Introduction
A consumer may compare bundles containing two goods—for example, units of tea and units of fruit—and judge some bundles equally desirable. An indifference curve collects the bundles that receive the same preference ranking as one chosen reference bundle. The idea is useful because it gives a visual way to represent preference comparisons without assuming that preferences have a numerical score.
When a question provides a table of stated rankings, the first task is to identify exactly which bundles are tied with the reference bundle. Those bundles belong to its indifference curve. A bundle ranked above or below the reference does not belong to that curve. And if the table does not state how a bundle compares with the reference, its status cannot be inferred from that table alone.
From a Preference Statement to a Set
Suppose the reference bundle is \(\bar{x}=(\bar{x}_1,\bar{x}_2)\). To decide whether a listed bundle \(x\) belongs to \(I(\bar{x})\), compare it with the reference in both directions. If the consumer regards each as at least as good as the other, they are indifferent, and \(x\) belongs to the set. If the consumer strictly prefers one to the other, they are not indifferent.
This definition distinguishes three kinds of information in a ranking table. A bundle may be stated to be equally preferred to the reference, strictly preferred to it, or strictly less preferred than it. Only the first category belongs to the reference bundle’s indifference curve. A table may also leave a bundle unclassified; that means its relation to the reference has not been specified, not that it is indifferent.
A preference ranking is ordinal: it tells us which bundles are preferred, tied, or less preferred. It does not, by itself, assign a measurable amount of satisfaction. In particular, a label such as “rank 2” is not a quantity that can be added or averaged. For plotting, the relevant information is simply which listed bundles share the reference bundle’s ranking.
Plotting Listed Equal-Ranked Bundles
For two goods, put the quantity of good 1 on the horizontal axis and the quantity of good 2 on the vertical axis. A bundle \(x=(x_1,x_2)\) is plotted at horizontal coordinate \(x_1\) and vertical coordinate \(x_2\). First plot the reference bundle. Then plot every other bundle explicitly stated to be indifferent to it. Do not include bundles ranked strictly above or below it.
A table gives comparisons for the bundles it lists. It does not automatically supply preference information about every point between listed bundles, or about bundles that are not listed. Accordingly, when the task is to plot from a table, plot the specified points. A line drawn between points may be a visual aid, but the table alone does not establish that every point on that line is indifferent to the reference.
Worked Example: Selecting the Bundles Tied With a Reference
A consumer compares bundles of notebooks and pens. The reference bundle is \(\bar{x}=(4,5)\). The table reports each listed bundle’s preference relation to the reference. Identify the bundles on \(I(\bar{x})\) and give their plotting coordinates. The first coordinate is notebooks; the second is pens.
| Bundle | Relation to \((4,5)\) |
|---|---|
| \((4,5)\) | Reference bundle |
| \((2,8)\) | Indifferent |
| \((6,3)\) | Indifferent |
| \((5,5)\) | Indifferent |
| \((3,7)\) | Strictly preferred |
| \((1,3)\) | Strictly less preferred |
Method. Select the reference bundle and every row marked indifferent. The reference belongs to its own indifference curve because \(\bar{x}\sim\bar{x}\). The other listed bundles tied with it are \((2,8)\), \((6,3)\), and \((5,5)\).
Plot. On axes with notebooks horizontal and pens vertical, plot \((4,5)\), \((2,8)\), \((6,3)\), and \((5,5)\). Each ordered pair gives the horizontal coordinate first and the vertical coordinate second.
The bundle \((3,7)\) is excluded because it is strictly preferred to the reference, while \((1,3)\) is excluded because it is strictly less preferred. The common mistake is to include every bundle that seems “similar” to the reference, or to include a strictly preferred bundle because it has more of one good. The correction is to use the stated preference relation, not a visual guess based on quantities.
Check. The set of plotted bundles consists of the reference plus the three rows explicitly marked indifferent; neither strict-ranking row is included.
Why the Two-Way Comparison Matters
The definition of indifference follows directly from the meaning of weak preference. A statement \(x\succsim\bar{x}\) alone says that \(x\) is at least as good as the reference; it does not say that the two bundles are equally preferred. To establish indifference, the reverse comparison must also hold.
This equivalence gives a practical test. When a table states both directions, the bundle is tied with the reference. When it states that the reference is strictly preferred, or that the candidate is strictly preferred, the two bundles are not indifferent. The test also explains why a single comparison should not be mistaken for an equality of rankings.
Worked Example: Reading Ranking Labels Carefully
A consumer’s reference bundle of rice and vegetables is \(\bar{x}=(3,6)\). A survey table uses the following labels: “same ranking as reference,” “preferred to reference,” and “less preferred than reference.” Determine which listed bundles belong to the reference indifference curve, then state the coordinates to plot.
| Bundle | Reported label |
|---|---|
| \((3,6)\) | Reference |
| \((5,4)\) | Same ranking as reference |
| \((2,9)\) | Same ranking as reference |
| \((4,7)\) | Preferred to reference |
| \((1,5)\) | Less preferred than reference |
Method. Translate “same ranking as reference” into indifference. By definition, those bundles are weakly preferred to the reference and the reference is weakly preferred to them. Include the reference itself as well.
Therefore, plot \((3,6)\), \((5,4)\), and \((2,9)\). Exclude \((4,7)\), which is preferred to the reference, and exclude \((1,5)\), which is less preferred. The plotting order is unchanged: the rice quantity is the first coordinate, and the vegetable quantity is the second.
