Profit Maximization
 16 minutes read  3369 wordsContext
 Profit maximization is ubiquitous in the economic theory of firms' behavior.
 In practice, many successful managers often divert from following this objective.
 Why is, then, profit maximization used so often in theory?
 Can we still learn something from it?
 How does it work?
Course Structure Overview
Lecture Structure and Learning Objectives
Structure
 The Bloody ROI (Case Study)
 Basic Concepts
 Examples
 Current Field Developments
Learning Objectives
 Explain why profit maximization has such a prominent role in modern applications.
 Explain apparent discrepancies between economic theory and business practice.
 Describe the solution to the profit maximization problem with a single input.
 Describe the solution to the profit maximization problem with multiple inputs.
 Analyze the relationship between demanded input factors and prices.
The Bloody ROI
Mr. Cook replied –with an uncharacteristic display of emotion–that a return on investment (ROI) was not the primary consideration on such issues. “When we work on making our devices accessible by the blind,” he said, “I don’t consider the bloody ROI.” It was the same thing for environmental issues, worker safety, and other areas that don’t have an immediate profit. The company does “a lot of things for reasons besides profit motive. We want to leave the world better than we found it.”
Why Tim Cook Doesn’t Care About ‘The Bloody ROI’, (Denning 2014)
Apple’s 2014 Shareholder Meeting
 One shareholder is the National Center for Public Policy Research (NCPPR) investor group.
 A conservative think tank.
 Does not own much stock in Apple (not in the top stockholders).
 Apple had many programs targeting environmental sustainability.
 NCPPR pushed a proposal about disclosing the costs of such sustainability programs.
 Argued for more transparency in actions towards the “amorphous concept of environmental sustainability”.
 Rejected by Apple shareholders, with only \(3\%\) voting in favor.
The Q & A incident
 Tim Cook was asked two questions by the NCPPR representative in the Q&A.
 Whether these “green actions” that the company had adopted were good for profitability?
 Whether the company would commit to only taking actions that were good for profitability?
 Reports say that an angry Tim Cook included in his reply that “If you want me to do things only for ROI reasons, you should get out of this stock.”
Doesn’t Tim Cook care about Apple’s profits?
 This was not the gist of Tim Cook’s answer!
 Apple was immensely profitable at the time of the incident!
 Still, many of its individual activities produce no profits.
 The 2020 McKinsey survey indicated that 33% of managers experienced tradeoffs between purpose and profit, while 72% percent of the employees wanted purpose to be more central (Gast et al. 2020).
Why is Profit Maximization so Prominent in Theory?
 In practice, only chasing profits can lead to tunnel vision.
 Shortsighted strategies do not always have the desired effect in the longrun.
 Sometimes companies pursue other strategic goals (e.g., social and environmental responsibility).
 Antiwar campaigns, environmental programs, and free software updates might not be profitable.
 However, they help create a customer ecosystem and build customer loyalty.
 This, in turn, can lead to sustainable, greater profits in the longrun.
 Nonetheless, ignoring profits for too long can lead to insolvency!
 Where is the golden ratio? Finding it distinguishes good and bad managers!
Profit Maximization

The profit maximization assumption is the most common approach in the economic modeling of the behavior of firms.

Firms choose prices, inputs, and outputs to maximize profits.

Profit = Revenue  Cost
\[\pi(q, K, L) = \underbrace{p q}_{Revenue}  \underbrace{(r K + w L)}_{Cost}\]

