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

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?

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: $$\frac{p_1 x_1}{p_0 x_1} \le \frac{e(p_1, u_1)}{e(p_0, u_1)} = \frac{e(p_1, u_0)}{e(p_0, u_0)}.$$