1.7 — Income & Substitution Effects

# 1.7 — Income & Substitution Effects
## ECON 306 • Microeconomic Analysis • Fall 2020
### 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/microF20"><i class="fa fa-github fa-fw"></i>ryansafner/microF20</a><br> <a href="https://microF20.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>microF20.classes.ryansafner.com</a><br>

---

# A Demand Function (Again)

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

- 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 `$x)$`
]
]

---

# The (Own) Price Effect

---

# The (Own) Price Effect

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

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

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

.center[
.hi[Price Effect] `$=$` .hi-purple[Real income effect] `$+$` .hi-purple[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 `$x$` falls to $5. Your now consume 20 units of `$x$`.

- This is the .hi[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 makes you much poorer
    - Price decrease makes you much 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

---

# 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 one good, `$x$` increases

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

- That's the .hi[substitution effect]
]

]
]

---

# Substitution Effect: Demonstration

- .hi[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-purple[Real income effect]: change in consumption due to change in real purchasing power
    - Can be positive (**normal goods**) or negative (**inferior goods**)
    - Lower price of `$x$` means you can buy more `$x$`, `$y$`, or *both* (depending on your preferences between `$x$` and `$y$`)

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

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

---

# Real Income and Substitution Effects, Graphically I

- Original optimal consumption `$(A)$`

]

]

---

# 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

]

]

---

# 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. `$(\uparrow$` `$x$`, `$\downarrow$` `$y)$`
    - Point B *must* be a *different* point on the original curve!

]

]

---

# 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. `$(\uparrow$` `$x$`, `$\downarrow$` `$y)$`
    - Point B *must* be a *different* point on the original curve!

]

]

---

# 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 more of `$x$` and/or `$y$`)
]

]

---

# 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 more 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 `$x$` falls, new optimal consumption at `$(C)$`

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

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

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

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

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

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

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

]

]

---

# Real Income and Substitution Effects, Graphically IV

- Original optimal consumption `$(A)$`

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

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

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

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

]

]

---

# Real Income and Substitution Effects: Inferior Good

]

---

# Real Income and Substitution Effects: Inferior Good

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

]

---

# Real Income and Substitution Effects: Inferior Good

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

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

]

---

# Real Income and Substitution Effects: Inferior Good

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

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

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

]

---

# Real Income and Substitution Effects: Inferior Good

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

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

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

- Price effect is *still* an `$\uparrow x$` from a `$\downarrow p_x$`!
    - Person would just prefer to spend more new purchasing power on other goods `$(y)$`!

- .hi-purple[The law of demand holds, even for inferior goods!]
    - Because subst. effect dominates real income effect
]
]
.pull-right[

]

---

# Violating the Law of Demand

---

# Recap: Real Income and Substitution Effects

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

- Exception in the theoretical .hi-purple[Giffen good]: negative R.I.E. `$>$` S.E.
    - An upward sloping demand curve!

---

---

# Demand Schedule

- .hi[Demand schedule] expresses the quantity of good a person would be willing to buy `$(q_D)$` at any given price `$(p_x)$`

- 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;"> 8 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 0 </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

- Law of Demand (price effect) `$\implies$` Demand curves always slope downwards

]

<img src="1.7-slides_files/figure-html/unnamed-chunk-22-1.png" width="504" />
]

---

# Demand Function

- .hi[Demand function] relates quantity to price

]

<img src="1.7-slides_files/figure-html/unnamed-chunk-23-1.png" width="504" />
]

---

# Inverse Demand Function

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

]

---

# Inverse Demand Function

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

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

]

<img src="1.7-slides_files/figure-html/unnamed-chunk-24-1.png" width="504" />
]

---

# 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="1.7-slides_files/figure-html/unnamed-chunk-25-1.png" width="504" />
]

---

# Deriving a Demand Curve Graphically

<img src="1.7-slides_files/figure-html/unnamed-chunk-27-1.png" width="504" />
]

]

- Demand curve for `$x$` relates consumer's 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="1.7-slides_files/figure-html/unnamed-chunk-29-1.png" width="504" />
]

]

- Demand curve for `$x$` relates consumer's 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="1.7-slides_files/figure-html/unnamed-chunk-31-1.png" width="504" />
]

]

---

# Deriving a Demand Curve Graphically

<img src="1.7-slides_files/figure-html/unnamed-chunk-33-1.png" width="504" />
]

]

---

# Deriving a Demand *Function* I

- I will always give you a (linear) demand function

- Today's class notes page shows how you can derive actual demand functions from utility functions

---

# Shifts in Demand I

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

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

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

]

]

---

# Shifts in Demand II

- A change in one of the .hi-purple["determinants of demand"] will **shift** demand curve!
    - Change in **income** `$m$`
    - Change in **price of other goods** `$p_y$` (substitutes or complements)
    - Change in **preferences** or **expectations** about good `$x$`
    - Change in **number of buyers**

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

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

]

]