Profits are usually measured in monetary units per time (e.g., Euros per week, Dollars per month, etc.).
Example: Profits with a Root Production Function
 Suppose production is only based on labor, i.e., \[q = \sqrt{L}.\]
 Then, profit is \[\pi(L) = p \sqrt{L}  w L\]
Example: Profits with a CobbDouglas Production Function
 Suppose production is given by \[q = K^{1/4} L^{1/4}.\]
 Then, profit is \[\pi(K, L) = p K^{1/4} L^{1/4}  r K  w L\]
Scaling Profits
 What happens to profit if we double all the input factors?
 We check what happens to the isoprofit curve.
 An isoprofit curve on the outputinput plane is a locus of points showing all the technologically efficient ways of combining production factors resulting in equal profits.
Example: Isoprofit and Profit Maximization
 Suppose that profit is \[\pi = p q  w L.\]
 Fix profit at \(\hat \pi\) and solve for \(q\) \[q = \frac{\hat \pi}{p} + \frac{w}{p}L\]
Factor Demand and Inverse Factor Demand
 Profit maximization induces some interesting relationships between input factors and prices.
Factor Demand
How much from each factor would a profit maximizing firm like to use?
 We check the factor demand function.
 The factor demand is a function that gives the profit maximizing input factor quantity for given input and output prices.
Example: Factor Demand for Root Production
\begin{align*} L(w) = \frac{1}{4} \frac{p^2}{w^2} \end{align*}
Inverse Factor Demand
For which price level is an input factor choice profit maximizing?
 We check the inverse factor demand function.
 The inverse factor demand is a function that gives the input price factor for which a factor choice maximizes profits.
Example: Inverse Factor Demand for Root Production
\begin{align*} w(L) = \frac{1}{2} \frac{p}{\sqrt{L}} \end{align*}
Revealed Profit Maximization Inequality

Suppose we observe choices \((q_{1}, K_{1}, L_{1})\) and \((q_{2}, K_{2}, L_{2})\) of a profit maximizing firm on dates \(1\) and \(2\).

Since \((q_{1}, K_{1}, L_{1})\) is profit maximizing under prices \((p_{1}, r_{1}, w_{1})\), then
\begin{align*} p_{1} q_{1}  r_{1} K_{1}  w_{1} L_{1} \ge p_{1} q_{2}  r_{1} K_{2}  w_{1} L_{2} . \end{align*}

Since \((q_{2}, K_{2}, L_{2})\) is profit maximizing under prices \((p_{2}, r_{2}, w_{2})\), then
\begin{align*} p_{2} q_{2}  r_{2} K_{2}  w_{2} L_{2} \ge p_{2} q_{1}  r_{2} K_{1}  w_{2} L_{1} . \end{align*}

Adding these two inequalities gives
\begin{align*} \Delta p \Delta q  \Delta r \Delta K  \Delta w \Delta L \ge 0 . \end{align*}
Interpretation
 If we find combinations of inputs, outputs, and prices for which the resulting inequality does not hold, then the firm was not maximizing profits on at least one of the dates.
 If all combinations of inputs, outputs, and prices in our data satisfy the resulting inequality, then we cannot reject the possibility of profit maximizing behavior.
Supplied quantities and prices

If the prices of input factors remain constant (i.e., \(\Delta r = \Delta w = 0\)), then
\begin{align*} \Delta p \Delta q \ge 0. \end{align*}

Changes in output quantities should have the same sign as changes in output prices.
Factor demanded quantities and prices

If the output price and the price of an input factor remain constant (say \(\Delta p = \Delta w = 0\)), then
\begin{align*} \Delta r \Delta K \le 0. \end{align*}

