**Homothetic preferences are those where the demand for goods in the ****consumer equilibrium**** increases or decreases in the same proportion as income varies.**

In other words, if the person’s disposable income doubles, for example, the quantity demanded for each product in the optimal basket will also double.

To explain it another way, suppose the consumer equilibrium is (x1, x2). If the income is multiplied by *t* , this factor being greater than 0, then the new point of greatest satisfaction will be ( *t* x1, *t* x2).

The above can be plotted as follows:

In the homothetic preferences, the **income effect** is simplified , posing an unrealistic but useful situation for the analysis. That is, it is unlikely that the income of the individual and the demand for the goods will increase in exactly the same proportion.

Another point to note is that this type of preference applies only to **normal goods** . These are those whose demand increases as income rises.

The opposite of a normal good is an **inferior good** , one whose consumption decreases when the person’s income is higher.

## Homothetic preferences in substitute goods

Homothetic preferences are also observed in **substitute goods** , that is, they do not always behave as in the graph shown above.

In the case of goods that are interchangeable, the consumer will allocate his entire budget to the one with the lowest price.

For example, having two goods, x1 and x2, if the price of the first (p1) is greater than that of the second (p2), only the cheapest product will be part of the demanded basket.

In summary, if R is the **budget constraint** :

If p1 <p2 then x1 = R / p1 and x2 = 0

If p2 <p1 then x2 = R / p2 and x1 = 0

Given the above, preferences are homothetic because if the consumer’s budget is doubled, for example, the quantity of the purchased good is also doubled.

In summary, when the budget constraint rises from R to 2R:

If p1 <p2 then x1 = 2R / p1 and x2 = 0

If p2 <p1 then x2 = 2R / p2 and x1 = 0

We can explain the above with an example. Suppose a person can choose between two substitute goods x1 and x2. His budget is US $ 10 and the price of the cheapest product, which is the second, is US $ 2. Therefore, he will buy 5 units of that merchandise.

If the budget constraint is US $ 20, 10 units of x2 will be purchased and the consumer will move to a better situation than the previous one, as we see in the following graph:

## Homothetic preferences of complementary goods

Homothetic preferences are also observed in **complementary goods** , for example, two products (A and B) that are necessary to prepare a plate of food.

The budget must be distributed to acquire both inputs, always in the same proportion, for example, 1 kilogram of A for each kilogram of B.

Let’s imagine that the person has US $ 10 and that A’s kilogram costs US $ 4, while B’s costs US $ 1. Then, 2 kilograms of each good will be purchased, as we observe below:

x = Quantity of A and B, which must be equal

4x + x = 10

5x = 10

x = 2

If the budget is US $ 20, the optimal basket will consist of 4 kg of each product.

4x + x = 20

5x = 20

x = 4