class: center, middle, inverse, title-slide # 1.4 — Utility Maximization ## 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> --- 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-purple[objective] *as best as they can* - Subject to .hi-turquoise[constraints]: limited resources (income, time, attention, etc) ] .pull-right[ .center[  ] ] --- # 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[  ] ] --- # Remember! .center[  ] --- # Constrained Optimization III .pull-left[ - All constrained optimization models have three moving parts: ] .pull-right[ .center[  ] ] --- # Constrained Optimization III .pull-left[ - All constrained optimization models have three moving parts: 1. **Choose:** .hi-purple[ < some alternative >] ] .pull-right[ .center[  ] ] --- # 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[  ] ] --- # 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[  ] ] --- # Constrained Optimization: Example I .pull-left[ .bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[ .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[  ] ] --- # Constrained Optimization: Example II .pull-left[ .bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[ .hi-green[**Example**:] How should FedEx plan its delivery route? 1. **Choose:** 2. **In order to maximize:** 3. **Subject to:** ] ] .pull-right[ .center[  ] ] --- # Constrained Optimization: Example III .pull-left[ .bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[ .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[  ] ] --- # Constrained Optimization: Example IV .pull-left[ .bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[ .hi-green[**Example**:] How do elected officials make decisions in politics? 1. **Choose:** 2. **In order to maximize:** 3. **Subject to:** ] ] .pull-right[ .center[  ] ] --- # The Utility Maximization Problem .pull-left[ - The individual's .hi[utility maximization problem] we've been modeling, finally, 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[  ] ] --- # The Utility Maximization Problem: Tools .pull-left[ .smallest[ - We now have the tools to understand individual choices: - .hi-red[Budget constraint]: individual’s **constraints** of income and market prices - How .red[market] trades off between goods - .red[.ul[Marginal cost]] (of good `\(x\)`, in terms of `\(y)\)` - .hi-green[Utility function]: individual’s **objective** to maximize, based on their preferences - How .green[individual] trades off between goods - .green[.ul[Marginal benefit]] (of good `\(x\)`, in terms of `\(y)\)` ] ] .pull-right[ .center[  ] ] --- # The Utility Maximization Problem: Verbally .pull-left[ - The .hi[individual's constrained optimization problem]: > choose a bundle of goods to maximize utility, subject to income and market prices ] .pull-right[ .center[  ] ] --- # The Utility Maximization Problem: Mathematically .pull-left[ `$$\max_{x,y \geq0} u(x,y)$$` `$$s.t. p_xx+p_yy=m$$` - This requires calculus to solve.^{.magenta[†]} We will look at **graphs** instead! ] .pull-right[ .center[  ] ] .footnote[^{.magenta[†]} See the [mathematical appendix](/content/1.4-content/#solving-the-constrained-optimization-problem-with-calculus) in today's class notes on how to solve it with calculus, and an example.] --- # The Individual's Optimum: Graphically .pull-left[ - .hi[Graphical solution]: **Highest indifference curve _tangent_ to budget constraint** - Bundle A! ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Graphically .pull-left[ - .hi[Graphical solution]: **Highest indifference curve _tangent_ to budget constraint** - Bundle A! - B or C spend all income, but a better combination exists ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Graphically .pull-left[ - .hi[Graphical solution]: **Highest indifference curve _tangent_ to budget constraint** - Bundle A! - B or C spend all income, but a better combination exists - D is higher utility, but *not affordable* at current income & prices ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Why Not B? .pull-left[ .smallest[ `$$\begin{align*} \color{#047806}{\text{indiff. curve slope}} &> \color{#D7250E}{\text{budget constr. slope}} \\ \end{align*}$$` ] ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Why Not B? .pull-left[ .smallest[ `$$\begin{align*} \color{#047806}{\text{indiff. curve slope}} &> \color{#D7250E}{\text{budget constr. slope}} \\ \color{#047806}{\frac{MU_x}{MU_y}} &> \color{#D7250E}{\frac{p_x}{p_y}} \\ \color{#047806}{2} &> \color{#D7250E}{0.5} \\\end{align*}$$` - .hi-green[Consumer] views MB of `\(x\)` is 2 units of `\(y\)` - Consumer’s “exchange rate:” .hi-green[2Y:1X] - .hi-red[Market]-determined MC of `\(x\)` is 0.5 units of `\(y\)` - Market exchange rate is .hi-red[0.5Y:1X] ] ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Why Not B? .pull-left[ .smallest[ `$$\begin{align*} \color{#047806}{\text{indiff. curve slope}} &> \color{#D7250E}{\text{budget constr. slope}} \\ \color{#047806}{\frac{MU_x}{MU_y}} &> \color{#D7250E}{\frac{p_x}{p_y}} \\ \color{#047806}{2} &> \color{#D7250E}{0.5} \\\end{align*}$$` - .hi-green[Consumer] views MB of `\(x\)` is 