Question

Show that if there are only two goods (Xand \(F\) ) to choose from, both cannot be inferior goods. If Xis inferior, how do changes in income affect the demand for \(\mathrm{F}\) ?

Short answer

Explain the effect on the demand for good F when good X is an inferior good. Answer: No, both goods X and F cannot be inferior goods in a two-good economy. When good X is an inferior good, good F must be a normal good. In this scenario, as the consumer's income increases, demand for good X decreases, and they will substitute it with good F. As a result, the demand for good F increases when the income increases.

Step-by-step solution

Define an inferior good

An inferior good is a good for which the demand decreases as the consumer's income increases. This is due to the fact that when income increases, consumers tend to replace inferior goods with more desired (normal) goods. Formally, an inferior good can be defined as one where the income elasticity of demand is negative.

Introduce the income-consumption curve

The income-consumption curve represents the relationship between a consumer's income and their optimal consumption bundle given their budget constraint and preferences. In a two-good economy consisting of goods X and F, as the consumer's income increases, they will adjust their consumption of X and F to maximize their utility.

Show that not both goods can be inferior

Suppose, for the sake of argument, that both goods X and F are inferior goods. In this case, as the consumer's income increases, they will buy less of both goods. However, this creates a contradiction, since the consumer's income must be spent on the two goods. As their income increases, they cannot decrease their consumption of both goods indefinitely because they have to spend their entire budget, and there are only two goods available. Therefore, we conclude that both goods X and F cannot be inferior goods.

Understand the scenario where good X is inferior

Now we consider the case where good X is an inferior good. In this scenario, as the consumer's income increases, they will decrease their consumption of good X and substitute it with another good, in this case, good F.

Determine the effect of changes in income on the demand for good F

As established in step 3, if good X is inferior, good F must be a normal good. Since F is a normal good, the income elasticity of demand for good F will be positive. Therefore, when income increases, consumers will substitute away from the inferior good X and increase their consumption of the normal good F. As a result, changes in income will have a positive effect on the demand for good F. In conclusion, it has been shown that in a two-good economy with goods X and F, both goods cannot be inferior. If good X is inferior, good F must be a normal good, and demand for good F will increase with an increase in income.