Chapter 3 Preferences Key Concept: characterize preferences by a binary comparison measuring at least as good as. Derive the indifference curves for a given preference. The marginal rate of substitution (MRS) measures the marginal willingness to pay for x in terms of y.

Chapter 3 Preferences Choose the “best” thing one can “afford.” How do you find the best?

Through cat-cat binary comparisons, we can find the cutest cat.

A binary relation : w (x1, x2) w (y1, y2) (x1, x2) is at least as good as (y1, y2)

(x1, x2) w (y1, y2) This binary relation w is complete and transitive.

Complete every two bundles can be compared for any (x1, x2), (y1, y2) either (x1, x2) w (y1, y2) (y1, y2) w (x1, x2) or both

Transitive for any (x1, x2), (y1, y2), (z1, z2) if (x1, x2) w (y1, y2) and (y1, y2) w (z1, z2) then (x1, x2) w (z1, z2)

Complete + Transitive = Rational preference

From this binary relation w, one can derive two other binary relations s and i.

(x1, x2) s (y1, y2) if and only if (x1, x2) w (y1, y2) and it is not the case that (y1, y2) w (x1, x2). Read this as the consumer strictly prefers (x1, x2) to (y1, y2).

(x1, x2) i (y1, y2) if and only if (x1, x2) w (y1, y2) and (y1, y2) w (x1, x2). Read this as the consumer is indifferent between (x1, x2) and (y1, y2).

Given a binary relation w and for (x1, x2), can list all the bundles that are at least as good as it -- the weakly preferred set. Similarly, can list all the bundles for which the consumer is indifferent to it -- the indifference curve.

Fig. 3.1

We don’t need to use the idea of utility. Preferences are enough.

Fig. 5.1

Two distinct indifference curves cannot cross.

Fig. 3.2

Perfect substitutes ten dollar coins and five dollar coins

Fig. 3.3

Perfect complements left shoe and right shoe

Fig. 3.4

Bads, neutrals

Fig. 3.5

Fig. 3.6

Satiation smart and tall

Fig. 3.7

Discrete goods

Fig. 3.8

Useful assumptions Monotonicity: the more, the better if x1 ≥ y1, x2 ≥ y2 and (x1, x2) ≠ (y1, y2), then (x1, x2) s (y1, y2) This implies indifference curves have negative slopes (examine).

Fig. 3.9

Convexity Averages are preferred to extremes. if (y1, y2) w (x1, x2) and (z1, z2) w (x1, x2), then for any weight t between 0 and 1, (ty1+(1-t)z1, ty2+(1-t)z2) w (x1, x2)

Fig. 3.10

We often assume it to get an interior solution instead of a corner solution.

Strict convexity: obviously stronger than convexity. if (y1, y2) w (x1, x2), (z1, z2) w (x1, x2), and (y1, y2) ≠ (z1, z2), then for any weight t strictly in between 0 and 1, (ty1+(1-t)z1, ty2+(1-t)z2) s (x1, x2)

Strict convexity rules out preferences such as the perfect substitute type.

To describe preferences, a useful way is to calculate the marginal rate of substitution (MRS).

The MRS measures the rate at which the consumer is “just” willing to substitute one thing for the other. Specify which for which and is evaluated where.

MRS1, 2 for a little of good 1, the amount of good 2 that the consumer is willing to give up to stay indifferent about this change ∆x2/ ∆x1

Fig. 3.11

The MRS1, 2 at a point is the slope of the indifference curve at that point (to stay put) and measures the marginal willingness to pay for good 1 in terms of good 2. If good 2 is money, then it is often called the marginal willingness to pay.

Useful assumption diminishing MRS when you have more of x1, it can substitute for x2 less

Chapter 3 Preferences Key Concept: characterize preferences by a binary comparison measuring at least as good as. Derive the indifference curves for a given preference. The marginal rate of substitution (MRS) measures the marginal willingness to pay for x in terms of y.