Budget Constraints, the Budget Line

Looking at the individual consumer’s demand the first question is how much a consumer is able to buy of a commodity at all? Everyday experience tells us that the available, disposable income constrains the attainable quantity of any particular product, and the higher the price of a particular commodity, the smaller the quantity we are able to purchase. However, we cannot neglect the fact, that the total disposable income is usually distributed among several commodities, and cannot be spent on only one thing. Thus, the constraints that limit our purchase intentions are the disposable income, the price of the commodity we want to purchase, and the prices of other commodities that we also wish to buy of our disposable income. The available income and the prices of the commodities that we may purchase impose a limitation on the purchase options, defining a budget constraint.

The decision problem and budget constraint described above is illustrated by a simple example. Suppose that the consumer – a student, – spents 2000 HUF each day in the canteen to buy two things: scones and sandwiches. The given income is spent every day, on these two

8 Following the logic of the main text it may seem reasonable to start the chapter by assessing the consumer’s subjective system of preferences, but the introduction of the budget constraint is easier to explain in a beginner’s text. For this reason the chapter begins with the explanation of the budget constraints and its influencing factors, and the more abstract topic of consumer preferences and utility assessment are discussed afterwards.

goods and nothing else. The question is how much the student can buy of each product, spending all the money he has.

To answer the question, besides the income we have to know the prices of the two goods One scone costs 100 HUF in the canteen, and the price of one sandwich is 250 HUF.

Knowing the prices and the income the student has many purchase options, as the following examples show:

• If all the income is spent on scones, then 20 units of scones can be purchased (20 × 100 =2000 HUF)

• If all the income is spent on sandwiches, then 8 units of them can be purchased (8 × 250 =2000 HUF)

• If the income is divided between the two goods, then – for example – 2 sandwiches may be bought for 500 HUF and the remaining 1500 HUF is enough for buying 15 scones.

• Similarly an attainable option is the purchase of 4 sandwiches (4 × 250 = 1000 HUF) and 10 scones (10 × 100 = 1000 HUF).

• The student may also buy the bundle of 6 sandwiches (6 × 250 =1500 HUF) and 5 scones (5 × 100 = 500 HUF).

• Furthermore, assuming the units are divisible, then 7 sandwiches (7×250=1750 HUF) and 2.5 scones (2.5 × 100 = 250 HUF) is another possible choice (of course, in the present example this is only a theoretical option).

Really, to identify the purchase options, the available income (2000 HUF) was divided into two parts, a sum spent on scones (which is the number of scones multiplied by the unit price of scones, 100 HUF) and a sum spent on sandwiches (that is, the number of sandwiches multiplied by the unit price of sandwiches, 250 HUF). Therefore:

2000 HUF = 250 HUF×the number of sandwiches + 100 HUF×the number of scones.

This relationship is called budget constraint, or budget line.

The budget constraint (budget line) is the set of all bundles of two goods that the consumer is able to purchase at given prices and a given income, assuming all the income is spent on these goods (Farkasné Fekete – Molnár, 2007).

The formula for the budget constraint (budget line) is the following:

I = p_x × x + p_y × y, - where I is the consumer’s income,

- x is the amount of the first product (sandwiches), and y is the amount of the second product (scones) purchased,

- px is the unit price of product x (the sandwiches) and py is the unit price of product y (the scones).

In our example the income and prices are I = 2000, px=250, py=100, and the formula for the budget line is: 2000=250 × x +100 × y

The equation for the budget line holds for exactly those (x ; y) combinations of goods that the consumer is able to buy at the given income and given prices, spending all the income on these goods. Thus, feasible product-combinations are the following sandwich-scone pairs (0;20), (8;0), (2;15), (4;10), (6;5) and even (7;2,5). The consumer’s income and the unit prices of the two commodities exactly define the budget constraint.

The budget line is graphed in a coordinate system that has the purchaseable quantities of the two products on the horizontal and the vertical axes, respectively. In our example the x-axis shows the amount of sandwiches, and the y-axis shows the amount of scones to be

bought, and each pair (or bundle) of attainable products is represented by a point in the budget line.

Note that the white boxes in the budget line of Figure 3.1 represent the (sandwich, scone)-combinations described in our example. The other points in the budget line also fit the budget constraint, but really such combinations are attainable only if the products are divisible, that is, both product x and y are sold and bought in fractional units, too.

