Chapter 3 Preferences Choose the “best” thing one can “afford” Start with consumption bundles (a complete list of the goods that is involved in consumer’s choice problem) A binary relation : w (x 1, x 2 ) w (y 1, y 2 ) is read as (x 1, x 2 ) is at least as good as (y 1, y 2 ) This binary relation w is complete, reflexive and transitive

Complete: for any (x 1, x 2 ), (y 1, y 2 ), either (x 1, x 2 ) w (y 1, y 2 ), (y 1, y 2 ) w (x 1, x 2 ) or both (every two bundles can be compared) Reflexive: for any (x 1, x 2 ), (x 1, x 2 ) w (x 1, x 2 ) (any bundle is at least as good as itself) Transitive: for any (x 1, x 2 ), (y 1, y 2 ), (z 1, z 2 ), if (x 1, x 2 ) w (y 1, y 2 ) and (y 1, y 2 ) w (z 1, z 2 ), then (x 1, x 2 ) w (z 1, z 2 ) Rational preference One can experimentally test whether these three axioms are satisfied. (kids, social preference)

From this binary relation w, one can derive two other binary relations s and i. (x 1, x 2 ) s (y 1, y 2 ) if and only if (x 1, x 2 ) w (y 1, y 2 ) and it is not the case that (y 1, y 2 ) w (x 1, x 2 ). Read this as the consumer strictly prefers (x 1, x 2 ) to (y 1, y 2 ). (x 1, x 2 ) i (y 1, y 2 ) if and only if (x 1, x 2 ) w (y 1, y 2 ) and (y 1, y 2 ) w (x 1, x 2 ). Read this as the consumer is indifferent between (x 1, x 2 ) and (y 1, y 2 ).

Given a binary relation w and for (x 1, x 2 ), 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 We don’t need to use the idea of utility. Preferences are enough.

Fig. 3.1

Two distinct indifference curves cannot cross. Perfect substitutes: ten dollar coins and five dollar coins Perfect complements: left shoe and right shoe Bads, neutrals Satiation Discrete goods

Fig. 3.2

Fig. 3.3

Fig. 3.4

Fig. 3.5

Fig. 3.6

Fig. 3.7

Fig. 3.8

Some useful assumptions Monotonicity: if x 1 ≥ y 1, x 2 ≥ y 2 and (x 1, x 2 ) ≠ (y 1, y 2 ), then (x 1, x 2 ) s (y 1, y 2 ) (the more, the better) (indifference curves have negative slopes) (Examine)

Convexity: if (y 1, y 2 ) w (x 1, x 2 ) and (z 1, z 2 ) w (x 1, x 2 ), then for any weight t between 0 and 1, (ty 1 +(1-t)z 1, ty 2 +(1-t)z 2 ) w (x 1, x 2 ) (averages are preferred to extremes) (Examine) (interior solution, non convex ref to circle)

Fig. 3.9

Fig. 3.10

Strict convexity: if (y 1, y 2 ) w (x 1, x 2 ), (z 1, z 2 ) w (x 1, x 2 ), and (y 1, y 2 ) ≠ (z 1, z 2 ), then for any weight t strictly in between 0 and 1, (ty 1 +(1-t)z 1, ty 2 +(1-t)z 2 ) s (x 1, x 2 )

The marginal rate of substitution (the MRS one thing for another thing, evaluated where) measures the rate at which the consumer is “just” willing to substitute one thing for the other

MRS 1, 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, ∆x 2 / ∆x 1 The MRS 1, 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.

Fig. 3.11

Useful assumption: diminishing MRS (when you have more of x 1, it can substitute for x 2 less)