# Problem Set 3

*Due by 11:59 PM Sunday, September 27, 2020 by PDF upload in Blackboard Assignments*

Please **type** your answers to the following questions in a document and **save as a PDF**In MS Word, or Pages, or most word processing software, File -> Save As -> PDF, or File -> Export -> PDF.

to **upload on Blackboard** under Assignments. You may handwrite answers if you will be able to scan/photograph & convert them **to a single PDF**, if they are easily readable, but this is *not preferred*. See my guide to making a PDF - an essential skill in the modern world. If you are handwriting answers, you may print the `pdf`

above and write on it, or just write on a piece of paper (we only need your answers).

*You are part of a massive class. It will be much easier for us to grade, annotate, and return to you in a timely manner if your files are PDFs, so please save us some trouble. Thank you!*

For the few questions that ask you to draw a **graph**, *try* to do so *on your computer* (use MS Paint, the drawing tools in MS Word or MS Powerpoint, plot points in MS Excel, drawing/notetaking apps, etc.), and save it as an image to include on your homework document. Again, they need not be perfect or to scale, just show that you understand the broad idea. Being able to understand and sketch the graphs is still a very important and useful skill! If all else fails, I will be lenient in grading graph questions if you are unable to technologically include a graph.

You may work together (and I highly encourage that) but you must turn in your own answers. Your TA, under my supervision, will grade homeworks 70% for completion, and for the remaining 30%, and one question will be graded for accuracy - so it is best that you try every problem, even if you are unsure how to complete it accurately.

## Concepts and Critical Thinking

Please answer the following questions briefly (1-3 sentences). Use examples as necessary. Be sure to label graphs fully, if appropriate.

### Question 1

Describe, in your own words, what the marginal rate of technical substitution means. How is it different from the slope of the isocost line?

### Question 2

Describe, in your own words, what is true at the least-cost input combination (the optimum) for a firm. Why is it the optimum? What does the equality of the slope of the isoquant curve and the slope of the isocost line *mean*, in English?

### Question 3

Explain the difference between the short run and the long run in production.

### Question 4

Describe, in your own words, what the law of diminishing marginal returns means. How can firms increase output?

## Problems

Show all work for calculations. You may lose points, even if correct, for missing work. Be sure to label graphs fully, if appropriate.

### Question 5

Suppose a firm can hire labor at $5/hour and rent capital for $20 per hour.

#### Part A

Write an equation for the total cost of the firm.

#### Part B

Suppose the firm wants to spend exactly $100. With labor on the horizontal axis and capital on the vertical axis, find the equation of the isocost line (in a graphable form), and graph it.

#### Part C

If the firm is completely automated (i.e. it uses *only* capital), how many units of capital can they employ for $100?

#### Part D

If the firm uses only labor, how many units of labor can they employ for $100?

#### Part E

What is the slope of the isocost line? What does it represent?

#### Part F

Suppose a tax on capital makes renting capital raises the price of capital to $25 per hour. What is the new (graphable) equation of the $100 isocost line? Graph the new isocost line on the same graph.

### Question 6

For each of the following production functions, identify whether the production process exhibits constant returns to scale, increasing returns to scale, or decreasing returns to scale. Be sure to show your work!

#### Part A

\[q=2L+4K\]

#### Part B

\[q=6L^{0.25}K^{0.75}\]

#### Part C

\[q=2L^{0.8}K^{0.4}\]

#### Part D

\[q=2L^{0.25}K^{0.25}\]

### Question 7

Jerry’s Berries is a small farm that has the following production function for strawberries using combinations of labor \((l)\) and land \((t)\): \[q = 2 \, lt\]

The marginal products (of labor, \(l\); and land, \(t)\) are:

\[\begin{align*} MP_l &= 2t\\ MP_t &= 2l\\ \end{align*}\]

Put labor, \(l\) on the horizontal axis and land, \(t\) on the vertical axis.

#### Part A

Write an equation for \(MRTS_{l,t}\).

#### Part B

Suppose the farm is currently using 4 units of labor and 1 unit of land. How much output (tons of strawberries) is the farm producing?

