To progress toward our goal of modeling optimal choice for consumers, we will need to discuss the constraints consumers face when making decisions. Here we will focus on monetary constraints. Since decision makers face conditions of scarcity, and no individual has an infinite amount of money to spend on goods and services, these constraints play a key role in understanding consumer choice.
The budget constraint is the primary constraint we use in the model of consumer decision making. Simply, the budget constraint says that a consumer cannot spend more dollars than she has on any bundle of goods. In this section, let’s focus on bundles of two goods, which, as we have seen, take the form of a pair \((x, y)\text{,}\) where \(x\) is the number of units of good \(x\) and \(y\) is the number of units of good \(y\text{.}\) If a consumer aims to purchase donuts (\(x\)) which cost $1 each, and cups of coffee (\(y\)) which cost $2 each, and our consumer has $4 to spend, we can identify which bundles are affordable to her. For example, the bundle \((2, 1)\) is affordable: it costs her $2 on donuts + $2 on coffee = $4 total. The bundles \((1, 1)\) and even \((0, 0)\) are affordable as well. However, the bundle \((1, 2)\) is not affordable: it would cost the consumer $5, which is above her budget.
We can model this a bit more formally by thinking of the components of the budget constraint. Her total spending on \(x\) and \(y\) must stay less than or equal to her income, which is $4. How much does she spend on donuts? Well, this depends on how many she buys. But since we know each donut costs exactly $1, her spending on donuts can be expressed as \(1*x\text{.}\) Similarly, her spending on coffee can be expressed as \(2y\text{.}\) Putting the pieces together, we can express the budget constraint in an equation:
\begin{equation*}
x + 2y \leq 4
\end{equation*}
The budget constraint allows us to plug in any \((x, y)\) bundle and determine whether the condition of affordability is met. Moreover, we can graph the budget constraint to visualize which bundles are affordable and which are not, by graphing the line \(x + 2y = 4\text{.}\) The collection of affordable bundles (the shaded area on the graph below) is the consumer’s budget set. Anything that lies above the line and outside of the shaded area is unaffordable!
Figure2.4.1.The shaded area is the budget set for the budget constraint \(x + 2y = 4\text{.}\)
Checkpoint2.4.2.
Consider a consumer who has a $20 budget. The price of a pizza slice is $5 and the price of bubble tea is $4. Which of the following bundles, where \(x\) is the quantity of pizza slices and \(y\) is the quantity of bubble tea, can this consumer afford with her budget?
\((x,y)\)
a
(2,2)
b
(3,1)
c
(3,2)
Answer.
To find this, all you need to do is to find out the cost of the bundle. For each good, find price multiplied by quantity, then add them up. So, for bundle a this is \(5*2+4*2=18\text{,}\) since this is less than her budget of $20, it is affordable. For b, \(5*3+4*1=19\text{,}\) also affordable. But with c, \(5*3+4*2=23>20\text{,}\) so this is an unaffordable bundle.
The budget constraint contains three demand parameters we have seen before - exogenous variables which will influence consumer decision making: \(P_x\text{,}\) the price of good \(x\); \(P_y\text{,}\) the price of good \(y\); and \(I\text{,}\) the consumer’s income. What happens to the budget constraint if one of these parameters changes? If consumer income were to change from 4 to 6, the new budget constraint becomes \(x + 2y = 6\text{!}\) This represents an outward shift in the budget constraint, expanding the collection of bundles the consumer can afford. [Similarly, a reduction in income would shift the budget constraint inward toward the origin.]
Figure2.4.3.The budget set expands when income increases from 4 to 6. The budget constraint changes from \(x + 2y = 4\) to \(x + 2y = 6\text{.}\)
What if the price of one of the goods changes? If good \(x\) becomes more expensive and \(P_x\) increases from 1 to 2, the new budget constraint becomes \(2x + 2y = 4\text{!}\) This can be seen by an inward tilt in the budget constraint, reducing the collection of affordable bundles. The distinction between a shift and a tilt in budget constraints is important. To see why, consider the two intercepts. The y-intercept on the budget set represents the number of units of good \(y\) the consumer can purchase if she spends all of her income on \(y\); similarly, the x-intercept represents the number of units of good \(x\) the consumer can purchase if she spends all of her income on \(x\text{.}\) You can think of each of these as the maximum number of each individual unit a consumer can purchase. Therefore, if income changes, the maximum number of units of each good changes. When income increases from 4 to 6, the consumer can spend those additional $2 on more \(x\text{,}\) more \(y\text{,}\) or some combination of \(x\) and \(y\text{.}\)
Figure2.4.4.The budget set contracts when \(P_x\) increases from 1 to 2. The budget constraint changes from \(x + 2y = 4\) to \(2x + 2y = 4\text{.}\)
However, a price change impacts budget differently than an income change. Primarily, this is because its impact is asymmetric. When the price of good \(x\) increases, the maximum affordable quantity of \(x\) declines, but the maximum affordable quantity of \(y\) does not! So while the x-intercept decreases, while the y-intercept stays the same. A useful interpretation of the price change is as a change in the relative price of \(x\) and \(y\) 1 . When \(P_x = 1\) and \(P_y = 2\text{,}\) one unit of \(y\) is “worth” two units of \(x\text{:}\) if you wanted to sell a unit of \(y\text{,}\) you would receive two dollars, which could then be spent on two units of \(x\text{.}\) But when \(P_x\) increases to 2, now one unit of \(y\) is worth exactly one unit of \(x\text{.}\) This makes \(y\) relatively less valuable and \(x\) relatively more valuable. Relative prices give us the rate at which the market asks consumers to trade off units of one good for units of another; knowledge of relative prices can allow us to more easily analyze the tradeoffs consumers face.