Figure 3.1: Plotting the budget line 2000=250 × x +100 × y

Source: Author’s own construction

Note that the points lying below the budget line represent product combinations that cost less than the consumer’s disposable income. Point A in the figure is like that (4 sandwiches and 5 scones cost only 1500 HUF), because with 4 sandwiches the consumer could buy as much as 10 scones without breaking the budget constraint. Choosing point A the consumer saves 500 HUF. On the other hand, the points lying above the budget line, – like point B in the figure – represent product combinations that are more expensive than the available income. Point B indicates a bundle of 20 scones and 2 sandwiches (that cost altogether 2500 HUF), although with 20 scones no sandwiches could be bought. For attaining the bundle represented by point B the consumer should borrow 500 HUF.

The transformation of the equation of the budget line I = px × x + py × y gives the equivalent formula: I - px × x = py × y .

Then expressing y we have the following form: I / p_y – ( p_x / p_y ) × x = y , therefore the budget line equation can be written in the following form: y = (I / p_y )- (p_x / p_y) × x .

For our example: y = (2000/100) – (250/100) × x, that is, y = 20 – 2.5 × x.

Comparing this latter formula to the graphical representation of the budget line we see, that the budget line intersects the vertical axis at y= 20 (that is, at I/p_y), and the slope of the line is -2.5, which is the negative of the ratio of the prices of the two commodities: - px/py.

How can the consumer’s budget constraint change? Can the currently less expensive point A, or the unattainable point B become points of the budget line under changed circumstances? How will the budget line respond to a change in the consumer’s income or the prices of the products?

Suppose, for example, that the consumer’s income is halved, that is, instead of 2000 HUF only 1000 HUF can be spent daily on scones and sandwiches, so I = 1000. Then the formula for the new budget line is 1000=250 × x + 100 × y, that is, y = 10 – 2.5 × x. This means that spending all the income on scones, 10 units can be bought instead of 20, and spending all the income on sandwiches only 4 units are attainable instead of the previous 8 ones. Similarly, for any formerly attainable product combinations the current income allows only half of the previous quantities (e.g. instead of the previous combination of 4 sandwiches

and10 scones now the attainable bundle contains 2 sandwiches and 5 scones). This means that the budget line shifts, downwards with decreasing incomes – and upwards with increasing incomes, proportionally to the change in the income. As the prices are unchanged, their ratio remains the same, so the slope of the budget line also remains the same, and the new budget line is parallel to the previous one (see, line (2) in Figure 3.2).

To understand the impact of price changes suppose, that with the previous income (I=2000 Ft) the unit price of sandwiches doubles in the example – while the price of scones does not change. When we spend all our income on scones, altogether 20 scones are attainable as before, but when spending all our money on sandwiches only 4 units can be bought instead of the previous 8 ones. The new budget line is described by the equation:

2000=500 × x + 100 × y, that is, y = 20 – 5 × x, and, as is shown by line (3) in Figure 3.2, the slope of the line has changed.

The impact of the change in scone prices – e.g. the doubling of scone prices (from 100 HUF to 200 HUF) – can be described similarly with unchanged income (2000 HUF) and unchanged unit price of sandwiches (250 HUF). Now the bundle of (8 sandwiches; 0 scones) is still attainable, but spending all our money on scones (buying 0 sandwiches), only 10 scones are attainable at the increased prices. The new budget line is: 2000=250 × x + 200 × y that is, y = 10 – 1.25 × x, as is shown by line (4) in Figure 3.2.

Figure 3.2: The impact of income or price changes on the budget line

Source: Author’s own construction

Finally suppose that with the same income the prices of the two products change in the same proportion, e.g. both of the prices double. When we spend all our money on sandwiches, we can buy only half of the previous amount, spending all our money on scones, again half of the previous amount is attainable, and choosing to buy any combination of scones and sandwiches, the amounts will be exactly half of the previously attainable bundle. The new budget line is: 2000=500 × x + 200 × y, or in the other form: y = 10 – 2.5 × x. This equation has been graphed earlier as line (2) in Figure 3.2, so the product bundles attainable for the consumer are exactly the same, as the ones that are attainable at unchanged product prices and halved income. Therefore it is no difference for the consumer to experience a general price rise of a certain proportion, or a fall in income in the same proportion, because the eventual

outcome is the decrease of the attainable product bundles - in other words, the consumer’s real income - by the same proportion. The rise in the consumer’s income, or the decrease in the product prices can be similarly interpreted as the increase in the consumer’s real income, because both will lead to the increase in the product bundles attainable for the consumer.

The real income is the amount of material goods and services that the consumer can purchase for his/her money income (nominal income) (Farkasné Fekete – Molnár, 2007.)

In the above paragraphs the consumer’s budget constraint (budget line) and its influencing factors were discussed for the case when the consumer’s income is spent on two goods. In the real world, however, the consumers divide their incomes among many products and services at the same time. The model of two goods is an idealistic, simpified situation, although this simple model can serve as a useful starting point to describe more general situations⁹.