Changes in the price of an input factor should have the opposite sign of changes in the corresponding demanded factor quantities.
Current Field Developments
 Modern Dynamic Stochastic General Equilibrium (DSGE) macroeconomic models used by central banks have extensive microfounded industries with profit maximizing firms under different competition structures.
 Market studies use microeconomic models with profit maximizing firms under different competition structures to obtain estimates of market power.
 Other firm objectives are not very commonly used in applications.
 Past profits are one of the most prominent (but not the only one) measures of a firm’s performance.
Concise Summary
 Profit maximization is ubiquitous in economic theory and modeling applications.
 Firms determine input (factor demand) and output levels to maximize profits.
 Very simple idea, but not always realistic.
 Managers divert from profit maximization to achieve other strategic goals.
 However, completely disregarding profit maximization for long periods can lead to business failures.
Further Reading
 Varian (2010, chap. 20)
 CORE Team (2017, secs. 7.37.5)
 Denning (2014)
 Gast et al. (2020)
Mathematical Details
Profits
The general profit function of a \(m\text{output}\), \(n\text{input}\) production process is given by \[\pi\left(q_{1},… , q_{m}, x_{1},…, x_{n}\right) = \underbrace{\sum_{i=1}^m p_{i}q_{i}}_{Revenue}  \underbrace{\sum_{i=1}^n w_{i}x_{i}}_{Cost}.\]
Isoprofit loci
The isoprofit loci give the combinations of inputs and outputs resulting in a given level of profit \(\hat \pi\), i.e., \[\left\{ (q, x) \in \mathbb{R}^{m}\times\mathbb{R}^{n} \colon\ \hat \pi = \sum_{i=1}^m p_{i}q_{i}  \sum_{i=1}^n w_{i}x_{i} \right\}.\] For the single output, single input case (\(m=1=n\)), we obtain the isoprofit line in the outputinput space by solving for \(q\). Thus, \[q = \frac{\hat \pi}{p} + \frac{w}{p}x .\]
Fixed and variable factors
If a factor is fixed, then there is a fixed cost component. E.g., say that \(x_{j}\) is fixed equal to \(\hat x_{j}\). Then \[\pi = \sum_{i=1}^n p_{i}q_{i}  \underbrace{\sum_{i\neq j} w_{i}x_{i}}_{Variable\ cost}  \underbrace{w_{j}\hat x_{j}}_{fixed\ cost}.\]
Single factor profit maximization
The firm maximizes its profit by choosing the level of the input factor
\begin{align*} \max_{x} &\left\{ p q  w x \right\} \\ s.t.\quad & q \le f(x). \end{align*}
Because profits are increasing in \(q\), if the production function is monotonic (the usual case), the problem simplifies to \[\max_{x} \left\{ p f(x)  w x \right\}.\] For nonboundary solutions, the marginal rates of revenue and cost are equalized at the maximum (why?), namely \[p f'(x) = w.\]
Factor demand
How much from each factor would a profit maximizing firm like to use? The profit maximization optimality condition determines the factor demand as \[x(w, p) = (f')^{1}\left(\frac{w}{p}\right)\] On some occasions, it is more convenient to avoid inverting \(f'\) and directly work with the optimality condition. This gives us the factor price as a function of the demanded factor quantity, so \[w(x) = p f'(x).\] We then say that we work with the inverse factor demand function.
Multiple factor profit maximization
The firm maximizes its profit by choosing the levels of the input factors
\begin{align*} \max_{x_{1}, x_{2}} &\left\{ p q  w_{1} x_{1}  w_{2} x_{2} \right\} \\ s.t.\quad & q \le f(x_{1}, x_{2}). \end{align*}
Because profit is increasing in \(q\), as long as the production function is monotonic (the usual case), the problem simplifies to \[\max_{x_{1}, x_{2}} \left\{ p f(x_{1}, x_{2})  w_{1} x_{1}  w_{2} x_{2} \right\}.\] For nonboundary solutions, the marginal rates of revenue and cost are equalized for each factor at the maximum, namely
\begin{align*} p \frac{\partial f(x_{1}, x_{2})}{\partial x_{1}} &= w_{1} \\ p \frac{\partial f(x_{1}, x_{2})}{\partial x_{2}} &= w_{2}. \end{align*}
Combining the first order conditions, we see that the marginal rate of technical substitution is equal to relative prices at any interior maximum, i.e., \[\mathrm{MRTS}(x_{1}, x_{2}) = \frac{\partial f(x_{1}, x_{2}) / \partial x_{1}}{\partial f(x_{1}, x_{2}) / \partial x_{2}} = \frac{w_{1}}{w_{2}}.\]
Nonzero profits are incompatible with constant returns to scales
Suppose that \((x_{1}, x_{2})\) maximizes profits under a constant returns to scales production technology. Then, for any \(t>1\),
\begin{align*} \pi\left(t x_{1}, t x_{2}\right) &= p f(t x_{1}, t x_{2})  w_{1} t x_{1}  w_{2} t x_{2} \\ & = t \left( p f(x_{1}, x_{2})  w_{1} x_{1}  w_{2} x_{2} \right) \\ & = t \pi\left(x_{1}, x_{2}\right). \end{align*}
If \(\pi\left(x_{1}, x_{2}\right)\) is positive, then \(t \pi\left(x_{1}, x_{2}\right) > \pi\left(x_{1}, x_{2}\right)\). Since \((x_{1}, x_{2})\) is profit maximizing, \(\pi\left(x_{1}, x_{2}\right)\) cannot be positive. Repeating the argument with \(0 < t < 1\), we can exclude the possibility of negative profits. Therefore, only zero maximizing profits are compatible with constant returns to scales.
Exercises
Group A

