How Individuals Make Choices Based on Their Budget Constraints

Understanding Budget Constraints

Budget constraints are easy to understand if you apply a little math. The appendix The Use of Mathematics in Principles of Economics explains all the math you are likely to need in this book. So if math is not your strength, you might want to take a look at the appendix.

Step 1: The equation for any budget constraint is:

$\text { Budget }=\mathrm{P}_{1} \times \mathrm{Q}_{1}+\mathrm{P}_{2} \times \mathrm{Q}_{2}$

where P and Q are the price and quantity of items purchased and Budget is the amount of income one has to spend.

Step 2. Apply the budget constraint equation to the scenario. In Alphonso's case, this works out to be:

$\begin{equation} \begin{aligned} \text { Budget } &=\mathrm{P}_{1} \times \mathrm{Q}_{1}+\mathrm{P}_{2} \times \mathrm{Q}_{2} \\ \$ 10 \text { budget } &=\$ 2 \text { per burger } \times \text { quantity of burgers }+\$ 0.50 \text { per bus ticket } \times \text { quantity of bus tickets } \\ \$ 10 &=\$ 2 \times \mathrm{Q}_{\text {burgers }}+\$ 0.50 \times \mathrm{Q}_{\text {bus tickets }} \end{aligned} \end{equation}$

Step 3. Using a little algebra, we can turn this into the familiar equation of a line:

$\mathrm{y}=\mathrm{b}+\mathrm{mx}$

$\$ 10=\$ 2 \times \mathrm{Q}_{\text {burgers }}+\$ 0.50 \times \mathrm{Q}_{\text {bus tickets }}$

Step 4. Simplify the equation. Begin by multiplying both sides of the equation by 2:

$\begin{aligned} 2 \times 10 &=2 \times 2 \times \mathrm{Q}_{\text {burgers }}+2 \times 0.5 \times \mathrm{Q}_{\text {bus tickets }} \\ 20 &=4 \times \mathrm{Q}_{\text {burgers }}+1 \times \mathrm{Q}_{\text {bus tickets }} \end{aligned}$

Step 5. Subtract one bus ticket from both sides:

$20-\mathrm{Q}_{\text {bus tickets }}=4 \times \mathrm{Q}_{\text {burgers }}$

Divide each side by 4 to yield the answer:

$\begin{aligned} 5-0.25 \times \mathrm{Q}_{\text {bus tickets }} &=Q_{\text {burgers }}\\                                    &\text { or }\\ \mathrm{Q}_{\text {burgers }} &=5-0.25 \times \mathrm{Q}_{\text {bus tickets }} \end{aligned}$

Step 6. Notice that this equation fits the budget constraint in Figure 2.2. The vertical intercept is 5 and the slope is –0.25, just as the equation says. If you plug 20 bus tickets into the equation, you get 0 burgers. If you plug other numbers of bus tickets into the equation, you get the results shown in Table 2.1, which are the points on Alphonso's budget constraint.

Step 7. Notice that the slope of a budget constraint always shows the opportunity cost of the good which is on the horizontal axis. For Alphonso, the slope is −0.25, indicating that for every four bus tickets he buys, Alphonso must give up 1 burger.

There are two important observations here. First, the algebraic sign of the slope is negative, which means that the only way to get more of one good is to give up some of the other. Second, the slope is defined as the price of bus tickets (whatever is on the horizontal axis in the graph) divided by the price of burgers (whatever is on the vertical axis), in this case $0.50/$2 = 0.25. So if you want to determine the opportunity cost quickly, just divide the two prices.