1.3 — Budget Constraint

# 1.3 — Budget Constraint
## ECON 306 • Microeconomic Analysis • Fall 2020
### Ryan Safner Assistant Professor of Economics <a href="mailto:safner@hood.edu">safner@hood.edu</a> <a href="https://github.com/ryansafner/microF20">ryansafner/microF20</a> <a href="https://microF20.classes.ryansafner.com"> microF20.classes.ryansafner.com</a>

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# Outline

### [Rational Choice Theory](#5)
### [Constrained Optimization](#8)
### [Consumer Behavior: Basic Framework](#23)
### [The Budget Constraint](#27)
### [Changes in Parameters](#42)

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# The Two Major Models of Economics as a "Science"

## Optimization

- Agents have .hi[objectives] they value

- Agents face .hi[constraints]

- Make .hi[tradeoffs] to maximize objectives within constraints

## Equilibrium

- Agents .hi[compete] with others over **scarce** resources

- Agents .hi[adjust] behaviors based on prices

- .hi[Stable outcomes] when adjustments stop

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class: inverse, center, middle

# Rational Choice Theory

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# Consumer Behavior

- How do people decide:
    - which products to buy
    - which activities to dedicate their time to
    - how to save or invest/plan for the future

- A model of behavior we can extend to most scenarios

- Answers to these questions are building blocks for .hi[demand curves]
]

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# Rational Choice Theory: Beyond Consumers

- *Everyone* is a consumer
  - "Goods and services" isn't just food, clothing, etc, but *anything* that you value!

- Consumers making purchasing decisions will be our paradigmatic *example*
  - But we are really talking about how **individuals** make choices in almost *any* context!
]

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# Constrained Optimization

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# Constrained Optimization I

- We model most situations as a .hi[constrained optimization problem]:

- People .hi[optimize]: make tradeoffs to achieve their .hi[objective] *as best as they can*

- Subject to .hi[constraints]: limited resources (income, time, attention, etc)

]

---

# Constrained Optimization II

- One of the most generally useful mathematical models

- *Endless applications*: how we model nearly every decision-maker

> consumer, business firm, politician, judge, bureaucrat, voter, dictator, pirate, drug cartel, drug addict, parent, child, etc

- .hi-purple[Key economic skill: recognizing how to apply the model to a situation]

]

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# Constrained Optimization III

- All constrained optimization models have three moving parts:

]

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# Constrained Optimization III

- All constrained optimization models have three moving parts:

1. **Choose:** .hi-purple[ < some alternative >]

]

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# Constrained Optimization III

- All constrained optimization models have three moving parts:

1. **Choose:** .hi-purple[ < some alternative >]

2. **In order to maximize:** .hi-green[< some objective >]

]

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# Constrained Optimization III

- All constrained optimization models have three moving parts:

1. **Choose:** .hi-purple[ < some alternative >]

2. **In order to maximize:** .hi-green[< some objective >]

3. **Subject to:** .hi-red[< some constraints >]

]

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# Constrained Optimization: Example I

.content-box-green[
.hi-green[**Example**:] A Hood student picking courses hoping to achieve the highest GPA while getting an Econ major.

1. **Choose:**

2. **In order to maximize:**

3. **Subject to:**

]
]
.pull-right[
.center[
![](../images/education.png)
]
]

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# Constrained Optimization: Example II

1. **Choose:**

2. **In order to maximize:**

3. **Subject to:**

]
]
.pull-right[
.center[
![](../images/routeoptimization.png)
]
]

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# Constrained Optimization: Example III

.content-box-green[
.hi-green[**Example**:] The U.S. government wants to remain economically competitive but reduce emissions by 25%.

