How to prove Paasche index is less or equal to CPI if preferences are homothetic?

Question

Given that $$CPI=\frac{e(p^1,u^0)}{e(p^0,u^0)}$$ and the Paasche index $$PPI=\frac{p^1\cdot x^1}{p^0\cdot x^1}$$ How do I show that $PPI\leq CPI$ if preferences are homothetic?

Here's what I've done: Want to show that \begin{align} &\frac{p^1\cdot x^1}{p^0\cdot x^1}\leq \frac{e(p^1,u^0)}{e(p^0,u^0)}\\ \implies& p^0x^1 e(p^1,u^0)\geq p^1 x^1 e(p^0,u^0)\\ \implies& p^0x(p^1,w^1) e(p^1,u^0)\geq p^1 x(p^1,w^1) e(p^0,u^0)\\ \implies& p^0w^1x(p^1,1) e(p^1,u^0)\geq p^1 w^1x(p^1,1) e(p^0,u^0)\\ \implies& p^0x(p^1,1) e(p^1,u^0)\geq p^1x(p^1,1) e(p^0,u^0) \end{align} Since $p^1x(p^1,1)=w_1=1$, and $e(p^0,u^0)=w^0$, we want to show that $$p^0x(p^1,1) e(p^1,u^0)\geq w^0$$

How exactly do I proceed?

Answer 1

Given that preferences are homothetic, we have that $e(p_1, u) = e(p_1,1) u$ and that $e(p_0, u) = e(p_0,1) u$ (the expenditure function is linear in $u$)

So, $$ \frac{e(p_1, u_0)}{e(p_0, u_0)} = \frac{e(p_1,1)}{e(p_0,1)}\frac{u_0 u_1}{u_0 u_1} = \frac{e(p_1, u_1)}{e(p_0, u_1)} $$ Next, $e(p_0, u_1)$ is the minimal expenditure at prices $p_0$ to reach utility level $u_1$. We also know that buying $x_1$ will give us at least utility level $u_1$. As such, $$ p_0 x_1 \ge e(p_0, u_1). $$ Finally, $e(p_0, u_0)$ is the minimal expenditure at prices $p_0$ to reach utilty level $u_0$. This is reached by buying bundle $x_0$ so, $$ p_0 x_0 = e(p_0, u_0). $$

Combining all this, we get: $$ \frac{p_1 x_1}{p_0 x_1} \le \frac{e(p_1, u_1)}{e(p_0, u_1)} = \frac{e(p_1, u_0)}{e(p_0, u_0)}. $$