Consider a $n$-good industry with a representative consumer with utility function for the differentiated goods given by,

$ U=\left(\sum^n_{i=1}q_i^\beta\right)^\theta $

Suppose that the representative consumer is endowed with income $I$.

a. Derive the inverse and direct demands;

My solution: I am writing the lagrange as:

$ L=\left(\sum^n_{i=1}q_i^\beta\right)^\theta+\lambda(I-\sum^n_{i=1}p_iq_i) $


$ \theta \left(\sum^n_{i=1}q_i^\beta\right)^{\theta-1}\beta q_i^{\beta-1}-\lambda p_i=0 $

So the inverse demand function is:

$ p_i=\frac{\theta \left(\sum^n_{i=1}q_i^\beta\right)^{\theta-1}\beta q_i^{\beta-1}}{\lambda} $

Is this right? Because the $\lambda$ seems a little bit strange. If yes, how do I isolate $q_i$, when there is this sum symbol to derive the demand function?


You stopped a bit too early.

Let me re-write the FOC for good $i$ as follows:

$$ \lambda=\frac{\theta \left(\sum^n_{i=1}q_i^\beta\right)^{\theta-1}\beta q_i^{\beta-1}}{p_i} $$

As this is identical across goods, equalise for good $i$ and good $j$, where, after simplification, you get:

$$ \frac{q_i^{\beta-1}}{p_i} = \frac{q_j^{\beta-1}}{p_j} $$

This relation hold for any pair of goods. This means you can re-write the demand for any good in terms of $q_i$ and relative prices. Then, you can use the budget constraint to obtain the final demand for good $i$, in terms of income $I$ and all the prices.

Hopefully this information is enough for you to move forward and solve the problem.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy