<p>This is because when the Marshallian demand <span class="math-container">$x_i(p,w)$</span> is constant in <span class="math-container">$w$</span> (i.e. no income effect), the Hicksian demand <span class="math-container">$h_i(p,u)$</span> is constant in <span class="math-container">$u$</span>. To see this, we only need to apply the relation
<span class="math-container">$$h_i(p,u)=x_i(p,e(p,u)).$$</span>
For <span class="math-container">$u^0\neq u^1$</span>, we have
<span class="math-container">$$h_i(p,u^0)=x_i(p,e(p,u^0))=x_i(p,e(p,u^1))=h_i(p,u^1).$$</span> </p>

<p>Therefore, 
<span class="math-container">$$\int_{p^0_1}^{p^1_1} \! h_i(p,u^0) \mathop{dp_1} = \int_{p^0_1}^{p^1_1} \! h_i(p,u^1) \, \mathop{dp_1}.$$</span></p>

CC BY-SA 4.0
2018-10-11T18:50:25.117Z
2018-10-11T18:50:25.117Z
Ziwei Wang
<p>In every textbook it says that it is easy to see that with no income effect, the integral</p>

<p><span class="math-container">$\int_{p^0_1}^{p^1_1} \! h(p,u_0) \mathrm{d}p_1. = \int_{p^0_1}^{p^1_1} \! h(p,u_1) \, \mathrm{d}p_1$</span></p>

<p>Could someone explain to me why?</p>

<microeconomics>
Proof that EV = CV when there is no income effect
Rainroad
# Proof that EV = CV when there is no income effect

(Asked by: username_0 on 2018-10-11)

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?

**Tags:** microeconomics


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## 1 answers

### Answer by username_1 on 2018-10-11. Score: 1 | Accepted answer

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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