8  Revealed Preference

For more information on these topics, see Varian Chapter 7: Revealed Preference.

8.1 Objective

In previous chapters, we explored what preferences can reveal about people’s behavior. However, preferences are not directly observable in practice. This chapter explains how to infer a consumer’s preferences based on their demand information.

8.2 Assumptions

In this chapter, for simplicity we’ll assume that preferences are monotonic and strictly convex.

8.3 The Principle of Revealed Preference

Directly revealed preferred: if bundle \((x_1, x_2)\) was chosen while bundle \((y_1, y_2)\) was available (that is, it was affordable), then \((x_1, x_2)\) was directly revealed preferred to \((y_1, y_2)\).

If \((x_1, x_2)\) is chosen, then it is directly revealed preferred to all bundles in the shaded blue area: that is, all bundles \((y_1, y_2)\) such that \(p_1 x_1 + p_2 x_2 \geq p_1 y_1 + p_2 y_2\).

Practice Question 1: If kale is $2, lettuce is $1, and I buy 3 kale and 1 lettuce, then \((3, 1)\) is directly revealed preferred to:







If one bundle is chosen when another is affordable, that indicates that the first bundle is not only directly revealed preferred, but also that it is preferred:

The principle of revealed preference: Let \((x_1, x_2)\) be the chosen bundle when prices are \((p_1, p_2)\), and let \((y_1, y_2)\) be some other bundle such that \(p_1 x_1 + p_2 x_2 \geq p_1 y_1 + p_2 y_2\). Then if the consumer is choosing the most preferred bundle she can afford, we must have \((x_1, x_2) \succ (y_1, y_2)\).

8.4 Indirectly Revealed Preferred

If \((x_1, x_2)\) is directly revealed preferred to \((y_1, y_2)\) and \((y_1, y_2)\) is directly revealed preferred to a third bundle \((z_1, z_2)\), then because \((x_1, x_2) \succ (y_1, y_2)\) and \((y_1, y_2) \succ (z_1, z_2)\), by transitivity we can also say \((x_1, x_2) \succ (z_1, z_2)\) and we’d say that \((x_1, x_2)\) is indirectly revealed preferred to \((z_1, z_2)\).


Practice Question 2: Consider two sets of prices for markers and notebooks: when markers are $5 and notebooks are $5, I buy one marker and 3 notebooks. When markers are $4 and notebooks are $8, I buy one marker and 2 notebooks. This indicates that (1, 3) markers and notebooks are indirectly revealed preferred to what?






8.5 Trapping the Indifference Curve

8.6 Classwork 8

All our analysis depends on the assumption that consumers have preferences and consistently choose the best bundle of goods they can afford. If consumers do not behave this way, our estimates of the indifference curves become meaningless. So, how can we determine if consumers are actually following this maximizing model? The answer is WARP:

Weak Axiom of Revealed Preference: If \((x_1, x_2)\) is directly revealed preferred to \((y_1, y_2)\), and the two bundles are not the same, then it cannot happen that \((y_1, y_2)\) is directly revealed preferred to \((x_1, x_2)\).

For example, the image below demonstrates a violation of WARP:

When faced with a flatter budget line, the consumer chooses the bundle \((x_1, x_2)\) even though the bundle \((y_1, y_2)\) is also affordable. However, when the budget line becomes steeper due to a change in prices, the consumer switches their choice to \((y_1, y_2)\) even though \((x_1, x_2)\) remains affordable.

This situation suggests one of two possibilities:

  1. The consumer is not selecting the optimal bundle they can afford.

  2. There are other changes in the choice problem that we have not observed.

In any case, this type of violation does not align with the model of consumer choice in a stable environment.

  1. When prices are \((p_1, p_2) = (1, 2)\) a consumer demands \((x_1, x_2) = (1, 2)\), and when prices are \((q_1, q_2) = (2, 1)\) the consumer demands \((y_1, y_2) = (2, 1)\). Is this behavior consistent with the model of maximizing behavior?

  2. When prices are \((p_1, p_2) = (2, 1)\) a consumer demands \((x_1, x_2) = (1, 2)\), and when prices are \((q_1, q_2) = (1, 2)\) the consumer demands \((y_1, y_2) = (2, 1)\). Is this behavior consistent with the model of maximizing behavior?

  3. In part b), which bundle is preferred by the consumer, the x-bundle or the y-bundle?