<p>The following argument assumes that we are dealing only with interior solutions. Let $(x_1,x_2,x_3)$ be an optimal demand bundle at prices $(p_1,p_2,p_3)$ and income $m$ and assume that $(x_1,x_2,x_3)$ is an optimal demand bundle at prices $(p_1',p_2,p_3)$ and income $m$. We want to show that one of the following three cases holds: </p>

<ol>
<li>$x_2=x_2'$ and $x_3=x_3'$,</li>
<li>$x_2&lt;x_2'$ and $x_3&lt;x_3'$, or</li>
<li>$x_2&gt;x_2'$ and $x_3&gt;x_3'$.</li>
</ol>

<p>What is not allowed to happen is that $x_2\leq x_2'$ and $x_3&gt;x_3'$ or $x_2&gt;x_2'$ and $x_3\leq x_3'$. Assume one of these cases happens, without loss of generality, we can assume it is the first.</p>

<p>By the strict concavity of $f_2$ and $f_3$, we have
$$MU_2(x_1,x_2,x_3)=\frac{\partial f_2(x_2)}{\partial x_2}\geq \frac{\partial f_2(x_2')}{\partial x_2}=MU_2(x_1',x_2',x_3')$$
and
$$MU_3(x_1,x_2,x_3)=\frac{\partial f_3(x_3)}{\partial x_3}&lt; \frac{\partial f_3(x_3')}{\partial x_3}=MU_3(x_1',x_2',x_3').$$ 
Therefore,
$$\frac{MU_2(x_1',x_2',x_3')}{p_2}\leq\frac{MU_2(x_1,x_2,x_3)}{p_2}=\frac{MU_3(x_1,x_2,x_3)}{p_3}&lt;\frac{MU_3(x_1',x_2',x_3')}{p_3},$$
in contradiction to the equimarginal principle.</p>

CC BY-SA 4.0
2018-09-09T19:39:50.540Z
2018-09-09T19:39:50.540Z
Michael Greinecker
<p>Suppose that consumer's utility function for three goods is separable, that is,
<br/>
$U(x_1, x_2, x_3) = f_1(x_1) + f_2(x_2) + f_3(x_3)$ ...(i)
<br/>
where $f_i$ is increasing and strictly concave, i=1,2,3. Show that
<br/></p>

<ul>
<li>$ df_i/dP_1 $ for i = 2,3 can be either positive or negative, but they must be of the same sign where $f_i$ is the demand function for good i and $P_1$ is price of good 1.</li>
</ul>

<p><br/>
i reasoned using the equimarginal principle where at optimum we must have 
<br/>
$ MU_1/P1 = MU_2/P2 = MU_3/P3 $.
<br/>
suppose good 1 is normal, if $P_1$ rises demand for good 1 falls, that is $df_1/dP_1$ &lt; 0. Due to law of diminishing marginal utility $MU_1$ rises, however to maintain equilibrium $ MU_2,  MU_3$ must also rise. So, $df_2/dp_1 &lt; 0$ and $ df_3/dP_1 &lt; 0$. But i couldn't do it more mathematically, taking the derivative of (i) w.r.t $P_1$ isn't of much use, please help.</p>

