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 $U\left(x,y\right)=T{x}^{\alpha }{y}^{\beta }$ for a given budget $B$, when the prices of goods are $p_x$ and $p_y$, i.e.

$$\underset{x,y}{\max <!--\; FUNCTION\; APPLICATION\; -->}T{x}^{\alpha }{y}^{\beta }\text{ under the condition that }x{p}_x+y{p}_y=B.$$

We form the Lagrange function

$$𝕃\left(x,y,\lambda \right)=T{x}^{\alpha }{y}^{\beta }+\lambda \left(x{p}_x+y{p}_y-B\right)$$

and the first order conditions:

$$\begin{array}{lll}\hfill \frac{d}{\mathit{dx}}\left(T{x}^{\alpha }{y}^{\beta }+\lambda \left(x{p}_x+y{p}_y-B\right)\right)& =\mathit{T\alpha }{x}^{\alpha -1}{y}^{\beta }+\lambda p_x=0& \hfill \text{(9.8)}\\ \hfill \frac{d}{\mathit{dy}}\left(T{x}^{\alpha }{y}^{\beta }+\lambda \left(x{p}_x+y{p}_y-B\right)\right)& =T{x}^{\alpha }\beta y^{\beta -1}+\lambda p_y=0& \hfill \text{(9.9)}\\ \hfill \frac{d}{\mathit{d\lambda }}\left(T{x}^{\alpha }{y}^{\beta }+\lambda \left(x{p}_x+y{p}_y-B\right)\right)& =x{p}_x+y{p}_y-B=0& \hfill \text{(9.10)}\end{array}$$ The FOC 3 represents the second order condition. We transform the other two by adding $-\lambda p_x$ and $-\lambda p_y$, respectively $$\begin{array}{lll}\hfill \mathit{T\alpha }{x}^{\alpha -1}{y}^{\beta }& =-\lambda p_x& \hfill \text{(9.11)}\\ \hfill T{x}^{\alpha }\beta y^{\beta -1}& =-\lambda p_y& \hfill \text{(9.12)}\end{array}$$

and then dividing the equations by each other. This cancels $\lambda$ and the right side is simplified.

$$\frac{\mathit{T\alpha }{x}^{\alpha -1}{y}^{\beta }}{T{x}^{\alpha }\beta y^{\beta -1}}=\frac{\mathit{\alpha y}}{\mathit{\beta x}}=\frac{p_x}{p_y}$$

(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 $\frac{y}{x}$ weighted with the ratio of the exponents (elasticities) $\frac{\alpha }{\beta }$.
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 $\tilde{U}\left(x,y\right)=\ln <!--\; FUNCTION\; APPLICATION\; -->\left(T{x}^{\alpha }{y}^{\beta }\right)=\alpha \ln <!--\; FUNCTION\; APPLICATION\; -->x+\beta \ln <!--\; FUNCTION\; APPLICATION\; -->y+\gamma$.
We summarize the two equations as follows, where we have slightly transformed both:

$$\begin{array}{llll}\hfill x{p}_x+y{p}_y& =B& \hfill & \\ \hfill \frac{\alpha }{\beta }y{p}_y& =x{p}_x& \hfill & \end{array}$$

We insert and get $B=\frac{\alpha }{\beta }{p}_{y}y+y{p}_y=\frac{\alpha +\beta }{\beta }y{p}_y$ or

$$\begin{array}{llll}\hfill y{p}_y& =\frac{\beta }{\alpha +\beta }B& \hfill & \\ \hfill x{p}_x& =\frac{\alpha }{\alpha +\beta }B& \hfill & \end{array}$$

This means that the expenditure for a good $x{p}_x$ 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 $\frac{x{p}_x}{y{p}_y}=\frac{\alpha }{\beta }$, 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 $p_x\uparrow$ of a good increases, the consumed quantity $x\downarrow$ ecreases by the same amount. However, the amount of expenditure $x{p}_x$ remains constant. However, this correlation only applies to certain special utility functions such as the Cobb-Douglas utility function.