Cobb–Douglas production function
In economics and econometrics, 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. The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas between 1927 and 1947;<ref>Template:Cite journal</ref> according to Douglas, the functional form itself was developed earlier by Philip Wicksteed.<ref>Template:Cite book</ref>
Formulation
In its most standard form for production of a single good with two factors, the function is given by:
- <math>Y(L,K)=AL^\beta K^\alpha</math>
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)Template:Clarify
- A = total factor productivity
- <math>0<\alpha<1</math> and <math>0<\beta<1</math> are the capital and labor elasticities of output, respectively. These values are constants determined by available technology.
Capital and labour are the two "factors of production" of the Cobb–Douglas production function.
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 Template:Math, previously used by Knut Wicksell, Philip Wicksteed, and Léon Walras, although Douglas only acknowledges Wicksteed and Walras for their contributions.<ref>Template:Cite book</ref> Not long after Knut Wicksell's death in 1926, Paul Douglas and Charles Cobb implemented the Cobb–Douglas function in their work covering the subject manner of producer theory for the first time.<ref>Template:Cite book</ref> 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.<ref name="doug">Template:Cite journal</ref>
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."<ref name="doug"/> 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.<ref name="doug"/> 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.<ref>Template:Cite journal</ref>
Capital and Labour Elasticity
The factor of production elasticity of output is the percentage change in output that follows from a 1% change in that factor of production, holding constant all the other factors of production as well as the total factor productivity.
In the Cobb–Douglas production function, the capital elasticity of output is <math>\alpha</math>, while the labor elasticity of output is <math>\beta</math>.
Marginal Products
The marginal product of capital is <math>MPK=\frac{\partial Y}{\partial K}=\alpha \frac{Y}{K}>0</math>.
The marginal product of labor is <math>MPL=\frac{\partial Y}{\partial L}=\beta \frac{Y}{L}>0</math>.
That is, increasing capital always leads to an increase in output, increasing labor always leads to an increase in output, and increasing the total factor productivity <math>A</math> increases the marginal product of capital and the marginal product of labor.
Law of diminishing returns
The Cobb–Douglas production function satisfies the law of diminishing returns; that is, the marginal product of capital, while always positive, is declining. As capital increases (holding labor and total factor productivity constant), the output increases but at a diminishing rate. An analogous result holds for labor.
In formulas:
- <math>\frac{\partial MPK}{\partial K} = \frac{\partial^2 Y}{\partial K^2} < 0</math>
- <math>\frac{\partial MPL}{\partial L} = \frac{\partial^2 Y}{\partial L^2} < 0</math>
Marginal Rate of Technical Substitution
The marginal rate of technical substitution equals:
- <math>MRTS(K,L) = \dfrac{MPK}{MPL} = \dfrac{\alpha}{\beta} \dfrac{L}{K}</math>
Elasticity of Substitution
The elasticity of substitution is constant and equal to 1.
Template:Hidden{\dfrac{d\ln(MRTS)}{dMRTS}} = \dfrac{1}{MRTS \frac{\beta}{\alpha}} \cdot \frac{\beta}{\alpha} \cdot MRTS = 1</math>
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Cross derivatives
An increase in labor raises the marginal product of capital, while an increase in capital raises the marginal product of labor.
In formulas: <math>\dfrac{\partial MPK}{\partial L}>0</math>; <math>\dfrac{\partial MPL}{\partial K}>0</math>.
Returns to scale
If <math>\alpha+\beta=1</math>, then returns to scale are constant, meaning that an increase of capital K and labor L by a factor k will produce an increase in output Y of a factor k, that is <math>Y(kL,kK)=kY(L,K)</math>.<ref name="Pearson Education">Template:Cite book</ref>
If <math>\alpha+\beta<1</math>, then returns to scale are decreasing, meaning that an increase of capital K and labor L by a factor k will produce an increase in output Y smaller than a factor k, that is <math>Y(kL,kK)<kY(L,K)</math>.<ref name="Pearson Education"/>
If <math>\alpha+\beta>1</math>, then returns to scale are increasing, meaning that an increase in capital K and labor L by a factor k produce an increase in output Y greater than a factor k, that is, <math>Y(kL,kK)>kY(L,K)</math>.<ref name="Pearson Education"/>
Remuneration under perfect competition
Under perfect competition, the factors of production are remunerated at their total marginal product.
Suppose that <math>Y=F(L,K) = L^\alpha K^{1-\alpha}</math> where <math>0<\alpha<1</math>. In this case
<math>MP_L=\alpha L^ {\alpha-1} K^ {1-\alpha} = \alpha (\frac{K}{L}) ^ {1-\alpha}</math> and <math>MP_K=(1-\alpha) L^{\alpha} K^ {-\alpha} = (1-\alpha) (\frac{K}{L}) ^ {-\alpha}</math>.
Therefore, <math>Y=L\cdot MP_L + K\cdot MP_K = \alpha L^\alpha K^{1-\alpha} + (1-\alpha)L^\alpha K^{1-\alpha} </math>. Dividing both sides by <math>Y=F(L,K) = L^\alpha K^{1-\alpha}</math> we obtain that the remuneration of labor is <math>\alpha</math> of the production and the remuneration of capital is <math>(1-\alpha)</math> of the production.
