1.5: Preferences II: MRS and Utility Functions

# 1.5: Preferences II: MRS and Utility Functions
## ECON 306 · Microeconomic Analysis · Fall 2019
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i> safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/microf19"><i class="fa fa-github fa-fw"></i> ryansafner/microf19</a><br> <a href="https://microF19.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i> microF19.classes.ryansafner.com</a><br>

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# Utility Functions and Indifference Curves I

- Indifference curve: all equally preferred bundles `$\iff$` same utility level

- Each indifference curve represents one level (or contour) of utility surface (function)

]

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# Utility Functions and Indifference Curves II

3-D Utility Function: `$u(x,y)=\sqrt{xy}$`
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]
.pull-right[
.center[
2-D Indifference Curve Contours: `$y=\frac{u^2}{x}$`

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]

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# MRS and Marginal Utility I

- Recall: .shout[marginal rate of substitution `\$MRS_{x,y}\$`] is the slope of the indifference curve
    - Rate at which `$y$` is given up for 1 more `$x$`

- It would be handy to be able to calculate MRS
    - Recall it changes (a curve, not a straxfight line)!
    - We can calculate it using something from the **utility function**

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# MRS and Marginal Utility II

- .shout[Marginal utility]: change in (total) utility from a one-unit increase in consumption of a good

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# MRS and Marginal Utility II

- .shout[Marginal utility]: change in (total) utility from a one-unit increase in consumption of a good

.content-box-green[
.green[**Marginal utility of `\$x\$`**]: `$MU_x = \frac{\Delta u(x,y)}{\Delta x}$`
]

]
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# MRS and Marginal Utility II

- .shout[Marginal utility]: change in (total) utility from a one-unit increase in consumption of a good

.content-box-green[
.green[**Marginal utility of `\$x\$`**]: `$MU_x = \frac{\Delta u(x,y)}{\Delta x}$`
]

.content-box-green[
.green[**Marginal utility of `\$y\$`**]: `$MU_y = \frac{\Delta u(x,y)}{\Delta y}$`
]

]
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]

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# MRS and Marginal Utility II

- .shout[Marginal utility]: change in (total) utility from a one-unit increase in consumption of a good

.content-box-green[
.green[**Marginal utility of `\$x\$`**]: `$MU_x = \frac{\Delta u(x,y)}{\Delta x}$`
]

.content-box-green[
.green[**Marginal utility of `\$y\$`**]: `$MU_y = \frac{\Delta u(x,y)}{\Delta y}$`
]

- Math (calculus): "*marginal*" means "*derivative with respect to*"
- I will always derive marginal utility functions for you

]
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# MRS and Marginal Utility: Example

`$$u(x,y) = x^2+y^3$$`

- Marginal utility of x: `$\quad MU_x = 2x$`
- Marginal utlity of y: `$\quad MU_y = 3y$`

]

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# MRS and Marginal Utility III

- How to relate `$MU$` and `$MRS$`?

- Moving along an indifference curve
    - `$X$` and `$Y$` will change
    - `$MU_x$` and `$MU_y$` will change
    - *Utility is constant* `$(\Delta u=0)$`

]
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]

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# MRS and Marginal Utility IV

- How to relate `$MU$` and `$MRS$`?

- Moving along an indifference curve
    - `$X$` and `$Y$` will change
    - `$MU_x$` and `$MU_y$` will change
    - *Utility is constant* `$(\Delta u=0)$`

`$$\begin{align*}
MU_x \Delta x+MU_y\Delta y&= \Delta u\\
\end{align*}$$`

]
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]

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# MRS and Marginal Utility IV

- How to relate `$MU$` and `$MRS$`?

- Moving along an indifference curve
    - `$X$` and `$Y$` will change
    - `$MU_x$` and `$MU_y$` will change
    - *Utility is constant* `$(\Delta u=0)$`

`$$\begin{align*}
MU_x \Delta x+MU_y\Delta y&= \Delta u\\
MU_x \Delta x+MU_y \Delta y &=0\\ \end{align*}$$`

]
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]

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# MRS and Marginal Utility IV

- How to relate `$MU$` and `$MRS$`?