Suppose that a firm produces output \(q\) using only labor \(L\) according to the production function \(q=\sqrt{L}\). The price of output is \(p=2\), and the price of labor is \(w=1\). Find the profit maximizing labor and the corresponding output. What is the optimal profit?
Maximize \(\pi(L) = 2\sqrt{L}  L\) to find \(L=1\) and \(q=1\). Then, the optimal profit is \(\pi=1\). 
Suppose that a firm produces output \(q\) using labor \(L\) and capital \(K\) according to the production function \(q=K^{1/4}L^{1/4}\). The price of output is \(p=2\), the wage is \(w=1\), and the interest rate is \(r=1\).
 What happens to output when capital and labor are doubled?
 Find the profit maximizing labor, capital, and output.
 Find the maximum profits.
 Is there a unique maximizing output if the production is given by \(q=K^{1/2}L^{1/2}\)? Why, or why not?
 The production function exhibits decreasing returns to scale.
 Maximize \(\pi(K, L) = 2K^{1/4}L^{1/4}  K  L\) to find \(L=K\). Substituting into the production function, we find that the maximum satisfies \(\pi(L) = 2L^{1/2}  2L\). Not all of these points are maximizing allocations. Maximizing \(\pi(L) = 2L^{1/2}  2L\) gives \(L=1/4\). Thus, \(K=1/4\) and \(q=1/2\).
 Substituting \(L=K=1/4\) into the profit function gives \(\pi=1/2\).
 If we attempt to follow the same process, we find \(\pi(L) = 2L  2L = 0\). Thus, profits are zero irrespective of the chosen allocation. Every choice of output produced by \(K=L\) constitutes a maximizing allocation. This is because the production function exhibits constant returns to scales.
Group B

Consider a firm with a two dimensional production function \(f\).
 What happens to the firm’s profits if \(f\) exhibits increasing returns to scale and the firm doubles its scale of operation while all prices remain fixed?
 What happens to the firm’s profits if \(f\) exhibits decreasing returns to scale and the firm halves its scale of operation while all prices remain fixed?

The profits of the firm more than double because the output more than doubles, while the cost doubles.
\begin{align*} \pi\left(2 x_{1}, 2 x_{2}\right) &= p f(2 x_{1}, 2 x_{2})  w_{1} 2 x_{1}  w_{2} 2 x_{2} \\ & > 2 \left( p f(x_{1}, x_{2})  w_{1} x_{1}  w_{2} x_{2} \right) \\ & = 2 \pi\left(x_{1}, x_{2}\right). \end{align*}

The profits of the firm less than halve because the output less than halves, while the cost halves.
\begin{align*} \pi\left(x_{1}, x_{2}\right) &= \pi\left(2 \frac{1}{2} x_{1}, 2 \frac{1}{2} x_{2}\right) \\ &= p f\left(2 \frac{1}{2} x_{1}, 2 \frac{1}{2} x_{2}\right)  w_{1} 2 \frac{1}{2} x_{1}  w_{2} 2 \frac{1}{2} x_{2} \\ & < 2 \left( p f\left(\frac{1}{2} x_{1}, \frac{1}{2} x_{2}\right)  w_{1} \frac{1}{2} x_{1}  w_{2} \frac{1}{2} x_{2} \right) \\ & = 2 \pi\left(\frac{1}{2} x_{1}, \frac{1}{2} x_{2}\right). \end{align*}
Hence,
\begin{align*} \pi\left(\frac{1}{2} x_{1}, \frac{1}{2} x_{2}\right) > \frac{1}{2} \pi\left(x_{1}, x_{2}\right). \end{align*}