Let us normalize the price of <math>Y</math> to 1. In a competitive equilibrium the value of marginal product of a production factor equals its price or <math>P_Y\cdot MP_K = MP_K=w </math> and similarly <math>MP_K= r </math> where <math>w </math> is the wage rate and <math>r</math> is the price of capital, the real interest rate (assuming that capital fully depreciates after one period, otherwise, the price of capital is <math>r+\delta</math> where <math>\delta</math> is the depreciation rate of capital).
The total production can be written as follows: <math>Y=L\cdot w + K\cdot r </math>. That is, the value of production is divides between renumeration for labor and renumeration for capital.
Generalized form
In its generalized form, the Cobb–Douglas function models more than two goods. The Cobb–Douglas function may be written as<ref name=Palgrave>Template:Cite book</ref>
- <math>f(x)=A \prod_{i=1}^n x_i^{\lambda_i}, \qquad x = (x_1, \ldots, x_n).</math>
where
- A is an efficiency parameter
- n is the total number of input variables (goods)
- Template:Math are the (non-negative) quantities of good consumed, produced, etc.
- <math>\lambda_i</math> is an elasticity parameter for good i
Criticisms
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. It is now widely accepted that labor share is declining in industrialized economies.<ref>Template:Cite report</ref><ref>Template:Cite journal</ref> The production function contains a principal assumption that may not always provide the most accurate representation of a country's productive capabilities and supply-side efficiencies. This assumption is a "constant share of labor in output," which may not be effective when applied to cases of countries whose labor markets are growing at significant rates.<ref>Template:Cite journal</ref> Another issue within the fundamental composition the Cobb–Douglas production function is the presence of simultaneous equation bias. When competition is presumed, the simultaneous equation bias has impact on all function types involving firm decisions – including the Cobb–Douglas function. In some cases this simultaneous equation bias doesn't appear. However, it is apparent when least squares asymptotic approximations are used.<ref>Template:Cite journal</ref>
However, many modern authorsTemplate:Who have developed models which give microeconomically based Cobb–Douglas production functions, including many New Keynesian models.<ref>Template:Cite book</ref> 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).<ref>Template:Citation</ref> The Cobb–Douglas production function is inconsistent with modern empirical estimates of the elasticity of substitution between capital and labor, which suggest that capital and labor are gross complements. A 2021 meta-analysis of 3186 estimates concludes that "the weight of evidence accumulated in the empirical literature emphatically rejects the Cobb–Douglas specification."<ref>Template:Citation</ref>
In a 1974 article,<ref>Template:Citation</ref> economist Anwar Shaikh demonstrates that any economic data, together with the assumption of a constant share of production between capital and labor and respecting the assumption of constant returns to scale, can be expressed in the form of a Cobb-Douglas production function; he shows that the Cobb-Douglas function is in fact governed by algebraic relationships concerning the distribution of value added between capital and labor, and that the production function therefore does not ultimately rely on any genuine assumption about production itself. To demonstrate this, Anwar Shaikh constructs a Cobb-Douglas function based on fictitious data (data tracing the word “Humbug”), which is strongly correlated with the underlying fictitious production function (R² = .993). In the same article, he also shows that Robert Solow’s article ‘Technical Change and the Aggregate Production Function’<ref>Template:Citation</ref> (1957) paved the way for the neoclassical approach in the economic analysis of growth, makes the same mistake."
Cobb–Douglas utilities
The Cobb–Douglas function is often used as a utility function.<ref>Template:Cite book</ref><ref name=Palgrave /> Utility <math>\tilde{u}</math> is a function of the quantities <math>x_i</math> of the <math>n</math> goods consumed:
- <math> \tilde{u}(x)= \prod_{i=1}^n x_i^{\lambda_i} </math>
Utility functions represent ordinal preferences and do not have natural units, unlike production functions. As the 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 <math> \lambda = \sum_{i=1}^n \lambda_i </math> and <math> \alpha_i = \frac{\lambda_i}{ \lambda}</math>, so <math> \sum_{i=1}^n \alpha_i = 1</math>, and write the utility function as:
- <math>u(x) = \prod_{i =1}^n x_i^{\alpha_i} </math>
The consumer maximizes utility subject to the budget constraint that the cost of the goods is less than her wealth <math>w</math>. Letting <math>p_i</math> denote the goods' prices, she solves:
- <math> \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 </math>
- The Marginal Rate of Substitution between each two goods is
- <math>MRS_{i,j}=\frac{\alpha_i}{\alpha_j}{x_j \over x_i}={p_i \over p_j}\Rightarrow
p_j x_j={p_i \alpha_j\over \alpha_i}x_i</math>
- By inserting to the budget constrain we obtain
- <math>p_i x_i + \textstyle \sum_{j\neq i}^n \displaystyle p_i x_i \frac{\alpha_j}{\alpha_i} = w</math>
- <math>\Rightarrow p_i x_i(1+\sum_{j\neq i}^n \frac{\alpha_j}{\alpha_i})=w
\Rightarrow p_i x_i\frac{(\alpha_i +\sum_{j\neq i}^n \alpha_j)} {\alpha_i}=w \Rightarrow p_i x_i\frac{1} {\alpha_i}=w
</math>
- <math>\Rightarrow x_i^*=\frac{\alpha_i w }{p_i} \forall i</math>
Note that <math> p_i x^*_i = \alpha_i w </math>, the consumer spends fraction <math>\alpha_i</math> of her wealth on good Template:Mvar.
