![]() ![]() Therefore, to find the optimal bundle, we will set the MRS equal to the price ratio and plug the result back into the budget constraint. Thus, we can rewrite equation (22) as the Cobb-Douglas. It’s easy to see that all the conditions for using the Lagrange method are met: the MRS is infinite when $x_1 = 0$, zero when $x_2 = 0$, and smoothly descends along any budget line. 1Although Cobb-Douglas does restrict the elasticity of substitution between the demand for labor. Thanks! Demand with Cobb-Douglas Utility Functionsįor a generic Cobb-Douglas utility function If the exponents don’t add up to 1, just divide each exponent by the sum of the exponents to get the ratio. If you catch a typo or error, or just have a suggestion, please submit a note here. Note: A beautiful property of the Cobb-Douglas demand function is that when the exponents add up to 1, each exponent corresponds to the ratio of the budget you’d spend on that specific good. This work is under development and has not yet been professionally edited. I'm trying to make them each a "standalone" treatment of a concept, but there may still be references to the narrative flow of the book that I have yet to remove. However, this correlation only applies to certain special utilityįunctions such as the Cobb-Douglas utility function.BETA Note: These explanations are in the process of being adapted from my textbook. Not influenced by the price of the good itself or by the consumptionīehavior with regard to the other good (price, quantity). ![]() For specific prices p x and p y and income M, the consumer has the budget. This implies in particular that the expenditure on one good is The Cobb-Douglas utility function has the form u(x, y) x a y 1 - a for 0 < a < 1. 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. More broadly, it offers a normative foundation, with a quasi-linear utility function, to the use of a Cobb-Douglas demand function of two attributes. 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 □ 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) In Figure 4.5B, we have illustrated the indifference curves for c 1/5, d 4/5. In Figure 4.5A, we have illustrated the indifference curves for c 1/2, d 1/2. The preferences represented by the Cobb-Douglas utility function have the general shape depicted in Figure 4.5. ![]() max x, y T x α y β under the condition that x p x + y p y = B. The input demand functions under LR profit maximization can be solved from the marginal revenue product. The Cobb-Douglas utility function will be useful in several examples. Now, we solve the known problem of the household optimum with a Cobb-Douglas utility ![]()
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