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.

$$\underset{x,y}{\mathrm{max}}T{x}^{\alpha}{y}^{\beta}\text{undertheconditionthat}x{p}_{x}+y{p}_{y}=B.$$ |

We form the Lagrange function

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

and the first order conditions:

$$\begin{array}{lll}\hfill \frac{d}{\mathit{dx}}\left(T{x}^{\alpha}{y}^{\beta}+\lambda \left(x{p}_{x}+y{p}_{y}-B\right)\right)& =\mathit{T\alpha}{x}^{\alpha -1}{y}^{\beta}+\lambda {p}_{x}=0\phantom{\rule{2em}{0ex}}& \hfill \text{(9.8)}\phantom{\rule{0.33em}{0ex}}\\ \hfill \frac{d}{\mathit{dy}}\left(T{x}^{\alpha}{y}^{\beta}+\lambda \left(x{p}_{x}+y{p}_{y}-B\right)\right)& =T{x}^{\alpha}\beta {y}^{\beta -1}+\lambda {p}_{y}=0\phantom{\rule{2em}{0ex}}& \hfill \text{(9.9)}\phantom{\rule{0.33em}{0ex}}\\ \hfill \frac{d}{\mathit{d\lambda}}\left(T{x}^{\alpha}{y}^{\beta}+\lambda \left(x{p}_{x}+y{p}_{y}-B\right)\right)& =x{p}_{x}+y{p}_{y}-B=0\phantom{\rule{2em}{0ex}}& \hfill \text{(9.10)}\phantom{\rule{0.33em}{0ex}}\end{array}$$ The FOC 3 represents the second order condition. We transform the other two by adding $-\lambda {p}_{x}$ and $-\lambda {p}_{y}$, respectively $$\begin{array}{lll}\hfill \mathit{T\alpha}{x}^{\alpha -1}{y}^{\beta}& =-\lambda {p}_{x}\phantom{\rule{2em}{0ex}}& \hfill \text{(9.11)}\phantom{\rule{0.33em}{0ex}}\\ \hfill T{x}^{\alpha}\beta {y}^{\beta -1}& =-\lambda {p}_{y}\phantom{\rule{2em}{0ex}}& \hfill \text{(9.12)}\phantom{\rule{0.33em}{0ex}}\end{array}$$

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
$\stackrel{\u0303}{U}\left(x,y\right)=\mathrm{ln}\left(T{x}^{\alpha}{y}^{\beta}\right)=\alpha \mathrm{ln}x+\beta \mathrm{ln}y+\gamma $.

We summarize the two equations as follows, where we have slightly transformed
both:

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

$$\begin{array}{llll}\hfill y{p}_{y}& =\frac{\beta}{\alpha +\beta}B\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x{p}_{x}& =\frac{\alpha}{\alpha +\beta}B\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$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.

(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