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?

up vote 1 down vote accepted

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, $$\int_{p^0_1}^{p^1_1} \! h_i(p,u^0) \mathop{dp_1} = \int_{p^0_1}^{p^1_1} \! h_i(p,u^1) \, \mathop{dp_1}.$$

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