9.4 The Lagrange formalism for the example of a Cobb-Douglas utility function

Now, we solve the known problem of the household optimum with a Cobb-Douglas utility function U x,y = Txαyβ for a given budget B, when the prices of goods are px and py, i.e.

max x,yTxαyβ under the condition that xp x + ypy = B.

We form the Lagrange function

𝕃 x,y,λ = Txαyβ + λ xp x + ypy B

and the first order conditions:

d dx Txαyβ + λ xp x + ypy B = xα1yβ + λp x = 0 (9.8) d dy Txαyβ + λ xp x + ypy B = Txαβyβ1 + λp y = 0 (9.9) d Txαyβ + λ xp x + ypy B = xpx + ypy B = 0 (9.10) The FOC 3 represents the second order condition. We transform the other two by adding λpx and λpy, respectively xα1yβ = λp x (9.11) Txαβyβ1 = λp y (9.12)

and then dividing the equations by each other. This cancels λ and the right side is simplified.

xα1yβ Txαβyβ1 = αy βx = px py (9.13)

The resulting equation represents the central point of the solution. It represents a relationship between the marginal utility ratio and the price ratio.
1) The marginal utility ratio in the Cobb-Douglas utility function is always the inverse ratio of the quantities of goods y x weighted with the ratio of the exponents (elasticities) α β.
2) The technique factor T is irrelevant. It does not influence the optimal choice of goods, but only the level of utility achieved.
3) You obtain the same solution when considering a monotone transformation (e.g. the logarithm) of the original utility function. Try it out by calculating it for Ũ x,y = ln Txαyβ = α ln x + β ln y + γ.
We summarize the two equations as follows, where we have slightly transformed both:

xpx + ypy = B α βypy = xpx

We insert and get B = α βpyy + ypy = α+β β ypy or

ypy = β α + βB xpx = α α + βB

This means that the expenditure for a good xpx immer einem festen Anteil am Budget entsprechen. always corresponds to a fixed share of the budget. The shares of expenditure behave like the exponents of the utility function xpx ypy = α β, i.e. for a good with a high relative utility (= large exponent) a lot is spent. This implies in particular that the expenditure on one good is not influenced by the price of the good itself or by the consumption behavior with regard to the other good (price, quantity). If the price px of a good increases, the consumed quantity x ecreases by the same amount. However, the amount of expenditure xpx remains constant. However, this correlation only applies to certain special utility functions such as the Cobb-Douglas utility function.

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de