Dynamic Competition
 16 minutes read  3336 wordsContext
 Competition in real markets is not a static phenomenon. Firms can change their choices from date to date and adapt their strategies based on past events and the reactions of competitive firms.
 In such fluid settings, some firms take the initiative and set the pace of competition in the market. Other firms follow.
 Do firms benefit from assuming a market leader role?
 How does the sequence of moves affect the market power?
 Why does market entry influence the behavior of incumbent firms?
Course Structure Overview
Lecture Structure and Learning Objectives
Structure
 The Model T (Case Study)
 The Stackelberg model
 Sequential Competition in Prices
 The Role of Market Entry in Competition
 Current Field Developments
Learning Objectives
 Describe the impact of move order on market power under quantity competition.
 Describe the impact of move order on market power under price competition.
 Highlight the welfare effects of leadership under various competition means.
 Illustrate the effect of free entry in dynamic competition.
 Illustrate the importance of credibility in entry deterrence.
The Model T
People can have the model T in any color—so long as it’s black.
Henry Ford
Ford vs General Motors
 Ford and General Motors are two of the largest automobile companies worldwide.
 They were both founded in the beginning of the \(20^{\text{th}}\) century.
 Since then, they have been competing for the lead in the US automobile market.
Third Time is a Charm
 Henry Ford attempted two times to start an automobile business and failed.
 His third attempt in 1903 was the Ford Motor Company.
 Automobiles were far from affordable back then.
 Until 1908 Ford sold only hundreds or a few thousands of cars per year.
Fordism
 Ford introduced the moving assembly line in 1908.
 The new model T was mass produced instead of hand assembled.
 The new production method drove the cost of model T down, making it affordable to more consumers.
 Model T sold millons over the next 20 years.
 Ford Motor Company transformed from a small startup to the leading automobile company.
Game On
 William Durant incorporated General Motors.
 There were about 45 different car companies in the US, most of them selling a handful of cars.
 General Motors was the opposite of Ford.
 It produced a wide variety of cars for a wide variety of consumer needs.
 In the first 2 years, General Motors cobbled together 30 companies, 11 of which where automakers.
The Market Leader
 Model T’s production innovations made Ford the leading car company from about 1910 to 1925.
 The reluctance of Ford to keep introducing innovations and the absence of flexibility changed the situation.
 In 1925, GM surpassed Ford in total revenue and kept ahead until 1986.
 Eventually in 1927, Ford shutdown for 6 months to update its production lines for Model A.
 GM also surpassed Ford in total sales during this period.
Leaderfollower Competition in Quantities
 The Stackelberg model of oligopoly describes a market structure with two or more firms such that
 the market does not suffer from any other market failure (imperfect information, externalities, etc),
 no other firms can enter the market,
 firms sell a homogeneous product,
 firms try to maximize their profits,
 consumers are price takers, firms choose the quantities that they produce, the leader chooses first, and the follower(s) choose afterward, and
 consumers try to maximize their utility.
Residual Demand
 The leader chooses the quantity it produces first.
 Thus, the follower does not face the entire market demand but only the demand remaining after the leader’s choice.
 The difference between the market demand and the aggregated supplied quantities of all competitive firms is a firm’s residual demand.
The Follower’s Problem
 We solve the problem with backward induction.
 The follower claims the residual demand.
 For any given quantity \(q_{1}\) produced by the leader, the follower’s problem is
\[\max_{q_{2}} \left\{ p(q_{1} + q_{2})q_{2}  c_{2}(q_{2}) \right\}.\]
 Interior optimal solutions require
\[p'(q_{1} + q_{2})q_{2} + p(q_{1} + q_{2}) = c_{2}'(q_{2}).\]
 From this condition, we obtain (solving for \(q_{2}\)) the best response of the follower, say
\[q_{2} = b_{2}(q_{1}).\]
The Leader’s Problem
 The leader anticipates the best response of the follower and solves
\[\max_{q_{1}} \left\{ p(q_{1} + b_{2}(q_{1}))q_{1}  c_{1}(q_{1}) \right\}.\]
 Its optimality condition is given by
\[p'(q_{1} + b_{2}(q_{1})) \left(1 + b_{2}'(q_{1})\right) q_{1} + p(q_{1} + b_{2}(q_{1})) = c_{1}'(q_{1}).\]
An Analytic Example

