# $\{\text{Giffen goods} \} \subsetneq \{\text{Inferior goods}\}$

I was asking myself why is a Giffen good an inferior good and I read the following post : Difference between Giffen and inferior goods. Why aren't all inferior goods Giffen goods?.

The 2nd comment of @Matthew Gunn presents things very well. A good is called inferior if you purchase less as your income increases: $$\frac{dq(p,w)}{dw}<0$$ where $$p$$ is its price and $$w$$ the income of the consumer.

A good is called a Giffen good if you purchase more as its price increases : $$\frac{dq(p,w)}{dp}>0$$.

I cannot intuitively and mathematically see why a Giffen good must be an inferior good : $$\frac{dq(p,w)}{dp}>0$$ does not imply that $$\frac{dq(p,w)}{dw}<0$$. Also, if we take an example of a very poor community that cannot afford meat so that potatoes become a Giffen good. Then if the incomes of a consumer increase, nothing guarantees that he will purchase less potatoes, right ? Maybe the increase in his income is so small that he will not be able to afford meat and keep buying more potatoes, so they are Giffen goods but not inferior.

Can someone prove that $$\{\text{Giffen goods} \} \subsetneq \{\text{Inferior goods}\}$$ ?

(I am new to microeconomics sorry if I am dealing with this the wrong way).

I will only answer the intuition part, since mathematically there already is an answer, but at the same time your question is not complete duplicate since the intuition part is important as well.

Intuition:

This is because Giffen goods by definition are goods for which there is positive relationship between price of the good and demand, all else equal.

For any normal good when income increases people will consume more of the good but that does not mean that there is positive relationship between price and quantity. People might consume more even when price is increasing (depending on relative income and price effects) but then you are changing 2 parameters income and price at the same time. Giffen effect has to come just from the price.

When it comes to Giffen good what generates the effect is the fact that income effect dominates the price effect, but this all happens while income is held constant. How is this possible? Because the income effect is generated by the price change itself. If you face constraint:

$$px+qy=m$$

Then increasing $$p$$ will create income effect even when $$m$$ is constant as it reduces the amount that constant income is able to buy. There is however absolutely no change in $$m$$ involved.

Now if $$m$$ is hold fixed, increase in $$p$$ will generate positive income effect (on quantity demanded of $$x$$) only if $$x$$ is inferior good. Otherwise, if $$x$$ is normal good $$p$$ increase will create an additional negative income effect that further reduces the consumption.

Again, when it comes to Giffen good only price has to change $$m$$ must remain constant. If you start changing $$m$$ you will get income effects but they will provide you with no indication whether good is Giffen or not since by definition Giffen goods are gods where the relationship between quantity demanded and price is positive, not relationships where increase in income generates extra demand while there is confounding change in price.

The rigorous proof has already been linked in 1muflon1's answer, but maybe this short version is more intutitive:

Consider the total effect (TE) on a consumer's demand for a good of an increase in the price of the good. This total effect can be additively split into the income effect (IE) and the substitution effect (SE), so TE = IE + SE. Rearranging, you get IE = TE - SE.

Now, by definition, for a Giffen good, TE > 0. Also, for any good, SE < 0. Thus, for a Giffen good, IE > 0. But this is just the definition of the good being an inferior good.