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?

  • 1
    Do you know how to derive the demand function? Do you know the definition of normal goods? – Michael Greinecker May 11 '15 at 21:08
  • 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 May 11 '15 at 21:12
  • 1
    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 May 11 '15 at 21:17
  • Ah, they discuss it here. youtu.be/kjI840VDW5I perhaps i can figure it out now. – Stan Shunpike May 11 '15 at 21:20
up vote 9 down vote accepted

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.

  • 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 May 11 '15 at 21:41
  • 1
    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 May 11 '15 at 21:55
  • @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 May 11 '15 at 22:30
  • @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 Dec 18 '17 at 14:26

Your Answer

 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.