Consider a firm with a single input factor production technology \(f\) having diminishing marginal product. The output and input prices, \(p\) and \(w\), are fixed. The firm produces at a positive output level for which \(p f'(x) > w\) holds and makes positive profits. Is this a profit maximizing firm? If not, and the firm wishes to increase its profits, should it increase or decrease the amount of the input factor?
Since the firm makes positive profits, it is better off than not producing at all, in which case its profits are nonpositive. However, the firm is not producing the profit maximizing output because the variational condition of interior solutions, namely \(p f'(x) = w\), is not satisfied. Since \(p\) and \(w\) are fixed, \(f'(x)\) needs to decrease for the variational condition to hold. This implies that the input factor level should increase because the marginal product is decreasing. 
Consider a firm with a production function \(f(x_{1}, x_{2}) = x_{1}^{1/2} x_{2}^{1/4}\). The output price is \(p\), and the input prices are \(w_{1}\) and \(w_{2}\). All prices are fixed.
 Show that, when maximizing profits, the marginal product of each factor is equal to the corresponding real factor price.
 Use your answer in the previous part to calculate the input factor demands.
 Suppose that \(p=4\), \(w_{1} = 2\), and \(w_{2} = 1\). What are the optimal demanded quantities of the input factors? What is the optimal supplied quantity? How much profit does the firm make?

The maximization problem of the firm is
\begin{align*} \max_{x_{1}, x_{2}} &\left\{ p x_{1}^{1/2} x_{2}^{1/4}  w_{1} x_{1}  w_{2} x_{2} \right\}, \end{align*}
with first order conditions
\begin{align*} \frac{w_{1}}{p} &= \frac{1}{2} x_{1}^{1/2} x_{2}^{1/4} \\ \underbrace{\frac{w_{2}}{p}}_{Real\ factor\ price} &= \underbrace{\frac{1}{4} x_{1}^{1/2} x_{2}^{3/4}}_{Marginal\ product} \end{align*}

Multiplying the two necessary conditions results in
\begin{align*} \frac{w_{1} w_{2}}{p^2} &= \frac{1}{8} x_{1}^{1/21/2} x_{2}^{1/43/4}, \end{align*}
and solving for \(x_{2}\) gives
\begin{align*} x_{2} = \frac{p^{4}}{64 w_{1}^{2} w_{2}^{2}}. \end{align*}
To obtain the factor demand for \(x_{1}\), we substitute the demand for \(x_{2}\) in the first variational condition to get
\begin{align*} \frac{w_{1}}{p} &= \frac{1}{2} x_{1}^{1/2} \frac{p}{2 \sqrt{2} w_{1}^{1/2} w_{2}^{1/2}}. \end{align*}
Then solving for \(x_{1}\) gives
\begin{align*} x_{1} &= \frac{p^{4}}{32 w_{1}^{3} w_{2}}. \end{align*}

Lastly, substituting the prices into the demands gives \(x_{1} = x_{2} = 1\). Substituting the optimal demanded quantities into the production function gives \(q = f(1,1) = 1\). Lastly, the firm’s profit is calculated as \(\pi = 4  2  1 = 1\).

Suppose that at the beginning of a given month, we observe that a firm produces an output level \(q=20\) at a price level \(p=2\). After a month, we observe that the same firm produces \(q = 15\) at a price of \(p=4\). No other changes are observed in the prices of the market. Does the firm maximize its profit?
Since all input prices remain constant, we have \(\Delta w_{i} = 0\) for all input factors between the two dates. This implies that the revealed profit maximization inequality reduces to \(\Delta p \Delta q \ge 0\). However, we observe \(\Delta p = 2\) and \(\Delta q = 5\). Thus, by the revealed profit maximization inequality, the firm did not maximize its profit on one of these two (or both) dates.
Group C

Consider a firm with a CobbDouglas production function \(f(x_{1}, x_{2}) = A x_{1}^{\alpha r} x_{2}^{(1  \alpha) r}\) for \(0 < \alpha < 1\) and \(r \neq 1\). Output and input prices, namely \(p\), \(w_{1}\), and \(w_{2}\), are fixed.
 Write the profit maximization problem of the firm.
 Calculate the first order conditions of the profit maximization problem.
 Calculate the supply function.
 Calculate the input factor demands.
 Calculate the profit of the firm.
 What happens to supply, input factor demands, and profit for \(r=1\)?

