class: center, middle, inverse, title-slide # 1.3 — Budget Constraint ## ECON 306 • Microeconomic Analysis • Fall 2020 ### Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microF20
microF20.classes.ryansafner.com
--- class: inverse # Outline ### [Rational Choice Theory](#5) ### [Constrained Optimization](#8) ### [Consumer Behavior: Basic Framework](#23) ### [The Budget Constraint](#27) ### [Changes in Parameters](#42) --- # The Two Major Models of Economics as a "Science" .pull-left[ ## Optimization - Agents have .hi[objectives] they value - Agents face .hi[constraints] - Make .hi[tradeoffs] to maximize objectives within constraints .center[ ![](../images/optimize.jpeg) ] ] -- .pull-right[ ## Equilibrium - Agents .hi[compete] with others over **scarce** resources - Agents .hi[adjust] behaviors based on prices - .hi[Stable outcomes] when adjustments stop .center[ ![](../images/equilibriumbalance.png) ] ] --- class: inverse, center, middle # Rational Choice Theory --- # Consumer Behavior .pull-left[ - 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] ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- # Rational Choice Theory: Beyond Consumers .pull-left[ - *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! ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- class: inverse, center, middle # Constrained Optimization --- # Constrained Optimization I .pull-left[ - 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) ] .pull-right[ .center[ ![:scale 100%](../images/optimize.jpeg) ] ] --- # Constrained Optimization II .pull-left[ - 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] ] .pull-right[ .center[ ![:scale 100%](../images/optimize.jpeg) ] ] --- # Constrained Optimization III .pull-left[ - All constrained optimization models have three moving parts: ] .pull-right[ .center[ ![:scale 100%](../images/optimize.jpeg) ] ] --- # Constrained Optimization III .pull-left[ - All constrained optimization models have three moving parts: 1. **Choose:** .hi-purple[ < some alternative >] ] .pull-right[ .center[ ![:scale 100%](../images/optimize.jpeg) ] ] --- # Constrained Optimization III .pull-left[ - All constrained optimization models have three moving parts: 1. **Choose:** .hi-purple[ < some alternative >] 2. **In order to maximize:** .hi-green[< some objective >] ] .pull-right[ .center[ ![:scale 100%](../images/optimize.jpeg) ] ] --- # Constrained Optimization III .pull-left[ - 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 >] ] .pull-right[ .center[ ![:scale 100%](../images/optimize.jpeg) ] ] --- # Constrained Optimization: Example I .pull-left[ .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) ] ] --- # Constrained Optimization: Example II .pull-left[ .content-box-green[ .hi-green[**Example**:] How should FedEx plan its delivery route? 1. **Choose:** 2. **In order to maximize:** 3. **Subject to:** ] ] .pull-right[ .center[ ![](../images/routeoptimization.png) ] ] --- # Constrained Optimization: Example III .pull-left[ .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) ] ] --- # Constrained Optimization: Example IV .pull-left[ .content-box-green[ .hi-green[**Example**:] How do elected officials make decisions in politics? 1. **Choose:** 2. **In order to maximize:** 3. **Subject to:** ] ] .pull-right[ .center[ ![](../images/veep.jpeg) ] ] --- # The Consumer's Problem .pull-left[ - The .hi[consumer's constrained optimization problem] is: ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- # The Consumer's Problem .pull-left[ - The .hi[consumer's constrained optimization problem] is: 1. **Choose:** .hi-purple[ < a consumption bundle >] ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- # The Consumer's Problem .pull-left[ - The .hi[consumer's constrained optimization problem] is: 1. **Choose:** .hi-purple[ < a consumption bundle >] 2. **In order to maximize:** .hi-green[< utility >] ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- # The Consumer's Problem .pull-left[ - 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 >] ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- class: inverse, center, middle # Consumer Behavior: Basic Framework --- # Consumption Bundles .pull-left[ - Imagine a (very strange) supermarket sells xylophones `\((x)\)` and yams `\((y)\)` - Your choices: amounts of `\(x, y\)` to buy as a .hi-purple[bundle] ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- # Consumption Bundles .pull-left[ - Represent bundles as a vector: `$$a=\begin{pmatrix} x \\ y\\ \end{pmatrix}$$` .content-box-green[ .hi-green[**Examples**:] `$$a=\begin{pmatrix} 4 \\ 12\\ \end{pmatrix}; \; b=\begin{pmatrix} 6 \\ 12\\ \end{pmatrix}; \; c=\begin{pmatrix} 21 \\ 0\\ \end{pmatrix}$$` ] ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- # Consumption Bundles: Graphically .pull-left[ - 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;" /> ] --- class: