16 Cost Minimization

For more information on these topics, see Varian Chapter 20: Cost Minimization.

Our goal is to study profit-maximizing firms in various market environments. While we previously examined profit maximization directly, we now take an indirect approach through cost minimization. These two perspectives—maximizing profit and minimizing costs—are dual problems. While profit maximization addresses the producer’s problem by considering the output price, cost minimization focuses on finding the least costly way to achieve a target output level. By focusing on cost minimization, we gain powerful insights into firm decision-making and market outcomes.

16.1 The Cost Minimization Problem

At its core, cost minimization is about finding the cheapest way to produce a desired level of output. Mathematically, we can express this as:

\[\begin{align} \min \ & w_1x_1 + w_2x_2 \\ \text{subject to} \ & f(x_1, x_2) = y \end{align}\]

Where:

$w_1$ and $w_2$ are the prices of two factors of production
$x_1$ and $x_2$ are the quantities of these factors
$f(x_1, x_2)$ is the production function
$y$ is the desired level of output

16.2 Parallels Between Cost Minimization and Utility Maximization

Utility Maximization Review

In utility maximization, consumers seek to maximize satisfaction subject to a budget constraint. We began with the budget line, which shows all affordable bundles given income and prices.

# Let m = 10, p1 = $4, and p2 = $1

library(tidyverse)

budgetline <- ggplot() +
  stat_function(fun = function(x1) 10 - 4 * x1) +
  xlim(0, 5) +
  ylim(0, 10)

print(budgetline)

Next, we added indifference curves, representing all bundles that provide the same level of satisfaction (utility).

# Let u = x1^.5 x2^.5
# The indifference curve representing a utility of 2 is: 2 = x1^.5 x2^.5
# Solving for x2 to graph it, I get: x2 = 4 / x1

indifferencecurves <- budgetline +
  stat_function(fun = function(x1) 4 / x1) +
  stat_function(fun = function(x1) 6.25 / x1) +
  stat_function(fun = function(x1) 9 / x1)
  
print(indifferencecurves)

The consumer maximizes utility by selecting the bundle on their budget line that touches the highest possible indifference curve, where the budget line and indifference curve are just tangent.

solution <- indifferencecurves +
  annotate(geom = "point", x = 5/4, y = 5)

print(solution)

We use the tangency condition (MRS = price ratio, or $\frac{\partial u / \partial x_1}{\partial u / \partial x_2} = \frac{p_1}{p_2}$) to algebraically solve for the consumer’s utility maximization solution.

Cost Minimization Overview

There are a lot of parallels between utility maximization and cost minimization. For the producer’s problem, we’ll start with drawing a few isocost curves: all the bundles of inputs that would cost the same. If input 1 costs $15 per unit and input 2 costs $10 per unit, we have:

\[\text{Total Cost} = 15 x_1 + 10 x_2\] For a total cost of $3,000:

\[3000 = 15 x_1 + 10 x_2\] Solve for $x_2$ to plot the curve:

\[x_2 = \frac{3000}{10} - \frac{15}{10} x_1\] Which is a straight line with a slope of input price ratio $\frac{w_1}{w_2}$. Here are isocost curves for total costs of $3,000, $4,000, and $5,000.

isocosts <- ggplot() +
  stat_function(fun = function(x1) 3000 / 10 - (15 / 10) * x1) +
  stat_function(fun = function(x1) 4000 / 10 - (15 / 10) * x1) +
  stat_function(fun = function(x1) 5000 / 10 - (15 / 10) * x1) +
  xlim(0, 400) +
  ylim(0, 400)

print(isocosts)

Now we’ll add the production function $f(x_1, x_2) = x_1^{0.6} x_2^{0.4}$. Level curves of the production function are called isoquants: this is all the bundles of inputs that will yield the same level of output. Consider an output of 200 units:

\[200 = x_1^{0.6} x_2^{0.4}\] Solve for $x_2$ so we can plot this isoquant:

\[x_2^{0.4} = \frac{200}{x_1^{0.6}}\] \[x_2 = \frac{200^{10/4}}{x_1^{3/2}}\]

isoquant <- isocosts + 
  stat_function(fun = function(x1) 200^(2.5) / (x1 ^ 1.5))

print(isoquant)