Suppose that the demand and cost functions are given by
\begin{align*} p(q) &= p_{0} + p_{1} q, \\ c_{1}(q_{1}) &= c q_{1}, \\ c_{2}(q_{2}) &= c q_{2}. \end{align*}
The Follower’s Problem
 The follower’s necessary condition becomes
\[p_{1}q_{2} + p_{0} + p_{1}(q_{1} + q_{2}) = c.\]
 Then, the best response of the follower is
\[b_{2}(q_{1}) = \frac{c  p_{0}  p_{1}q_{1}}{2 p_{1}}.\]
The Leader’s Problem
 The leader solves
\[\max_{q_{1}} \left\{ p\left(q_{1} + \frac{c  p_{0}  p_{1}q_{1}}{2 p_{1}}\right)q_{1}  c(q_{1}) \right\}.\]
 Its optimality condition is given by
\[\frac{p_{1}}{2}q_{1} + p_{0} + \frac{c  p_{0}}{2} + \frac{p_{1}}{2}q_{1} = c.\]
 Therefore, the leader produces
\[q_{1}^{\ast} = \frac{c  p_{0}}{2 p_{1}}.\]
Market Quantity and Price
 Given the leader’s quantity, the follower produces
\[q_{2}^{\ast} = b_{2}(q_{1}^{\ast}) = \frac{c  p_{0}}{2 p_{1}}  \frac{1}{2}\frac{c  p_{0}}{2 p_{1}} = \frac{c  p_{0}}{4 p_{1}}.\]
 The market quantity is
\[q^{\ast} = q_{1}^{\ast} + q_{2}^{\ast} = 3\frac{c  p_{0}}{4 p_{1}}.\]
 The market price is
\[p(q^{\ast}) = p_{0} + p_{1} q^{\ast} = \frac{p_{0} + 3 c}{4}.\]
Profits
 The follower makes profit
\[\pi_{2} = \left(p(q^{\ast})  c\right) q_{2}^{\ast} =  \frac{\left(p_{0}  c\right)^{2}}{16p_{1}}.\]
 The leader’s profit is
\[\pi_{1} = \left(p(q^{\ast})  c\right) q_{1}^{\ast} =  \frac{\left(p_{0}  c\right)^{2}}{8p_{1}}.\]
First Move Advantage
 The firms' profits in the Cournot model with symmetric costs are
\[\pi_{c} =  \frac{\left(p_{0}  c\right)^{2}}{9p_{1}}.\]
 Compared to the firms' profits in the Cournot model, we have
\[\pi_{1} > \pi_{c} > \pi_{2}.\]
 The leader makes more profit compared to Cournot competition, while the follower makes less profit.
 However, aggregate profits in the market are less compared to Cournot, i.e.,
\[\pi_{1} + \pi_{2} < 2 \pi_{c}.\]
LeaderFollower Competition in Prices

Consider a variation of the Stackelberg game in which firms are price setters.

This is the sequential version of the Bertrand model.

There are two firms \((i, j \in \{1,2\})\) with marginal cost equal to \(c\).

Firm \(1\) is the leader (moves first), and firm \(2\) is the follower (moves second).

The demand for firm \(i\) is
\begin{align*} d_{i}(p_{i}, p_{j}) = \left\{\begin{aligned} &10  \frac{1}{2}p_{i} & p_{i} < p_{j} \\ &5  \frac{1}{4}p_{i} & p_{i}=p_{j} \\ &0 & p_{i} > p_{j} \end{aligned}\right.. \end{align*}

The firm with the lowest price gets all the demand.

