpha L^\left\{1-alpha\right\}$

Proof:

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

Apply l'Hôpital's rule: $lim_\left\{gammarightarrow 0\right\} ln\left(Y\right) = ln\left(A\right) + alpha ln\left(K\right) + \left(1-alpha\right) ln\left(L\right)$

Therefore, $Y=AK^alpha L^\left\{1-alpha\right\}$