1.6 — Income & Substitution Effects

# 1.6 — Income & Substitution Effects
## ECON 306 • Microeconomic Analysis • Spring 2022
### 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/microS22"><i class="fa fa-github fa-fw"></i>ryansafner/microS22</a><br> <a href="https://microS22.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>microS22.classes.ryansafner.com</a><br>

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# Outline

### [The (Own) Price Effect](#4)
### [(Real) Income Effect](#10)
### [Substitution Effect](#15)
### [Putting the Effects Together](#18)
### [What About Inferior Goods?](#34)
### [On to Demand Curves](#49)

---

# A Demand Function (Again)

.pull-left[
.smallest[
- A consumer’s .hi[demand] (for good x) depends on .hi-turquoise[current prices & income]:

`$$q_x^D = q_x^D(m, p_x, p_y)$$`

- .hi-turquoise[How does demand (for x) change?]

1. .hi-purple[Income effects] `$\left(\frac{\Delta q_x^D}{\Delta m}\right)$`: how `$q_x^D$` changes with changes in income
2. .hi-purple[Cross-price effects] `$\left(\frac{\Delta q_x^D}{\Delta p_y}\right)$`: how `$q_x^D$` changes with changes in prices of *other* goods (e.g. `$y)$`
3. .hi-purple[(Own) Price effects] `$\left(\frac{\Delta q_x^D}{\Delta p_x}\right)$`: how `$q_x^D$` changes with changes in price (of `$x)$`
]
]
.pull-right[
.center[
![](../images/choices.jpg)
]
]
---

# The (Own) Price Effect

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# The (Own) Price Effect

- .hi-purple[Price effect]: change in optimal consumption of a good associated with a change in its price, holding income and other prices constant

`$$\frac{\Delta q_x^D}{\Delta p_x} < 0$$`

.hi-purple[The law of demand]: as the price of a good rises, people will tend to buy less of that good (and vice versa)
  - i.e. **the price effect is negative!**

]
.pull-right[
.center[
![](../images/pricetag.jpg)
]
]

---

# Decomposing the Price Effect

The .hi-purple[price effect] (law of demand) is actually the **net result of two effects**

1. .hi-green[(Real) income effect]: change in consumption due to change in real purchasing power

2. .hi-orange[Substitution effect]: change in consumption due to change in relative prices

.center[
.hi-purple[Price Effect] `$=$` .hi-green[Real income effect] `$+$` .hi-orange[Substitution Effect]
]

---

---

# (Real) Income Effect: Demonstration

- Suppose there is only 1 good to consume, `$x$`. You have a $100 income, and the price of `$x$` is $10. You consume 10 units of `$x$`

- Suppose the price of `$x$` rises to $20. You now consume 5 units of `$x$`.

- This is the .hi-green[real income effect]
]

]

---

# (Real) Income Effect: Demonstration

- .hi[Real income effect]: your consumption mix changes because of the change in the price of `$x$` changes your .hi-purple[real income] or .hi-purple[purchasing power] (the amount of goods you can buy)

- Note your **_actual_ (nominal) income** ($100) **never changed!**
]

]

---

# (Real) Income Effect: Size

- The *size* of the income effect depends on how large a *portion of your budget* you spend on the good

- **Large-budget items**:
    - e.g. Housing/apartment rent, car prices
    - Price increase/decreases makes you much poorer/wealthier

]

]
]

---

# (Real) Income Effect: Size

- The *size* of the income effect depends on how large a *portion of your budget* you spend on the good

- **Small-budget items**:
    - e.g. pencils, toothpicks, candy
    - Price changes don’t have much of an effect on your wealth or change your behavior much

]

![:scale 100%](../images/pencil.png)

]
]

---
class: inverse, center, middle
# Substitution Effect

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# Substitution Effect: Demonstration

- Suppose there are 1000’s of goods, none of them a major part of your budget
  - So real income effect is insignificant

- Suppose the price of good `$x$` increases

- You would consume *less* of `$x$` relative to other goods because `$x$` is now *relatively* more expensive

