Budget Constraint

Difference between Budget Set and Budget Constraint

There is a difference between budget set and budget constraint.

Let's contrast the two terms below so that it becomes clearer!The budget constraint represents all the possible combinations of two or more goods that a consumer can purchase, given current prices and their budget. Note that the budget constraint line will show all the combinations of goods you can buy given that you spend all the budget you allocate for these particular goods.It is easier to think about it in two goods scenario. Imagine you can buy only apples or bananas and have only $2. The price of an apple is 1$, and the cost of a banana is $2. If you only have $2, then all the possible combinations of goods that represent your budget constraint are as follows:

Table 1 - Budget constraint exampleThese two choices are illustrated in Figure 1 below.

Fig. 1 - Budget constraint example

Figure 1 shows a budget constraint line for a scenario depicted in Table 1. Because you cannot buy half an apple or half a banana, the only practically feasible points are A and B. At point A, you buy 2 apples and 0 bananas; at point B, you buy 1 banana and 0 apples.

In theory, all the points along the budget constraint represent the possible combinations of apples and bananas you could buy. One such point - point C, where you buy 1 apple and half a banana to spend your $2 is shown in Figure 1 above. However, this consumption combination is unlikely to be achieved in practice.

Because of the ratio of the two prices and the limited income, you are induced into choosing to trade off 2 apples for 1 banana. This trade-off is constant and results in a linear budget constraint with a constant slope.

• Properties of the budget constraint line:
  • The slope of the budget line reflects the trade-off between the two goods represented by the ratio of the prices of these two goods.
  • A budget constraint is linear with a slope equal to the negative ratio of the prices of the two goods.

Let's now look at how a budget set differs from the budget constraint. A budget set is more like a consumption opportunity set that a consumer faces, given their limited budget. Let's clarify by looking at Figure 2 below.

Fig. 2 - Budget set example

Figure 2 above shows a budget set represented by the green area within the budget constraint. All the points within that area, including the ones that lie on the budget constraint, are theoretically possible consumption bundles as they are the ones you can afford to purchase. This set of possible consumption bundles is what the budget set is.

Budget Constraint Line

What is the budget constraint line? The budget constraint line is a graphical representation of the budget constraint. Consumers who choose a consumption bundle that lies on their budget constraints utilize all of their income.Let's consider a hypothetical scenario in which a consumer must allocate all their income between the necessities of food and clothing. Let's denote the food price as \(P_1\) and the quantity chosen as \(Q_1\). Let the clothing price be \(P_2\), and the quantity of clothing be \(Q_2\). Consumer income is fixed and denoted by \(I\).What would the budget constraint line formula be?

Budget constraint formula

The formula for the budget constraint line would be:\(P_1 \times Q_1 + P_2 \times Q_2 = I\)Let's plot this equation to see the budget constraint line graph!

Fig. 3 - Budget constraint line

Figure 3 above shows a general budget constraint line graph that works for any two goods with any prices and any given income. The general slope of the budget constraint is equal to the ratio of the two product prices \(-\frac{P_1}{P_2}\).

The budget constraint line intersects the vertical axis at point \(\frac{I}{P_2}\); the horizontal axis intersection point is \(\frac{I}{P_1}\). Think about it: when the budget constraint intersects the vertical axis, you are spending all your income on good 2, and that is exactly the coordinate of that point! Conversely, when the budget constraint intersects the horizontal axis, you are spending all your income on good 1, and so the intersection point in units of that good is your income divided by the price of that good!

Budget Constraint Example

Let's go over an example of a budget constraint!Imagine Anna, who has a weekly income of $100. She can spend this income on either food or clothing. The price of food is $1 per unit, and the price of clothing is 2$ per unit.As the budget constraint line represents a few of the consumption combinations that would take up her entire income, we can construct the following table.

Table 2 - Consumption combinations example

Table 2 above shows the possible market baskets A, B, C, and D that Anna can choose to spend her income on. If she buys basket D, she spends all her income on food. Conversely, if she purchases basket A, she spends all her income on clothing and has nothing left to buy food, as clothing per unit costs $2. Market baskets B and C are possible intermediate consumption baskets between the two extremes.

Let's plot Anna's budget constraint!

Fig. 4 - Budget constraint example

Figure 4 above shows Anna's weekly budget constraint for food and clothing. Points A, B, C, and D represent the consumption bundles from Table 2.

What would the equation of Anna's budget constraint line be?

Let's denote the food price as \(P_1\) and the quantity that Anna chooses to buy weekly as \(Q_1\). Let the clothing price be \(P_2\), and the quantity of clothing that Anna chooses \(Q_2\). Anna's weekly income is fixed and denoted by \(I\).

The general formula for the budget constraint:\(P_1 \times Q_1 + P_2 \times Q_2 = I\)

Anna's budget constraint:

\(\$1 \times Q_1 + \$2 \times Q_2 = \$100\)

Simplifying:

\(Q_1 + 2 \times Q_2 = 100\)

What would the slope of Anna's budget constraint be?

We know the slope of the line is the ratio of the prices of the two goods:

\(Slope=-\frac{P_1}{P_2}=-\frac{1}{2}\).

We can also check the slope by re-arranging the equation in terms of \(Q_2\):

\(Q_1 + 2 \times Q_2 = 100\)

\(2 \times Q_2= 100 - Q_1\)

\(Q_2= \frac{1}{2} \times(100 - Q_1)\)

\(Q_2= 50-\frac{1}{2} Q_1\)

The coefficient in front of \(Q_1\) is equal to \(-\frac{1}{2}\) which is the same as the slope of the budget line!