class: center, middle, inverse, title-slide # 2.5 — Short Run Profit Maximization ## ECON 306 • Microeconomic Analysis • Spring 2022 ### Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com
--- class: inverse # Outline ### [Revenues](#3) ### [Profits](#16) ### [Comparative Statics](#26) ### [Calculating Profit](#30) ### [Short-Run Shut-Down Decisions](#42) ### [The Firm's Short-Run Supply Decision](#53) --- class: inverse, center, middle # Revenues --- # Revenues for Firms in *Competitive* Industries I .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" style="display: block; margin: auto;" /> ] .pull-right[ <img src="2.5-slides_files/figure-html/industry-graph-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Revenues for Firms in *Competitive* Industries I .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" style="display: block; margin: auto;" /> ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" style="display: block; margin: auto;" /> ] -- - .blue[Demand for a firm’s product] is **perfectly elastic** at the market price -- - Where did the .red[supply curve] come from? You’ll know today --- # Revenues for Firms in *Competitive* Industries II .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" style="display: block; margin: auto;" /> ] .pull-right[ - .hi[Total Revenue] `\(R(q)=pq\)` ] --- # Average and Marginal Revenues - .hi[Average Revenue]: revenue per unit of output `$$AR(q)=\frac{R}{q}$$` - `\(AR(q)\)` is **by definition** equal to the price! (Why?) -- - .hi[Marginal Revenue]: change in revenues for each additional unit of output sold: `$$\color{#e64173}{MR(q) = \frac{\Delta R(q)}{\Delta q} \approx \frac{R_2-R_1}{q_2-q_1}}$$` - Calculus: first derivative of the revenues function - .hi-purple[For a .ul[*competitive*] firm (only), MR(q) = p, i.e. the price!] --- # Average and Marginal Revenues: Example .bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[ .green[**Example**]: A firm sells bushels of wheat in a very competitive market. The current market price is .blue[$10/bushel]. ] -- .pull-left[ For the 1<sup>st</sup> bushel sold: - What is the total revenue? - What is the average revenue? ] -- .pull-right[ For the 2<sup>nd</sup> bushel sold: - What is the total revenue? - What is the average revenue? - What is the marginal revenue? ] --- # Total Revenue, Example: Visualized .pull-left[ .quitesmall[ | `\(q\)` | `\(R(q)\)` | |----:|-------:| | `\(0\)` | `\(0\)` | | `\(1\)` | `\(10\)` | | `\(2\)` | `\(20\)` | | `\(3\)` | `\(30\)` | | `\(4\)` | `\(40\)` | | `\(5\)` | `\(50\)` | | `\(6\)` | `\(60\)` | | `\(7\)` | `\(70\)` | | `\(8\)` | `\(80\)` | | `\(9\)` | `\(90\)` | | `\(10\)` | `\(100\)` | ] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Average and Marginal Revenue, Example: Visualized .pull-left[ .quitesmall[ | `\(q\)` | `\(R(q)\)` | `\(AR(q)\)` | `\(MR(q)\)` | |----:|-------:|--------:|--------:| | `\(0\)` | `\(0\)` | `\(-\)` | `\(-\)` | | `\(1\)` | `\(10\)` | `\(10\)` | `\(10\)` | | `\(2\)` | `\(20\)` | `\(10\)` | `\(10\)` | | `\(3\)` | `\(30\)` | `\(10\)` | `\(10\)` | | `\(4\)` | `\(40\)` | `\(10\)` | `\(10\)` | | `\(5\)` | `\(50\)` | `\(10\)` | `\(10\)` | | `\(6\)` | `\(60\)` | `\(10\)` | `\(10\)` | | `\(7\)` | `\(70\)` | `\(10\)` | `\(10\)` | | `\(8\)` | `\(80\)` | `\(10\)` | `\(10\)` | | `\(9\)` | `\(90\)` | `\(10\)` | `\(10\)` | | `\(10\)` | `\(100\)` | `\(10\)` | `\(10\)` | ] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" style="display: block; margin: auto;" /> ] --- class: inverse, center, middle # Profits --- # Recall: The Firm's Two Problems .pull-left[ .smallest[ 1<sup>st</sup> Stage: .hi[firm's profit maximization problem]: 1. **Choose:** .hi-blue[ < output >] 2. **In order to maximize:** .hi-green[< profits >] 2<sup>nd</sup> Stage: .hi[firm's cost minimization problem]: 1. **Choose:** .hi-blue[ < inputs >] 2. **In order to _minimize_:** .hi-green[< cost >] 3. **Subject to:** .hi-red[< producing the optimal output >] - Minimizing costs `\(\iff\)` maximizing profits ] ] .pull-right[ .center[ ![](../images/management.jpg) ] ] --- # Visualizing Total Profit As `\(R(q)-C(q)\)` .pull-left[ - `\(\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}\)` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-7-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Visualizing Total Profit As `\(R(q)-C(q)\)` .pull-left[ - `\(\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}\)` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Visualizing Total Profit As `\(R(q)-C(q)\)` .pull-left[ - `\(\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}\)` - Graph: find `\(q^*\)` to max `\(\pi \implies q^*\)` where max distance between `\(R(q)\)` and `\(C(q)\)` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Visualizing Total Profit As `\(R(q)-C(q)\)` .pull-left[ - `\(\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}\)` - Graph: find `\(q^*\)` to max `\(\pi \implies q^*\)` where max distance between `\(R(q)\)` and `\(C(q)\)` - Slopes must be equal: `$$\color{blue}{MR(q)}=\color{red}{MC(q)}$$` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Visualizing Total Profit As `\(R(q)-C(q)\)` .pull-left[ - `\(\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}\)` - Graph: find `\(q^*\)` to max `\(\pi \implies q^*\)` where max distance between `\(R(q)\)` and `\(C(q)\)` - Slopes must be equal: `$$\color{blue}{MR(q)}=\color{red}{MC(q)}$$` .smallest[ - At `\(q^*=5\)`: - `\(\color{blue}{R(q)=50}\)` - `\(\color{red}{C(q)=40}\)` - `\(\color{green}{\pi(q)=10}\)` ] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Visualizing Profit Per Unit As `\(MR(q)\)` and `\(MC(q)\)` .pull-left[ - At low output `\(q<q^*\)`, can increase `\(\pi\)` by producing *more*: `\(\color{blue}{MR(q)}>\color{red}{MC(q)}\)` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Visualizing Profit Per Unit As `\(MR(q)\)` and `\(MC(q)\)` .pull-left[ - At high output `\(q>q^*\)`, can increase `\(\pi\)` by producing *less*: `\(\color{blue}{MR(q)}<\color{red}{MC(q)}\)` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Visualizing Profit Per Unit As `\(MR(q)\)` and `\(MC(q)\)` .pull-left[ - `\(\pi\)` is *maximized* where `\(\color{blue}{MR(q)}=\color{red}{MC(q)}\)` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-14-1.png" width="504" style="display: block; margin: auto;" /> ] --- class: inverse, center, middle # Comparative Statics --- # If Market Price Changes I .pull-left[ - Suppose the market price **increases** - Firm (always setting .blue[MR]=.red[MC]) will respond by **producing more** ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-15-1.png" width="504" style="display: block; margin: auto;" /> ] --- # If Market Price Changes II .pull-left[ - Suppose the market price **decreases** - Firm (always setting .blue[MR]=.red[MC]) will respond by **producing less** ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-16-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Firm’s Supply Curve .pull-left[ - .hi-purple[The firm’s marginal cost curve is its supply curve]<sup>.magenta[‡]</sup> `$$\color{red}{p=MC(q)}$$` - How it will supply the optimal amount of output in response to the market price - .hi-purple[Firm always sets its price equal to its marginal cost] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-17-1.png" width="504" style="display: block; margin: auto;" /> ] .footnote[<sup>.magenta[‡]</sup> Mostly...there is an important **exception** we will see shortly!] --- class: inverse, center, middle # Calculating Profit --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - Profit is `$$\pi(q)=R(q)-C(q)$$` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-18-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - Profit is `$$\pi(q)=R(q)-C(q)$$` - Profit per unit can be calculated as: `$$\begin{align*} \frac{\pi(q)}{q}&=\color{blue}{AR(q)}-\color{orange}{AC(q)}\\ &=\color{blue}{p}-\color{orange}{AC(q)}\\ \end{align*}$$` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-19-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - Profit is `$$\pi(q)=R(q)-C(q)$$` - Profit per