Cost Minimization

Context

• In contrast to profit maximization, cost minimization is less controversial. Economists and managers agree that minimizing production costs is good practice.

• In fact, cost minimization arguments frequently appear in political discussions about minimum wages.
  • How is cost minimization different from profit maximization?
  • How is it similar to profit maximization?
  • How is it used in political debates?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• Minimum Wage in Germany (Case Study)
• Basic Concepts
• Examples
• Analysis of the Minimum Wage Argument and its Limitations
• Current Field Developments

Learning Objectives

• Explain why cost maximization has such a prominent role in modern applications.
• Describe the solution of the cost minimization problem with a single input.
• Describe the solution of the cost minimization with multiple inputs.
• Analyze the relationship between conditionally demanded input factors and prices.
• Apply the concept of cost minimization to analyze the minimum wage policy.

Minimum Wage in Germany

• The minimum wage (Mindestlohn) law was introduced on January 1st, 2015.
• Its introduction was one of the most controversial topics during the 2013 elections.
• The SPD, the greens, and the left party were in favor. The FDP and CDU were less keen.

Why are Minimum Wage Laws so Controversial?

• The minimum wage increases the labor cost of production \(\rightarrow\)
• Firms have an incentive to substitute labor with other input factors (e.g., capital) while keeping production constant (based on the MRTS) \(\rightarrow\)
• This can decrease labor demand (conditional factor demand for labor) \(\rightarrow\)
• Lead to job losses and unemployment.

Pessimistic predictions

• The Ifo Institute of Munich was painting a horror scenario at the time (2014).
• It calculated that a minimum wage of \(8.50\) Euros would jeopardize \(0.9\) million jobs!
• Its predictions in 2010 were even worse, with \(1.2\) million jobs on the line!
• The reports of Ifo cannot be found online anymore.
• But you can follow the story in 20/03/2014: ifo Institut, (Krüsemann, 2018)

A Less Harsh Reality

Statista

• The reality was much less catastrophic.
• Evidence suggests that the policy did not lead to job losses.
• In fact, they suggest that the policy led to some reduction in inequalities across individuals and regions (Ahlfeldt, Roth, & Seidel, 2018).

Why some Predictions were so off?

• The cost minimization channel is sensible!
• However, it is not the only channel affecting labor demand.
• Migration affects the number of available workers and changes the labor supply.
• Changes in demand for exports affect labor demand.
• What happens in the market is a combination of demand and supply.
• These effects may reinforce or cancel out each other.

Cost Minimization

• Cost minimization is a process in economic modeling and business practice by which firms choose the input factors producing a given level of output at the lowest cost.

Production Expenditures

• The total production expenditures for using capital \(K\) and labor \(L\) are calculated by

\[E(K, L) = r K + w L\]

• Expenditures are measured in monetary units (e.g., Euros, Dollars, etc.).

• They do not directly depend on production technology.
• They do not directly depend on the produced quantity.
• They are the same if we use \(K=2\) and \(L=1\) to produce \(q=1\) or \(q=2\).
• Not very helpful.
• How can we find the minimum cost of producing a given quantity?

Example: Cost Minimization with a Root Production Function

• Suppose production is only based on labor, i.e.,

\[q = \sqrt{L}.\]

• We want to find the minimum cost of producing \(q_{0}=2\) when the wage is \(w=1\).
• This means that \(L=4\).
• So, the minimum cost is \(w L = 4\).

Isocost Lines

• When there are many input factors, many different combinations of inputs can result in the same cost.
• An isocost curve is a locus of points showing all the input combinations resulting in equal production costs.
• For example, we fix \(\hat c = E(K,L)\) and solve for capital to get

\[K = \frac{1}{r}\hat c - \frac{w}{r} L\]

Example: Cost Minimization with a Cobb-Douglas Production Function

Conditional Factor Demand

• Substituting the partial derivatives in the optimally condition gives

\[\frac{L}{K} = \frac{r}{w} \implies L = \frac{r}{w} K\]

• Substituting in the production function yields

\[q = \left(\frac{r}{w}\right)^{1/2} K \implies K = \left(\frac{w}{r}\right)^{1/2} q\]

• Similarly, we obtain (Exercise)

\[L = \left(\frac{r}{w}\right)^{1/2} q\]

• The conditional factor demand is a function that gives the cost minimizing input factor quantity for a given output quantity and input prices.