1. **Choose:**

2. **In order to maximize:**

3. **Subject to:**

]
]
.pull-right[
.center[
![:scale 100%](../images/pollution.png)
]
]

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# Constrained Optimization: Example IV

1. **Choose:**

2. **In order to maximize:**

3. **Subject to:**

]
]
.pull-right[
.center[
![](../images/veep.jpeg)
]
]

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# The Consumer's Problem

- The .hi[consumer's constrained optimization problem] is:

]

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# The Consumer's Problem

- The .hi[consumer's constrained optimization problem] is:

1. **Choose:** .hi-purple[ < a consumption bundle >]

]

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# The Consumer's Problem

- The .hi[consumer's constrained optimization problem] is:

1. **Choose:** .hi-purple[ < a consumption bundle >]

2. **In order to maximize:** .hi-green[< utility >]

]

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# The Consumer's Problem

- The .hi[consumer's constrained optimization problem] is:

1. **Choose:** .hi-purple[ < a consumption bundle >]

2. **In order to maximize:** .hi-green[< utility >]

3. **Subject to:** .hi-red[< income and market prices >]

]

---

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# Consumption Bundles

- Imagine a (very strange) supermarket sells xylophones `$(x)$` and yams `$(y)$`

- Your choices: amounts of `$x, y$` to buy as a .hi-purple[bundle]

]

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# Consumption Bundles

- Represent bundles as a vector:

`$$a=\begin{pmatrix}
			x \\
			y\\
			\end{pmatrix}$$`

`$$a=\begin{pmatrix}
			4 \\
			12\\
			\end{pmatrix}; \; b=\begin{pmatrix}
			6 \\
			12\\
			\end{pmatrix}; \; c=\begin{pmatrix}
			21 \\
			0\\
			\end{pmatrix}$$`
]
]

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# Consumption Bundles: Graphically

- We can represent bundles graphically

- We'll stick with 2 goods `$(x, y)$` in 2-dimensions
]

.pull-right[
<img src="1.3-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" style="display: block; margin: auto;" />
]

---

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# Affordability

- If you had $100 to spend, what bundles of goods `$\{x, y\}$` would you buy?

- Only those bundles that are .hi-purple[affordable]

- Denote prices of each good as `$\{\color{#e64173}{p_x}, \color{#e64173}{p_y}\}$`

- Let `$\color{#e64173}{m}$` be the amount of income a consumer has

]

.pull-right[
.center[
![:scale 100%](https://www.dropbox.com/s/dv8ce57s6azr8e8/groceryaisle.jpg?raw=1)
]
]

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# Affordability

- If you had $100 to spend, what bundles of goods `$\{x, y\}$` would you buy?

- Only those bundles that are .hi-purple[affordable]

- Denote prices of each good as `$\{\color{#e64173}{p_x}, \color{#e64173}{p_y}\}$`

- Let `$\color{#e64173}{m}$` be the amount of income a consumer has

- A bundle `$\{x, y\}$` is .hi-purple[affordable] at given prices `$\{p_x,p_y\}$` when:

`$$p_x x+p_y y \leq m$$`
]

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# The Budget Set

- The set of *all* affordable bundles that a consumer can choose is called the .hi[budget set] or .hi[choice set]

`$$p_x x+p_y y \leq m$$`

]

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# The Budget Set & the Budget Constraint

- The set of *all* affordable bundles that a consumer can choose is called the .hi[budget set] or .hi[choice set]

`$$p_x x+p_y y \leq m$$`

- The .hi[budget *constraint*] is the set of all bundles that spend *all income* `$m$`:.hi[†]

`$$p_x x+p_y y = m$$`

]

.footnote[.hi[†] Note the difference (the in/equality), budget *constraint* is the **subset** of the *budget set* that *spends all income*.]

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# The Budget Constraint, Graphically

- For 2 goods, `$(x, y)$`

`$$p_xx+p_yy=m$$`

]

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# The Budget Constraint, Graphically

- For 2 goods, `$(x, y)$`

`$$p_xx+p_yy=m$$`

- Solve for `$y$` to graph

`$$y=\frac{m}{p_y}-\frac{p_x}{p_y}x$$`

]

<img src="1.3-slides_files/figure-html/BC-plot0-1.png" width="432" style="display: block; margin: auto;" />
]

