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Table & Graph Problem - You have been hired to advise Float's Boats on issues of production. They have some questions for you. Using the information below to answer the following questions.
1. How many workers should the company hire to maximize it total product?
2. What worker will be the most productive - point of diminishing marginal product?
3. Why does total product continue to go up even though marginal product goes down? (Hint - this related to utilizing capital?)
4. 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.
5. Now it is time to plan for the long-run. Float wants to rent more capital. He tells you that his marginal product of labor is 140 and the wage rate he pays his workers is $7. The capital he is looking at renting has a marginal product of 100. How much should he be willing to pay for this capital?
Calculus Application Problem: Sprocket’s Rockets makes mechanical rockets. They want your help in determining the optimal number of workers to hire. This is Sprocket’s Rockets production function (Video of Problem Being Solved): 1. Is this a short-run or long-run production function? Explain how you know this?
2. How many workers will Sprocket’s Rockets employ if it wants to maximize its production?
3. At what worker does production shift from increasing returns to diminishing returns? (i.e. what is the point of diminishing marginal returns?)
4. Why does the marginal product (productivity) of each worker go down after the point of diminishing marginal returns?
5. Sprocket’s Rockets 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 cost $1 and capital costs $2?
7. Based on the information in the previous problem – what level of marginal productivity would workers need to achieve in order for all of the workers to keep their jobs?
6. Sprocket’s Rockets 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 costs remain the same - labor cost $1 and capital costs $2. What should the company do?
Another Calculus Application Problem - You have been hired to advise the Bumpy Joe’s Shock Absorbers to give advice about production. Using the production function (equation) listed to the right to answer the following questions. 1. Is this a short-run or long-run production function? How do you know?
2. How many workers will Bumpy Joe need to hire to reach produce the maximum total product?
3. Which worker will be the most productive - point of diminishing marginal returns?
4. Bumpy Joe is currently debating whether to add additional workers or rent more capital. If the company chooses to hire more workers, the marginal product this labor will be 30 and the workers will be paid $10. If the company chooses to hire capital, the current marginal product of capital will be 90. This company should choose to rent more capital, instead of hiring workers if the rental rate of capital is…
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