2 units of `\(y\)` - Consumer’s “exchange rate:” .hi-green[2Y:1X] - .hi-red[Market]-determined MC of `\(x\)` is 0.5 units of `\(y\)` - Market exchange rate is .hi-red[0.5Y:1X] - Can .hi-purple[spend less on y, more on x] for .hi-purple[more utility!] ] ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Why Not C? .pull-left[ .smallest[ `$$\begin{align*} \color{#047806}{\text{indiff. curve slope}} &< \color{#D7250E}{\text{budget constr. slope}} \\ \\\end{align*}$$` ] ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-7-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Why Not C? .pull-left[ .smallest[ `$$\begin{align*} \color{#047806}{\text{indiff. curve slope}} &< \color{#D7250E}{\text{budget constr. slope}} \\ \color{#047806}{\frac{MU_x}{MU_y}} &< \color{#D7250E}{\frac{p_x}{p_y}} \\ \color{#047806}{0.125} &< \color{#D7250E}{0.5} \\\end{align*}$$` - .hi-green[Consumer] views MB of `\(x\)` is 0.125 units of `\(y\)` - Consumer’s “exchange rate:” .hi-green[0.125Y:1X] - .hi-red[Market]-determined MC of `\(x\)` is 0.5 units of `\(y\)` - Market exchange rate is .hi-red[0.5Y:1X] ] ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Why Not C? .pull-left[ .smallest[ `$$\begin{align*} \color{#047806}{\text{indiff. curve slope}} &< \color{#D7250E}{\text{budget constr. slope}} \\ \color{#047806}{\frac{MU_x}{MU_y}} &< \color{#D7250E}{\frac{p_x}{p_y}} \\ \color{#047806}{0.125} &< \color{#D7250E}{0.5} \\\end{align*}$$` - .hi-green[Consumer] views MB of `\(x\)` is 0.125 units of `\(y\)` - Consumer’s “exchange rate:” .hi-green[0.125Y:1X] - .hi-red[Market]-determined MC of `\(x\)` is 0.5 units of `\(y\)` - Market exchange rate is .hi-red[0.5Y:1X] - Can .hi-purple[spend less on y, more on x] for .hi-purple[more utility!] ] ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Why A? .pull-left[ .smallest[ `$$\begin{align*} \color{#047806}{\text{indiff. curve slope}} &= \color{#D7250E}{\text{budget constr. slope}} \\\end{align*}$$` ] ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Why A? .pull-left[ .smallest[ `$$\begin{align*} \color{#047806}{\text{indiff. curve slope}} &= \color{#D7250E}{\text{budget constr. slope}} \\ \color{#047806}{\frac{MU_x}{MU_y}} &= \color{#D7250E}{\frac{p_x}{p_y}} \\ \color{#047806}{0.5} &= \color{#D7250E}{0.5} \\\end{align*}$$` - .hi-green[Marginal benefit] = .hi-red[Marginal cost] - .hi-green[Consumer] exchanges at same rate as .hi-red[market] - .hi-purple[*No other combination* of (x,y) *exists* that could increase utility!]^{.magenta[†]} ] .footnote[^{.magenta[†]} At *current* income and market prices!] ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Two Equivalent Rules .pull-left[ ## Rule 1 `$$\frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$` - Easier for calculation (slopes) ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: Two Equivalent Rules .pull-left[ ## Rule 1 `$$\frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$` - Easier for calculation (slopes) ## Rule 2 `$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y}$$` - Easier for intuition (next slide) ] .pull-right[ <img src="1.4-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Individual's Optimum: The Equimarginal Rule `$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y} = \cdots = \frac{MU_n}{p_n}$$` - .hi[Equimarginal Rule]: consumption is optimized where the **marginal utility per dollar spent** is **equalized** across all `\(n\)` possible goods/decisions - Always choose an option that gives higher marginal utility (e.g. if `\(MU_x < MU_y)\)`, consume more `\(y\)`! - But each option has a different price, so weight each option by its price, hence `\(\frac{MU_x}{p_x}\)` --- # An Optimum, By Definition .pull-left[ - Any .hi[optimum] in economics: no better alternatives exist under current constraints - No possible change in your consumption that would increase your utility ] .pull-right[ .center[  ] ] --- # Practice I .smaller[ .bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[ .green[**Example**]: You can get utility from consuming bags of Almonds `\((a)\)` and bunches of Bananas `\((b)\)`, according to the utility function: `$$\begin{align*} u(a,b)&=ab\\ MU_a&=b \\ MU_b&=a \\ \end{align*}$$` You have an income of $50, the price of Almonds is $10, and the price of Bananas is $2. Put Almonds on the horizontal axis and Bananas on the vertical axis. 1. What is your utility-maximizing bundle of Almonds and Bananas? 2. How much utility does this provide? [Does the answer to this matter?] ] ] --- # Practice II, Cobb-Douglas! .smaller[ .bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[ .green[**Example**]: You can get utility from consuming Burgers `\((b)\)` and Fries `\((f)\)`, according to the utility function: `$$\begin{align*} u(b,f)&=\sqrt{bf} \\ MU_b&=0.5b^{-0.5}f^{0.5} \\ MU_f&=0.5b^{0.5}f^{-0.5} \\ \end{align*}$$` You have an income of $20, the price of Burgers is $5, and the price of Fries is $2. Put Burgers on the horizontal axis and Fries on the vertical axis. 1. What is your utility-maximizing bundle of Burgers and Fries? 2. How much utility does this provide? ] ]