---

# The Budget Constraint, Graphically

`$$p_xx+p_yy=m$$`

- Solve for `$y$` to graph

`$$y=\frac{m}{p_y}-\frac{p_x}{p_y}x$$`

- `$y$`-intercept: `$\frac{m}{p_y}$`
- `$x$`-intercept: `$\frac{m}{p_x}$`
]

<img src="1.3-slides_files/figure-html/BC-plot1-1.png" width="432" style="display: block; margin: auto;" />
]

# The Budget Constraint, Graphically

- For 2 goods, `$(x, y)$`

`$$p_xx+p_yy=m$$`

- Solve for `$y$` to graph

`$$y=\frac{m}{p_y}-\frac{p_x}{p_y}x$$`

- `$y$`-intercept: `$\frac{m}{p_y}$`
- `$x$`-intercept: `$\frac{m}{p_x}$`
- slope: `$-\frac{p_x}{p_y}$`

]

.pull-right[
<img src="1.3-slides_files/figure-html/BC-plot2-1.png" width="432" style="display: block; margin: auto;" />
]

---

# The Budget Constraint, Graphically

- For 2 goods, `$(x, y)$`

`$$p_xx+p_yy=m$$`

- Solve for `$y$` to graph

`$$y=\frac{m}{p_y}-\frac{p_x}{p_y}x$$`

- `$y$`-intercept: `$\frac{m}{p_y}$`
- `$x$`-intercept: `$\frac{m}{p_x}$`
- slope: `$\frac{p_x}{p_y}$`

]

.pull-right[
<img src="1.3-slides_files/figure-html/BC-plot3-1.png" width="432" style="display: block; margin: auto;" />
]

---

# The Budget Constraint: Example

.content-box-green[
.hi-green[**Example**]: Suppose you have an income of $50 to spend on lattes `$(l)$` and burritos `$(b)$`. The price of lattes is $5 and the price of burritos is $10. Let `$l$` be on the horizontal axis and `$b$` be on the vertical axis.

1. Write an equation for the budget constraint (in graphable form).

2. Graph the budget constraint.
]

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# Interpreting the Budget Constraint

- Points .hi-purple[on] the line spend all income
  - A: \$$5(0x)+$10(5y) = $50\$ 
  - B: \$$5(10x)+$10(0y) = $50\$
  - C: \$$5(2x)+$10(4y) = $50\$ 
  - D: \$$5(6x)+$10(2y) = $50\$

]

<img src="1.3-slides_files/figure-html/BC2-plot1-1.png" width="432" style="display: block; margin: auto;" />
]

---

# Interpreting the Budget Constraint

- Points .hi-purple[on] the line spend all income
  - A: \$$5(0x)+$10(5y) = $50\$ 
  - B: \$$5(10x)+$10(0y) = $50\$
  - C: \$$5(2x)+$10(4y) = $50\$ 
  - D: \$$5(6x)+$10(2y) = $50\$

- Points .hi-green[beneath] the line are .hi-green[affordable] but don't use all income
  - E: \$$5(3x)+$10(2y) = $35\$

]

<img src="1.3-slides_files/figure-html/BC2-plot2-1.png" width="432" style="display: block; margin: auto;" />
]

---

# Interpreting the Budget Constraint

- Points .hi-purple[on] the line spend all income
  - A: \$$5(0x)+$10(5y) = $50\$ 
  - B: \$$5(10x)+$10(0y) = $50\$
  - C: \$$5(2x)+$10(4y) = $50\$ 
  - D: \$$5(6x)+$10(2y) = $50\$

- Points .hi-green[beneath] the line are .hi-green[affordable] but don't use all income
  - E: \$$5(3x)+$10(2y) = $35\$

- Points .hi-red[above] the line are .hi-red[unaffordable] (at current income and prices)
  - F: \$$5(6x)+$10(4y) = $70\$

]

<img src="1.3-slides_files/figure-html/BC2-plot3-1.png" width="432" style="display: block; margin: auto;" />
]

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# Interpretting the Slope

- **Slope**: market-rate of .hi-purple[tradeoff] between `$x$` and `$y$`

- .hi-purple[Relative price] of `$x$` or its .hi-purple[opportunity cost]:

