# Budget Line and Indifference Curve Practice Problems

## Using Indifference Curve and Budget Line Graphs to Solve Economics Problems

In microeconomic theory, an indifference curve generally refers to a graph that illustrates different levels of utility, or satisfaction, of a consumer who has been presented with assorted combinations of goods. That is to say that at any point on the graphed curve, the consumer holds no preference for one combination of goods over another.

In the following practice problem, however, we will be looking at indifference curve data as it relates to the combination of hours that can be allotted to two workers in a hockey skate factory. The indifference curve created from that data will then plot the points at which the employer presumably should have no preference for one combination of scheduled hours over another because the same output is met. Let's take a glimpse at what that looks like.

## Practice Problem Indifference Curve Data

The following represents the production of two workers, Sammy and Chris, showing the number of completed hockey skates they can produce over the course of a regular 8-hour day:

From this indifference curve data, we have created 5 indifference curves, as shown in our indifference curve graph. Each line represents the combination of hours we can assign to each worker in order to get the same number of hockey skates assembled. The values of each line are as follows:

1. Blue - 90 Skates Assembled
2. Pink - 150 Skates Assembled
3. Yellow - 180 Skates Assembled
4. Cyan - 210 Skates Assembled
5. Purple - 240 Skates Assembled

This data provides the starting point for data-driven decision making regarding the most satisfactory or efficient schedule of hours for Sammy and Chris based on output. To accomplish this task, we will now add a budget line to the analysis to show how these indifference curves can be used to make the best decision.

## Introduction to Budget Lines

A consumer's budget line, like an indifference curve, is a graphical depiction of assorted combinations of two goods that the consumer can afford based upon their current prices and his or her income. In this practice problem, we will be graphing the employer's budget for employee's salaries against the indifference curves that depict various combinations of scheduled hours for those workers.

## Practice Problem 1 Budget Line Data

For this practice problem, assume that you have been told by the chief financial officer of the hockey skate factory that you have \$40 to spend on salaries and with that you are to assemble as many hockey skates as possible. Each of your employees, Sammy and Chris, both make a wage of \$10 an hour. You write the following information down:

Budget: \$40
Chris's Wage: \$10/hr
Sammy's Wage: \$10/hr

If we spent all of our money on Chris, we could hire him for 4 hours. If we spent all of our money on Sammy, we could hire him for 4 hours in Chris' place. In order to construct our budget curve, we jot down two points on our graph. The first (4,0) is the point at which we hire Chris and give him the total budget of \$40. The second point (0,4) is the point at which we hire Sammy and give him the total budget instead. We then connect those two points.

I've drawn my budget line in brown, as seen here on the Indifference Curve vs. Budget Line Graph. Before moving forward, you may want to keep that graph open in a different tab or print it out for future reference, as we will be examining it closer as we move along.

## Interpreting the Indifference Curves and Budget Line Graph

First, we must understand what the budget line is telling us. Any point on our budget line (brown) represents a point at which we will spend our entire budget. The budget line intersects with the point (2,2) along the pink indifference curve indicating that we can hire Chris for 2 hours and Sammy for 2 hours and spend the full \$40 budget, if we so choose. But the points that lie both below and above this budget line also have significance.

## Points Below the Budget Line

Any point below the budget line is considered feasible but inefficient because we can have that many hours worked, but we would not spend our entire budget. For instance, the point (3,0) where we hire Chris for 3 hours and Sammy for 0 is feasible but inefficient because here we would only spend \$30 on salaries when our budget is \$40.

## Points Above the Budget Line

Any point above the budget line, on the other hand, is considered infeasible because it would cause us to go over our budget. For instance, the point (0,5) where we hire Sammy for 5 hours is infeasible as it would cost us \$50 and we only have \$40 to spend.

## Finding the Optimal Points

Our optimal decision will lie on our highest possible indifference curve. Thus, we look at all the indifference curves and see which one gives us the most skates assembled.

If we look at our five curves with our budget line, the blue (90), pink (150), yellow (180), and cyan (210) curves all have portions that are on or below the budget curve meaning that they all have portions that are feasible. The purple (250) curve, on the other hand, is at no time feasible since it is always strictly above the budget line. Thus, we remove the purple curve from consideration.

Out of our four remaining curves, cyan is the highest and is the one that gives us the highest production value, so our scheduling answer must be on that curve. Note that many points on the cyan curve are above the budget line. Thus not any point on the green line is feasible. If we look closely, we see that any points between (1,3) and (2,2) are feasible as they intersect with our brown budget line. Thus according to these points, we have two options: we can hire each worker for 2 hours or we can hire Chris for 1 hour and Sammy for 3 hours. Both scheduling options result in the highest possible number of hockey skates based on our worker's production and wages and our total budget.

## Complicating the Data: Practice Problem 2 Budget Line Data

On page one, we solved our task by determining the optimal number of hours we could hire our two workers, Sammy and Chris, based on their individual production, their wage, and our budget from the company CFO.

Now the CFO has some new news for you. Sammy has gotten a raise. His wage is now increased to \$20 an hour, but your salary budget has stayed the same at \$40. What should you do now? First, you jot down the following information:

Budget: \$40
Chris's Wage: \$10/hr
Sammy's New Wage: \$20/hr

Now, if you give the entire budget to Sammy you can only hire him for 2 hours, while you can still hire Chris for four hours using the entire budget. Thus, you now mark the points (4,0) and (0,2) on your indifference curve graph and draw a line between them.

I've drawn a brown line between them, which you can see on Indifference Curve vs. Budget Line Graph 2. Once again, you may want to keep that graph open in a different tab or print it out for reference, as we will be examining it closer as we move along.

## Interpreting the New Indifference Curves and Budget Line Graph

Now the area beneath our budget curve has shrunk. Notice the shape of the triangle has also changed. It's much flatter, since the attributes for Chris (X-axis) haven't changed any, while Sammy's time (Y-axis) has become much more expensive.

As we can see. now the purple, cyan, and yellow curves are all above the budget line indicating that they are all unfeasible. Only the blue (90 skates) and pink (150 skates) have portions that are not above the budget line. The blue curve, however, is completely below our budget line, meaning all the points represented by that line are feasible but inefficient. So we will disregard this indifference curve as well. Our only options left are along the pink indifference curve. In fact, only points on the pink line between (0,2) and (2,1) are feasible, thus we can either hire Chris for 0 hours and Sammy for 2 hours or we can hire Chris for 2 hours and Sammy for 1 hour, or some combination of factions of hours that fall along those two points on the pink indifference curve.

## Complicating the Data: Practice Problem 3 Budget Line Data

Now for another change to our practice problem. Since Sammy has become relatively more expensive to hire, the CFO has decided to increase your budget from \$40 to \$50. How does this impact your decision? Let's write down what we know:

New Budget: \$50
Chris's Wage: \$10/hr
Sammy's Wage: \$20/hr

We see that if you give the entire budget to Sammy you can only hire him for 2.5 hours, while you can hire Chris for five hours using the entire budget if you wish. Thus, you can now mark down the points (5,0) and (0,2.5) and draw a line between them. What do you see?

If drawn correctly, you'll note that the new budget line has moved upward. It has also moved parallel to the original budget line, a phenomenon that occurs whenever we increase our budget. A decrease in budget, on the other hand, would be represented by a parallel shift downward in the budget line.

We see that the yellow (150) indifference curve is our highest feasible curve. To make the must select a point on that curve on the line between (1,2), where we hire Chris for 1 hour and Sammy for 2, and (3,1) where we hire Chris for 3 hours and Sammy for 1.

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