Also note that each good is affected solely by its own price. That is, any two goods are not substitute goods nor complementary goods. Namely, their cross elasticity equals to zero and the cross demand function of any good is described by a vertical line.
Finally, note that when the income increase by some percent the demand for the good increase by the same percent. That is, the elasticity of the demand with respect to income equals 1 and therefore, the Engel curve is a straight line starting from the origin.
Note that this is the solution for either <math>u(x)</math> or <math>\tilde{u}(x),</math> since the same preferences generate the same demand.
The indirect utility function can be calculated by substituting the demands <math>x_i</math> into the utility function. Define the constant <math>K= \prod_{i =1}^n \alpha_i^{\alpha_i}</math> and we get:
- <math> v(p,w) = \prod_{i =1}^n \left( \frac{w \alpha_i}{p_i} \right)^{\alpha_i}
= \frac{ \prod_{i =1}^n w^{\alpha_i}\cdot \prod_{i =1}^n \alpha_i^{\alpha_i} }{ \prod_{i =1}^n p_i ^{\alpha_i}} =K \left(\frac{ w}{\prod_{i =1}^n p_i ^{\alpha_i}} \right)</math>
which is a special case of the Gorman polar form. The expenditure function is the inverse of the indirect utility function:<ref>Template:Cite Varian Microeconomic Analysis 3</ref>Template:Rp
- <math>e(p, u) = (1/K)\prod_{i=1}^n p_i^{\alpha_i} u</math>
- The Marshallian demand function that Cobb-Douglas utility function
Various representations of the production function
The Cobb–Douglas function form can be estimated as a linear relationship using the following expression:
- <math> \ln(Y) = a_0 + \sum_i a_i \ln(I_i) </math>
where
- <math> Y = \text{output}</math>
- <math> I_i = \text{inputs}</math>
- <math> a_i = \text{model coefficients}</math>
The model can also be written as
- <math> Y = e^{a_0} (I_1)^{a_1} \cdot (I_2)^{a_2} \cdots </math>
As noted, the common Cobb–Douglas function used in macroeconomic modeling is
- <math> Y = K^\alpha L^\beta </math>
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.
Relationship to the CES production function
The constant elasticity of substitution (CES) production function (in the two-factor case) is
- <math> Y = A \left ( \alpha K^\gamma + (1-\alpha) L^\gamma \right )^{1/\gamma},</math>
in which the limiting case Template:Math corresponds to a Cobb–Douglas function, <math>Y=AK^\alpha L^{1-\alpha},</math> with constant returns to scale.<ref>Template:Cite book</ref>
To see this, the log of the CES function:
- <math> \ln(Y) = \ln(A) + \frac{1}{\gamma} \ln \left (\alpha K^\gamma + (1-\alpha) L^\gamma \right )</math>
can be taken to the limit by applying L'Hôpital's rule:
- <math> \lim_{\gamma\to 0} \ln(Y) = \ln(A) + \alpha \ln(K) + (1-\alpha) \ln(L).</math>
Therefore, <math>Y=AK^\alpha L^{1-\alpha}</math>.
Translog production function
The translog production function is an approximation of the CES function by a second-order Taylor polynomial in the variable <math>\gamma</math> about <math>\gamma = 0</math>, i.e. the Cobb–Douglas case.<ref>Template:Cite journal</ref><ref>Template:Cite book</ref> The name translog stands for "transcendental logarithmic." It is often used in econometrics for the fact that it is linear in the parameters, which means ordinary least squares could be used if inputs could be assumed exogenous.
In the two-factor case above the translog production function is
- <math>\begin{align}
\ln(Y) &= \ln(A) + \alpha \ln(K) + (1-\alpha) \ln(L) + \frac{1}{2} \gamma \alpha (1 - \alpha) \left[ \ln(K) - \ln(L) \right]^2 \\ &= \ln(A) + a_K \ln(K) + a_L \ln(L) + b_{KK} \ln^2(K) + b_{LL} \ln^{2}(L) + b_{KL} \ln(K) \ln(L) \end{align} </math> where <math>a_K</math>, <math>a_L</math>, <math>b_{KK}</math>, <math>b_{LL}</math>, and <math>b_{KL}</math> are defined appropriately. In the three factor case, the translog production function is:
- <math>\begin{align}
\ln(Y) & = \ln(A) + a_L\ln(L) + a_K\ln(K) + a_M\ln(M) + b_{LL}\ln^2(L) +b_{KK}\ln^2(K) + b_{MM}\ln^2(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{align}</math> where <math>A</math> = total factor productivity, <math>L</math> = labor, <math>K</math> = capital, <math>M</math> = materials and supplies, and <math>Y</math> = output.