Firm Supply

Context

• Competition is a frequent topic in political and economic discussion. Competition can constrain firms to act as price takers.

• Markets are not perfectly competitive, and firms are typically not purely price takers in reality. Nevertheless, such arguments remain central in economics, finance, and business.
  • Why is price taking behavior still so relevant?
  • How does competition restrain the behavior of firms?
  • How do firms decide how much to produce in competitive markets?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• Microsoft’s Pricing Strategies (Case Study)
• Basic Concepts
• Price Taking Examples
• Profit Maximization under Price Taking Exercise
• Current Field Developments

Learning Objectives

• Illustrate the implications of competition on individual firm supply.
• Explain how price taking firms decide how much to produce.
• Explain the shutdown decisions of competitive firms.
• Describe and compare the concepts of profit and producer’s surplus.

Microsoft’s Pricing Strategies

• In the early 1980s, several companies were competing in the operating system market of IBM-compatible PCs.
• In the 1990s and 2010s, Microsoft dominated the operating system market.
• In 2020s Microsoft’s dominance stopped, and its operating system is nowadays the second most used.

• How did Microsoft manage to dominate the operating system market?
• How did it lose its primacy?

MS-DOS

• In the early 1980s, the common practice of operating system companies was to charge hardware manufacturers for each operating system copy installed in a computer.

• Microsoft offered an alternative plan.
  • Charge computer manufacturers based on (the past number of) built computers.
  • The manufacturer was paying a general licensing fee and then could install the operating system in all the computers it produced.
  • Microsoft was offering low-priced licensing contracts making their operating system (MS-DOS) very attractive to manufacturers.

The Impact of Microsoft’s Early Pricing Strategy

• Effectively, manufacturers could purchase Microsoft’s operating system at much lower prices than the operating systems of other software companies.
  • A manufacturer had to pay \($50\ -\ $100 \) for installing an alternative operating system on an additional machine.
  • It cost nothing to install MS-DOS on an additional machine once a licensing contract with Microsoft has been signed.
  • MS-DOS ended up being the default operating system

Android

The Impact of Android on Microsoft’s Pricing Strategy

• Android is free. Anyone can install the operating system on her device after accepting the terms and conditions.
• For mobile device manufacturers, Android is a cheaper operating system alternative for their products.
• This leads to more competitive prices for consumers too.
  • Microsoft’s operating systems lost their primacy in the overall operating system market in \(2017\).
  • Microsoft’s operating systems are still dominant in less portable devices, such as desktop PCs and Laptops.
  • The rise of Android has also impacted Microsoft’s pricing strategies for its desktop operating systems.
  • Licensed users were able to upgrade to the last two versions of Microsoft’s operating system without paying for a new license.

Price taking

• One prerequisite axiom of competitive market structure is price taking behavior.
• A firm (or a consumer) is a price taker if it cannot influence the market price on its own. Instead, price takers consider market prices as exogenous.

The Supply Function

• Supply is one of the two fundamental elements, with the other one being demand, determining the market state.
• The firm’s supply function is a mapping that gives how much a profit maximizing firm would like to produce at a particular price.

The Shutdown Condition

• In the long-run, a firm operates if it makes non-negative profit.
• In the short-run, a firm might also keep producing while it makes losses.
• A firm shutdowns if the market price is below its average variable cost.
• Equivalently, a firm shutdowns if its revenue is not enough to cover, besides variable cost, at least some part of its fixed cost.

Price Taking Supply

• A price taking firm maximizes its profit when it produces at the point where its marginal cost is equal to the market price.
• For quantities where its marginal cost is above the market price,
  • the firm is making losses for the last unit of output it produces, and
  • it can increase its profit by lowering production.
• For quantities where its marginal cost is below the market price,
  • the firm is still making a profit for the last unit of output it produces, and
  • it can increase its profit even further by increasing production.

Inverse Supply

• The firm’s inverse supply function is a mapping that gives at which price would a profit maximizing firm produce a particular quantity.
• It can be obtained by the first order condition of the firm’s profit maximization problem.

Example: Price Taking Firm Supply

Economic Profit

• The (economic) profit or loss is the difference between the total revenue and the total cost of produced output.
• It is given by \[\pi(q) = p q - c(q)\]

Example: Profit

Exercise: Profits of Price Taking Firms

• Suppose that the firm’s cost function is \(c(q) = 2 q^{2}\).
• Suppose that the firm is a price taker, and the market price is \(p=8\).
• The firm maximizes its profit

\[\max_{q} \left\{ 8 q - 2 q^{2} \right\}.\]

• It solves the above to get

\[8 - 4 q \overset{!}{=} 0 \implies q = 2.\]

• The firm makes profit

\[\pi = 16 - 8 = 8.\]

Producer’s Surplus

• The producer’s surplus is the accumulated amount that a producer benefits from selling each produced unit at a market price higher than its marginal cost (i.e., the least price the producer is willing to sell).
• It can be calculated by the cumulative difference between the marginal revenue and cost of each produced unit.
• It can also be calculated by the difference between total revenue and variable cost. \[\Pi_{s}(q) = p q - \mu(q)\]

Example: Producer’s Surplus

Producer’s Surplus vs Profit

• If the production cost has a fixed component, a firm’s profit differs from its producer’s surplus by exactly this component.

\begin{align*} \pi(q) &= p q - c(q) \\ \Pi_{s}(q) &= p q - \mu(q) \\ \Pi_{s}(q) - \pi(q) &= \sigma(q) = c(0) \ge 0 \end{align*}

Current Field Developments

• The majority of models in economics and finance examine firm supply under perfect competition terms.
• Industrial organization is a field of economics that uses market structures diverging from perfect competition, which can be more conducive.
• We will examine alternative market structures where firms make more strategic decisions in the subsequent topics.

