# Proof that EV = CV when there is no income effect

In every textbook it says that it is easy to see that with no income effect, the integral

$$\int_{p^0_1}^{p^1_1} \! h(p,u_0) \mathrm{d}p_1. = \int_{p^0_1}^{p^1_1} \! h(p,u_1) \, \mathrm{d}p_1$$

Could someone explain to me why?

This is because when the Marshallian demand $$x_i(p,w)$$ is constant in $$w$$ (i.e. no income effect), the Hicksian demand $$h_i(p,u)$$ is constant in $$u$$. To see this, we only need to apply the relation $$h_i(p,u)=x_i(p,e(p,u)).$$ For $$u^0\neq u^1$$, we have $$h_i(p,u^0)=x_i(p,e(p,u^0))=x_i(p,e(p,u^1))=h_i(p,u^1).$$
Therefore, $$\int_{p^0_1}^{p^1_1} \! h_i(p,u^0) \mathop{dp_1} = \int_{p^0_1}^{p^1_1} \! h_i(p,u^1) \, \mathop{dp_1}.$$