## One additional assumption on preference relations

• Continuity:

• A small change in a bundle can only cause small changes to the preference rankings and level.

• Put differently, if A is preferred to B, then situations suitably close to A must also be preferred to B.

## Utility Function

• Main idea: instead of preference relations, we can describe a consumers' preferences using utility functions.

• Intuition: A utility function assigns a "satisfaction level" (a number) to each alternative or bundle.

• Instead of saying: $iPhone7 \succ Galaxy8$, we say $U(iPhone7) > U(Galaxy8)$

• Instead of saying: $(2 \textrm{ Beers}, 3 \textrm{ Pizzas} ) \succ (1 \textrm{ Beers}, 4 \textrm{ Pizzas} )$, we say $U(2, 3) > U(1, 4)$

## Utility Function

• But, how does a preference relations and a utility function connect?

• If a consumer's preference relations are:
1. complete,
2. transitive,
3. continuous
• Then, these preferences can be represented by a (continuous) utility function.

• Note on reflexivity, monotonicity, convexity.

## Utility Function

• When do we say that we a utility function represents someone's preference relations?

• A utility function $U(x)$ represents a preference relation $\succsim$ when:

• $x’ \succ x’’ \iff U(x’) > U(x’’), ~~$ for any $x'$ and $x''$

• $x’ \succsim x’’ \iff U(x’) \geq U(x’’) ~~$, for any $x'$ and $x''$

• $x’ \sim x’’ \iff U(x’) = U(x’), ~~$ for any $x'$ and $x''$

• Intuitively, $U()$ represents a preference relation if it yields the same ranking of alternatives.

## Utility Function

• That is, like preference relations, utility is an ordinal (i.e. ordering) concept.

• If, for example, $U(x) = 6$ and $U(y) = 2$ then bundle $x$ is strictly preferred to bundle $y$. But we cannot say: " $x$ is preferred three times as much as is $y$ ".

## Utility Functions & Indifference Curves

• An indifference curve contains equally preferred bundles.

• In terms of utility, all bundles in an indifference curve give the same utility level:

• If $x \sim y$, $\iff$ $U(x) = U(y)$.
• In other words, indifference curves are the level curves of utility functions.

## Utility Functions & Indiff. Curves - Example!

$u(x_1, x_2) = x_1 ~ x_2$

## Utility Functions & Indifference Curves

• A more complete collection of indifference curves describes more fully the consumer’s preferences (aka indifference map).

## Utility functions (are not unique)

• The preferences of a certain consumer can be described/represented by more than one utility function.

• That is, there is no unique utility function for a preference relation.

• Two utility functions $U()$ and $V()$ represent the same preferences, if we can obtain $V()$ by applying a strictly increasing function on U():

• That is, if we can write $V = f( U() )$ where $f()$ is strictly increasing.

## Utility functions (are not unique) - Example 1!

• Suppose $U(x_1,x_2) = x_1 x_2$ represents some preferences.

• You have bundles $(4,1), (2,3)$ and $(2,2)$.

• Under this utility function: $U(2,3) = 6 > U(4,1) = U(2,2) = 4$;

• Therefore, we know that: $(2,3) \succ (4,1) \sim (2,2)$.

## Utility functions (are not unique) - Example 1!

• Suppose we define $V = U^2$.

• Then, $V(x_1,x_2) = (x_1 x_2)^2 = x_1^2 x_2^2 \quad$

• ...and so, $V(2,3) = 36 > V(4,1) = V(2,2) = 16$

• SO! again $(2,3) \succ (4,1) \sim (2,2)$.

• That is, $V()$ preserves the same order as $U()$ and therefore represents the same preferences.

## Utility functions (are not unique) - Example 2!

• Suppose we define $W = 2U + 10$.

• Then, $W(x_1,x_2) = 2 (x_1 x_2) + 10$

• ...and so, $W(2,3) = 22 > W(4,1) = W(2,2) = 18$

• SO! again! $(2,3) \succ (4,1) \sim (2,2)$.

• That is, $W()$ preserves the same order as $U()$ and therefore represents the same preferences.

• A good is a commodity unit which increases utility (gives a more preferred bundle).

• A bad is a commodity unit which decreases utility (gives a less preferred bundle).

• A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).