- Moving along an indifference curve
    - `$X$` and `$Y$` will change
    - `$MU_x$` and `$MU_y$` will change
    - *Utility is constant* `$(\Delta u=0)$`

`$$\begin{align*}
MU_x \Delta x+MU_y\Delta y&= \Delta u\\
MU_x \Delta x+MU_y \Delta y &=0\\
MU_y \Delta y&= -MU_{x} \Delta x \\ \end{align*}$$`

]
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]

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# MRS and Marginal Utility IV

- How to relate `$MU$` and `$MRS$`?

- Moving along an indifference curve
    - `$X$` and `$Y$` will change
    - `$MU_x$` and `$MU_y$` will change
    - *Utility is constant* `$(\Delta u=0)$`

`$$\begin{align*}
MU_x \Delta x+MU_y\Delta y&= \Delta u\\
MU_x \Delta x+MU_y \Delta y &=0\\
MU_y \Delta y&= -MU_{x} \Delta x \\
\underbrace{\frac{\Delta y}{\Delta x}}_{MRS} &= -\frac{MU_{x}}{MU_{y}}\\ \end{align*}$$`

]
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# MRS and Preferences

`$$MRS=\underbrace{\frac{ \Delta y}{\Delta x}}_{slope} = -\frac{MU_{x}}{MU_{y}}$$`

- Observing the choices that consumers make, given their options, give us insight into their preferences

- Represented in indifference curves and MRS

- .whisper[Steepness] of indifference curves tells us how consumers trade off between goods

- Let's look at extremes first

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# MRS and Preferences: Steepness I

.pull-left[
<img src="05-slides_files/figure-html/unnamed-chunk-15-1.png" width="504" style="display: block; margin: auto;" />

.font90[
- Vertical curves `$\implies$` indifference between having more or less Downloads
- Downloads are a .shout[neutral] (neither good nor bad)
- `$MRS_{T,D}=\infty$`; give up `$\infty$` (or undefined) Downloads to get more Tickets
]
]
--

.pull-right[
<img src="05-slides_files/figure-html/unnamed-chunk-16-1.png" width="504" style="display: block; margin: auto;" />

.font90[
- Horizontal curves `$\implies$` indifference between having more or less Tickets
- Tickets are a .shout[neutral] (neither good nor bad)
- `$MRS_{T,D}=0$`;  give up `$0$` Downloads to get more Tickets
]
]
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# MRS and Preferences: Steepness II

.pull-left[
<img src="05-slides_files/figure-html/unnamed-chunk-17-1.png" width="504" style="display: block; margin: auto;" />

- Flatter `$\rightarrow$` willing to give up *few* Downloads per Ticket (and vice versa)
- `$MRS_{T,D}$` is small

]

.pull-right[
<img src="05-slides_files/figure-html/unnamed-chunk-18-1.png" width="504" style="display: block; margin: auto;" />

- Steeper `$\rightarrow$` willing to give up *more* Downloads per Ticket (and vice versa)
- `$MRS_{T,D}$` is large

]

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# MRS and Preferences: Goods, Bads, Neutrals

- Now we have better ways to classify objects:

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# MRS and Preferences: Goods, Bads, Neutrals

- Now we have better ways to classify objects:

- A .shout[good] enters utility function positively
    - `$\uparrow$` good `$\implies$` `$\uparrow$` utility (and vice versa)
    - Willing to pay (give up other goods) to *acquire more* (monotonic)
    
- A .shout[bad] enters utility function negatively
    - `$\uparrow$` good `$\implies$` `$\downarrow$` utility (and vice versa)
    - Willing to pay (give up other goods) to *get rid of*
]

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# MRS and Preferences: Goods, Bads, Neutrals

- Now we have better ways to classify objects:

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# MRS and Preferences: Curvature

