Cobb Douglas relation with uncompensated law of demand Does a Cobb Douglas or homothetic function satisfy the uncompensated law of demand?
Cobb-Douglas
Short answer
The Cobb-Douglass utility function is additively separable (take logs). Additively separable utility functions have normal goods (i.e. demand increases with income). Normal goods have negative own price effects, so they satisfy the law of demand.
Long answer
Consider the Cobb-Douglas utility function $u(x_1, \ldots x_n) = \prod_{i = 1}^n (x_i)^{\alpha_i}$ with $\alpha_i \ge 0$ and assume wlog that $\sum_i \alpha_i = 1$.
Then demand for good $i$ is given by:
$$
x_i = \frac{\alpha_i y}{p_i}
$$
Take two price vectors $p_1 = \begin{bmatrix} p_{1,i}\\ \vdots \\ p_{1,n} \end{bmatrix}$ and $p_2 = \begin{bmatrix} p_{2,i} \\ \vdots \\ p_{2,n}\end{bmatrix}$ with demands $x_1$ and $x_2$. We want to show that $(p_1 - p_2)' (x_1 - x_2) \le 0$.
We have:
$$
\begin{align*}
(p_1 - p_2)'(x_1 - x_2) &amp;= p_1' x_1 - p_1' x_2 - p_2' x_1 + p_2' x_2,\\
&amp;= y - \sum\frac{p_{1,i}}{p_{2,i}}\alpha_i y - \sum \frac{p_{2,i}}{p_{1,i}} \alpha_i y + y,\\
&amp;= 2 y - y \sum_i \alpha_i\left(\frac{p_{1,i}}{p_{2,i}} + \frac{p_{2,i}}{p_{1,i}}\right),\\
&amp;= 2 y - y \sum_i \alpha_i \frac{(p_{1,i})^2 + (p_{2,i})^2}{p_{1,i}p_{2,i}},\\
&amp;= 2 y - y \sum_i \alpha_i \frac{(p_{1,i} - p_{2,i})^2 + 2 p_{1,i}p_{2,i}}{p_{1,i}p_{2,i}},\\
&amp;= 2 y - y \sum_i \alpha_i \left(\frac{(p_{1,i} - p_{2,i})^2}{p_{1,i}p_{2,i}} + 2\right),\\
&amp;= 2 y - 2 y - y \sum_i \alpha_i \frac{(p_{1,i} - p_{2,i})^2}{p_{1,i}p_{2,i}} \le 0
\end{align*}
$$
Homothetic preferences
Demand functions from homothetic preferences are linear in income:
$$
x(p,m) = \alpha(p) m.
$$
where $\alpha(p)$ is the unit income demand. i.e.:
$$
x(p,1) = \alpha(p) \ge 0.
$$
Then:
$$
\frac{\partial x(p,m)}{\partial m} = \alpha(p) \ge 0,
$$
so homothetic preferences generate normal demands. This means that their won price effect is negative, so they also satisfy the law of demand.