2.4 — Costs of Production — Class Content

Overview
Readings
Slides
Assignments
Problem Set 3 Due Mon Oct 18
Appendix

Wednesday, October 13, 2021

Problem Set 3 is due by the end of the day Monday October 18.

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)!

Download as PDF

Assignments

Problem Set 3 Due Mon Oct 18

Problem set 3 (on classes 2.1-2.3) is due by the end of the day Monday, October 18 by upload to Blackboard Assignments.

Appendix

The Relationship Between Returns to Scale and Costs

There is a direct relationship between a technology’s returns to scale¹ and its cost structure: the rate at which its total costs increase² and its marginal costs change³. 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 $q = l^{α} k^{β}$ can be shown with some very tedious algebra to be:

$C (w, r, q) = [{(\frac{α}{β})}^{\frac{β}{α + β}} + {(\frac{α}{β})}^{\frac{- α}{α + β}}] w^{\frac{α}{α + β}} r^{\frac{β}{α + β}} q^{\frac{1}{α + β}}$

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

$\frac{\partial C (w, r, q)}{\partial q} = M C (q) = \frac{1}{α + β} (w^{\frac{α}{α + β}} r^{\frac{β}{α + β}}) q^{(\frac{1}{α + β}) - 1}$

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

$\frac{\partial^{2} C (w, r, q)}{\partial q^{2}} = \frac{1}{α + β} (\frac{1}{α + β} - 1) (w^{\frac{α}{α + β}} r^{\frac{β}{α + β}}) q^{(\frac{1}{α + β}) - 2}$

Three possible cases:

If $\frac{1}{α + β} > 1$ , this is positive $⟹$ decreasing returns to scale

Production function exponents $α + β < 1$

If $\frac{1}{α + β} < 1$ , this is negative $⟹$ increasing returns to scale

Production function exponents $α + β > 1$

If $\frac{1}{α + β} = 1$ , this is constant $⟹$ constant returns to scale

Production function exponents $α + β = 1$

Example (Constant Returns)

Let $q = l^{0.5} k^{0.5}$ .

$\begin{aligned} C (w, r, q) & = [{(\frac{0.5}{0.5})}^{\frac{0.5}{0.5 + 0.5}} + {(\frac{0.5}{0.5})}^{\frac{- 0.5}{0.5 + 0.5}}] w^{\frac{0.5}{0.5 + 0.5}} r^{\frac{0.5}{0.5 + 0.5}} q^{\frac{1}{0.5 + 0.5}} \\ C (w, r, q) & = [1^{0.5} + 1^{- 0.5}] w^{0.5} r^{0.5} q^{0.5} \\ C (w, r, q) & = w^{0.5} r^{0.5} q^{1} \end{aligned}$

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

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

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

Marginal costs would be

$M C (q) = \frac{\partial C (q)}{\partial q} = 15$

Average costs would be

$M C (q) = \frac{C (q)}{q} = \frac{15 q}{q} = 15$

Example (Decreasing Returns)

Let $q = l^{0.25} k^{0.25}$ .

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

If $w = 9$ , $r = 25$ :

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

Marginal costs would be

$M C (q) = \frac{\partial C (q)}{\partial q} = 30 q$

Average costs would be

$A C (q) = \frac{C (q)}{q} = \frac{15 q^{2}}{q} = 15 q$

Increasing, decreasing, or constant↩︎
At a decreasing rate, at an increasing rate, or at a constant rate, respectively↩︎
Decreasing, increasing, or constant, respectively↩︎

Last updated on Oct 9, 2021