If prices are equal, demand is equally split.
Non Equilibrium Prices
 Suppose that firm \(1\) sets a price \(p_{1}\) that is greater than the marginal cost of firm \(2\) (i.e., \(c\)).
 Firm \(2\) can undercut by a small amount and grab all the market. For instance, set price \(p_{2} = \frac{c + p_{1}}{2}\).
 Thus, firm \(1\) can only set a price equal to firm \(2\)’s marginal cost.
Equilibrium
 The only possible equilibrium is both firms to set a price equal to the (common) marginal cost.
 Firms do not have any incentive to deviate.
 Setting lower prices leads to losses.
 Setting higher prices leads to zero profits.
 The order with which firms move does not affect the market outcome.
A Game of Market Entry
 Consider a market with one firm already operating and a potential entrant.
 The entrant decides whether to enter the market.
 The incumbent decides whether to follow aggressive or complying competition behavior.
Entry Deterrence with Credible Threat
 \(SPE = \left\{ \left(Stay\ out, Fight \right) \right\}\)
Entry with NonCredible Threat
 \(SPE = \left\{ \left(Enter, Don’t\ fight \right) \right\}\)
Quantity Competition with Costly Entry
 We extend the Cournot duopoly model by allowing firms to choose whether they enter the market.
 Firms enter the market when they can make profits.
 Firms incur a fixed cost \(\bar c\) upon entry.
Market and Dynamics
 There is an infinite number of potential entrants.
 The market’s inverse demand is
\[p(q) = p_{0} + p_{1}q \quad(p_{0}>0,\ p_{1}<0).\]
 All firms have costs
\[c(q) = c_{1}q \quad(c_{1}>0).\]
 The firms that enter the market choose their quantities at the second date simultaneously.
 At the first date, the entrants decide how many of them enter the market.
Unregulated Entry and Exit
 When \(n\) firms enter the market, each firm makes profit
\[\pi_{i}(n) = \frac{(c_{1}  p_{0})^{2}}{(n + 1)^{2} p_{1}}  \bar c.\]
 In equilibrium
\[\pi_{i}(n^{e}) = 0 \iff n^{e} = \frac{p_{0}  c_{1}}{\sqrt{ \bar c p_{1}}}  1.\]
The Social Welfare
 The social welfare is calculated by
\[W(n) = \int_{0}^{n} p(s q(s)) \left(q(s) + sq'(s)\right) \mathrm{d} s  n c(q(n))  n \bar c.\]
The First Best Solution
 Aggregate profit is maximized when
\[p(n q(n))\left(q(n) + n q'(n)\right)  c(q(n))  n c'(q(n))q'(n) = \bar c .\]
 This implies that
\[n^{*} = \sqrt[3]{\frac{(p_{0}  c_{1})^{2}}{p_{1} \bar c}}  1 .\]
 Comparing with the unregulated entry equilibrium, we have
\[ n^{*} < n^{e} .\]
 With quantity competition, unregulated entry results in excessive entry of firms.
Current Field Developments
 Market dynamics and entry constitute important components of competition policy.
 Mergers and acquisitions are dynamic phenomena modifying the number of firms in the market and their market power.
 Merger cases are central from both an economic and competition law perspective.
 In the US and EU, antitrust laws aim to reduce or avoid excessive market concentration.
 Dynamic market models are commonly used to assess mergers and their impact on market power in court cases.
Concise Summary
 The sequence with which firms act in a market can influence market power.
 If firms compete in quantities, then the leader has a first move advantage.
 If firms compete in prices, then the sequence of moves is irrelevant.
 Competition forces do not concern only firms who already entered in a market.
 Incumbent firms strategically interact with potential competitors looking to enter the market.
 Free entry is a characteristic of market competition that can reduce firms' profits and result to low prices.
 On some occasions (e.g. entry costs), excessive entry can result to economic inefficiencies.
Further Reading
 Watson (2008, chaps. 15, 16)
 Belleflamme and Peitz (2010, secs. 4.1, 4.2.13)
 Varian (2010, secs. 28.2, 29.8)
Mathematical Details
Leaderfollower duopoly with quantity competition and asymmetric costs
