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Not influenced by the price of the good itself or by the consumptionīehavior with regard to the other good (price, quantity). In economics and econometrics, the CobbDouglas 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. This implies in particular that the expenditure on one good is for a good with a high relative utility (= large exponent) a lot is

The shares of expenditure behave like the exponents of the utility function This means that the expenditure for a goodĮinem festen Anteil am Budget entsprechen. We insert and get B = α β p y y + y p y = α + β β y p y We summarize the two equations as follows, where we have slightly transformed U ̃ x, y = ln ⁡ T x α y β = α ln ⁡ x + β ln ⁡ y + γ. the logarithm) of the original utility function. It does not influence the optimal choice of goods, but only the levelģ) You obtain the same solution when considering a monotone transformation Weighted with the ratio of the exponents (elasticities) Represents a relationship between the marginal utility ratio and the priceġ) The marginal utility ratio in the Cobb-Douglas utilityįunction is always the inverse ratio of the quantities of goods The resulting equation represents the central point of the solution. Tα x α − 1 y β T x α β y β − 1 = αy βx = p x p y Tα x α − 1 y β = − λ p x (9.11) T x α β y β − 1 = − λ p y (9.12)Īnd then dividing the equations by each other. The FOC 3 represents the second order condition. We form the Lagrange function 𝕃 x, y, λ = T x α y β + λ x p x + y p y − Bĭ dx T x α y β + λ x p x + y p y − B = Tα x α − 1 y β + λ p x = 0 (9.8) d dy T x α y β + λ x p x + y p y − B = T x α β y β − 1 + λ p y = 0 (9.9) d dλ T x α y β + λ x p x + y p y − B = x p x + y p y − B = 0 (9.10) max ⁡ x, y T x α y β under the condition that x p x + y p y = B. Now, we solve the known problem of the household optimum with a Cobb-Douglas utility