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Classwork # 1 - Use of Calculus in Economics
Classwork # 1 - Use of Calculus in Economics
Reading - Calculus in Economics - click here
Video on Calculus in Economics - click here
Calculus is use in economics to examine the rate of change between two things. The term "marginal" is used in economics to describe the way the change in one thing affect a change in something else. In mathematics, the slope shows this relationship.
Finding the slope for a straight line is simple because it is constant across the whole length of the line. The ratio of the rise in height to the run of the distance describes the slope. In economics, linear (straight line) relationships are shown in supply and demand curve.
Finding the slope for a nonlinear (curved line) relationship is more difficult because it changes. Calculus can be used to determine the slope of a curve at any point along the curve. There are many situations in economics where calculus can be used to analyze the changing relationship between to variables, such as with utility, production and cost functions.
The power rule in calculus is a simple rule for finding slope of some types of nonlinear equations. This is the formula for using the power rule:
Problem - The equation below shows a total utility function. Use the power rule to find the marginal utility equation and then use this information to find the quantity that will produce the maximum total utility.
Homework # 1 - Production
Homework # 1 - Production
Reading - Production - click here
Class Power Point on Production - click here
Real World Application - Robot Rentals - Bloomberg: The robots are coming—and not just to big outfits like automotive or aerospace plants. They’re increasingly popping up in smaller U.S. factories, warehouses, retail stores, farms, and even construction sites.
…a nascent trend of offering robots as a service—similar to the subscription models offered by software makers, wherein customers pay monthly or annual use fees rather than purchasing the products—is opening opportunities to even small companies. That financial model is what led Thomson to embrace automation. The company has robots on 27 of its 89 molding machines and plans to add more. It can’t afford to purchase the robots, which can cost $125,000 each, says Chief Executive Officer Steve Dyer. Instead, Thomson pays for the installed machines by the hour, at a cost that’s less than hiring a human employee—if one could be found, he says. “We just don’t have the margins to generate the kind of capital necessary to go out and make these broad, sweeping investments,” he says. “I’m paying $10 to $12 an hour for a robot that is replacing a position that I was paying $15 to $18 plus fringe benefits.”
…Formic is offering to set up robots and charge as little as $8 an hour, aiming first at the most tedious tasks, such as packing and unpacking products and feeding materials into existing machines. The potential market is huge, and it will only grow as the robots become more sophisticated, Farid says. (from Marginal Revolution)
Real World Application - AI and Radiology - Noah Smith on Substack
Radiology is a field optimized for human replacement, where digital inputs, pattern recognition tasks, and clear benchmarks predominate…But demand for human labor is higher than ever. In 2025, American diagnostic radiology residency programs offered a record 1,208 positions across all radiology specialties, a four percent increase from 2024, and the field’s vacancy rates are at all-time highs. In 2025, radiology was the second-highest-paid medical specialty in the country, with an average income of $520,000, over 48 percent higher than the average salary in 2015.
AI models don’t perform nearly as well once they get out in the real world and have to go far beyond their training data. And humans don’t trust the AI, so regulators and health insurers sometimes insist on human radiologists. But there are other reasons for the persistence of the radiology profession that are even more profound — and that should serve as important reminders about how little we know about the economic effect of AI.
First, AI model makers don’t actually know what radiologists do. They know radiologists read scans, but they don’t know all the other stuff they do:
Radiologists are useful for more than reading scans; a study that followed staff radiologists in three different hospitals in 2012 found that only 36 percent of their time was dedicated to direct image interpretation. More time is spent on overseeing imaging examinations, communicating results and recommendations to the treating clinicians and occasionally directly to patients, teaching radiology residents and technologists who conduct the scans, and reviewing imaging orders and changing scanning protocols. This means that, if AI were to get better at interpreting scans, radiologists may simply shift their time toward other tasks. This would reduce the substitution effect of AI. Finally, AI increases productivity, which reduces cost, which increases the number of patients who can be served. This increases the demand for radiologists’ labor:
Problem # 1 - You have been hired to advise Float's Toy Boats on issues of production. They have some questions for you. Using the information in the chart to the right to answer the following questions:
A. How many workers should the company hire to maximize its total product?
B. What worker will be the most productive - point of diminishing marginal product?
C. Why does total product continue to go up even though marginal product goes down? (Hint - this related to utilizing capital)
D. Float wants to double his production of boats and thinks that he could do that by hiring 12 workers. Why would this not work and why can he only double his production in the long run.
