<p>The answer comes from looking at the Hicksian compensated demand:</p>
<p>Since, WLOG, EV for only one good price change can be written as
<span class="math-container">$$EV(p_1,p_0,u) = \int_{p_1}^{p_0} h(p,u_1)dp$$</span>
and CV as
<span class="math-container">$$CV(p_1,p_0,u) = \int_{p_1}^{p_0} h(p,u_0)dp,$$</span>
and we know that <span class="math-container">$\frac{\partial h}{\partial u} \leq 0$</span> when the good is inferior (using the fact that <span class="math-container">$\frac{\partial x}{\partial w}=\frac{\partial h}{\partial u}\frac{\partial v}{\partial w} \leq 0$</span> and <span class="math-container">$\frac{\partial v}{\partial w}\geq0$</span>, coming from the identity <span class="math-container">$x(p,w)=h(p,v(p,w))$</span>), we can use the monotonicity of the integral to establish that
<span class="math-container">$$EV(p_1,p_0,u) = \int_{p_1}^{p_0} h(p,u_1)dp \leq \int_{p_1}^{p_0} h(p,u_0)dp = CV(p_1,p_0,u)$$</span>
when <span class="math-container">$p_1&lt;p_0$</span>.</p>
<p>For the case when <span class="math-container">$p_1 &gt; p_0$</span> just notice that <span class="math-container">$u_1 \leq u_0$</span>, where <span class="math-container">$u_i = v(p_i,w)$</span>, <span class="math-container">$\implies$</span> <span class="math-container">$h(p,u_1) \geq h(p,u_1)$</span>, so that
<span class="math-container">$$-EV(p_1,p_0,u) = \int_{p_0}^{p_1} h(p,u_1) dp \geq \int_{p_0}^{p_1} h(p,u_0) dp = - CV(p_1,p_0,u).$$</span></p>
<p><span class="math-container">$\square$</span></p>

CC BY-SA 4.0
2021-11-07T21:54:14.877Z
2022-09-12T17:29:23.577Z
2022-09-12T17:29:23.577Z
RAGMS
<p>I believe Marshallian demands are less steep than Hicksian demands because we reverse the y and x axis in economics. Thus a larger derivative of x with respect to p will be less steep since p is on the vertical and x is on the horizontal.<a href="https://i.sstatic.net/XXCpf.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XXCpf.png" alt="enter image description here"></a></p>

<p>(image from <a href="https://www.youtube.com/watch?v=T4GAFG8TVt0" rel="nofollow noreferrer">here</a>)</p>

CC BY-SA 3.0
2016-11-02T14:23:20.357Z
You are completely right but does that imply a shift in the welfare inequalities if the good is inferior? (Turning EV>CS>CV into CV>CS>CV)
RAGMS
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2016-11-02T00:06:54.473Z
2017-06-30T10:12:28.997Z
2017-06-30T10:12:28.997Z
MHall
<p>Through the Slutsky equation I know that if the good is inferior the marshallian demand function is steeper than the hicksian demand but I cannot understand why the Compensating variation is higher than the equivalent variation. Is this because the increase in utility shifts the hicksian to the left instead of shifting it to the right (in a graph with prices as y and demand as x)?</p>

CC BY-SA 3.0
2016-11-01T20:55:11.730Z
No, EV>(ConsumerSurplus>)CV if the good is normal and the other way around if it is inferior. I am just struggling with justifying this last assertion.
RAGMS
CC BY-SA 3.0
2016-11-01T22:04:55.913Z
I believe you are mistaken. In the symmetric Cobb-Douglas case $U(x,y) = x \cdot y$ none of the goods are inferior. Yet if you have income $m > 0$, prices $p_x = p_y = 1$ and altered price $p_x' = 4$ you will have $CV = m > \frac{m}{2} = EV$.
Giskard
CC BY-SA 3.0
2016-12-02T00:49:58.927Z
I think it's possible to have EV=CV=CS, when there is no wealth effect. Let me mess around with a utility function that's quasilinear in one good to double-check.
Theoretical Economist
<microeconomics><consumer-theory><welfare><consumer-surplus>
Why is EV<CV if the good is inferior?
RAGMS
# Why is EV<CV if the good is inferior?

(Asked by: username_0 on 2016-11-01)

Through the Slutsky equation I know that if the good is inferior the marshallian demand function is steeper than the hicksian demand but I cannot understand why the Compensating variation is higher than the equivalent variation. Is this because the increase in utility shifts the hicksian to the left instead of shifting it to the right (in a graph with prices as y and demand as x)?

**Tags:** microeconomics, consumer-theory, welfare, consumer-surplus

> **Comments:**
> **username_0**: No, EV>(ConsumerSurplus>)CV if the good is normal and the other way around if it is inferior. I am just struggling with justifying this last assertion.  
> **username_1**: I believe you are mistaken. In the symmetric Cobb-Douglas case $U(x,y) = x \cdot y$ none of the goods are inferior. Yet if you have income $m > 0$, prices $p_x = p_y = 1$ and altered price $p_x' = 4$ you will have $CV = m > \frac{m}{2} = EV$.  
> **username_2**: I think it's possible to have EV=CV=CS, when there is no wealth effect. Let me mess around with a utility function that's quasilinear in one good to double-check.  

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## 2 answers

### Answer by username_0 on 2021-11-07. Score: 1 | Accepted answer

The answer comes from looking at the Hicksian compensated demand:

Since, WLOG, EV for only one good price change can be written as
$$EV(p\_1,p\_0,u) = \int\_{p\_1}^{p\_0} h(p,u\_1)dp$$
and CV as
$$CV(p\_1,p\_0,u) = \int\_{p\_1}^{p\_0} h(p,u\_0)dp,$$
and we know that $\frac{\partial h}{\partial u} \leq 0$ when the good is inferior (using the fact that $\frac{\partial x}{\partial w}=\frac{\partial h}{\partial u}\frac{\partial v}{\partial w} \leq 0$ and $\frac{\partial v}{\partial w}\geq0$, coming from the identity $x(p,w)=h(p,v(p,w))$), we can use the monotonicity of the integral to establish that
$$EV(p\_1,p\_0,u) = \int\_{p\_1}^{p\_0} h(p,u\_1)dp \leq \int\_{p\_1}^{p\_0} h(p,u\_0)dp = CV(p\_1,p\_0,u)$$
when $p\_1<p\_0$.

For the case when $p\_1 > p\_0$ just notice that $u\_1 \leq u\_0$, where $u\_i = v(p\_i,w)$, $\implies$ $h(p,u\_1) \geq h(p,u\_1)$, so that
$$-EV(p\_1,p\_0,u) = \int\_{p\_0}^{p\_1} h(p,u\_1) dp \geq \int\_{p\_0}^{p\_1} h(p,u\_0) dp = - CV(p\_1,p\_0,u).$$

$\square$


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### Answer by username_3 on 2016-11-02. Score: 2

I believe Marshallian demands are less steep than Hicksian demands because we reverse the y and x axis in economics. Thus a larger derivative of x with respect to p will be less steep since p is on the vertical and x is on the horizontal.[![enter image description here](https://i.sstatic.net/XXCpf.png)](https://i.sstatic.net/XXCpf.png)

(image from [here](https://www.youtube.com/watch?v=T4GAFG8TVt0))

> **Comments:**
> **username_0**: You are completely right but does that imply a shift in the welfare inequalities if the good is inferior? (Turning EV>CS>CV into CV>CS>CV)  

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