Preferences

Intermediate Microeconomics (Econ 100A)

Kristian López Vargas

UCSC - Fall 2017

Rationality in Economics - Behavioral Postulates

  • A decision maker knows what he/she likes/enjoys and chooses his/her most preferred alternative among the available ones.

  • To say something about his/her behavior, we must model decision makers’ preferences.

Basics of Preferences Relations

John: apple better than Mango, apple better than banana, mango better than banana.

Basics of Preferences Relations

Alí, Bob, Carlos, ... , John, ... ,Wei

...

Basics of Preferences Relations

  • Preferences are a personal ranking of alternatives.

  • Preferences are a personal assignment of satisfaction level (utility).

Preferences Relations

Preference Relations

Comparing two different consumption bundles, $ x $ and $ y $ in the consumption space:

  • Strict preference "$ x \succ y $" : x is strictly more preferred than is y

  • Weak preference "$ x \succsim y $" : x is as at least as preferred as is y

  • Indifference "$ x \sim y $" : x is equally preferred as is y

Assumptions on Preference Relations

Assumptions on Preference Relations (1): Completeness

  • Completeness: For any two bundles x and y it is always possible to make the statement that either

  • $ x \succsim y $ or $ y \succsim x $

Assumptions on Preference Relations (2): Transitivity

  • Transitivity:
    • If x is at least as preferred as y, and
    • y is at least as preferred as z, then:
    • x is at least as preferred as z.
  • That is, if $ x \succsim y $ and $ y \succsim z $ implies $ x \succsim z $

Preferences in the Commodity Space

  • Recap: the Commodity Space is the positive quadrant of the n-dimensional plane ($ \Re_{+}^n $) where these baskets or bundles live.

Indifference Curves or Indifference Sets

  • Indifference Curves or Indifference Sets (of consumer i):

  • A set of bundles that a consumer regards as equal.

  • Take bundle $ x $. The set of all bundles equally preferred to $ x $ makes the "indifference curve" containing $ x $. We denote this set by $ I(x) $.

  • All the bundles $ y $ in this set have this property: $ y \sim x $.

  • Since an indifference “curve” is not necessarily a "curve", we might want to call it indifference “set”.

Indifference Curve (example)

  • E.g.: $ (3, 4) \sim (1, 12) $

Weakly preferred set WP(x)

  • WP(3,4) is the shaded area

Assumption on Preferences (3): More is better (monotoniticy)

More is Better / Monotonicity: * All else the same, more of a “good” commodity is better than less. * $ (5.01, 20) \succ (5, 20) $

Assumption on Preferences (3): More is better (monotoniticy)

  • This assumption implies that indifference sets are:
    • Curves! (not thick bands)
    • Downward sloped! (think about it)

Is there only one indifference curve?

  • No! Typically, there are infinite.

  • In most cases it makes sense we talk and draw several ("the indifference map").

Goods Vs. Bads Vs. Neutrals

Assume $ x_2 $ is a good: more is better.

Draw and IC for each case:

  • $ x_1 $ is a good.

  • $ x_1 $ is a bad.

  • $ x_1 $ is a neutral.

Home exercise:

  • Can two distinct indifference curves cross each other?

Assumption on Preferences (4): Convexity

(Weak) Convexity:

  • Mixtures of bundles are (weakly) preferred to the bundles themselves.

  • Example: If the 50-50 mixture of the bundles $ x $ and $ y $ is formed like this $ z = (0.5) x + (0.5)y $. Then $ z $ is at least as preferred as $ x $ OR $ y $.

Assumption on Preferences (5): Convexity

Assumption on Preferences (5): Convexity

  • Example of preferences that do not satisfy convexity

Slope of an Indifference Curve

  • The slope of an indifference curve is its marginal rate-of-substitution or MRS.

  • MRS is the rate at which the consumer is only just willing to exchange/substitute commodity 2 for a small amount of commodity 1.

  • $ MRS = \frac{d x_2} {d x_1} $ along one indifference curve.

Types of Preferences: Perfect Substitutes

  • If a consumer always regards units of commodities 1 and 2 as equivalent (or equivalent up to a fixed ratio), then these commodities are regarded as perfect substitutes for the consumer.

    • Example: if you like Coke and Pepsi exactly equally, the total amount of bottles is what matter for the consumer. Another example: Agave - Sugar

Types of Preferences: Perfect Complements

  • If a consumer always consumes commodities 1 and 2 in fixed proportion (e.g. one-to-one), then the commodities are perfect complements to the consumer.

  • Only the number of pairs in the fixed proportion matter to the consumer. Examples?

MRS

  • Think about the MRS in Perfect Substitutes

  • Think about the MRS in Perfect Complements