Firms choose their strategies sequentially. The leader chooses first, and the follower chooses afterward. Both firms choose their supplied quantities. The oligopoly of sequential quantity competition is known as the Stackelberg model.
The follower’s problem
The leader anticipates the follower’s actions and takes them into account when optimizing. We use this idea to solve the problem using backward induction. The follower’s problem is \[\max_{q_{2}} \left\{ p(q_{1} + q_{2}) q_{2}  c_{2}(q_{2}) \right\}.\]
The follower’s best response
Using a variational argument, we get \[p'(q_{1} + q_{2}) q_{2} + p(q_{1} + q_{2}) = c_{2}'(q_{2}).\] This condition determines the best response of the follower when the leader strategy is given by \(q_{2}\). Say that this relation is given by \(b_{2}(q_{1})\).
The leader’s problem
The leader solves \[\max_{q_{1}} \left\{ p(q_{1} + b_{2}(q_{1})) q_{1}  c_{1}(q_{1}) \right\}.\] Its optimality condition is given by \[p'(q_{1} + b_{2}(q_{1})) (1 + b_{2}'(q_{1})) q_{1} + p(q_{1} + b_{2}(q_{1})) = c_{1}'(q_{1}).\]
An affine demand, linear costs example
For inverse demand and costs given by
\begin{align*} p(q) &= p_{0} + p_{1} q \\ c_{1}(q) &= \kappa_{1} q \\ c_{2}(q) &= \kappa_{2} q, \end{align*}
the best response function of the follower becomes \[b_{2}(q_{1}) = \frac{\kappa_{2}  p_{0}  p_{1} q_{1}}{2 p_{1}}.\] The leader optimally chooses \[q_{1} = \frac{2\kappa_{1}  \kappa_{2}  p_{0}}{2 p_{1}}.\] Thus, the follower subsequently chooses \[q_{2} = \frac{3\kappa_{2}  2\kappa_{1}  p_{0}}{4 p_{1}}.\]
Market prices and quantities
The total quantity in the market is
\begin{align*} q &= q_{1} + q_{2} \\ &= \frac{2\kappa_{1}  \kappa_{2}  p_{0}}{2 p_{1}} + \frac{3\kappa_{2}  2\kappa_{1}  p_{0}}{4 p_{1}} \\ &= \frac{2 \kappa_{1} + \kappa_{2}  3 p_{0}}{4 p_{1}}. \end{align*}
The price can be calculated by the inverse demand function as \[p(q) = p_{0} + p_{1} \frac{2 \kappa_{1} + \kappa_{2}  3 p_{0}}{4 p_{1}} = \frac{2 \kappa_{1} + \kappa_{2} + p_{0}}{4}.\]
Profits
The leader has profit \[\pi_{1} = \left( \frac{2 \kappa_{1} + \kappa_{2} + p_{0}}{4}  \kappa_{1} \right) q_{1} =  \frac{\left(2 \kappa_{1}  \kappa_{2}  p_{0}\right)^{2}}{8 p_{1}},\] and the follower has profit \[\pi_{2} = \left( \frac{2 \kappa_{1} + \kappa_{2} + p_{0}}{4}  \kappa_{2} \right) q_{2} =  \frac{\left(3 \kappa_{2}  2\kappa_{1}  p_{0}\right)^{2}}{8 p_{1}}.\]
A Stackelberg Competition Example
Suppose that there are two firms in the market. Firm \(1\) is the leader, and firm \(2\) is the follower.Let the cost function of both firms be \[c_{i}(q_{i}) = 4 q_{i}.\] Let market inverse demand be \[p(q_{1} + q_{2}) = 28  2 \left(q_{1} + q_{2}\right).\]
The follower’s problem is \[\max_{q_{2}} \left\{ \left(28  2 \left(q_{1} + q_{2}\right)\right) q_{2}  4 q_{2} \right\},\] and the necessary condition gives \[28  2 q_{1}  4 q_{2} = 4.\] This condition determines the best response of the follower when the leader produces \(q_{1}\). \[q_{2} = \frac{24  2q_{1}}{4}.\] Then, the leader solves \[\max_{q_{1}} \left\{ \left(28  2 \left(q_{1} + \frac{24  2q_{1}}{4}\right) \right) q_{1}  4 q_{1} \right\}.\] Its optimality condition is given by \[12  2 q_{1} = 0.\]
Solving for \(q_{1}\), we find that the leader is optimally producing \(q_{1} = 6\). We can substitute \(q_{1} = 6\) back to the follower’s best response to get the follower’s quantity \[q_{2} = \frac{24  2q_{1}}{4} = 3.\] The market quantity is the sum of the firms' quantities \[q = q_{1} + q_{2} = 9.\] The market price can then be calculated by the inverse demand function \[p = p(q) = 10.\]
Finally, the leader has profit \[\pi_{1} = P q_{1}  4 q_{1} = 36,\] and the follower’s profit is \[\pi_{2} = P q_{2}  4 q_{2} = 18.\] The leader has a first move advantage and its profit is greater than the follower’s.
Exercises
Group A

