Firm Supply
 9 minutes read  1899 wordsContext
 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 IBMcompatible 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?
MSDOS
 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 lowpriced licensing contracts making their operating system (MSDOS) 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 MSDOS on an additional machine once a licensing contract with Microsoft has been signed.
 MSDOS ended up being the default operating system
Android
 Android is a community (open source) operating system for mobile devices based on the Linux kernel.
 The wide use of smartphones and tablets drastically changed the operating system market.
 Although Microsoft offered an operating system suitable for smartphones and tablets, it did not manage to keep its primacy.
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 longrun, a firm operates if it makes nonnegative profit.
 In the shortrun, 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.
Comprehensive 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 optout 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
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 nonboundary 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 singleprice 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

Consider a pricetaking, profit maximizing firm with cost function \(c(q) = 1000 + 10 q^{2}\).
 Calculate the supply function.
 Find the output level that minimizes the average cost.
 Calculate the profit function.
 What is the marginal effect of a price change on profit?
 What is the minimum price for which the firm makes non negative profit?

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

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 \(q = 10\).

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*}

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

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

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*}