Concise Summary

• Price taking behavior is not realistic, but it has predictive power in many competitive markets.
• Two lessons can be useful rules of thumb for business decisions in markets with intense competition:
  • Firms maximize their profits by choosing to produce so that the marginal cost equals the market price (marginal revenue).
  • They opt-out if the market price is below the average variable cost.
• The profit and producer’s surplus measure the operational performance of firms.
• In the presence of fixed costs, profit and producer surplus are not identical.

Further Reading

• Varian (2010, chap. 23)
• CORE Team (2017, secs. 8.3, 8.8)

Mathematical Details

The supply function

The firm is willing to supply at prices covering its average variable cost. The cost function incorporates the technological constraints faced by the firm. Using the cost function, the profit maximization problem can be rewritten as

\begin{align*} \max_{q} \left\{ p q - c(q) \right\}. \end{align*}

The profit can be decomposed into two parts:

• revenue \(pq\) and
• cost \(c(q)\).

For non-boundary solutions, the marginal revenue and marginal cost are equalized at the maximum (why?).

\begin{align*} p = c'(q) \end{align*}

The above condition is necessary but not sufficient. The condition gives a maximum if

\begin{align*} c''(q) > 0 . \end{align*}

Therefore, marginal cost should be increasing (locally) at the maximum.

The firm opts out if for every level of output \(q>0\)

\begin{align*} \bar{\mu}(q) = \frac{\mu(q)}{q} = \frac{c(q) - c(0)}{q} \ge p . \end{align*}

The solution to the profit maximization problem determines the supply of the firm. Supply \(s(p)\) gives the quantity that a profit maximizing firm would like to produce and sell in the market. On some occasions, it is easier to work with the inverse supply mapping \(p(q)\), which gives the price for which the supplied quantity \(q\) is profit maximizing. The inverse supply can be directly obtained from the variational condition \(p=c'(q)\).

A cubic cost example

Consider the cubic cost function

\begin{align*} c(q) = q^{3} + c_{2} q^{2} + c_{1} q + c_{0} . \end{align*}

The inverse supply of a profit maximizing firm is

\begin{align*} p(q) = \left\{ \begin{matrix} &\left[0, c_{1} - \frac{c_{2}^{2}}{4}\right) & q = 0\\ &3 q^{2} + 2 c_{2} q + c_{1} & q \ge -\frac{c_{2}}{2} \end{matrix} \right. . \end{align*}

Producer’s surplus

The surplus of the producer is the cumulative difference between the marginal revenue and cost of every produced unit. In a single-price market (such as in the case of perfect competition), the revenue of each unit (marginal revenue) is equal to the market price. The cost per unit of production is the marginal cost. Then, we can calculate

\begin{align*} \Pi_{s}(q) &= \int_{0}^{q} \left( p - c'(z) \right) dz \\ &= \left[ pz - c(z) \right]_{0}^{q} \\ &= p q - \left( c(q) - c(0) \right) \\ &= p q - \mu(q) \end{align*}

The case of constant returns to scale

If production exhibits constant returns to scale, then \(c(q) = c_{1} q\). Therefore, a profit maximizing firm has supply

\begin{align*} s(p) = \left\{ \begin{matrix} 0 & p < c_{1}\\ [0, \infty) & p = c_{1} \\ \infty & p > c_{1} \end{matrix} \right. . \end{align*}

Its inverse supply is horizontal, constantly equal to \(c_{1}\) for all positive levels of output.

Exercises

Group A

1. Consider a price-taking, profit maximizing firm with cost function \(c(q) = 1000 + 10 q^{2}\).

   1. Calculate the supply function.
   2. Find the output level that minimizes the average cost.
   3. Calculate the profit function.
   4. What is the marginal effect of a price change on profit?
   5. What is the minimum price for which the firm makes non negative profit?

   1. The first order condition of the profit maximization problem is \(20 q = p\). Solving for \(q\) results in the supply function \(s(p) = p / 20\).

   2. The average cost is determined by

      \begin{align*} \bar{c}(q) = \frac{1000}{q} + 10 q . \end{align*}

      We can calculate the minimum average cost by solving the equation \(\bar c(q) = c'(q)\), which is equivalent to

      \begin{align*} \frac{1000}{q} + 10 q = 20 q. \end{align*}

      This results to \(q = 10\).

   3. The profit function is given by

      \begin{align*} \pi(p) &= p s(p) - c\left(s(p)\right) \\ &= \frac{p^{2}}{20} - 1000 - 10 \frac{p^{2}}{400} \\ &= - 1000 + \frac{p^{2}}{40} . \end{align*}

   4. The effect of a marginal price change on profit is \(\pi'(p) = p / 20\).

   5. The firm makes non negative profit for prices such that \(\pi(p) \ge 0\), which implies that \(p \ge 200\).

2. Suppose that a firm has a supply function given by \(s(p) = 100 + 4 p\). Calculate the inverse supply function.

   We solve for \(p\) to get

   \begin{align*} p(q) &= - 25 + \frac{q}{4}. \end{align*}

References

References

Topic's Concepts

• producers surplus
• loss
• profit
• firms inverse supply function
• firms supply function
• price taker