- That’s the .hi-orange[substitution effect]
]

]
]

---

# Substitution Effect: Demonstration

- .hi-orange[Substitution effect]: consumption mix changes because of a change in **relative prices**

- Buy more of the (now) relatively cheaper items

- Buy less of the (now) relatively more expensive item `$(x)$`
]

]
]

---

---

# Putting the Effects Together

- .hi-green[Real income effect]: change in consumption due to change in real purchasing power
    - **Different directions**: positive (**normal goods**) or negative (**inferior goods**)
    - Higher price of `$x$` means you must buy less `$x$`, `$y$`, or *both* (depending on your preferences)

- .hi-orange[Substitution effect]: change in consumption due to change in relative prices
    - If `$x$` gets more expensive relative to `$y$`, consume `$\downarrow x$` (and `$\uparrow y$`)
    - Always the same direction: `$(\downarrow$` relatively expensive goods, `$uparrow$` relatively cheaper goods)
    - This is why demand curves slope downwards!
    
--

.center[
.hi-purple[Price Effect] `$=$` .hi-green[Real income effect] `$+$` .hi-orange[Substitution Effect]
]

---

# Real Income and Substitution Effects, Graphically I

- Original optimal consumption `$(A)$`

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Real Income and Substitution Effects, Graphically I

- Original optimal consumption `$(A)$`

- .purple[**(Total) price effect:** `\$A \rightarrow C\$`]

- Let's decompose this into the two effects

]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" style="display: block; margin: auto;" />

]

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# Real Income and Substitution Effects, Graphically II

- .orange[**Substitution effect:**] what you would choose under the **new exchange rate** to **remain indifferent** as before the change

]

]

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# Real Income and Substitution Effects, Graphically II

- .orange[**Substitution effect:**] what you would choose under the **new exchange rate** to **remain indifferent** as before the change

- Graphically: shift *new* budget constraint inwards until tangent with *old* indifference curve

- `$A \rightarrow B$` on same I.C. `$(\udownarrow$` `$x$`, `$\uparrow$` `$y)$`
  - .slate[Note: Point B *must* be a *different* point on the original curve! Why?]

]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Real Income and Substitution Effects, Graphically III

- .green[**(Real) income effect:**] change in consumption due to the **change in purchasing power** from the change in price

]

]

---

# Real Income and Substitution Effects, Graphically III

- .green[**(Real) income effect:**] change in consumption due to the **change in purchasing power** from the change in price

- `$B \rightarrow C$` to new budget constraint (can buy less of `$x$` and/or `$y$`)
]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

- Price of `$x$` rises, new optimal consumption at `$(C)$`

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

- Price of `$x$` rises, new optimal consumption at `$(C)$`

- .orange[**Substitution effect:** `\$A \rightarrow B\$`] on same I.C. `$(\downarrow$` more expensive `$x$` and `$\uparrow$` `$y)$`

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

- Price of `$x$` rises, new optimal consumption at `$(C)$`

- .orange[**Substitution effect:** `\$A \rightarrow B\$`] on same I.C. `$(\downarrow$` more expensive `$x$` and `$\uparrow$` `$y)$`

- .green[**(Real) income effect:** `\$B \rightarrow C\$`] to new budget constraint (can buy less of `$x$` and/or `$y$`)

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

- Price of `$x$` rises, new optimal consumption at `$(C)$`

- .orange[**Substitution effect:** `\$A \rightarrow B\$`] on same I.C. `$(\downarrow$` more expensive `$x$` and `$\uparrow$` `$y)$`

- .green[**(Real) income effect:** `\$B \rightarrow C\$`] to new budget constraint (can buy less of `$x$` and/or `$y$`)

- .purple[**(Total) price effect:** `\$A \rightarrow C\$`]

]

]

---

---

# What About Inferior Goods?

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# Real Income and Substitution Effects, Graphically I

- Original optimal consumption `$(A)$`

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Real Income and Substitution Effects, Graphically I