Cost Function

• Using the conditional factor demands, we can calculate the minimum expenditure required for producing \(q\)

\[c(q) = r \left(\frac{w}{r}\right)^{1/2} q + w \left(\frac{r}{w}\right)^{1/2} q = 2 \left(r w\right)^{1/2} q\]

• The cost function gives the minimum cost required for producing an output quantity given input prices.

Application: What Happens when Wages Increase?

• Consider the Cobb-Douglas case and suppose that wages increase (\(w \quad\uparrow\)).

• Conditional demand for labor decreases (\(L = \left(\frac{r}{w}\right)^{1/2} q \quad\downarrow\))
• Conditional demand for capital increases (\(K = \left(\frac{w}{r}\right)^{1/2} q \quad\uparrow\))
• Production cost increases (\(c(q) = 2 \left(r w\right)^{1/2} q \quad\uparrow\))

Substitution Effect

Profit Maximization vs Cost Minimization

Special Type of Costs

Fixed Costs

• Sometimes, production involves costs that cannot be adjusted (at least in the short-run).
• For example, the offices rented by a firm cannot change on short notice.
• Then, one component of the cost function is fixed, i.e.,

\[c(q) = \alpha + \beta q\]

• Fixed costs are business costs that do not depend on the level of production. They are the same for every amount of produced output.

Sunk Costs

• Fixed costs may either be recoverable or not.
  • Suppose that a firm owns the property where its offices are located. If production ceases at some point, it can liquidate the property and recover some of its costs.
  • Suppose that a firm rents the property where its offices are located. If production ceases at some point, it cannot recover the rents that it paid during its operation.
• Sunk costs are fixed business costs that cannot be recovered if production stops.
• Bygones are bygones: Sunk costs do not affect the future choices of (rational decision making) firms.

Average cost function

• In business practice, it is more useful to talk about the average production cost.
• The average cost is the mean production cost of a cost minimizing firm producing an output quantity under given input prices.

\[\bar{c}(q, w_1, w_2) = \frac{Total\ cost}{\#Produced\ units} = \frac{c(q)}{q}\]

Revealed Cost Minimization Inequality

• Suppose that we observe the choices \((K_{1}, L_{1})\) and \((K_{2}, L_{2})\) of a cost minimizing firm on dates \(1\) and \(2\).
• Since \((K_{1}, L_{1})\) is cost minimizing under prices \((r_{1}, w_{1})\), it should hold \[r_{1} K_{1} + w_{1} L_{1} \le r_{1} K_{2} + w_{1} L_{2}.\]
• Since \((K_{2}, L_{2})\) is cost minimizing under prices \((r_{2}, w_{2})\), it should hold \[r_{2} K_{2} + w_{2} L_{2} \le r_{2} K_{1} + w_{2} L_{1}.\]
• Adding these two inequalities gives \[\Delta r \Delta K + \Delta w \Delta L \le 0.\]
• How can we interpret this inequality?

Current Field Developments

• Cost minimization processes are often used in applications.
• In business, cost minimization is often used by firms because tracking (marginal) costs is easier than estimating marginal revenues (which is required for profit maximization).
• Other cost minimization applications focus on
  • Inventory management problems
  • Transportation problems (with applications going beyond economics)
  • Data-center (i.e., data storage) management

Comprehensive Summary

• Cost minimization is a process with many useful applications in economic modeling and business practice.
• Firms determine the inputs minimizing their costs conditional on producing a given output level.
• Cost minimization does not require information on marginal revenue (because the output level is given).
• In principle, it is easier for firms to follow because less information is required compared to profit maximization.
• Very simple idea with many practical applications.
• However, oversimplifications should be avoided as they can lead to inaccurate predictions.

Further Reading

• Varian, 2010, Chapter 21
• CORE-Team, 2017, sec. 7.3
• Krüsemann, 2018
• Ahlfeldt et al., 2018

Mathematical Details

Production Expenditures

A firm employing input factors \(\left(x_{1}, \dots, x_{n}\right)\) under prices \(\left(w_{1}, \dots, w_{n}\right)\) has expenditures \[E\left(x_{1}, \dots, x_{n}\right) = \sum_{j=1}^{n} x_{j} w_{j}\] If a factor is fixed, then there is a fixed component to expenditures. For instance if \(x_{1}\) is fixed to \(\hat x_{1}\), we have \[c\left(x_{2}, \dots, x_{n}\right) := c\left(\hat x_{1}, x_{2}, \dots, x_{n}\right) = \underbrace{\hat x_{1} w_{1}}_{fixed\ cost,\ say\ w_{0}} + \sum_{j=2}^{n} x_{j} w_{j}\]

