Suppose there are two people living together. They have the same utility function (CES) $$u(x,h)=(x^\rho+h^\rho)^\frac{1}{\rho}$$ where $x$ is a type of good, $h$ is housing. The price of good is $p$, the unit of rent is $r$. Their income is both $w$. The maximization problem is (if I have it correctly) \begin{align*} &\max u(x_1,h)+u(x_2,h)\\ &\text{s.t. }x_1p+x_2p+rh=2w \end{align*} I want to determine how $\rho$ affects the per capita consumption of the $x$. Since there are 3 goods in this model, I'm finding it difficult to start. Any ideas are appreciated.
How do I determine the degree of substitution affect the change in per capita consumption of a good?
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Given the great amount of symmetry in your problem, it can actually be reduced to a problem in only two variables. Let us first show that in the optimum $x_1 = x_2$. Towards a contradiction, assume that $(x_1, x_2, h)$ is optimal but $x_1 \ne x_2$.
Consider the allocation $(\tilde x, \tilde x, h)$ where $\tilde x = \frac{x_1 + x_2}{2}$. This bundle is feasible: $$ p \tilde x + p \tilde x + r h = p x_1 + p x_2 + rh \le 2w. $$
Also, by strict concavity of $u(x,h)$, we have: $$ u(\tilde x, h) + u(\tilde x, h) > \frac{1}{2} u(x_1,h) + \frac{1}{2} u(x_2,h) + \frac{1}{2} u(x_1,h) + \frac{1}{2} u(x_2,h) = u(x_1, h) + u(x_2,h), $$ which contradicts optimality of $(x_1, x_2, h)$.
Now, given that $x_1 = x_2$ in the optimum, we can rewrite the optimisation problem as: $$ \max_{x,h} 2 u(x,h) \text{ s.t. } 2p x + rh \le 2w. $$ The factor of 2 in the objective function can be ignored as it will not change the solution of this maximisation problem. As such, the optimal bundle will be given by the usual demands for a CES utility function where the prices are $2p$ and $r$ and the income level is $2w$.