E. Float tells you that he sells each boat for $5 and that it costs $10 to hire each worker. How many workers should he hire? How much are his labor costs and how much revenue does he make?
F. Suppose he has to raise the wages he pays his workers to $15, but cannot change the price that he sells boats? How many workers should he now hire? Why does he need to change the number of workers he hires?
Problem # 2 - Bobby's Baubles makes wants your help in determining the optimal number of workers to hire. This is Bobby's Baubles production function:
A. Is this a short-run or long-run production function? Explain how you know this?
B. How many workers will Bobby's Baubles employ if it wants to maximize its production?
C. At what worker does production shift from increasing returns to diminishing returns? (i.e. what is the point of diminishing marginal returns?)
D. Why does the marginal product (productivity) of each worker go down after the point of diminishing marginal returns?
E. Suppose the price of baubles is $0.2 and Bobby pays her workers $1. Should she hire six or seven workers?
F. Bobby's Baubles is now trying to decide how many workers to hire and how much capital to rent. The chart to the right shows the marginal product of labor and capital. In this situation, the company wants to keep its cost of production below $10. What ratio of labor and capital should a company employ if labor costs $1 and capital costs $2?
G. How much money will Bobby's Baubles make if the price of baubles is $0.2 if it employs this level of workers and capital?
Bobby's Baubles learns that new machines have been invented that are 50% more productive (the new levels of marginal product are shown in the chart to the right). The company want to keep costs at $10 and costs remain the same - labor costs $1 and capital costs $2.
H. How many workers and machines should the company employ?
I. How much money will Bobby's Baubles make if the price of baubles is $0.2 if it employs this level of workers and capital?
J. Should the company expand its production further? What will happen to the profits if it hires another worker and another machine?
K. Suppose the previous workers want to keep their jobs, what do the workers need to do?
Optional Videos Explaining Production Functions, Short-Run Production and Long-Run Production
Printable Copy of Homework Assignment - click here
Problem # 1 - You have been hired to advise Sea Shanty Candy to give advice about production. Using the production function (equation) listed to the right to answer the following questions.
A. Is this a short-run or long-run production function? How do you know?
B. How many workers will Sea Shanty Candy need to hire to produce the maximum total product?
C. The equation in the box to the right shows the relationship between the price of a good, the marginal product of the last worker and the wage rate. Use the total product equation to determine the amount of candy produced by the worker where maximum total product is reached. Then use this information and a wage rate of $2 to calculate the lowest price that Sea Shanty Candy can sell candy.
D. Which worker will be the most productive - point of diminishing marginal returns? Why are there diminishing returns for short-run production?
The table to the right shows the long-run production choices for Sea Shanty Candy. Sea Shanty Candy can hire unskilled workers (L1), skilled workers (L2), basic machines (K1) and advanced machines (K2). Use this information to answer the following questions.
E. The equation from part C that shows the relationship between the price of a good, the marginal product of the last worker and the wage rate. Based on this information, how much of each type of labor and capital should Sea Shanty Candy employ if candy sells for $0.25?
F. How much candy will Sea Shanty Candy produce?
G. How much will Sea Shanty Candy make in profits?
H. How will Sea Shanty Candy change the amount of unskilled workers (L1), skilled workers (L2), basic machines (K1) and advanced machines (K2) it produces if the price of candy goes up to $0.33? (Use closest approximations) What is the total amount of candy produced?
I. How does the information on price and quantity produced demonstrate the law of supply? Explain and draw a graph to support your answer.