Isocost lines

The isocost loci give the combinations of inputs resulting in a given level of costs \(\hat c\), i.e., \[\left\{ x \in \mathbb{R}^{n} \colon\ \hat c = \sum_{j=1}^n w_{j}x_{j} \right\}. \] For the case of two inputs (\(n=2\)), we obtain the isocost line in the input plane by solving for \(x_{2}\). Thus, \[x_{2} = \frac{\hat c}{w_{2}} - x_{1} \frac{w_{1}}{w_{2}}\]

Single factor cost minimization

The problem statement is

\begin{align*} \min_{x} &\left\{ w_{0} + w x \right\} \\ s.t.\quad & f(x) \ge q . \end{align*}

If the production function is (strictly) monotonic (for instance, a root production function) and the factor price \(w\) is positive, then the solution to the problem is obtained by merely inverting the production function. Thus,

\begin{align*} x(q) &= f^{-1}\left( q \right) , \\ c(q, w) &= w_{0} + w x(q) . \end{align*}

Multiple factor cost minimization

The problem is given by

\begin{align*} \min_{x_{1}, x_{2}} &\left\{ w_{1} x_{1} + w_{2} x_{2} \right\} \\ s.t.\quad & f(x_{1}, x_{2}) \ge q. \end{align*}

For non-boundary solutions, a variations arguments shows that the ratio of marginal products is equal to the ratio of factor prices at interior, cost minimizing allocations. This means

\begin{align*} \mathrm{MRTS}(x_{1}, x_{2}) = \frac{\partial f(x_{1}, x_{2}) / \partial x_{1}}{\partial f(x_{1}, x_{2}) / \partial x_{2}} = \frac{w_{1}}{w_{2}}. \end{align*}

Conditional Factor Demand

The last equation typically results in a relationship between the two input factors \(x_{1}\) and \(x_{2}\). One can usually eliminate one of the two factors and express the remaining factor as a function of the produced quantity by combining this relationship with the problem’s constraint (the production function inequality). This determines the conditional (on the output level) factor demands \[x_{i}(q, w_1, w_2) \quad (i = 1,2),\] which show how much from each factor would a cost minimizing firm aiming to produce \(q\) under prices \(q\) would like to hire. When the discussion focuses on the relationship of demanded input factors with output, prices are typically omitted from the argument list, and we simply write \[x_{i}(q) \quad (i = 1,2).\]

Cost function

The cost function can be obtained by \[c(q, w_1, w_2) = w_{1} x_{1}(q, w_1, w_2) + w_{2} x_{2}(q, w_1, w_2).\] Similarly to the conditional factor demand, prices are often omitted from the cost function’s arguments, i.e., \[c(q) = w_{1} x_{1}(q) + w_{2} x_{2}(q)\]

Implications of returns to scales on costs

If the production technology exhibits constant returns to scale, then \[ c(t q) = t c(q) \quad \quad (t>0).\] If the production technology exhibits increasing returns to scale, then \[ c(t q) < t c(q) \quad \quad (t>1).\] Lastly, for production technologies exhibiting decreasing returns to scale, we have \[ c(t q) > t c(q) \quad \quad (t>1).\]

Exercises

Group A

1. Consider a firm with a production function \(f(x_{1}, x_{2}) = 4 x_{1}^{1/2} x_{2}^{1/2}\). The input prices are \(w_{1} = 40\) and \(w_{2} = 10\).

   1. What is the slope of the isocost lines with respect to the first input factor?
   2. What is the cost minimizing marginal rate of technical substitution between \(x_{1}\) and \(x_{2}\)?
   3. Find the cost minimizing conditional factor demanded quantities and the minimum production cost when the firm produces an output level \(q\).
   4. Calculate the cost minimizing conditional factor demanded quantities and the minimum production cost when \(q=40\).

   1. For a fixed level of cost \(\hat c\), we have \(\hat c = 40 x_{1} + 10 x_{2}\). Solving for \(x_{2}\) gives \[ x_{2} = \frac{1}{10}\hat c - 4 x_{1}, \] so the slope of the isocost with respect to \(x_{1}\) is equal to \(-4\).

   2. From the cost minimizing condition, one can calculate \[ \mathrm{MRTS}(x_{1}, x_{2}) = \frac{w_{1}}{w_{2}} = 4. \] The firm substitutes a unit of \(x_{1}\) with \(4\) units of \(x_{2}\) at the minimizing cost level of production.