The maximization problem of the firm is
\begin{align*} \max_{x_{1}, x_{2}} &\left\{ p A x_{1}^{\alpha r} x_{2}^{(1  \alpha) r}  w_{1} x_{1}  w_{2} x_{2} \right\} \end{align*}

The necessary conditions for interior solutions are
\begin{align*} w_{1} &= p A \alpha r x_{1}^{\alpha r1} x_{2}^{(1  \alpha)r} = p \alpha r \frac{f(x_{1}, x_{2})}{x_{1}} \\ w_{2} &= p (1  \alpha) r x_{1}^{\alpha r} x_{2}^{(1  \alpha)r  1} = p (1  \alpha) r \frac{f(x_{1}, x_{2})}{x_{2}} \end{align*}

To shorten the notation, let \(q = f(x_{1}, x_{2})\). With this notation, we can rewrite the first order conditions as
\begin{align*} x_{1} &= p \alpha r \frac{q}{w_{1}} \\ x_{2} &= p (1  \alpha) r \frac{q}{w_{2}}. \end{align*}
Substituting in the production function gives
\begin{align*} q &= f(x_{1}, x_{2}) \\ &= A \left(p \alpha r \frac{q}{w_{1}}\right)^{\alpha r} \left(p (1  \alpha) r \frac{q}{w_{2}}\right)^{(1  \alpha) r} \\ &= A (p r q)^{r} \left(\frac{\alpha}{w_{1}}\right)^{\alpha r} \left(\frac{1  \alpha}{w_{2}}\right)^{(1  \alpha) r}, \end{align*}
which implies
\begin{align*} q(w_{1}, w_{2}) &= A^{\frac{1}{1r}} (p r)^{\frac{r}{1r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{\alpha r}{1r}} \left(\frac{1  \alpha}{w_{2}}\right)^{\frac{(1  \alpha) r}{1r}}. \end{align*}

We can obtain the input factor demands by replacing the supply function in the first order conditions. This gives
\begin{align*} x_{1} &= p \alpha r \frac{A^{\frac{1}{1r}} (p r)^{\frac{r}{1r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{\alpha r}{1r}} \left(\frac{1  \alpha}{w_{2}}\right)^{\frac{(1  \alpha) r}{1r}}}{w_{1}} \\ &= (A p r)^{\frac{1}{1r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{1  (1  \alpha)r}{1r}} \left(\frac{1  \alpha}{w_{2}}\right)^{\frac{(1  \alpha) r}{1r}}, \end{align*}
and analogously
\begin{align*} x_{2} &= (A p r)^{\frac{1}{1r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{\alpha r}{1r}} \left(\frac{1  \alpha}{w_{2}}\right)^{\frac{1  \alpha r }{1r}}. \end{align*}

The easiest way to calculate the profit of the firm is to use the first order conditions of part 2 and the supply function. We then have
\begin{align*} \pi(p, w_{1}, w_{2}) &= p q  w_{1} p \alpha r \frac{q}{w_{1}}  w_{2} p (1  \alpha) r \frac{q}{w_{2}} \\ &= p q\left(1  \alpha r  (1  \alpha) r \right) \\ &= p (1  r) q \\ &= (p A)^{\frac{1}{1r}} \left(1  r \right) r^{\frac{r}{1r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{\alpha r}{1r}} \left(\frac{1  \alpha}{w_{2}}\right)^{\frac{(1  \alpha) r}{1r}} . \end{align*}

If \(r=1\), the production technology has constant returns to scale. Then, the firm has zero profit, and it is indifferent about its level of supply.