#### Part C

From its current production, how much *more* output would the farm get by utilizing 1 more unit of labor? What about 1 more unit of land (instead of labor)?

#### Part D

From its current production, how many units of land would the farm be willing to forgo in order to use one more unit of labor and still produce the same output as before? How many units of labor would the farm be willing to forgo in order to use one more unit of land and still produce the same output as before?

#### Part E

Suppose the farm can choose between input combinations of \(a = (4,1)\), \(b=(2,2)\), \(c=(2,1)\), and \(d=d(3,2)\). What outputs does each combination yield?

#### Part F

Sketch a graph, plotting bundles \(a, b, c\), and \(d\). Indicate any isoquant curve(s) they are on, and how much output each provides.

### Question 8

Mad Max’s Road Warriors fix potholes in interstate highways. Max’s road crews fill potholes using workers and shovels, always in 1 to 1 combinations. Using 1 worker with 1 shovel can fill 10 potholes in a day. A worker with 2 shovels can still only fill 10 potholes, as can 2 workers with 1 shovel.

#### Part A

What is the relationship between workers and shovels as inputs?

#### Part B

Draw three production isoquants: one corresponding to filling 10 potholes, one for 20 potholes, and one for 30 potholes.

#### Part C

Mad Max receives a state contract to fill in 30 potholes. If the price of shovels is $10, and wages are $5, what is the lowest cost Mad Max can fulfill the contract at? Plot this isocost line on the graph.

#### Part D

Add two more isocost lines to the graph, one for a total cost of $15 and one for $30. Hint: you’ll need to find the equations, and nicely, the slope doesn’t change.

#### Part E

If the cost of buying a shovel rises from $10 to $20, what will happen to the rate at which Max Max combines workers and shovels to fill the potholes? Why?

### Question 9

Your firm builds aircraft engines using both labor (L) and power tools (K). The production function for aircraft engines is \(q=10 L^{0.5}K^{0.5}\)

#### Part A

Can you completely mechanize the production of aircraft engines (i.e. produce using all capital, no labor)?

#### Part B

Suppose you currently have 100 power tools. In the short run, you cannot buy or sell power tools. Derive your short run production function.

#### Part C

Find the (i) total product of labor, (ii) average product of labor, and (iii) marginal product of labor for using 0, 1, 2, 3, 4, and 5 workers.Hint: make a table!

Round to the nearest engine. Does labor experience diminishing returns?

#### Part D

Using your answer from Part C, sketch two graphs (roughly), one of total product, and one of marginal and average product, each with labor on the horizontal axis.

#### Part E

In the long run, does the production function exhibit constant returns to scale, increasing returns to scale, or decreasing returns to scale?

### Question 10

Dunder Mifflin paper company produces reams of paper each week according to the production function: \[\begin{align*}q &=10 l^{0.5} k^{0.5}\\ MP_l& = 5l^{-0.5}k^{0.5}\\ MP_k& = 5l^{0.5}k^{-0.5}\\ \end{align*}\]

They have determined that they need to ship 1,000 reams of paper this week to Scranton, PA. Using capital costs $20, whereas labor costs $10.

#### Part A

What is the cost-minimizing combination of labor and capital that will yield 1,000 reams of paper? Round each to the nearest whole number.

#### Part B

What is the total cost of using this combination of inputs?

#### Part C

Now suppose that they need to double their output this week, and need to produce 2,000 reams of paper. How does their optimal combination of inputs change?Hint: neither the equation for MRTS nor any prices are changing!

#### Part D

What is the total cost of this new level of output?

#### Part E

Suppose management at Dunder Mifflin develops a new program that magically makes everyone at the firm more productive, such that the firm’s new production function becomes: \[\begin{align*}q &=20 l^{0.5} k^{0.5}\\ MP_l& = 10l^{-0.5}k^{0.5}\\ MP_k& = 10l^{0.5}k^{-0.5}\\ \end{align*}\]

Still needing to supply 2,000 reams of paper this week at the same input prices, what is their new optimal combination of labor and capital?

#### Part F

How much does this combination cost? What does this show you about technological improvement (or “total factor productivity”)?