library(tidyverse)
# ggplot() +
# stat_function(fun = function(q) ___) +
# xlim(0, 20)5.3 Profit Maximization
Definition: A firm’s profit is equal to their total revenue minus their total costs:
\[\pi = TR - TC\]
Suppose you’re starting a tutoring business. If the market price of tutoring is $20 per hour (that is, students can get tutoring for $20/hour lots of places), then you won’t get any business if you set your price higher than $20. You could set your price lower, but why would you? You can do as much tutoring as you want at a rate of $20/hour, so if you set your rate lower, you’d just be making less money.
Multiple Choice: What is your total revenue? Let \(q\) be the number of hours you spend tutoring each week (quantity sold).
- \(TR = 20 + q\)
- \(TR = 20/q\)
- \(TR = 20 - q\)
- \(TR = 20 \times q\)
Answer:
Suppose your total costs are given by \(TC = 5 + 4q + q^2\). This might involve direct costs like flyers for advertising your tutoring business, or it might involve opportunity costs: if you spend too much time tutoring each week, you can’t spend as much time studying for your own classes or taking time for leisure.
- Use ggplot and stat_function to visualize how your total costs increase as your hours spent tutoring increase.
Answer:
- Now use your answer to question 1 to write down your profit function. Visualize it using ggplot.
Answer:
- Your visualization in question 3 should show you a downward facing parabola: profits increase for low quantities of tutoring per week, then profit hits a maximum, and then starts falling as marginal costs outweigh marginal benefits. By eyeballing, what is the profit maximizing quantity of tutoring to provide each week, and how much profit do you earn?
Answer:
MB = MC
Consider the profit function \(\pi = TR - TC\). When profit is maximized (the top of the upside down parabola), the slope of the tangent line is equal to zero. That is, the derivative of \(\pi\) with respect to \(q\) is 0.
Here I take the derivative of the profit function and set it equal to zero:
\[\begin{align} \pi &= TR - TC\\ \frac{d \pi}{d q} &= \frac{d TR}{dq} - \frac{d TC}{dq} = 0\\ \frac{d TR}{dq} &= \frac{d TC}{dq} \end{align}\]
So at the profit maximizing quantity, the derivative of total revenue is equal to the derivative of total cost. The derivative of total revenue tells you how much your total revenue increases when you increase \(q\) by a little bit: this is your marginal revenue (AKA marginal benefit). Similarly, the derivative of total cost tells you how much your total cost increases when you increase \(q\) by a little bit: this is your marginal cost.
We’ve just proved that in order to profit maximize, you set MB = MC, just like we talked about in assignment 5.2.
- Take the total revenue function from question 1 and take the derivative to find marginal revenue.
Answer:
\[\begin{align} TR &= \\ MR &= \end{align}\]
- Take the total cost function from question 2 and take the derivative to find marginal cost.
Answer: \[\begin{align} TC &= \\ MC &= \end{align}\]
- Set marginal revenue equal to marginal cost and solve for \(q\). You should get the same profit maximizing level of \(q\) that you found by eyeballing in question 4.
Answer:
\[\begin{align} MR &= MC\\ \end{align}\]
More Profit Maximization Practice
- Continue to assume the firm is a price-taker. If the market price for the firm’s output good is $6 and the firm’s total cost function is given by \(TC = 400 + 2 q + \frac{q^2}{300}\), find the firm’s profit maximizing level of output and the total profit they generate. Show your calculations.
Answer:
\[\begin{align} \end{align}\]
- If the firm is a price-taker, the market price for the firm’s output good is $250, and the firm’s total cost function is given by \(TC = 200 + 50 q + 10 q^2\), find the firm’s profit maximizing level of output and the total profit they generate. Show your calculations.
Answer:
\[\begin{align} \end{align}\]