Why must a Giffen good (which has a positively sloped demand curve) be an inferior good?
You have to consider the substitution and income effects. As you’ve said, a Giffen good has a positively sloped demand curve, what means that if price increases then demanded quantity will raise too. But, by definition, the substitution effect is always negative, in the sense that such a change in price will make demanded quantity fall. Then, in order to have a final increase in demand, the income effect must be positive and greater than substitution effect. Moreover, taking monetary income constant, an increase in price causes a fall in your real income (the lower quantity of goods you can purchase). Therefore, because of a reduction in income there is a greater demand. This implies the income effect is positive and the Giffen good is inferior, provided that a reduction in income increases its demand, ceteris paribus.