- Original optimal consumption `$(A)$`

- .purple[**(Total) price effect:** `\$A \rightarrow C\$`]

- Let's decompose this into the two effects

]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-14-1.png" width="504" style="display: block; margin: auto;" />

]

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# Real Income and Substitution Effects, Graphically II

- .orange[**Substitution effect:**] what you would choose under the **new exchange rate** to **remain indifferent** as before the change

]

]

---

# Real Income and Substitution Effects, Graphically II

- .orange[**Substitution effect:**] what you would choose under the **new exchange rate** to **remain indifferent** as before the change

- Graphically: shift *new* budget constraint inwards until tangent with *old* indifference curve

- `$A \rightarrow B$` on same I.C. `$(\downarrow$` `$x$`, `$\uparrow$` `$y)$`

]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-16-1.png" width="504" style="display: block; margin: auto;" />
]

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# Real Income and Substitution Effects, Graphically III

- .green[**(Real) income effect:**] change in consumption due to the **change in purchasing power** from the change in price

]

]

---

# Real Income and Substitution Effects, Graphically III

- .green[**(Real) income effect:**] change in consumption due to the **change in purchasing power** from the change in price

- `$B \rightarrow C$` to new budget constraint (can buy less of `$x$` and/or `$y$`)
]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

- Price of `$x$` rises, new optimal consumption at `$(C)$`

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

- Price of `$x$` rises, new optimal consumption at `$(C)$`

- .orange[**Substitution effect:** `\$A \rightarrow B\$`] on same I.C. `$(\downarrow$` more expensive `$x$` and `$\uparrow$` `$y)$`

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

- Price of `$x$` rises, new optimal consumption at `$(C)$`

- .orange[**Substitution effect:** `\$A \rightarrow B\$`] on same I.C. `$(\downarrow$` more expensive `$x$` and `$\uparrow$` `$y)$`

- .green[**(Real) income effect:** `\$B \rightarrow C\$`] to new budget constraint (can buy less `$x$` and/or `$y$`)

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

- Price of `$x$` rises, new optimal consumption at `$(C)$`

- .orange[**Substitution effect:** `\$A \rightarrow B\$`] on same I.C. `$(\downarrow$` more expensive `$x$` and `$\uparrow$` `$y)$`

- .green[**(Real) income effect:** `\$B \rightarrow C\$`] to new budget constraint (can buy less `$x$` and/or `$y$`)

- .purple[**(Total) price effect:** `\$A \rightarrow C\$`]

]

]

---

<img src="1.6-slides_files/figure-html/unnamed-chunk-24-1.png" width="576" style="display: block; margin: auto;" />
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# Violating the Law of Demand

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: What would it take to violate the law of demand?
]

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# A Giffen Good

- .hi-purple[Giffen good]: theoretical good that violates law of demand

.smallest[
  1. Inferior good (negative .green[real income effect])
  2. .hi-green[real income effect] `$>$` .hi-orange[substitution effect]
]

- Price increase (decrease) causes person to buy *more* (less)
]

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# Recap: Real Income and Substitution Effects

.center[
.hi-purple[Price Effect] `$=$` .hi-green[Real income effect] `$+$` .hi-orange[Substitution Effect]
]

- .hi-orange[Substitution effect]: is always in the direction of the cheaper good

- .hi-green[Real Income effect]: can be positive (normal) or negative (inferior)

- .hi-purple[Law of Demand]/Demand curves slope downwards (.hi-purple[Price effect]) mostly because of the substitution effect
    - Even (inferior) goods with negative real income effects overpowered by substitution effect

- Theoretical **Giffen good** exception: .green[negative R.I.E.] `$>$` .orange[S.E.]

