class: center, middle, inverse, title-slide # 1.5 — Solving the Consumer’s Problem ## 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> --- # The Consumer's Problem: Review .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[  ] ] --- # The Consumer's Problem: Tools .pull-left[ - We now have the tools to understand consumer choices: - .hi-red[Budget constraint]: consumer's **constraints** of income and market prices - How the .red[market] trades off between two goods - .hi-green[Utility function]: consumer's **preferences** to maximize - How the .green[consumer] trades off between two goods ] .pull-right[ .center[  ] ] --- # The Consumer's Problem: Verbally .pull-left[ - The .hi[consumer's constrained optimization problem]: > choose a bundle of goods to maximize utility, subject to income and market prices ] .pull-right[ .center[  ] ] --- # The Consumer's Problem: Mathematically .pull-left[ `$$\max_{x,y} u(x,y)$$` `$$s.t. p_xx+p_yy=m$$` - This requires calculus to solve^.red[1]. We will look at **graphs** instead! ] .pull-right[ .center[  ] ] .footnote[^.red[1] See the mathematical appendix in today's class notes on how to solve it with calculus, and an example.] --- # The Consumer's Optimum: Graphically .pull-left[ - .hi[Graphical solution]: **Highest indifference curve _tangent_ to budget constraint** - Bundle A! ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" /> ] --- # The Consumer'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 - Averages `\(\succ\)` extremes! ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" /> ] --- # The Consumer'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 - Averages `\(\succ\)` extremes! - D is higher utility, but *not affordable* at current income & prices ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" /> ] --- # The Consumer's Optimum: Why Not B? .pull-left[ `$$\begin{align*} \text{indiff. curve slope} &> \text{budget constr. slope} \\\end{align*}$$` ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" /> ] --- # The Consumer's Optimum: Why Not B? .pull-left[ `$$\begin{align*} \text{indiff. curve slope} &> \text{budget constr. slope} \\ | MRS_{x,y} | &> | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &> | \frac{p_x}{p_y} | \\ | -2 | &> | -0.5 | \\\end{align*}$$` - .hi-green[Consumer] would exchange at .hi-green[2Y:1X] - .hi-red[Market] exchange rate is .hi-red[0.5Y:1X] ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" /> ] --- # The Consumer's Optimum: Why Not B? .pull-left[ `$$\begin{align*} \text{indiff. curve slope} &> \text{budget constr. slope} \\ | MRS_{x,y} | &> | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &> | \frac{p_x}{p_y} | \\ | -2 | &> | -0.5 | \\\end{align*}$$` - .hi-green[Consumer] would exchange at .hi-green[2Y:1X] - .hi-red[Market] exchange rate is .hi-red[0.5Y:1X] - Can .hi-purple[spend less on y more on x] and get .hi-purple[more utility!] ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" /> ] --- # The Consumer's Optimum: Why Not C? .pull-left[ `$$\begin{align*} \text{indiff. curve slope} &< \text{budget constr. slope} \\\end{align*}$$` ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-7-1.png" width="504" /> ] --- # The Consumer's Optimum: Why Not C? .pull-left[ `$$\begin{align*} \text{indiff. curve slope} &< \text{budget constr. slope} \\ | MRS_{x,y} | &< | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &< | \frac{p_x}{p_y} | \\ | -0.125 | &< | -0.5 | \\\end{align*}$$` - .hi-green[Consumer] would exchange at .hi-green[0.125Y:1X] - .hi-red[Market] exchange rate is .hi-red[0.5Y:1X] ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" /> ] --- # The Consumer's Optimum: Why Not C? .pull-left[ `$$\begin{align*} \text{indiff. curve slope} &< \text{budget constr. slope} \\ | MRS_{x,y} | &< | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &< | \frac{p_x}{p_y} | \\ | -0.125 | &< | -0.5 | \\\end{align*}$$` - .hi-green[Consumer] would exchange at .hi-green[0.125Y:1X] - .hi-red[Market] exchange rate is .hi-red[0.5Y:1X] - Can .hi-purple[spend less on x, more on y] and get .hi-purple[more utility!] ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" /> ] --- # The Consumer's Optimum: Why A? .pull-left[ `$$\begin{align*} \text{indiff. curve slope} &= \text{budget constr. slope} \\\end{align*}$$` ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" /> ] --- # The Consumer's Optimum: Why A? .pull-left[ `$$\begin{align*} \text{indiff. curve slope} &= \text{budget constr. slope} \\ | MRS_{x,y} | &= | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &= | \frac{p_x}{p_y} | \\ | -0.5 | &= | -0.5 | \\\end{align*}$$` - .hi-green[Consumer] would exchange at same rate as .hi-red[market] - .hi-purple[*No other combination* of (x,y) *exists* at current prices & income that could increase utility!] ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" /> ] --- # The Consumer'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="1.5-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" /> ] --- # The Consumer'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="1.5-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" /> ] --- # Visualizing the Equimarginal Rule .pull-left[ - Compare `\(MU_x\)` per $1 spent vs. `\(MU_y\)` per $1 spent - Graphs on right are *not* indifference curves! ] .pull-right[ <!-- --> ] --- # Visualizing the Equimarginal Rule .pull-left[ - Suppose you consume 4 of `\(x\)` and 12.5 of `\(y\)` (points B) `$$\frac{MU_x}{p_x} > \frac{MU_y}{p_y}$$` ] .pull-right[ <!-- --> ] --- # Visualizing the Equimarginal Rule .pull-left[ - Suppose you consume 4 of `\(x\)` and 12.5 of `\(y\)` (points B) `$$\frac{MU_x}{p_x} > \frac{MU_y}{p_y}$$` - More "bang for your buck" with `\(x\)` than `\(y\)` - Consume more `\(x\)`, less `\(y\)`! ] .pull-right[ <!-- --> ] --- # Visualizing the Equimarginal Rule .pull-left[ - At points A, consuming 10 of `\(x\)` and 5 of `\(y\)` `$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y}$$` - No change (more `\(x\)`, less `\(x\)`, more `\(y\)`, less `\(y\)`) that could increase your utility! - The optimum! Cost-adjusted marginal utilities are equalized ] .pull-right[ <!-- --> ] --- # The Consumer's Optimum: The Equimarginal Rule I `$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y} = \cdots = \frac{MU_n}{p_n}$$` - .hi[Equimarginal Rule]: consumption is optimized where the **marginal utility per dollar spent** is **equalized** across all `\(n\)` possible goods/decisions - You will always choose an option that gives higher marginal utility (e.g. if `\(MU_x > MU_y)\)` - But each option has a different cost, so we weight each option by its cost, hence `\(\frac{MU_x}{p_x}\)` --- # The Consumer's Optimum: The Equimarginal Rule II .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[  ] ] --- # Markets Equalize Everyone's MRS I .pull-left[ - .hi-purple[Markets make it so everyone faces the _same_ relative prices] - Budget constraint. slope, `\(-\frac{p_x}{p_y}\)` - Note individuals' incomes, `\(m\)`, are certainly different! - A person's optimal choice `\(\implies\)` they make same tradeoff as the market - Their MRS `\(=\)` relative price ratio - .hi-purple[markets equalize everyone's MRS] ] .pull-right[ .center[  ] ] --- # Markets Equalize Everyone's MRS II Two people will very different income and preferences face the same market prices, and choose optimal consumption (points A and A') at an exchange rate of `\(0.5Y:1X\)` .pull-left[ <img src="1.5-slides_files/figure-html/unnamed-chunk-18-1.png" width="504" /> ] .pull-right[ <img src="1.5-slides_files/figure-html/unnamed-chunk-19-1.png" width="504" /> ] --- # Optimization and Equilibrium .pull-left[ - If people can *learn* and *change* their behavior, they will always switch to a higher-valued option - If a person has no *better* choices (under current constraints), they are at an .hi[optimum] - **If everyone is at an optimum**, the **system** is in .hi[equilibrium] ] .pull-right[ .center[   ] ] --- # Practice I .smaller[ .content-box-green[ .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[ .content-box-green[ .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? ] ]