• For home: Draw indifference curves for each of them

## Drawing an Indifference Curve

1. Identify the utility function: $U(x_1, x_2)$.

2. Set the utility level to a constant level $k$: $U(x_1, x_2) = k$.

3. Solve for $x_2$ in the previous equation to obtain a generic indifference curve.

4. Give $k$ an arbitrary value and draw the curve.

5. To draw another curve, set $k$ equal to another value and draw again.

## Drawing an Indifference Curve - Example!

1. Utility: $U(x_1, x_2) = x_1^{0.5} x_2$

2. Utility at a constant level: $x_1^{0.5} x_2 = k$

3. Solve for x2: $x_2 = \frac{k}{x_1^{0.5}}$.

4. Give $k$ a value: $x_2 = \frac{ 10 }{x_1^{0.5}}$. Draw!

5. Give $k$ another value: $x_2 = \frac{ 20 }{x_1^{0.5}}$. Draw!

• Try with other functions... for example: $U(x_1, x_2) = x_1 + x_2^{0.5}$

## Typical utility fns: perfect substitutes

• General: $~ U(x_1,x_2) = \alpha x_1 + \beta x_2 ~~$, for $\alpha, \beta > 0$

• Example: $~ U(x_1,x_2) = x_1 + x_2$

## Typical utility fns: perfect complements

• General: $~ U(x_1,x_2) = \textrm{min} \{ \frac{x_1}{\alpha}, \frac{x_2}{\beta} \} ~~$, for $\alpha, \beta > 0$.

• Example: $~ U = \textrm{min} \{ x_1, x_2 \} ~~$

## Typical utility fns: Cobb-Douglas

General $U(x_1,x_2) = x_1^a x_2^b$ with a > 0 and b > 0

Examples:

• $U(x_1,x_2) = x_1^{0.5} x_2^{0.5} \quad$ (that is, $a = b = 1/2$)

• $U(x_1,x_2) = x_1 x_2^3 \quad$ (that is, $a = 1, b = 3$)

• $U(x_1,x_2) = x_1^{2} x_2^{0.75} \quad$ (that is, $a = 2, b = 0.75$)

## Marginal Utility

• Marginal means “incremental”.

• The marginal utility of commodity $x_1$ is the rate-of-change of total utility as the quantity of commodity $x_1$ consumed increases.

• $MU_{x_1} = \frac{\partial U}{\partial x_1}$ and $MU_{x_2} = \frac{\partial U}{\partial x_2}$

• To know whether $MU_{x_1}$ is increasing or decreasing in $x_1$, we take the derivative of $MU_{x_1}$ with respect to $x_1$. That is, we calculate: $\frac{\partial MU_{x_1}}{\partial x_1}$

## Marginal Utilities - Cobb-Douglas Example!

Utility: $U(x_1,x_2) = x_1^{0.5} x_2^2$

Marginal Utilities:

$MU_{x_1} = \frac{\partial U}{\partial x_1} = 0.5 x_1^{-0.5} x_2^2$

$MU_{x_2} = \frac{\partial U}{\partial x_2} = 2 x_1^{0.5} x_2^1$

## Marginal Utilities - Perf. Substitutes Example!

Utility: $U(x_1,x_2) = a x_1 + b x_2$

Marginal Utilities:

$MU_{x_1} = \frac{\partial U}{\partial x_1} = a$

$MU_{x_2} = \frac{\partial U}{\partial x_2} = b$

## Marginal Utilities - Perf. Complements Example!

• Utility: $U(x_1,x_2) = min \{ x_1 , x_2 \}$

• Not differentiable at the kinks.

## Marginal Utilities and the MRS

Claim: $MRS \equiv \frac{d x_2}{d x_1} = - \frac{ MU_{x_1} }{ MU_{x_2} }$

Here is the math:

1. Remember, the equation for an indifference curve is: $U(x_1,x_2) = k$, where satisfaction, $k$, is a constant.

2. Do total differentiation: $\frac{\partial U}{\partial x_1} d x_1 + \frac{\partial U}{\partial x_2} d x_2 = d k$,

3. But $dk = 0$ since $k$ is a constant, and $\frac{\partial U}{\partial x_i}$ can be replaced by $MU_{x_i}$. That is: $MU_{x_1} d x_1 + MU_{x_2} d x_2 = 0$

4. We solve for $\frac{d x_2}{d x_1}$ and we get: $\frac{d x_2}{d x_1} = - \frac{ MU_{x_1} }{ MU_{x_2} }$

## Marginal Utilities and the MRS - Examples:

• If $U = x_1^{2} x_2$,
• Then: $MRS = - \frac{MU_{x_1}}{MU_{x_2}} =$ $- \frac{2 x_1 x_2}{ x_1^2 } =$ $- 2 \frac{ x_2 }{ x_1 }$

• If $U = x_1 x_2^{0.5}$,
• Then: $MRS = - \frac{MU_{x_1}}{MU_{x_2}} =$ $- \frac{x_2^{0.5}}{ x_1 0.5 x_2^{-0.5}} =$ $- 2 \frac{ x_2 }{ x_1 }$

• If $U = x_1 + 5 x_2$,
• Then: $MRS = - 1/5$