CC BY-SA 4.0
2018-09-09T15:07:37.387Z
You cannot "suppose good 1 is normal" or use "law of diminishing marginal utility". You are given a specific mathematical optimization problem (utility maximization w.r.t. budget constraint). Solve that, then reason from the solution.
Giskard
CC BY-SA 4.0
2018-09-09T16:14:05.373Z
Hint: Suppose $\mathrm df_2/\mathrm dp_1$ and $\mathrm df_3/\mathrm dp_1$ have different signs. See if you can deduce a contradiction, which would show that it's wrong to suppose the two derivatives have different signs.
Herr K.
CC BY-SA 4.0
2018-09-09T17:59:47.087Z
The question is odd. It is impossible for all derivatives to be positive. If one good becomes more expensive, you cannot suddenly buy more of all goods, at least if your income is independent of prices.
Michael Greinecker
CC BY-SA 4.0
2018-09-09T18:08:06.533Z
@denesp I can't suppose that good 1 is normal, but I can prove it (if a good is normal, when it's not inferior), since when consumer's utility function for goods are separable, it can be proved that none of the goods can be inferior. As for the use of law of diminishing marginal utility, it's given that $f_i$ is strictly concave, so I thought it was reasonable. I tried optimizing like you said, from the FOCs I arrived at the equimarginal principal, since the demand functions '$f_i(x_i)$ have not been explicitly mentioned, I couldn't proceed further.
Chd
CC BY-SA 4.0
2018-09-09T18:11:43.933Z
@michael greinecker not all goods I think, it asks to show that when price of good 1 changes, demand for good 2 and 3 move in same direction.
Chd
CC BY-SA 4.0
2018-09-09T18:13:28.943Z
@Clarissa Yes, but then it cannot be that this direction "can be either positive or negative." In particular, it cannot be positive.
Michael Greinecker
CC BY-SA 4.0
2018-09-09T18:35:24.820Z
@michael greinecker don't you think it might be possible if goods 1 & 2 and 1 & 3 are substitutes and (if good 1 becomes expensive) rise in price of good 1 does not substantially reduce real income.
Chd
CC BY-SA 4.0
2018-09-09T18:40:39.493Z
@Clarissa There are two things: One is showing that the direction of changes in demand for each good is the same for changes in the price for good $1$. The other is that this change "can be either positive or negative". Indeed, the change in the demand of good 1 must be negative if good 1 is normal, and, as you correctly noted, with separable utility functions (with strictly concave factors), all goods must be normal.
Michael Greinecker
CC BY-SA 4.0
2018-09-09T18:46:19.950Z
@michael greinecker, you are right, "changes in demand for each good is same for changes in price of good 1", here each should include only goods 2 & 3 as I've mentioned in the bulleted point, " i=2,3" and not 1.
Chd
CC BY-SA 4.0
2018-09-09T18:47:40.960Z
Ooops. I should read more carefully.
Michael Greinecker
CC BY-SA 4.0
2018-09-09T18:54:36.043Z
One more remark: The $f_i$ notation seems to be overladed, they are both demand functions and factors in the utility function.
Michael Greinecker
<microeconomics><utility><self-study>
separable utility function and cross price effect
Chd
# separable utility function and cross price effect

(Asked by: username_0 on 2018-09-09)

Suppose that consumer's utility function for three goods is separable, that is,
  
$U(x\_1, x\_2, x\_3) = f\_1(x\_1) + f\_2(x\_2) + f\_3(x\_3)$ ...(i)
  
where $f\_i$ is increasing and strictly concave, i=1,2,3. Show that

* $ df\_i/dP\_1 $ for i = 2,3 can be either positive or negative, but they must be of the same sign where $f\_i$ is the demand function for good i and $P\_1$ is price of good 1.

i reasoned using the equimarginal principle where at optimum we must have
  
$ MU\_1/P1 = MU\_2/P2 = MU\_3/P3 $.
  
suppose good 1 is normal, if $P\_1$ rises demand for good 1 falls, that is $df\_1/dP\_1$ < 0. Due to law of diminishing marginal utility $MU\_1$ rises, however to maintain equilibrium $ MU\_2, MU\_3$ must also rise. So, $df\_2/dp\_1 < 0$ and $ df\_3/dP\_1 < 0$. But i couldn't do it more mathematically, taking the derivative of (i) w.r.t $P\_1$ isn't of much use, please help.