Homework # 3 - Videos on Productivity
Homework # 3 - Videos on Productivity
Printable Copy of Homework Assignment - click here
Class Power Point - click here
Video - How Employers Kill Worker Motivation - click here
Article - Thinking Outside the (Big) Box - click here
Homework # 4 - Cost
Homework # 4 - Cost
Reading - Cost - click here
Optional Videos Explaining How to Graph Cost Curve and the Concept of Returns to Scale
Classwork # 4 - Problems with Cost
Classwork # 4 - Problems with Cost
Answers to Classwork - click here
Problem # 1 - Enchanted Mirror Company is trying to decide how many enchanted mirrors to produce to succeed in the mirror market in both the short run and long run.
A. What is the fixed cost for this company? What does this imply about the company’s ability to change capital?
B. What is the lowest price the Enchanted Mirror Company can charge for enchanted mirrors?
C. Suppose the market price for exercise machines is $5. How many mirrors should the Enchanted Mirror Company make? What is the relationship between the marginal cost and the price at that quantity?
Problem # 2 - Big Belly Chair Company - The manager at the Big Belly Chair Company would like your help in determining if the company can be profitable in the chair market. The manager shows you the total cost curve for the Big Belly Chair Company in the hope that you can use this one piece of information to help the company:
A. What is the equation for the marginal cost curve? What does the marginal cost curve show?
B. What is the equation for the average cost curve? Why is it important to know information about average cost?
C. If you were running this company, what would be the minimum price that you could sell chairs at (assuming you could live without a profit)?
D. Assume that the production function (and costs) for Big Belly Chair Company is similar to other chair companies. Based on this, do you think it would be easy for new chair companies to enter the market?
F. What effect do you think this will have on the ability of Big Belly Chair Company to make a profit in the long run?
Video Showing Solution to Cost Problem - Click Here
Problem # 3 - Elaine’s Keychains is trying to make some production and marketing decisions. Use the equation for its daily total cost to answer the following questions.
A. Does this Elaine’s Keychains have any fixed cost? If so, what are they?
B. Elaine thinks she is producing at her most cost competitive point when she is producing 71.5 keychains a day? Is this correct?
C. What is the “breakeven” price that she can charge for her keychains?
D. What is the lowest price that she can sell keychains at and still stay in business for the short-run?
Homework # 5 - Problems with Cost (Answers)
Homework # 5 - Problems with Cost (Answers)
Printable Copy of Assignment - click here
Problem # 1 -Strega Nona’s Spaghetti Restaurant is trying to make some decisions about the price and quantity of plates of spaghetti that it produces and sells. Use the cost information to answer the following questions.
A. Does this Strega Nona’s Spaghetti Restaurant have any fixed cost? How do you know it has fixed costs?
B. At what price does Strega Nona’s Spaghetti Restaurant begin to earn a profit? How many plates of spaghetti will it produce at the price?
C. If the price is below the breakeven price, what is the lowest price that Strega Nona’s Spaghetti Restaurant can charge for a plate of spaghetti and stay in business? How many plates of spaghetti will it produce at that price?
D. Based on the last two questions, what is the relationship between the price of plates of spaghetti and the quantity of plates of spaghetti that sells on the market? How does the marginal cost curve fit in this relationship and why might that be significant?
Problem # 2 - Brightest Bulb Tutoring Services is trying to figure out how many students it should have as clients and how much it should charge for an hour of tutoring. The equation to the right represents its cost equation:
A. What is Brightest Bulb’s average cost equation?
B. What is Brightest Bulb’s marginal cost equation?
C. How many students should it tutor to have its lowest average cost?
D. What is the lowest price it should sell its tutoring services (if it wants to break even)?
E. Brightest Bulb faces two competitors in the market for tutoring services: Engineering Epiphany and Guaranteed Genius. The cost equations for both companies are shown to the right. Based on this information, which company will be most price competitive, and therefore most likely to succeed in the market?
Printable Copy of Assignment - click here
Extra Test Review
Extra Problems for Production and Cost These are old test questions with the answers. Below are two videos. - One shows how to solve the production problems and the other shows how to solve the cost problems. The problems presented in the videos are different from the ones on the "extra problems" but the process for solving the problems is the same.
Extra Problems - Click Here
Answers to Extra Problems - Click Here
Calculus - Production.MP4Calculus in Economics - Cost.MP4Page updated
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