The cost-minimizing input bundle is where the isoquant for 200 units is tangent to the lowest possible isocost curve.

solution <- isoquant + 
  annotate(geom = "point", x = 200, y = 200)

print(solution)

At this tangency, the slopes of the isoquant and isocost curves are equal. The slope of the isoquant is the Marginal Rate of Technical Substitution (MRTS), while the slope of the isocost curve is the input price ratio. The tangency condition is:

\[\frac{MP_{x_1}}{MP_{x_2}} = \frac{w_1}{w_2}\] Where $MP$ refers to the marginal product of the input and $w$ refers to that input’s price. Rearranging, we have:

\[\frac{MP_{x_1}}{w_1} = \frac{MP_{x_2}}{w_2}\] To interpret this equation: suppose $x_1$ refers to labor and $x_2$ refers to capital equipment. The firm solves its cost minimization problem when an extra dollar spent on workers has the same benefit as an extra dollar spent on capital.

This is actually obvious: if an extra dollar spent on workers earns a higher return than an extra dollar spent on capital so that $\frac{MP_{x_1}}{w_1} \gt \frac{MP_{x_2}}{w_2}$, then the producer can produce more at a lower cost by spending more on workers. When the producer is doing the best they can, the returns on an extra dollar spent on any input good is equivalent.

Now think back to the utility maximization tangency condition: $\frac{\partial u / \partial x_1}{\partial u / \partial x_2} = \frac{p_1}{p_2}$. This equation says that the rate at which the consumer is willing to trade off one good for another (the marginal rate of substitution) must equal the rate at which they can trade the goods, given their prices.

We can rearrange this equation to focus on the marginal utility (MU) per dollar spent on each good:

\[\frac{MU_{x_1}}{p_1} = \frac{MU_{x_2}}{p_2}\]

This means that at the utility-maximizing solution, the additional satisfaction (or utility) from spending an extra dollar on good 1 is equal to the additional satisfaction from spending an extra dollar on good 2.

Note that if this condition didn’t hold, the consumer could increase their total utility by reallocating their spending. For instance, if spending an extra dollar on good 1 brought more utility than spending it on good 2, the consumer should buy more of good 1 and less of good 2.

16.3 Classwork 16: Cost Minimization

Tangency Condition: A firm uses labor ($L$) and capital ($K$) to produce output ($Q$) according to the production function: $Q = L^{0.7} K^{0.3}$.

The wage rate (w) is $20 per unit of labor, and the rental rate of capital (r) is $12 per unit of capital.
1. Write down the cost minimization problem for this firm if producers want their output to be 300 units (minimize ____ subject to ____).
2. Use the tangency condition to show that the optimal ratio of L to K is $L = 1.4 K$. Note: because the log transformation is a monotonic transformation, the cost minimization solution for $Q$ is the same as it is for $\log(Q)$. So, you can still take logs if you want to simplify your calculations for marginal products.
3. If the firm wants to produce 300 units of output, find the optimal quantities of L and K.
4. Calculate the total cost of production for 300 units of output.
Cobb-Douglas Shortcut: Now, let’s verify if the Cobb-Douglas shortcut works for the problem in part 1.
1. According to the Cobb-Douglas shortcut, what fraction of total cost should be spent on labor? On capital?
2. Use the Cobb-Douglas shortcut to show that $L^* = 0.035 TC$ and $K^* = 0.025 TC$.
3. Eliminate $TC$ to find the relationship between L and K: $L^* = 1.4 K^*$.
4. Using the production function and the relationship you found in part c, solve for L and K when $Q = 300$.
5. Compare your results from Problems 1 and 2. Does the Cobb-Douglas shortcut work in this case?
Reflection: Explain in your own words why the tangency condition $\frac{MP_L}{w} = \frac{MP_K}{r}$ leads to cost minimization. What units are $\frac{MP_L}{w}$ measured in?