Thus, as the input changes, the scale of production also changes because of the proportionality. All the factors in the long run behave like variable and they are not fixed. This is a long term concept, which means that this concept is defined for a time period after which the production function is determined on the basis of variable factors only. This is because defining differentiability on boundary points requires extra care thats usually unnecessary for the issue at hand (such as defining concavity of a function via the Hessian matrix). To concentrate attention on what happens to Q / L or output per worker (and hence, unless the employment ratio changes, output per capita), Solow rewrote the Cobb-Douglas production function in what we shall refer to as per capita form: Q / L A K a L b - 1 A K a / L 1 - b since multiplying by L b - 1 is the same as dividing by L 1 - b. With the proportionate increase in the input factors, the. Douglas is a linear homogeneous production function, which implies, that the factors of production can be substituted for one another up to a certain extent only. These are the change in output as a result of change in the factor inputs, in the same proportion. Definition: The Cobb-Douglas Production Function, given by Charles W. Therefore, the minus sign is added in order to make MRTS a positive number. Perhaps the most common form of production function in economics, the CobbDouglas function has a range of attractive properties. Mathematically, it is represented as M R T S L = − Δ K Δ L MRT Δ L Δ K represents the slope of the isoquant curve and is always negative for Cobb Douglas Production function. Marginal Rate of Technical Substitution (MRTS) is the rate of change of one input with respect to other input along an isoquant. Cobb - Douglass utility function u(x 1 x 2) x 1 x 2 (1 ) sub. 1) Marginal Rate of Technical Substitution: The Cobb-Douglas utility function is a special case where this \income e ect' exactly cancels out the substitution e ect, so the consumption of one good is independent of the price of the other goods.
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