unit can be calculated as: `$$\begin{align*} \frac{\pi(q)}{q}&=\color{blue}{AR(q)}-\color{orange}{AC(q)}\\ &=\color{blue}{p}-\color{orange}{AC(q)}\\ \end{align*}$$` - Multiply by `\(q\)` to get total profit: `$$\pi(q)=q\left[\color{blue}{p}-\color{orange}{AC(q)} \right]$$` ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-20-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - At market price of .blue[p* = $10] - At q* = 5 (per unit): - At q* = 5 (totals): ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-21-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - At market price of .blue[p* = $10] - At q* = 5 (per unit): - .blue[AR(5) = $10/unit] - At q* = 5 (totals): - .blue[R(5) = $50] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-22-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - At market price of .blue[p* = $10] - At q* = 5 (per unit): - .blue[AR(5) = $10/unit] - .orange[AC(5) = $7/unit] - At q* = 5 (totals): - .blue[R(5) = $50] - .red[C(5) = $35] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-23-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - At market price of .blue[p* = $10] - At q* = 5 (per unit): - .blue[AR(5) = $10/unit] - .orange[AC(5) = $7/unit] - .green[A`\\(\pi\\)`(5) = $3/unit] - At q* = 5 (totals): - .blue[R(5) = $50] - .red[C(5) = $35] - .green[`\\(\pi\\)` = $15] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-24-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - At market price of .blue[p* = $2] - At q* = 1 (per unit): - At q* = 1 (totals): ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-25-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - At market price of .blue[p* = $2] - At q* = 1 (per unit): - .blue[AR(1) = $2/unit] - At q* = 1 (totals): - .blue[R(1) = $2] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-26-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - At market price of .blue[p* = $2] - At q* = 1 (per unit): - .blue[AR(1) = $2/unit] - .orange[AC(1) = $10/unit] - At q* = 1 (totals): - .blue[R(1) = $2] - .red[C(1) = $10] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-27-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Calculating (Average) Profit as AR(q)-AC(q) .pull-left[ - At market price of .blue[p* = $2] - At q* = 1 (per unit): - .blue[AR(1) = $2/unit] - .orange[AC(1) = $10/unit] - .green[A`\\(\pi\\)`(1) = -$8/unit] - At q* = 1 (totals): - .blue[R(1) = $2] - .red[C(1) = $10] - .green[`\\(\pi\\)`(1) = -$8] ] .pull-right[ <img src="2.5-slides_files/figure-html/unnamed-chunk-28-1.png" width="504" style="display: block; margin: auto;" /> ] --- class: inverse, center, middle # Short-Run Shut-Down Decisions --- # Short-Run Shut-Down Decisions .pull-left[ - What if a firm's profits at `\(q^*\)` are **negative** (i.e. it earns **losses**)? - .hi-purple[Should it produce at all?] ] .pull-right[ .center[ ![:scale 90%](../images/emptywarehouse.jpg) ] ] --- # Short-Run Shut-Down Decisions .pull-left[ - Suppose firm chooses to produce **nothing** `\((q=0)\)`: - If it has **fixed costs** `\((f>0)\)`, its profits are: `$$\begin{align*} \pi(q)&=pq-C(q)\\ \end{align*}$$` ] .pull-right[ .center[ ![:scale 90%](../images/emptywarehouse.jpg) ] ] --- # Short-Run Shut-Down Decisions .pull-left[ - Suppose firm chooses to produce **nothing** `\((q=0)\)`: - If it has **fixed costs** `\((f>0)\)`, its profits are: `$$\begin{align*} \pi(q)&=pq-\color{red}{C(q)}\\ \pi(q)&=pq-\color{red}{f-VC(q)}\\ \end{align*}$$` ] .pull-right[ .center[ ![:scale 90%](../images/emptywarehouse.jpg) ] ] --- # Short-Run Shut-Down Decisions .pull-left[ - Suppose firm chooses to produce **nothing** `\((q=0)\)`: - If it has **fixed costs** `\((f>0)\)`, its profits are: `$$\begin{align*} \pi(q)&=pq-C(q)\\ \pi(q)&=pq-f-VC(q)\\ \pi(0)&=-f\\ \end{align*}$$` i.e. it (still) pays its fixed costs ] .pull-right[ .center[ ![:scale 90%](../images/emptywarehouse.jpg) ] ] --- # Short-Run Shut-Down Decisions .pull-left[ - A firm should choose to produce **no output** `\((q=0)\)` only when: `$$\begin{align*} \pi \text{ from producing} &< \pi \text{ from not producing}\\ \end{align*}$$` ] --- # Short-Run Shut-Down Decisions .pull-left[ - A firm should choose to produce **no output** `\((q=0)\)` only when: `$$\begin{align*} \pi \text{ from producing} &< \pi \text{ from not producing}\\ \pi(q) &< -f \\ \end{align*}$$` ] --- # Short-Run Shut-Down Decisions .pull-left[ - A firm should choose to produce **no output** `\((q=0)\)` only when: `$$\begin{align*} \pi \text{ from producing} &< \pi \text{ from not producing}\\ \pi(q) &< -f \\ pq-VC(q)-f &<-f\\ \end{align*}$$` ] --- # Short-Run Shut-Down Decisions .pull-left[ - A firm should choose to produce **no output** `\((q=0)\)` only when: `$$\begin{align*} \pi \text{ from producing} &< \pi \text{ from not producing}\\ \pi(q) &< -f \\ pq-VC(q)-f &<-f\\ pq-VC(q) &< 0\\ \end{align*}$$` ] --- # Short-Run Shut-Down Decisions .pull-left[ - A firm should choose to produce **no output** `\((q=0)\)` only when: `$$\begin{align*} \pi \text{ from producing} &< \pi \text{ from not producing}\\ \pi(q) &< -f \\ pq-VC(q)-f &<-f\\ pq-VC(q) &< 0\\ pq &< VC(q)\\ \end{align*}$$` ] --- # Short-Run Shut-Down Decisions .pull-left[ - A firm should choose to produce **no output** `\((q=0)\)` only when: `$$\begin{align*} \pi \text{ from producing} &< \pi \text{ from not producing}\\ \pi(q) &< -f \\ pq-VC(q)-f &<-f\\ pq-VC(q) &< 0\\ pq &< VC(q)\\ \color{red}{p} & \color{red}{<} \color{red}{AVC(q)}\\ \end{align*}$$` - .hi[Shut down price]: firm will shut down production *in the short run* when `\(p<AVC(q)\)` ] .pull-right[ .center[ ![:scale 90%](../images/shutitdown.jpg) ] ] --- class: inverse, center, middle # The Firm’s Short Run Supply Decision --- # The Firm’s Short Run Supply Decision .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-29-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Firm's Short Run Supply Decision .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-30-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Firm's Short Run Supply Decision .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-31-1.png" width="504" style="display: block; margin: auto;" /> ] --- # The Firm's Short Run Supply Decision .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-32-1.png" width="504" style="display: block; margin: auto;" /> ] .pull-right[ ] --- # The Firm's Short Run Supply Decision .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-33-1.png" width="504" style="display: block; margin: auto;" /> ] .pull-right[ ] --- # The Firm's Short Run Supply Decision .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-34-1.png" width="504" style="display: block; margin: auto;" /> ] .pull-right[ .center[ Firm’s .red[short run supply curve]: ] `$$\begin{cases} p=MC(q) & \text{if } p \geq AVC \\ q=0 & \text{If } p < AVC\\ \end{cases}$$` ] --- # The Firm's Short Run Supply Decision .pull-left[ <img src="2.5-slides_files/figure-html/unnamed-chunk-35-1.png" width="504" style="display: block; margin: auto;" /> ] .pull-right[ .center[ Firm’s .red[short run supply curve]: ] `$$\begin{cases} p=MC(q) & \text{if } p \geq AVC \\ q=0 & \text{If } p < AVC\\ \end{cases}$$` ] --- # Summary: **1. Choose `\(q^*\)` such that `\(MR(q)=MC(q)\)`** -- **2. Profit `\(\pi=q[p-AC(q)]\)`** -- **3. Shut down if `\(p<AVC(q)\)`** -- .center[ Firm's short run (inverse) supply: ] `$$\begin{cases} p=MC(q) & \text{if } p \geq AVC\\ q=0 & \text{If } p < AVC\\ \end{cases}$$` --- # Choosing the Profit-Maximizing Output `\(q^*\)`: Example .bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[ .smaller[ .green[**Example**]: Bob’s barbershop gives haircuts in a very competitive market, where barbers cannot differentiate their haircuts. The current market price of a haircut is $15. Bob’s daily short run costs are given by: `$$\begin{align*} C(q) &= 0.5q^2\\ MC(q) &=q\\ \end{align*}$$` ] ] .smaller[ 1. How many haircuts per day would maximize Bob’s profits? 2. How much profit will Bob earn per day? 3. Find Bob’s shut down price. 4. Write an equation for Bob’s short-run supply curve ]