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.

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

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

...

*Preferences*are a personal ranking of alternatives.*Preferences*are a personal assignment of*satisfaction level*(**utility**).

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*

**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 $

**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 $

- 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 (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”*.

- E.g.: $ (3, 4) \sim (1, 12) $

- WP(3,4) is the shaded area

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

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

No! Typically, there are infinite.

In most cases it makes sense we talk and draw several ("the

*indifference map*").

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.

- Can two distinct
*indifference curves*cross each other?

**(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 $.

- Example of preferences that do not satisfy
**convexity**

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.

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

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?

Think about the MRS in Perfect Substitutes

Think about the MRS in Perfect Complements