Consumer Theory question You plan to use the following specification for an empirical study:
$$e_i = \alpha_i + \sum_{j=1}^{n} \beta_{ij}p_i + \gamma_iy +\delta_i, 
i=1,...,n$$ 
where $e_i$ is the consumer's expenditure on good $i$, $p_i$ is the price of good $i$, $y$ is the income and the $\alpha_i, \beta_{ij}, \gamma_i, \delta_i$ are parameters.

You want the specification to be consistent with standard consumer theory and three friends offer the following opinions: 

1) “the equation will work for any values of the parameters;” 

(2) “you need to impose the  $\beta_{ij}=0,\sum_{i=1}^{n}, \delta_{i}=1$";

(3)“you need to impose restrictions such as $\beta_ij = -\gamma_i\alpha_j, \gamma_i \geq 0,\sum_{i=1}^{n}, \gamma{i}=1,\delta_i=0 $

(a) Explain which of the friends gave sensible advice.
(b) Why should you really not listen to the advice of the other friends?
You could use the fact that $\sum_{i=1}^{n}e_i=y$ (adding-up property) to see which parameter values are compatible with the above specification and this identity.
Before proceeding, you should replace on the RHS $\sum_{i=1}^{n}\beta_{ij}p_i$ by $\sum_{j=1}^{n}\beta_{ij}p_j$.