inverse, center, middle # The Budget Constraint --- # Affordability .pull-left[ - 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) ] ] --- # Affordability .pull-left[ - 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: `$$p_x x+p_y y \leq m$$` ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- # The Budget Set .pull-left[ - The set of *all* affordable bundles that a consumer can choose is called the .hi[budget set] or .hi[choice set] `$$p_x x+p_y y \leq m$$` ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] --- # The Budget Set & the Budget Constraint .pull-left[ - 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* `\(m\)`:<sup>.hi[†]</sup> `$$p_x x+p_y y = m$$` ] .pull-right[ .center[ ![:scale 100%](../images/groceryaisle.jpg) ] ] .footnote[<sup>.hi[†]</sup> Note the difference (the in/equality), budget *constraint* is the **subset** of the *budget set* that *spends all income*.] --- # The Budget Constraint, Graphically .pull-left[ - For 2 goods, `\((x, y)\)` `$$p_xx+p_yy=m$$` ] --- # The Budget Constraint, Graphically .pull-left[ - For 2 goods, `\((x, y)\)` `$$p_xx+p_yy=m$$` - Solve for `\(y\)` to graph `$$y=\frac{m}{p_y}-\frac{p_x}{p_y}x$$` ] .pull-right[ <img src="1.3-slides_files/figure-html/BC-plot0-1.png" width="432" style="display: block; margin: auto;" /> ] --- # The Budget Constraint, Graphically .pull-left[ - 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: `\(\frac{m}{p_x}\)` ] .pull-right[ <img src="1.3-slides_files/figure-html/BC-plot1-1.png" width="432" style="display: block; margin: auto;" /> ] # The Budget Constraint, Graphically .pull-left[ - 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 .pull-left[ - 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. ] --- # Interpreting the Budget Constraint .pull-left[ - 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\\) ] .pull-right[ <img src="1.3-slides_files/figure-html/BC2-plot1-1.png" width="432" style="display: block; margin: auto;" /> ] --- # Interpreting the Budget Constraint .pull-left[ - 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\\) ] .pull-right[ <img src="1.3-slides_files/figure-html/BC2-plot2-1.png" width="432" style="display: block; margin: auto;" /> ] --- # Interpreting the Budget Constraint .pull-left[ - 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\\) ] .pull-right[ <img src="1.3-slides_files/figure-html/BC2-plot3-1.png" width="432" style="display: block; margin: auto;" /> ] --- # Interpretting the Slope .pull-left[ - **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;" /> ] --- # Interpretting the Slope .pull-left[ - **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;" /> ] --- class: inverse, center, middle # Changes in Parameters --- # Changes in Parameters .pull-left[ `$$\begin{aligned} \color{#e64173}{m} &= \color{#e64173}{p_x} x + \color{#e64173}{p_y} y\\ y &= \frac{\color{#e64173}{m}}{\color{#e64173}{p_y}}-\frac{\color{#e64173}{p_x}}{\color{#e64173}{p_y}}x\\ \end{aligned}$$` - 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] ] .pull-right[ .center[ ![](../images/choices.jpg) ] ] --- # Changes in Income, `\(m\)` .pull-left[ - Changes in .hi-purple[income]: a *parallel* shift in budget constraint .content-box-green[ .hi-green[**Example**]: An increase in income - Same slope (relative prices don't change!) - .hi-green[Gain of affordable bundles] ] ] .pull-right[ <img src="1.3-slides_files/figure-html/BC-m-change-plot-1.png" width="432" style="display: block; margin: auto;" /> ] --- # 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. ] --- # Changes in Relative Prices, `\(p_x\)` or `\(p_y\)` .pull-left[ - Changes in .hi-purple[relative prices]: *rotate* the budget constraint .content-box-green[ .hi-green[**Example**]: An increase in the price of `\(x\)` - Slope steepens: `\(-\frac{p_x'}{p_y}\)` - .hi-red[Loss of affordable bundles] ] ] .pull-right[ <img src="1.3-slides_files/figure-html/BC-x-change-plot-1.png" width="432" style="display: block; margin: auto;" /> ] --- # Changes in Relative Prices, `\(p_x\)` or `\(p_y\)` .pull-left[ - Changes in .hi-purple[relative prices]: *rotate* the budget constraint .content-box-green[ .hi-green[**Example**]: A decrease in the price of `\(y\)` - Slope flattens: `\(-\frac{p_x}{p_y'}\)` - .hi-green[Gain of affordable bundles] ] ] .pull-right[ <img src="1.3-slides_files/figure-html/BC-y-change-plot-1.png" width="432" style="display: block; margin: auto;" /> ] --- # Economics is About (Changes in) *Relative* Prices .pull-left[ - .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 ] .pull-right[ .center[ ![:scale 80%](../images/scale.png) ] ] --- # Economics is About (Changes in) *Relative* Prices .pull-left[ - **"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"... ] .pull-right[ .center[ ![:scale 80%](../images/scale.png) ] ] --- # 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. ]