<p>A good is <em>normal</em> if its demand is increasing in income. So let $p_x$ and $p_y$ be the price of the goods with quantities $x$ and $y$ and let $m$ be income.</p>

<p>Suppose $ax&gt;by$. Then $\min\{ax,by\}=by$. By slightly reducing $x$ by and spending the saved money on $y$, one gets a better bundle. For an optimal bundle, this cannot be.  </p>

<p>Similarly, it cannot be optimal that $by&gt;ax$. So in the optimal consumption bundle, it must be the case that $ax=by$. It is also not that hard to see that the consumer will spend all her income. So rewrite the condition as 
$$y=\frac{a}{b}x$$ and plug it into the budget equation $$p_x x+p_y y=m$$ to get
$$p_x x+ p_y\frac{a}{b}x=m=x\Big(p_x+p_y\frac{a}{b}\Big).$$
Therefore, we get the demand function given by $$x(p_x,p_y,m)=\frac{m}{p_x+p_y\frac{a}{b}},$$ which is clearly increasing in $m$. Similarly, one shows that the other good is normal too.</p>

<p>Pedantic remark: A differentiable function can be increasing at every point without the derivative being strictly positve everywhere. The function given by $x\mapsto x^3$ has derivative $0$ at $0$ but is everywhere increasing.</p>

CC BY-SA 3.0
2017-12-18T14:26:14.067Z
@snoram: I suspect it does not have to be too complicated.  A utility function like $U(x,y,z)=\sqrt{\min\{a{x},b{y}\}} + cz^2$ might be enough
Henry
CC BY-SA 3.0
2015-05-11T21:41:03.617Z
How does your last remark apply here? Are you saying $$\frac{\partial x}{\partial m}$$ here is not increasing everywhere in this case?
Stan Shunpike
CC BY-SA 3.0
2015-05-11T21:55:50.407Z
An implicit assumption here is that since the utility is a function of only two goods (in this case perfect complements) no other goods exists in the economy and in this case as Michael showed they are normal goods, but I guess with a much more complicated utility function of many goods where two are perfect complements, there can be perfect complements which are NOT normal goods.
snoram
CC BY-SA 3.0
2015-05-11T22:30:31.240Z
@StanShunpike Taking derivatives works nicely here, the point is that the definition of a good being normal is slightly different than demand having a positive derivative with respect to income. I think it is generally a good idea to separate the concept from specific mathematical tools that are convenient to work with.
Michael Greinecker
CC BY-SA 3.0
2015-05-11T21:33:34.070Z
2015-05-11T21:33:34.070Z
Michael Greinecker
<p>Two goods $x,y$ are perfect complements if they have the utility function 
$$U(x,y) = \min \lbrace ax,by \rbrace $$ 
$$a,b \in \Bbb{Q}^+$$
My professor said $x,y$ have to be normal goods but didn't explain why well enough that I could understand. </p>

<p><strong>My Question:</strong> </p>

<p>Are perfect complements always normal goods? If so, why?</p>

CC BY-SA 3.0
2015-05-11T21:08:02.590Z
Do you know how to derive the demand function? Do you know the definition of normal goods?
Michael Greinecker
CC BY-SA 3.0
2015-05-11T21:12:32.033Z
Normal good: $$\frac{\partial x}{\partial m} > 0$$ Perfect Complements Demand Function: $$x = \frac{\bar{U}}{a}$$ Not sure about the latter, but that's what I have in my notes.
Stan Shunpike
CC BY-SA 3.0
2015-05-11T21:17:04.843Z
Oh, I see your point. I should just take the derivative of demand wrt to $m$ and I might learn more about its behavior. But my demand function is stated in terms of utility. That's why it didn't occur to me to take the derivative wrt to $m$. I am not sure how to derive the perfect complements demand function wrt $m$.
Stan Shunpike
CC BY-SA 3.0
2015-05-11T21:20:11.373Z
Ah, they discuss it here. http://youtu.be/kjI840VDW5I perhaps i can figure it out now.
Stan Shunpike
<consumer-theory><perfect-complements>
Do perfect complements have to be normal goods? If so, why?
Stan Shunpike
# Do perfect complements have to be normal goods? If so, why?

(Asked by: username_0 on 2015-05-11)

Two goods $x,y$ are perfect complements if they have the utility function
$$U(x,y) = \min \lbrace ax,by \rbrace $$
$$a,b \in \Bbb{Q}^+$$
My professor said $x,y$ have to be normal goods but didn't explain why well enough that I could understand.

**My Question:**

Are perfect complements always normal goods? If so, why?

**Tags:** consumer-theory, perfect-complements

> **Comments:**
> **username_1**: Do you know how to derive the demand function? Do you know the definition of normal goods?  
> **username_0**: Normal good: $$\frac{\partial x}{\partial m} > 0$$ Perfect Complements Demand Function: $$x = \frac{\bar{U}}{a}$$ Not sure about the latter, but that's what I have in my notes.  
> **username_0**: Oh, I see your point. I should just take the derivative of demand wrt to $m$ and I might learn more about its behavior. But my demand function is stated in terms of utility. That's why it didn't occur to me to take the derivative wrt to $m$. I am not sure how to derive the perfect complements demand function wrt $m$.  
> **username_0**: Ah, they discuss it here. http://youtu.be/kjI840VDW5I perhaps i can figure it out now.  

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## 1 answers

### Answer by username_1 on 2015-05-11. Score: 12 | Accepted answer

A good is *normal* if its demand is increasing in income. So let $p\_x$ and $p\_y$ be the price of the goods with quantities $x$ and $y$ and let $m$ be income.

Suppose $ax>by$. Then $\min\{ax,by\}=by$. By slightly reducing $x$ by and spending the saved money on $y$, one gets a better bundle. For an optimal bundle, this cannot be.

Similarly, it cannot be optimal that $by>ax$. So in the optimal consumption bundle, it must be the case that $ax=by$. It is also not that hard to see that the consumer will spend all her income. So rewrite the condition as
$$y=\frac{a}{b}x$$ and plug it into the budget equation $$p\_x x+p\_y y=m$$ to get
$$p\_x x+ p\_y\frac{a}{b}x=m=x\Big(p\_x+p\_y\frac{a}{b}\Big).$$
Therefore, we get the demand function given by $$x(p\_x,p\_y,m)=\frac{m}{p\_x+p\_y\frac{a}{b}},$$ which is clearly increasing in $m$. Similarly, one shows that the other good is normal too.

Pedantic remark: A differentiable function can be increasing at every point without the derivative being strictly positve everywhere. The function given by $x\mapsto x^3$ has derivative $0$ at $0$ but is everywhere increasing.

> **Comments:**
> **username_0**: How does your last remark apply here? Are you saying $$\frac{\partial x}{\partial m}$$ here is not increasing everywhere in this case?  
> **username_3**: An implicit assumption here is that since the utility is a function of only two goods (in this case perfect complements) no other goods exists in the economy and in this case as Michael showed they are normal goods, but I guess with a much more complicated utility function of many goods where two are perfect complements, there can be perfect complements which are NOT normal goods.  
> **username_1**: @StanShunpike Taking derivatives works nicely here, the point is that the definition of a good being normal is slightly different than demand having a positive derivative with respect to income. I think it is generally a good idea to separate the concept from specific mathematical tools that are convenient to work with.  
> **username_2**: @username_3: I suspect it does not have to be too complicated.  A utility function like $U(x,y,z)=\sqrt{\min\{a{x},b{y}\}} + cz^2$ might be enough  

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