6.b The Budget Constraint
In the previous section we looked at indifference curves, which showed what the consumer would like to
have with respect to two goods X and Y. In this section we will look at what the consumer can actually
afford. A budget constraint is an equation, graph, or table that shows the bundles available to the
consumer. If we again assume there are only two goods available X and Y, a budget constraint has the
equation
I = Px·X + Py·Y.
In this equation I is the total money income, P x is the price of good X, and Py is the price of good Y. Since
there are only two goods, the left side of this equation represents total expenditure. Let's go over an
example of a budget constraint. Let total income I = $100, price of good X P x = $10 and price of good Y
Py = $5. Table 6.b.1 shows the bundles that can be acquired under this budget constraint.
X
0
1
2
Y
20
18
16
10
0
Table 6.b.1 shows the
bundles that can be acquired
under the budget constraint
100 = 10X + 5Y.
If the consumer buys none of good X, then the budget allows for 20 units of good Y to be bought. If the
consumer buys one unit of good X, then what is left in the budget is enough to buy 18 units of good Y. As
the consumer buys more of X, less of Y can be bought all the way until the consumer buys 10 units of good
X when nothing is left in the budget to by good Y. Plotting these bundles on a graph gives us the budget
line. Diagram 6.b.1 shows the budget line corresponding to table 6.b.1.
Diagram 6.b.1
Y
20
Vertical axis intercept = I/Py
Budget line 100 = 10X + 5Y
Slope of Budget line = Px / Py
Y
Diagram 6.b.1 shows the
budget line for 100=10X+5Y.
If all of the income was spent
on good Y, then 20 unitsof Y
would be bought. This gives
the vertical axis intercept.
Similarly, 10 units of X gives
our horizontal axis intercept.
Horizontal axis intercept = I/Px
X
X
10
We find the slope of a line by dividing the change in Y by the change in X, Y/X. Since the vertical axis
intercept is I/Py and the horizontal intercept is I/Px we can substitute these in for Y and X respectively.
I / Py
Slope of budget line =
= P x / Py
I / Px
The slope of the budget line is the price of good X divided by the price of good Y. This is also referred to
as the relative price of good X. This is because it shows the number of good Y the consumer must give up
in order to get one more unit of X.
Shifts in Budget lines
Budget lines can shift in three ways. The first is a parallel shift. This is where the budget line
shift outward or inward in a parallel fashion. This happens when the Income I changes or when the prices
of good X and Y change in a similar manner. If the income of a consumer is cut in half, then the consumer
can buy half as many of goods X and Y as before. Similarly, if income stays the same but the price of
goods X and Y both double, then again the consumer can only buy half as many of goods X and Y.
Diagram 6.b.2 shows these situations.
Y
Case A: I = $100
Px =$10
Py =$5
Diagram 6.b.2
20
Budget line #1
Cases A & D
Case B: I = $50
Px =$10
Py =$5
10
Budget line #2
Cases B & C
X
5
Case C: I = $100
Px =$20
Py =$10
Case D: I = $50
Px =$5
Py =$2.5
10
Diagram 6.b.2. Shows four different cases which correspond to two different budget lines. Case A has the
initial values for income and prices and corresponds to budget line #1. Then, income is cut in half, as
shown in case B. This corresponds to a parallel shift in the budget line with case B represented by budget
line #2. If income was not cut in half, but instead both the price of good X and good Y doubled as in case
C, then we would have a similar shift in the budget line. Case D shows if income and prices were cut in
half. In this situation, nothing real has changed; the consumer still has the same purchasing power and the
corresponding budget line is budget line #1.
The next type of shift in budget lines is called a pivot. The budget line pivots when the relative
price of good X changes. In other words, a pivot is caused by a change one of the prices, either P x or Py. A
per unit subsidy to the consumer for one of the goods would be an example of a pivot of a budget line.
Diagram 6.b.3 shows a pivot in the budget line due to a $5 per unit subsidy for good X.
Y
20
Initial condition: I = $100
Px = $10
Py = $5
Diagram 6.b.3
With $5/unit X subsidy:
I = $100
Px = $5
Py = $5
X
10
20
Diagram 6.b.3 a five dollar per unit subsidy for good X causes the budget line to pivot.
The third type of budget line shift is called a kinked budget line. This is where the budget line
changes slope at some point. An example of a kinked budget line is the food stamp program. Lets say tat
good X is food and the consumer receives a food stamp worth 5 units of food. The first 5 units of food are
then free to the consumer, however the consumer must pay for every unit after 5. This situation is shown in
diagram 6.b.4.
Y
Diagram 6.b.4
Original
budget line
20
Budget line
with food
stamp
Diagram 6.b.4 shows a kinked budget
line. A food stamp worth 5 units allows
the consumer to get up to 5 units of food
and still acquire 20 units of Y. If the
consumer wished to get a sixth unit of
food, then the consumer would have to
sacrifice a certain amount of good Y,
representing the downward sloping part
of the budget line.
X food
5
10
15