# Quasilinear preferences

Quasilinear preferences are those where, to achieve his greatest satisfaction, the individual buys only up to a certain quantity of one of the two goods (x1 and x2) that make up his basket. That is, in the equilibrium of the consumer, the demand for one of the goods has a limit.

In other words, when a person presents these types of preferences, the increase in his disposable income will not always raise the demand for x1 and x2. Thus, the income effect will be observed in only one of the goods.

Quasilinear preferences are different from homothetic preferences. These are those where the quantity demanded of x1 and x2 always increases or decreases in the same proportion as the budget constraint.

## Graphical representation of quasilinear preferences

The graphical representation of the quasilinear preferences must correspond to a map where all the indifference curves are equal, as in the following image:

In other words, the same indifference curve will shift vertically as income increases.

For example, if the utility function is as follows:

We calculate the Marginal Profit (MU) of each good:

Next, we find the marginal rate of substitution (RMS), which is interpreted as the number of units of the good x1 that the consumer is willing to give up to obtain an additional unit of x2. All this, while maintaining the same level of satisfaction for the buyer.

Given the above, if the amount obtained from x2 increases, the RMS also rises. That is, the more the individual has of good x2, the greater will be his interest in exchanging it for good x1.

This type of preferences applies, for example, when a person is going to finish equipping their kitchen. Let's imagine that with your budget you have to buy the refrigerator and cutlery. Of the first good, you only need one, but of the second you can buy many units.

## Quasilinear Preferences Example

Let's see an example of quasilinear preferences where we have the following utility function:

Now suppose the budget constraint is $ 100, with the price of x1 and x2 being $ 5 and $ 3, respectively.

To solve for consumer equilibrium we must first find the slope of the balance line.

The subtraction of the two equations (E1-E2) is equal to zero if they correspond to the same budget constraint.

Next, we set this slope equal to the RMS, which —as explained above — is equal to -x2.

Therefore, for any value of R the optimal amount of x2 holds. If the budget is US $ 100, we can find x1 by solving its value in the equation of the balance line:

Likewise, if the budget goes up to US $ 200, it only increases the consumption of x1 by 20 units.

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