Ritu Tiwari

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. The Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1927–1947. In its most standard form for production of a single good with two factors, the function is 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 ? and ? 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 ? = 0.45, a 1% increase in capital usage would lead to approximately a 0.45% increase in output. Further, if ? + ? = 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 ? + ? < 1, returns to scale are decreasing, and if ? + ?> 1, returns to scale are increasing. Assuming perfect competition and ? + ? = 1, ? and ? 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. Nonetheless, the Cobb–Douglas function has been applied to many other contexts besides production. It can be applied to utility[7] as follows: 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: \tilde{u}(x)=\prod_{i=1}^L x_i^{\lambda_{i}}, \qquad x=(x_1, \cdots, x_L). Let ? = ?1 + ... + ?L, since the function x \mapsto x^\frac1\lambda is strictly monotone for x > 0, it follows that u(x)=\tilde{u}(x)^{\frac1\lambda} represents the same preferences. Setting ?i = ?i/? it can be shown that 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 \ln u(x)=\sum_{i=1}^L {\alpha_{i}}\ln x_i which makes the consumer's optimization problem: \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: \forall j: \qquad x_j^\star=\frac{w \alpha_j}{p_j}. An interpretation of this solution is that the per-unit fraction of the consumers incomes used in purchasing good j is exactly the marginal term ?j The Cobb–Douglas function form can be estimated as a linear relationship using the following expression: \ln(Y) = a_0 + \sum_i a_i \ln(I_i) Where: Y = \text{Output} I_i = \text{Inputs} a_i = \text{Model coefficients} The model can also be written as Y = (I_1)^{a_1} * (I_2)^{a_2} \cdots As noted, the common Cobb–Douglas function used in macroeconomic modeling is 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. The name translog stands for 'transcendental logarithmic'. The three factor translog production function is: \begin{align} \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{align} where A = total factor productivity, L = labor, K = capital, M = materials and supplies, and Y = output. The constant elasticity of substitution (CES) function is Y = A \left ( \alpha K^\gamma + (1-\alpha) L^\gamma \right )^{\frac{1}{\gamma}}, in which the limiting case ? = 0 corresponds to a Cobb–Douglas function, Y=AK^\alpha L^{1-\alpha}, with constant returns to scale. To see this, the log of the CES function, \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: \lim_{\gamma\to 0} \ln(Y) = \ln(A) + \alpha \ln(K) + (1-\alpha) \ln(L). Therefore, Y=AK^\alpha L^{1-\alpha} .