   3. Using the cost minimizing condition, we obtain \[ 4 = \mathrm{MRTS}(x_{1}, x_{2}) = \frac{f_{x_{1}}(x_{1}, x_{2})}{f_{x_{2}}(x_{1}, x_{2})} = \frac{2 x_{1}^{-1/2} x_{2}^{1/2}}{2 x_{1}^{1/2} x_{2}^{-1/2}} = \frac{x_{2}}{x_{1}}. \] Solving for \(x_{2}\) and substituting into the production function gives \[ q = 4 x_{1}^{1/2} x_{2}^{1/2} = 4 x_{1}^{1/2} 2 x_{1}^{1/2} = 8 x_{1}. \] The last equation implies that \(x_{1} = q / 8\) and \(x_{2} = q / 2\). Hence, the cost function is \[ c(q) = w_{1} x_{1} + w_{2} x_{2} = 40 \frac{q}{8} + 10 \frac{q}{2} = 10 q. \]

   4. Substituting the given output level results in \(x_{1} = 5\) and \(x_{2} = 20\). Moreover, the minimum cost is \(400\).

2. Consider a firm with a production technology \(f\) having two factors that exhibit diminishing marginal rates of technical substitution. The input factor prices, \(w_{1}\) and \(w_{2}\), are fixed. The firm produces an output level \(q\) using input factors quantities such that \(\mathrm{MP}(x_{1}; x_{2}) / w_{1} > \mathrm{MP}(x_{2}; x_{1}) / w_{2}\) holds. Is this a cost minimizing firm? If not, how can the firm reduce its cost?

   The firm does not minimize its cost because the variational condition \(\mathrm{MRTS}(x_{1}, x_{2}) = w_{1} / w_{2}\) is not satisfied. Since \(w_{1}\) and \(w_{2}\) are fixed, \(\mathrm{MRTS}(x_{1}, x_{2})\) needs to decrease for the variational condition to hold. Since the marginal rate of technical substitution is decreasing, the firm can reduce its cost by increasing the input factor level of \(x_{1}\) and decreasing that of \(x_{2}\).

3. Suppose that at the beginning of a given month, we observe that a firm produces an output level \(q\) using input factors \(x_{1}=1\) and \(x_{2}=3\) with corresponding prices \(w_{1}=3\) and \(w_{2}=1\). After a month, we observe that the same firm produces the same level of output using \(x_{1}=5\) and \(x_{2}=1\) under prices \(w_{1}=5\) and \(w_{2}=3\). Does the firm minimize its cost?

   We calculate \(\Delta w_{1}=2\), \(\Delta w_{2}=2\), \(\Delta x_{1} =4\), and \(\Delta x_{2} =-2\). Therefore \[ \Delta w_{1} \Delta x_{1} + \Delta w_{2} \Delta x_{2} = 2 \cdot 4 + 2 \cdot (-2) = 4 > 0. \] Since the revealed cost minimization inequality is not satisfied, the firm does not minimize its cost at one of these two (or both) dates.

Group B

1. Prove that if a firm maximizes its profits, then it minimizes its costs.

   Suppose that a firm maximizes its profits under prices \(p\), \(w_{1}\), \(w_{2}\) when producing \(\hat q\) and demanding \(\hat x_{1}\) and \(\hat x_{2}\). Towards contradiction, suppose that \(\hat x_{1}\) and \(\hat x_{2}\) do not minimize the firm’s cost for producing \(\hat q\) under the given prices. Then, there exist \(\tilde x_{1}\) and \(\tilde x_{2}\), different than \(\hat x_{1}\) and \(\hat x_{2}\), that produce \(\hat q\) with less cost. Then

   \begin{align*} w_{1} \hat x_{1} + w_{2} \hat x_{2} &> w_{1} \tilde x_{1} + w_{2} \tilde x_{2} \implies \\ \hat \pi = p \hat q - w_{1} \hat x_{1} - w_{2} \hat x_{2} &< p \hat q - w_{1} \tilde x_{1} - w_{2} \tilde x_{2}, \end{align*}

   which implies that \(\hat q\), \(\hat x_{1}\), and \(\hat x_{2}\) are not profit maximizing.

2. What are the differences between the conditional factor demand and the factor demand? How are they related?

   The conditional factor demand is obtained by the cost minimization problem, while the factor demand is obtained by the profit maximization problem. The conditional factor demand depends on the output quantity, say \(q\), and factor prices, say \(w_{i}\). Instead, the factor demand depends on the output price, say \(p\), and factor prices.