**Tags:** microeconomics, utility, self-study

> **Comments:**
> **username_1**: You cannot "suppose good 1 is normal" or use "law of diminishing marginal utility". You are given a specific mathematical optimization problem (utility maximization w.r.t. budget constraint). Solve that, then reason from the solution.  
> **username_2**: Hint: Suppose $\mathrm df_2/\mathrm dp_1$ and $\mathrm df_3/\mathrm dp_1$ have different signs. See if you can deduce a contradiction, which would show that it's wrong to suppose the two derivatives have different signs.  
> **username_3**: The question is odd. It is impossible for all derivatives to be positive. If one good becomes more expensive, you cannot suddenly buy more of all goods, at least if your income is independent of prices.  
> **username_0**: @denesp I can't suppose that good 1 is normal, but I can prove it (if a good is normal, when it's not inferior), since when consumer's utility function for goods are separable, it can be proved that none of the goods can be inferior. As for the use of law of diminishing marginal utility, it's given that $f_i$ is strictly concave, so I thought it was reasonable. I tried optimizing like you said, from the FOCs I arrived at the equimarginal principal, since the demand functions '$f_i(x_i)$ have not been explicitly mentioned, I couldn't proceed further.  
> **username_0**: @username_3 not all goods I think, it asks to show that when price of good 1 changes, demand for good 2 and 3 move in same direction.  
> **username_3**: @Clarissa Yes, but then it cannot be that this direction "can be either positive or negative." In particular, it cannot be positive.  
> **username_0**: @username_3 don't you think it might be possible if goods 1 & 2 and 1 & 3 are substitutes and (if good 1 becomes expensive) rise in price of good 1 does not substantially reduce real income.  
> **username_3**: @Clarissa There are two things: One is showing that the direction of changes in demand for each good is the same for changes in the price for good $1$. The other is that this change "can be either positive or negative". Indeed, the change in the demand of good 1 must be negative if good 1 is normal, and, as you correctly noted, with separable utility functions (with strictly concave factors), all goods must be normal.  
> **username_0**: @username_3, you are right, "changes in demand for each good is same for changes in price of good 1", here each should include only goods 2 & 3 as I've mentioned in the bulleted point, " i=2,3" and not 1.  
> **username_3**: Ooops. I should read more carefully.  
> **username_3**: One more remark: The $f_i$ notation seems to be overladed, they are both demand functions and factors in the utility function.  

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## 1 answers

### Answer by username_3 on 2018-09-09. Score: 5 | Accepted answer

The following argument assumes that we are dealing only with interior solutions. Let $(x\_1,x\_2,x\_3)$ be an optimal demand bundle at prices $(p\_1,p\_2,p\_3)$ and income $m$ and assume that $(x\_1,x\_2,x\_3)$ is an optimal demand bundle at prices $(p\_1',p\_2,p\_3)$ and income $m$. We want to show that one of the following three cases holds:

1. $x\_2=x\_2'$ and $x\_3=x\_3'$,
2. $x\_2<x\_2'$ and $x\_3<x\_3'$, or
3. $x\_2>x\_2'$ and $x\_3>x\_3'$.

What is not allowed to happen is that $x\_2\leq x\_2'$ and $x\_3>x\_3'$ or $x\_2>x\_2'$ and $x\_3\leq x\_3'$. Assume one of these cases happens, without loss of generality, we can assume it is the first.

By the strict concavity of $f\_2$ and $f\_3$, we have
$$MU\_2(x\_1,x\_2,x\_3)=\frac{\partial f\_2(x\_2)}{\partial x\_2}\geq \frac{\partial f\_2(x\_2')}{\partial x\_2}=MU\_2(x\_1',x\_2',x\_3')$$
and
$$MU\_3(x\_1,x\_2,x\_3)=\frac{\partial f\_3(x\_3)}{\partial x\_3}< \frac{\partial f\_3(x\_3')}{\partial x\_3}=MU\_3(x\_1',x\_2',x\_3').$$
Therefore,
$$\frac{MU\_2(x\_1',x\_2',x\_3')}{p\_2}\leq\frac{MU\_2(x\_1,x\_2,x\_3)}{p\_2}=\frac{MU\_3(x\_1,x\_2,x\_3)}{p\_3}<\frac{MU\_3(x\_1',x\_2',x\_3')}{p\_3},$$
in contradiction to the equimarginal principle.


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