*Monday, March 21, 2021*

## Overview

Today we cover costs before we put them together next class with revenues to solve the firm’s (first stage) *profit maximization problem*. While it seems we are adding quite a bit, and learning some new math problems with calculating costs, we will practice it more next class, when put together with revenues.

## Readings

- Ch. 7 in Goolsbee, Levitt, and Syverson, 2019

## Slides

Below, you can find the slides in two formats. Clicking the image will bring you to the html version of the slides in a new tab. Note while in going through the slides, you can type `h` to see a special list of viewing options, and type `o` for an outline view of all the slides.

The lower button will allow you to download a PDF version of the slides. I suggest printing the slides beforehand and using them to take additional notes in class (*not everything* is in the slides)!

## Assignments

## Problem Set 3 Due Mon Mar 21

Problem set 3 (on classes 2.1-2.3) is due by the end of the day Monday, March 21 by upload to Blackboard Assignments.

## Appendix

### Marginal Cost and Variable Cost

Marginal cost is defined as the change in total costs from a change in output: \[MC(q)=\frac{\Delta C(q)}{\Delta q}\]

Recall that total cost is the sum of fixed and variable costs

\[C(q)=f+VC(q)\] However, since fixed costs never change, any change in total cost is a change in variable cost \[\Delta C(q) = \Delta VC(q)\]

Thus, marginal cost actually measures the change in *variable costs* from a change in output:
\[MC(q)=\frac{\Delta C(q)}{\Delta q}=\frac{\Delta VC(q)}{\Delta q}\]
Thus, **fixed cost has no effect on marginal cost**, and **marginal cost is always measuring the change in variable costs** with additional output.

Furthermore, because of this relationship with marginal cost measuring the change in variable cost from additional output, for any specific quantity of output, e.g. \(q_1\), the variable cost of producing \(q_1\) can be seen on the graph below as the total area under the marginal cost curve to the left of \(q_1\):

### The Relationship Between Returns to Scale and Costs

There is a direct relationship between a technology’s returns to scale^{1} and its cost structure: the rate at which its total costs increase^{2} and its marginal costs change^{3}. This is easiest to see for a single input, such as our assumptions of the short run (where firms can change \(l\) but not \(\bar{k})\):

\[q=f(\bar{k},l)\]

### Constant Returns to Scale:

### Decreasing Returns to Scale

### Increasing Returns to Scale

### Cobb-Douglas Cost Functions

The total cost function for Cobb-Douglas production functions of the form
\[q=l^{\alpha}k^{\beta}\]
can be shown with some *very tedious* algebra to be:

\[C(w,r,q)=\left[\left(\frac{\alpha}{\beta}\right)^{\frac{\beta}{\alpha+\beta}} + \left(\frac{\alpha}{\beta}\right)^{\frac{-\alpha}{\alpha+\beta}}\right] w^{\frac{\alpha}{\alpha+\beta}} r^{\frac{\beta}{\alpha+\beta}} q^{\frac{1}{\alpha+\beta}}\]

If you take the first derivative of this (to get marginal cost), it is:

\[\frac{\partial C(w,r,q)}{\partial q}= MC(q) = \frac{1}{\alpha+\beta} \left(w^{\frac{\alpha}{\alpha+\beta}} r^{\frac{\beta}{\alpha+\beta}}\right) q^{\left(\frac{1}{\alpha+\beta}\right)-1}\]

How does marginal cost change with increased output? Take the second derivative:

\[\frac{\partial^2 C(w,r,q)}{\partial q^2}= \frac{1}{\alpha+\beta} \left(\frac{1}{\alpha+\beta} -1 \right) \left(w^{\frac{\alpha}{\alpha+\beta}} r^{\frac{\beta}{\alpha+\beta}}\right) q^{\left(\frac{1}{\alpha+\beta}\right)-2}\]

Three possible cases:

- If \(\frac{1}{\alpha+\beta} > 1\), this is positive \(\implies\)
*decreasing*returns to scale

- Production function exponents \(\alpha+\beta < 1\)

- If \(\frac{1}{\alpha+\beta} < 1\), this is negative \(\implies\)
*increasing*returns to scale

- Production function exponents \(\alpha+\beta > 1\)

- If \(\frac{1}{\alpha+\beta} = 1\), this is constant \(\implies\)
*constant*returns to scale

- Production function exponents\(\alpha+\beta = 1\)

#### Example (Constant Returns)

Let \(q=l^{0.5}k^{0.5}\).

\[\begin{align*} C(w,r,q)&=\left[\left(\frac{0.5}{0.5}\right)^{\frac{0.5}{0.5+0.5}} + \left(\frac{0.5}{0.5}\right)^{\frac{-0.5}{0.5+0.5}}\right] w^{\frac{0.5}{0.5+0.5}} r^{\frac{0.5}{0.5+0.5}} q^{\frac{1}{0.5+0.5}}\\ C(w,r,q)&= \left[1^{0.5}+1^{-0.5} \right]w^{0.5}r^{0.5}q^{0.5}\\ C(w,r,q)&= w^{0.5}r^{0.5}q^{1}\\ \end{align*}\]

Consider input prices of \(w=\$9\) and \(r=\$25\):

\[\begin{align*}C(w=9,r=25,q)&=9^{0.5}25^{0.5}q \\ & =3*5*q\\ & =15q\\\end{align*}\]

That is, total costs (at those given input prices, and technology) is equal to 15 times the output level, \(q\):

Marginal costs would be

\[MC(q) = \frac{\partial C(q)}{\partial q} = 15\]

Average costs would be

\[MC(q) = \frac{C(q)}{q} = \frac{15q}{q} = 15\]

#### Example (Decreasing Returns)

Let \(q=l^{0.25}k^{0.25}\).

\[\begin{align*} C(w,r,q)&=\left[\left(\frac{0.25}{0.25}\right)^{\frac{0.25}{0.25+0.25}} + \left(\frac{0.25}{0.25}\right)^{\frac{-0.25}{0.25+0.25}}\right] w^{\frac{0.25}{0.25+0.25}} r^{\frac{0.25}{0.25+0.25}} q^{\frac{1}{0.25+0.25}}\\ C(w,r,q)&= \left[1^{0.5}+1^{-0.5} \right]w^{0.5}r^{0.5}q^{2}\\ C(w,r,q)&= w^{0.5}r^{0.5}q^{2}\\ \end{align*}\]

If \(w=9\), \(r=25\):

\[\begin{align*}C(w=9,r=25,q)&=9^{0.5}25^{0.5}q^2 \\ & =3*5*q^2\\ & =15q^2\\\end{align*}\]

Marginal costs would be

\[MC(q) = \frac{\partial C(q)}{\partial q} = 30q\]

Average costs would be

\[AC(q) = \frac{C(q)}{q} = \frac{15q^2}{q} = 15q\]