4  Preferences

For more information on these topics, see Varian Chapter 3: Preferences.

Objective

In the last chapter, I explained how consumers choose the best bundle of goods they can afford, detailing what “afford” means in precise terms. In this chapter, I will provide a similarly detailed explanation of what might constitute the “best bundle.”

But first: one last thought about budget lines.

4.1 Final Note on Budget Lines

Practice Question: If all prices doubled, you would be no better or worse off compared to prices staying the same and your income going from m to:






4.2 Preference Notation

The idea of preference is based on a consumer’s behavior. If a consumer never chooses Bundle 2 when Bundle 1 is available, that indicates a preference for Bundle 1.

For example, consider these two bundles:

  • Bundle 1: 3 graham crackers and 1 hot cocoa packet
  • Bundle 2: 1 graham cracker and 3 hot cocoa packets

If you choose Bundle 1 when both options are available, it means you prefer Bundle 1 to Bundle 2, or at least you are indifferent between them. In precise terms, Bundle 1 is at least as good as Bundle 2 to you.

Preference Notation: \(\succ\), \(\succeq\), \(\sim\)

  • \((x_1, x_2) \succ (y_1, y_2)\): This means that \((x_1, x_2)\) is strictly preferred over \((y_1, y_2)\). For example, if you would only choose Bundle 2 if Bundle 1 was not available, that indicates you strictly prefer Bundle 1.

  • \((x_1, x_2) \succeq (y_1, y_2)\): This means that \((x_1, x_2)\) is at least as good as \((y_1, y_2)\). If you choose Bundle 1, it indicates that Bundle 1 is at least as good as Bundle 2, even if you might be indifferent between them.

  • \((x_1, x_2) \sim (y_1, y_2)\): This means that the consumer is exactly indifferent between \((x_1, x_2)\) and \((y_1, y_2)\). If you have no preference between 3 graham crackers and 1 hot cocoa packet versus 1 graham cracker and 3 hot cocoa packets, then you are indifferent between the two bundles.


Practice Question: If \((x_1, x_2) \succeq (y_1, y_2)\) but it is not true that \((x_1, x_2) \sim (y_1, y_2)\), then:






4.3 Indifference Curves

An Indifference Curve through a consumption bundle \((x_1, x_2)\) consists of all other bundles \((y_1, y_2)\) such that \((x_1, x_2) \sim (y_1, y_2)\).


Consider this example while answering the practice question below:

Practice Question: Which of the following is not supported by the indifference curves above?






4.4 Marginal Rate of Substitution

That practice question eluded to the fact that the slope of the indifference curve represents a person’s marginal rate of subtitution (MRS): the rate at which you are just willing to substitute one good for the other.

If you are indifferent between these two bundles:

  • Bundle 1: 2 instant noodles and 1 jalapeno
  • Bundle 2: 1 instant noodle and 4 jalapenos

Then you are just willing to substitute 1 instant noodle to gain 3 jalapenos; in other words, your MRS is \(3/-1 = -3\).

Practice Question: If your indifference curve is the straight line between these two bundles:

  • Bundle 1: 3 kiwis and 1 lime
  • Bundle 2: 1 kiwi and 3 limes

What is your marginal rate of substitution?






4.5 Classwork 4

  1. Fill in the blanks to write a proof (by contradiction) of the fact that indifference curves can’t cross.

    Consider two indifference curves \(I_1\) and \(I_2\). Bundle \(A\) is on \(I_1\) and not on \(I_2\), and Bundle \(C\) is on \(I_2\) and not on \(I_1\). The two indifference curves intersect at bundle \(B\), so bundle \(B\) is on both \(\underline{\hspace{2cm}}\). Because \(A\) and \(B\) are both on \(I_1\), we know that \(A \sim B\). And because \(B\) and \(C\) are both on \(\underline{\hspace{2cm}}\), we know that \(\underline{\hspace{2cm}}\). Transitivity indicates we should therefore expect that \(\underline{\hspace{2cm}}\). But this is a contradition, because \(A\) is on \(I_1\), \(C\) is not, and \(I_1\) contains all the bundles \(X\) such that \(A \sim X\). Therefore, two indifference curves can never intersect.

  2. Perfect substitutes are goods that can always be used interchangeably. For instance, if I’m just as happy using butter or margarine to make cookies, then butter and margarine are perfect substitutes for me. Suppose I need a cup of butter or margarine to make a batch of cookies (and for example, .93 cups to make .93 batches of cookies). Assume I’d always prefer to be able to make more cookies than less.

    I’m indifferent between having .25 cups of margarine and .25 cups of butter to having .5 cups of margarine and \(\underline{\hspace{2cm}}\) cups of butter.

    Sketch my indifference curves for butter and margarine.

    Compare your sketch to the standard “C”-shape indifference curves where the more of one good you have, the more you’re willing to give it up to get more of the other good. Do perfect substitutes have this property of a diminishing marginal rate of substitution?

  3. Perfect Complements are goods that are always consumed together in fixed proportions, like right shoes and left shoes. Having an extra left shoe does me no good, but I’d rather have more pairs of shoes than less.

    I’m indifferent between having:

    • 2 right shoes and 2 left shoes
    • 3 right shoes and \(\underline{\hspace{2cm}}\) left shoes
    • \(\underline{\hspace{2cm}}\) right shoes and 3 left shoes

    Sketch my indifference curves for right and left shoes. What letter of the alphabet does the shape of this indifference curve remind you of?

  4. Bads are commodities that the consumer doesn’t like. If I love fruit but hate fruit flies, suppose I’m indifferent between getting an extra piece of fruit, given it has a fly on it.

    I’m indifferent between having:

    • 3 fruit and 3 flies
    • 4 fruit and \(\underline{\hspace{2cm}}\) flies

    And I’m indifferent between having:

    • 3 fruit and 1 fly
    • 4 fruit and \(\underline{\hspace{2cm}}\) flies

    Sketch my indifference curves for fruit and flies. What’s different about indifference curves for bads?