> Consuming 1 more unit of `$x$` requires giving up `$\frac{p_x}{p_y}$` units of `$y$`

]

.pull-right[
<img src="1.3-slides_files/figure-html/BC-slope-plot-1.png" width="432" style="display: block; margin: auto;" />
]

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# Interpretting the Slope

- **Slope**: market-rate of .hi-purple[tradeoff] between `$x$` and `$y$`

- .hi-purple[Relative price] of `$x$` or its .hi-purple[opportunity cost]:

> Consuming 1 more unit of `$x$` requires giving up `$\frac{p_x}{p_y}$` units of `$y$`

- Foreshadowing:

> Is **your** valuation of the tradeoff between `$x$` and `$y$` the same as the market rate?

]

.pull-right[
<img src="1.3-slides_files/figure-html/unnamed-chunk-2-1.png" width="432" style="display: block; margin: auto;" />
]

---

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# Changes in Parameters

`$$\begin{aligned}
\color{#e64173}{m} &= \color{#e64173}{p_x} x + \color{#e64173}{p_y} y\\
y &= \frac{\color{#e64173}{m}}{\color{#e64173}{p_y}}-\frac{\color{#e64173}{p_x}}{\color{#e64173}{p_y}}x\\
\end{aligned}$$`

- Budget constraint is a function of specific .hi-purple[parameters]
  - `$\color{#e64173}{m}$`: income
  - `$\color{#e64173}{p_x}, \color{#e64173}{p_y}$`: market prices

- Economics: .hi-purple[how changes in constraints affect people's choices]

]

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# Changes in Income, `$m$`

- Changes in .hi-purple[income]: a *parallel* shift in budget constraint

- Same slope (relative prices don't change!)

- .hi-green[Gain of affordable bundles]
]
]

<img src="1.3-slides_files/figure-html/BC-m-change-plot-1.png" width="432" style="display: block; margin: auto;" />
]

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# Changes in Income, `$m$`: Example

.content-box-green[
**Example**: Continuing the lattes and burritos example, (income is $50, lattes are $5, burritos are $10), suppose your income doubles to $100.

1. Find the equation of the new budget constraint (in graphable form).

2. Graph the new budget constraint.
]

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# Changes in Relative Prices, `$p_x$` or `$p_y$`

- Changes in .hi-purple[relative prices]: *rotate* the budget constraint

- Slope steepens: `$-\frac{p_x'}{p_y}$`

- .hi-red[Loss of affordable bundles]
]
]

<img src="1.3-slides_files/figure-html/BC-x-change-plot-1.png" width="432" style="display: block; margin: auto;" />
]

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# Changes in Relative Prices, `$p_x$` or `$p_y$`

- Changes in .hi-purple[relative prices]: *rotate* the budget constraint

- Slope flattens: `$-\frac{p_x}{p_y'}$`

- .hi-green[Gain of affordable bundles]
]
]

<img src="1.3-slides_files/figure-html/BC-y-change-plot-1.png" width="432" style="display: block; margin: auto;" />
]

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# Economics is About (Changes in) *Relative* Prices

- .hi-purple[Economics is about (changes in) *relative* prices]

- Budget constraint slope is `$\left(\frac{p_x}{p_y}\right)$`

- Only **"real"** changes in *relative* prices (from changes in market valuations) change consumer constraints

]

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# Economics is About (Changes in) *Relative* Prices

- **"Nominal"** prices are often meaningless!

- .hi-green[**Example**]: Imagine yourself in a strange country. All you know is that the price of bread is "6"...
]

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# Changes in Relative Prices: Example

.content-box-green[
**Example**: Continuing the lattes and burritos example (income is $50, lattes are $5, burritos are $10).

1. Suppose the price of lattes doubles from $5 to $10. Find the equation of the new budget constraint and graph it.

2. Return to the original price of lattes ($5) and suppose the price of burritos falls from $10 to $5. Find the equation of the new budget constraint and graph it.
]