A tempting error is to interpret “preferred to reference” as “on the same curve but higher.” That is incorrect: a strict preference means the bundles do not have the same ranking. The label does not identify how far apart the bundles are in any numerical sense; it only rules out indifference.
Check. Exactly two non-reference bundles are labeled as having the same ranking, so the plotted set contains three points including the reference.
Using Tables Without Adding Unsupported Information
A table may report only some comparisons. If a bundle is absent, or its relation is described as unknown, it should not be added to the indifference curve on the basis of that table. This is a limit on what can be concluded from the supplied information, not a claim that the bundle cannot in fact be indifferent. Other preference information could settle the comparison, but it must be provided or derived from stated assumptions.
The same care applies when a question gives a ranking number. If the table explicitly defines equal numbers as ties, bundles assigned the reference’s number are equally ranked under that table. If it merely gives ordered positions with no statement that a tie means indifference, do not assume that two bundles share a ranking. Read the table’s instructions before interpreting its labels.
When constructing a plot, label the axes and preserve the order of the coordinates. If the first coordinate is the quantity of good 1, the point \((2,7)\) is two units of good 1 and seven units of good 2—not the reverse. This simple check prevents a correct preference classification from being displayed at the wrong location.
Worked Example: Separating Ties From Unreported Comparisons
A hypothetical household compares bundles of heating fuel and insulation panels. Its reference is \(\bar{x}=(6,2)\), with fuel as the first coordinate and panels as the second. The table includes direct statements about some bundles and says that another bundle has “no comparison reported.” Identify the points that can be plotted on the reference indifference curve from this information.
| Bundle | Information relative to \((6,2)\) |
|---|---|
| \((6,2)\) | Reference |
| \((4,5)\) | At least as good as the reference, and the reference is at least as good as this bundle |
| \((8,1)\) | The reference is strictly preferred |
| \((5,3)\) | Candidate is strictly preferred |
| \((3,6)\) | No comparison reported |
Method. Apply the two-way test. For \((4,5)\), the table states both \((4,5)\succsim(6,2)\) and \((6,2)\succsim(4,5)\). Thus \((4,5)\sim(6,2)\), so it belongs to \(I(\bar{x})\). The reference itself also belongs.
For \((8,1)\), the reference is strictly preferred, so the two bundles are not indifferent. For \((5,3)\), the candidate is strictly preferred, so it is not indifferent either. The table gives no comparison for \((3,6)\); therefore, its status cannot be determined from the stated information.
Plot only \((6,2)\) and \((4,5)\) as points established to be on the curve. Do not plot \((3,6)\) as an equal-ranked point merely because it is unclassified. The common mistake is to treat “unknown” as “tied”; the correction is to distinguish missing information from an explicit two-way preference comparison.
Check. There is one non-reference bundle with both weak-preference statements, so the curve’s listed points are exactly \((6,2)\) and \((4,5)\).
Common Mistakes and a Reliable Procedure
A preference table is a set of statements, not a picture of a curve. Convert each relevant statement into the definition before plotting. In particular, “at least as good as” in one direction is insufficient for indifference; “strictly preferred” in either direction excludes a candidate; and an unreported comparison does not count as a tie.
Write down the reference bundle and determine which good is represented by each coordinate.
Use the table’s stated comparison to decide whether it is tied with, strictly preferred to, strictly less preferred than, or unclassified relative to the reference.
Include the reference and each bundle for which both weak-preference directions hold. Do not turn strict or unreported comparisons into ties.
Put the first good on the horizontal axis and the second good on the vertical axis, unless the question specifies a different convention.
The main exam trap is to classify bundles by their quantities rather than by the rankings that are given. A bundle with more of one good need not be assumed to be tied with or preferred to the reference based on quantity alone. Use the stated ranking information and make no additional preference claim unless it follows from the assumptions provided.
Check Your Understanding
For each problem, identify only the bundles established to be on the reference bundle’s indifference curve. Use the first coordinate for good 1 and the second for good 2.
- The reference is \((2,5)\). A table states that \((4,3)\) is indifferent to it, \((1,7)\) is strictly preferred, and \((3,2)\) is strictly less preferred. Which listed points belong to the curve? (Answer: \((2,5)\) and \((4,3)\).)
- The reference is \((5,1)\). For \((3,4)\), the table states both that it is weakly preferred to the reference and that the reference is weakly preferred to it. For \((6,2)\), only the first statement is given. Which bundles are established to be indifferent to the reference? (Answer: \((5,1)\) and \((3,4)\).)
- A ranking table says the reference \((4,4)\) is tied with \((2,7)\) and \((6,3)\); it says \((3,6)\) is strictly preferred and gives no comparison for \((5,5)\). List the points established to belong to the indifference curve. (Answer: \((4,4)\), \((2,7)\), and \((6,3)\).)
- The reference is \((1,8)\). A table assigns it and \((3,5)\) to the same explicitly defined ranking class, while assigning \((4,6)\) to a strictly higher class and \((2,3)\) to a strictly lower class. Which points should be plotted as equal-ranked bundles? (Answer: \((1,8)\) and \((3,5)\).)
- For reference bundle \((7,2)\), a table states that \((5,5)\) is weakly preferred to \((7,2)\), and \((7,2)\) is weakly preferred to \((5,5)\). It also states that \((8,3)\) is strictly preferred, while no relation is given for \((4,7)\). Give the established indifference-curve points and explain what to do with the unclassified bundle. (Answer: Plot \((7,2)\) and \((5,5)\); do not classify \((4,7)\) as indifferent because its comparison is unreported.)