   Let \(x_{i}^{c}\) and \(x_{i}\) correspondingly denote the conditional factor and factor demands. By exercise 1, if a firm is maximizing its profits, then its minimizes it costs. Therefore, if \(q(p, w_{1}, w_{2})\) is the supply function obtained by the profit maximization problem, then the two demands are interrelated by

   \begin{align*} x_{i}^{c}\left(q(p, w_{1}, w_{2}), w_{1}, w_{2}\right) = x_{i}\left(p, w_{1}, w_{2}\right). \end{align*}

3. Consider a firm with a Cobb-Douglas production function \(f(x_{1}, x_{2}) = A x_{1}^{\alpha r} x_{2}^{(1 - \alpha) r}\) for \(0 < \alpha < 1\) and \(r \neq 1\). Input prices, namely \(w_{1}\), and \(w_{2}\), are fixed.

   1. Write the Lagrangian of the cost minimization problem of the firm.
   2. Calculate the first order conditions of the cost minimization problem.
   3. Calculate the conditional input factor demands.
   4. Calculate the cost function of the firm.

   1. The Lagrangian of the problem is obtained by subtracting from the objective of the firm the production constraint multiplied by the Lagrange multiplier \(\lambda\). Specifically,

      \begin{align*} \mathcal{L} = w_{1} x_{1} + w_{2} x_{2} - \lambda \left(A x_{1}^{\alpha r} x_{2}^{(1 - \alpha) r} - q\right). \end{align*}

   2. Differentiating the Lagrangian of the problem gives the first order conditions

      \begin{align*} w_{1} &= \lambda A \alpha r x_{1}^{\alpha r - 1} x_{2}^{(1 - \alpha) r} = \lambda \alpha r \frac{q}{x_{1}}, \\ w_{2} &= \lambda A (1 - \alpha) r x_{1}^{\alpha r} x_{2}^{(1 - \alpha) r - 1} = \lambda (1 - \alpha) r \frac{q}{x_{2}}. \end{align*}

   3. Solving the first order conditions for \(x_{1}\) and \(x_{2}\), respectively, and substituting to the production function results in

      \begin{align*} q = A \left(\lambda \alpha r \frac{q}{w_{1}}\right)^{\alpha r} \left(\lambda (1 - \alpha) r \frac{q}{w_{2}}\right)^{(1 - \alpha) r} = A \lambda^{r} r^{r} \left(\frac{\alpha}{w_{1}}\right)^{\alpha r} \left(\frac{1 - \alpha}{w_{2}}\right)^{(1 - \alpha) r} q^{r}, \end{align*}

      from which we obtain

      \begin{align*} \lambda = A^{-\frac{1}{r}} r^{-1} \left(\frac{\alpha}{w_{1}}\right)^{-\alpha} \left(\frac{1 - \alpha}{w_{2}}\right)^{-(1 - \alpha)} q^{\frac{1-r}{r}}, \end{align*}

      Substituting back to the first order conditions, we can eliminate \(\lambda\), and solve for \(x_{1}\), \(x_{2}\) to obtain the conditional demand functions

      \begin{align*} x_{1} &= \lambda \alpha r \frac{q}{w_{1}} = \left(\frac{w_{2}}{w_{1}}\frac{\alpha}{1 - \alpha}\right)^{1-\alpha} \left(\frac{q}{A}\right)^{\frac{1}{r}}, \\ x_{2} &= \lambda (1 - \alpha) r \frac{q}{w_{2}} = \left(\frac{w_{1}}{w_{2}}\frac{1 - \alpha}{\alpha}\right)^{\alpha} \left(\frac{q}{A}\right)^{\frac{1}{r}}. \end{align*}

   4. The easiest way to calculate the cost function of the firm is to use the first order conditions of part 2 and the objective function. We then have

      \begin{align*} c(q, w_{1}, w_{2}) &= w_{1} \lambda \alpha r \frac{q}{w_{1}} + w_{2} \lambda (1 - \alpha) r \frac{q}{w_{2}} \\ &= \lambda r q \\ &= \left(\frac{\alpha}{w_{1}}\right)^{-\alpha} \left(\frac{1 - \alpha}{w_{2}}\right)^{-(1 - \alpha)} \left(\frac{q}{A}\right)^{\frac{1}{r}}. \end{align*}

References

References

Topic's Concepts

• average cost
• sunk costs
• fixed costs
• cost function
• conditional factor demand
• isocost
• cost minimization