---

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# Deriving a Demand Curve Graphically

<img src="1.6-slides_files/figure-html/unnamed-chunk-26-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-27-1.png" width="504" style="display: block; margin: auto;" />

]

.smallest[
- Demand curve for `$x$` relates optimal consumption of `$x$` ("quantity") as price of `$x$` changes
- At `$p_x=4$`, consumer buys 2 `$x$`
]

---

# Deriving a Demand Curve Graphically

<img src="1.6-slides_files/figure-html/unnamed-chunk-28-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-29-1.png" width="504" style="display: block; margin: auto;" />

]

.smallest[
- Demand curve for `$x$` relates optimal consumption of `$x$` ("quantity") as price of `$x$` changes
- At `$p_x=4$`, consumer buys 2 `$x$`; at `$p_x=2$`, consumer buys 5 `$x$`

]
---

# Deriving a Demand Curve Graphically

<img src="1.6-slides_files/figure-html/unnamed-chunk-30-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-31-1.png" width="504" style="display: block; margin: auto;" />

]

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# Deriving a Demand Curve Graphically

<img src="1.6-slides_files/figure-html/unnamed-chunk-32-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-33-1.png" width="504" style="display: block; margin: auto;" />

]

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# From Individual Demand to Market Demand

- Note so far we have been talking about *an individual person’s* demand

- In principles, you learned about the entire .hi-purple[market demand]

]

]

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# From Individual Demand to Market Demand

- Note so far we have been talking about *an individual person’s* demand

- In principles, you learned about the entire .hi-purple[market demand]

- This is simply the sum of all individuals’ demands

]

]

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# Demand Schedule (For Individual Or Market)

- .hi[Demand schedule] expresses the quantity of good a person(s) would be willing to buy `$(q_D)$` at any given price `$(p_x)$`
  - Holding constant all other prices `$(p_y)$` and income `$(m)$`! (.hi[“ceterus paribus”])

- Note: .hi-purple[each of these is a consumer's optimum at a given price!]
]

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> price </th>
   <th style="text-align:right;"> quantity </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 5 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 7 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 8 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 9 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
</tbody>
</table>
]
]

---

# Demand Curve

- .hi[Demand curve] graphically represents the demand schedule

- Also measures a person's .hi-purple[maximum willingness to pay (WTP)] for a given quantity

- .hi[Law of Demand (price effect)] `$\implies$` demand curves always slope downwards

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-37-1.png" width="504" style="display: block; margin: auto;" />
]

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# Demand Function

- .hi[Demand function] relates quantity to price

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: 
`$$q=10-p$$`
]
- Not graphable (wrong axes)!

]

]

---

# Inverse Demand Function

- .hi[*Inverse* demand function] relates price to quantity
    - Take demand function and solve for `$p$`

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: 
`$$p=10-q$$`
]
- Graphable (price on vertical axis)!

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-38-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Inverse Demand Function

- .hi[*Inverse* demand function] relates price to quantity
    - Take demand function and solve for `$p$`

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: 
`$$p=10-q$$`
]

- Vertical intercept (.hi["Choke price"]): price where `\$q_D=0\$` ($10), just high enough to discourage *any* purchases

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-39-1.png" width="504" style="display: block; margin: auto;" />
]

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# Inverse Demand Function

- Read two ways:

- Horizontally: at any given price, how many units person wants to buy

- Vertically: at any given quantity, the .hi[maximum willingness to pay (WTP)] for that quantity
    - This way will be very useful later

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-40-1.png" width="504" style="display: block; margin: auto;" />
]

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# Shifts in Demand I

- Note a  simple (inverse) demand function only relates (own) **price** and **quantity**

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: `$q=10-p$` or `$p=10-q$`
]

- What about all the other .hi-purple["determinants of demand"] like income and other prices?

- They are captured in the vertical intercept (choke price)!

]

]

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# Shifts in Demand II

- A change in one of the .hi-purple["determinants of demand"] will **shift** demand curve!
    1. Change in **income** `$m$`
    2. Change in **price of other goods** `$p_y$`
    3. Change in **preferences** or **expectations** about good `$x$`

- Shows up in (inverse) demand function by a **change in intercept (choke price)**!

- See my [Visualizing Demand Shifters](https://ryansafner.shinyapps.io/Demand/)

]

]