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Does a Cobb Douglas or homothetic function satisfy the uncompensated law of demand?

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    $\begingroup$ How do you define the uncompensated law of demand? What do you think about the answer? $\endgroup$
    – Bertrand
    May 15 at 8:38
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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: $$ \begin{align*} (p_1 - p_2)'(x_1 - x_2) &= p_1' x_1 - p_1' x_2 - p_2' x_1 + p_2' x_2,\\ &= y - \sum\frac{p_{1,i}}{p_{2,i}}\alpha_i y - \sum \frac{p_{2,i}}{p_{1,i}} \alpha_i y + y,\\ &= 2 y - y \sum_i \alpha_i\left(\frac{p_{1,i}}{p_{2,i}} + \frac{p_{2,i}}{p_{1,i}}\right),\\ &= 2 y - y \sum_i \alpha_i \frac{(p_{1,i})^2 + (p_{2,i})^2}{p_{1,i}p_{2,i}},\\ &= 2 y - y \sum_i \alpha_i \frac{(p_{1,i} - p_{2,i})^2 + 2 p_{1,i}p_{2,i}}{p_{1,i}p_{2,i}},\\ &= 2 y - y \sum_i \alpha_i \left(\frac{(p_{1,i} - p_{2,i})^2}{p_{1,i}p_{2,i}} + 2\right),\\ &= 2 y - 2 y - y \sum_i \alpha_i \frac{(p_{1,i} - p_{2,i})^2}{p_{1,i}p_{2,i}} \le 0 \end{align*} $$

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.

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  • $\begingroup$ Why do additively separable utility functions have normal goods? $\endgroup$
    – Giskard
    May 15 at 13:48
  • $\begingroup$ The notation here is somewhat unfortunate. $x_i$ is supposed to be demand for good $i$, but you seem to denote two different consumption bundles by $x_1$ and $x_2$. $\endgroup$
    – Giskard
    May 15 at 14:22
  • $\begingroup$ And what about the second part of the question, where $u$ is not Cobb-Douglas type, just homothetic? :) $\endgroup$
    – Giskard
    May 15 at 14:23
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    $\begingroup$ @Giskard If I'd use $x^1$ and $x^2$ some would complain that they are confusing as they are not powers ;-). I'll add a part on homothetic preferences. $\endgroup$
    – tdm
    May 15 at 15:57

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