`$$MRS=\underbrace{\frac{ \Delta y}{\Delta x}}_{slope} = -\frac{MU_{x}}{MU_{y}}$$`

- .whisper[Curvature] of indifference curves tells us how goods are related

- Relatively **straight** curves: goods are more .shout[substitutable] for one another

- Relatively **bent** curves: goods are more .shout[complementary] with one another

- Look at extreme cases first to get the idea

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# MRS and Preferences: Curvature II

.content-box-green[
.green[**Example**]: Consider 1-Liter bottles of coke and 2-Liter bottles of coke 
]

- Always willing to substitute Two 1-Ls:One 2-L

- .shout[Perfect substitutes]: goods that can be substituted at same fixed rate and yield same utility

- `$MRS_{1L,2L}=-0.5$` (a constant!)

]

.pull-right[
<img src="05-slides_files/figure-html/unnamed-chunk-20-1.png" width="504" style="display: block; margin: auto;" />
]

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# MRS and Preferences: Curvature III

`$$u(x,y)=w_xx+w_yy$$`

- `$w_i$`: relative weight (intensity of relative preference)

- Known as **linear preferences**
]

- `$MRS_{L_{1},L_{2}}=-\frac{w_x}{w_y} = -\frac{1}{2}$`
]
]

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<img src="05-slides_files/figure-html/unnamed-chunk-21-1.png" width="504" style="display: block; margin: auto;" />
]

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# MRS and Preferences: Curvature IV

- Always consume together in fixed proportions (1:1)

- .shout[Perfect complements]: goods that can be consumed together in same fixed proportion and yield same utility

- MRS: ?

]

.pull-right[
<img src="05-slides_files/figure-html/unnamed-chunk-22-1.png" width="504" style="display: block; margin: auto;" />
]

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# MRS and Preferences: Curvature V

`$$u(x,y)=min\{w_xx,w_yy\}$$`

- `$w_i$`: relative weight (intensity of relative preference)

- MRS `$=0, \infty$`, or undefined

- Known as **Leontief preferences**
]

]
]

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<img src="05-slides_files/figure-html/unnamed-chunk-23-1.png" width="504" style="display: block; margin: auto;" />
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# MRS and Preferences: Curvature VI

.pull-left[
<img src="05-slides_files/figure-html/unnamed-chunk-24-1.png" width="504" style="display: block; margin: auto;" />

- Straighter `$\rightarrow$` more substitutable

]
--

.pull-right[
<img src="05-slides_files/figure-html/unnamed-chunk-25-1.png" width="504" style="display: block; margin: auto;" />

- Curved `$\rightarrow$` more complementary

]

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# Cobb-Douglas Utility Functions

- A very common functional form in economics is .shout[Cobb-Douglas]

`$$u(x,y)=x^ay^b$$`

- Where `$a, b >0$` (and very often `$a+b=1$`)
- Extremely useful, you will see it often!
    - Strictly convex and monotonic indifference curves
    - Other nice properties (we'll see later)
    - See the appendix in [today's class page](http://microf19.classes.ryansafner.com/class/05-class)
]

]

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# Practice

.content-box-green[
.green[**Example**]: Suppose you can consume apples `$(a)$` and broccoli `$(b)$`, and earn utility according to the utility function:
`$$u(a,b)=2ab$$`

Where your marginal utilities are:

`$$\begin{align*}
MU_a&=2b\\
MU_b&=2a\\
\end{align*}$$`

1. Put `$a$` on the horizontal axis and `$b$` on the vertical axis. Write an equation for `$MRS_{a,b}$`.

2. Would bundles of `$(1, 4)$` and `$(2, 2)$` be on the same indifference curve?

3. Sketch a graph of the indifference curve from part 2. (Bonus: find the equation of this indifference curve. See the appendix in [today's class page](http://microf19.classes.ryansafner.com/class/05-class).)

4. Is this curve convex? Hint: Does `$MRS_{a,b} \downarrow$` as `$a \uparrow$`?
]