We can express the budget constraint when the demand parameters all have general values as:
\begin{equation*}
P_x x + P_y y \leq I
\end{equation*}
As we have seen, \(P_x x\) is the consumer’s spending on good \(x\text{,}\)\(P_y y\) is the consumer’s spending on good \(y\text{,}\) and \(I\) is consumer income. Even at this level of generality, we can find both of the budget constraint intercepts. The \(x\) intercept - which, remember, gives the maximum number of units of \(x\) the consumer can afford - is expressed as \(\frac{I}{P_x}\text{.}\) For example, if the consumer has $10 in income, and a unit of \(x\) costs $2, then if she spends all of her income on \(x\text{,}\) she can afford \(\frac{10}{2} = 5\) units of \(x\text{.}\) The \(y\) intercept is expressed similarly as \(\frac{I}{P_y}\text{.}\)
Figure2.4.5.Budget constraint with general parameters: \(P_x x + P_y y = I\text{.}\)
Checkpoint2.4.6.
Consider a consumer with Income of $300 (\(I=300\)) that consumes only beer (\(x\)) and donuts (\(y\)). Each beer costs $2 (\(P_{x}=2\)) and each donut costs $1 (\(P_{y}=1\)).
If this consumer only buys beer, what is the maximum he could afford? What if he only buys donuts?
Hint.
If someone spends all of their income on one good, that means all other quantities are zero, right? Think about this and the budget constraint function above.
Answer.
So, the simplest way to see this is to just think it through in terms of the logic. If he has $300 and each beer is $2, if we divide 300 by 2, that gives us the affordable highest quantity of beer, \(x=150\text{.}\) Doing the same thing for donuts, 300 divided by 1 gives \(y=300\text{.}\)
We can also think of these in terms of the budget constraint function \(P_x x + P_y y = I\text{.}\) We want the value of x when y is zero, given these prices and income. \(2 x + 1 (0) = 300\) Solving for x by dividing by 2, we get the same answer as above.
In terms of the graph, what are these values? They are the x and y intercepts, where the budget constraint graph hits the relevant axis. This will always be \(\frac{I}{P_{x}}\) for the x axis and \(\frac{I}{P_{y}}\) for the y axis.
We can find the slope of the budget constraint most easily by solving the budget constraint \(P_x x + P_y y = I\) for \(y\text{:}\)
\begin{equation*}
P_x x + P_y y = I
\end{equation*}
\begin{equation*}
P_y y = - P_x x + I
\end{equation*}
\begin{equation*}
y = - \frac{P_x}{P_y} x + \frac{I}{P_y}
\end{equation*}
Once the budget constraint is in \(y = mx + b\) form, we can see that the slope of the budget constraint is \(- \frac{P_x}{P_y}\text{.}\) We can interpret this ratio of the prices of the two goods as the relative prices of the goods. In the example above, recall that \(P_x = 1\) and \(P_y = 2\text{,}\) which gives the budget constraint a slope of \(-\frac{1}{2}\text{.}\) The \(\frac{1}{2}\) tell us that at these prices, 1 unit of \(x\) can be exchanged for exactly \(\frac{1}{2}\) unit of \(y\text{,}\) while the negative sign tells us that the 1 unit of \(x\) has to be given up in order to get the \(\frac{1}{2}\) unit of \(y\text{.}\) Therefore, the slope of the budget constraint gives us the relative prices of the goods \(x\) and \(y\text{:}\) the number of units of \(y\) a consumer could receive in the market for giving up a unit of \(x\text{.}\) 2
Checkpoint2.4.7.
Take the situation in Checkpoint 2.4.6 above and use that as a starting point for this question. As a reminder, \(I=300\text{,}\)\(P_{x}=2\text{,}\) and \(P_{y}=1\text{.}\)
First, graph the budget constraint. Then, show what happens when \(P_{x}\) goes up to $3.
Answer.
So our first budget constraint, with the original prices, has the x and y intercepts of 150 and 300, as shown in Figure 2.4.8 When the price of beer rises, the intercept changes on the x axis to 100, rotating the graph inward.
This represents a change in not just beer possibilities, but so long as the consumer is not spending all of their money on donuts, the affordable bundles containing both goods are reduced, even though the consumer still has the same amount of (nominal) income. Because the price change reduced the set of affordable bundles, we say there’s been a change in real income for this consumer.
Figure2.4.8.Budget constraint when \(P_{x}\) increases
