# Demand derived from Cobb-Douglas utility, interpretation, check

I derived demand, given a Cobb-Douglas utility function but I am not really sure if I did it correctly. I am especially struggling with the sum signs and the subscripts of $$i$$ & $$j$$. It would be really great if someone could check. I want to maximize utility for 2 goods, here $$j$$ and $$i$$.

$$\ u(x_i)=\prod_{i=1}^n x^a_i$$

$$\ s.t.:M=\sum_{j=1}^n p_jx_j$$

$$\ L= \sum_{i=1}^n a_ilogx_i+\lambda(M-\sum_{j=1}^np_jx_j)$$

$$(1)\frac{\partial L}{\partial x_i} = \frac{a_i}{ x_i}-\lambda p_i=0$$

$$(2)\frac{\partial L}{\partial x_j} = \frac{a_j}{ x_j}-\lambda p_j=0$$

$$(3)\frac{\partial L}{\partial \lambda} = M-\sum_{j=1}^np_jx_j=0$$

from (1) and (2) it follows:

$$\frac{p_i}{x_i} = \frac{a_i/x_i}{a_j/x_j}=\frac{a_ix_j}{a_jx_i}$$

$$x_j = \frac{p_ja_jx_i}{p_ja_j}$$

$$x_j$$ into (3)

$$M = \sum_{j=1}^np_j(\frac{p_ja_jx_i}{p_j/a_j})=0$$

$$x_i= \frac{a_iM}{\sum_{j=1}^na_jp_j}$$

Furthermore I would like to interpret what happens if we have an efficency shock for good $$i$$. Meaning, good $$i$$ becomes cheaper. This leads to an increase in relative income. So $$M$$ increases which leads to an increased demand for good $$x_i$$, rest hold constant. Is that correct?

• Sorry. Yes, a drop in $p_i$. And of course $M$ stays constant, but normally we say then that the relative income increases, since the household spends less money for each unit of $x_i$. So again, $p_i$ drops, this will increase the amount of $x_i$. – Pete Mar 15 '19 at 16:59
• His derivation is incorrect (due to a few algebra mistakes). The Cobb-Douglas utility function leads to a set of demand functions in which demand for each good depends only on it's own price. Check the two-good case for reference. See my solution below. – dlnB Mar 15 '19 at 17:05
• @dlnB: You're right. I misread $p_j$ in the denominator as $p_i$. – Herr K. Mar 15 '19 at 17:10

From (1) and (2) you get $$\frac{x_j}{x_i}=\frac{a_j p_i}{a_i p_j},$$ or equivalently, $$x_j =\frac{a_j p_i}{a_i p_j} x_i.$$ Substituting this into equation 3 for $$j=2,...,n$$ and $$i=1$$ (solving for the demand function for good 1) we get $$M=p_1x_1 + \sum_{j=2}^n p_j \frac{a_j p_1}{a_1 p_j} x_1$$ $$M=p_1x_1 + \sum_{j=2}^n \frac{a_j p_1}{a_1} x_1$$ $$M=p_1x_1 + \frac{p_1}{a_1} x_1\sum_{j=2}^n a_j$$ $$M=p_1x_1(1 + \frac{1}{a_1}\sum_{j=2}^n a_j).$$ Finally, solving for $$x_1$$ we get $$x_1^* = \frac{M}{p_1}(1+ \frac{\sum_{j=2}^n a_j}{a_1})^{-1}$$ $$x_1^* =\frac{a_1}{\sum_{j=1}^n a_j} \frac{M}{p_1}.$$
Analogously, $$x_i^*=\frac{a_i}{\sum_{j=1}^n a_j} \frac{M}{p_i},$$ for $$i=1,...,n$$.
As is always the case for demand functions derived from a Cobb-Douglas utility function, the consumer spends a constant share of income on each good. To see this, rearrange the previous equation to get $$p_ix_i^*=\frac{a_i}{\sum_{j=1}^n a_j} M,$$ for $$i=1,...,n$$.