Consider an endogenous timing game in which firms simultaneously choose whether to play \(Early\) or to play \(Late\) . If both firms make the same choice, then they compete as in the Bertrand competition. If the firms make different choices, then they compete as in the Stackelberg model (sequential competition in quantities), where the leader is the firm that chose \(Early\) and the follower is the firm that chose \(Late\). Suppose that firms have cost functions to \(c(q)=q\), and market demand is given by \(p(q) = 5  q\).
 Calculate the profits of the firms under the different means of competition.
 Draw the extensive form of the game.
 Calculate all the Nash Equilibria.
 If the firms compete simultaneously using prices, they set \(p_{1} = p_{2} = 1\) and make zero profits. If they compete sequentially using quantities, the follower makes profit \[\pi_{f} = \frac{(51)^{2}}{16} = 1,\] and the leader makes profit \[\pi_{l} = \frac{(51)^{2}}{8} = 2.\]
 The extensive form of the game is
 There are two pure strategy Nash equilibria, namely \[\left\{\left\{Early, Late\right\}, \left\{Late, Early\right\}\right\},\] and a mixed strategy equilibrium in which each firm chooses \(Early\) with probability \(p=\frac{2}{3}\).

Suppose that there are two firms in a market with an affine inverse demand function \(p(q) = p_{0} + p_{1}q\), where \(p_{0}>0\) and \(p_{1}<0\). The firms have constant marginal costs equal to \(c\) and no fixed production cost. Show that the leader in a leaderfollower duopoly with quantity competition has a first move advantage. Specifically, show that the leader’s profit is always greater than the profit it would attain in a market with simultaneous quantity competition.
In the leaderfollower problem, the leader makes profit
\begin{align*} \pi_{l} =  \frac{(c_{1}  p_{0})^2}{8 p_{1}} . \end{align*}
In the simultaneous move game, each firm makes profit
\begin{align*} \pi_{s} = \frac{(c_{1}  p_{0})^{2}}{9 p_{1}}. \end{align*}
One can calculate the ratio of the profits
\begin{align*} \frac{\pi_{l}}{\pi_{s}} = \frac{(c_{1}  p_{0})^2}{8 p_{1}} \frac{9 p_{1}}{(c_{1}  p_{0})^{2}} = \frac{9}{8} > 1, \end{align*}
which implies that \(\pi_{l}>\pi_{s}\).

Consider a market with inverse market demand function \(p(q) = 100  2q\). The total cost function for any firm in the market is given by c(q) = 4q. Suppose that the market consists of two firms that compete by choosing their supplied quantities. Firm 1 is the market leader, while firm 2 is the follower. Calculate the firms' optimally supplied quantities, total market output, and market price.
The follower takes prices as given and solves
\begin{align*} \max_{q_{2}} \left\{ p(q_{1} + q_{2}) q_{2}  c(q_{2}) \right\}, \end{align*}
which determines the best response of the follower as
\begin{align*} b_{2}(q_{1}) = \frac{48  q_{1}}{2}. \end{align*}
Then, the leader solves
\begin{align*} \max_{q_{1}} \left\{ p\left(q_{1} + \frac{48  q_{1}}{2}\right) q_{1}  4 q_{1} \right\} = \max_{q_{1}} \left\{ \left(48  q_{1}\right) q_{1} \right\}, \end{align*}
which determines the optimal supplied quantity of the leader as \(q_{1} = 24\). Substituting into the best response of the follower results in \(q_{2} = 12\). The market quantity is \(q_{s} = q_{1} + q_{2} = 36\), and the market price is
\begin{align*} p(q_{s}) = 100  72 = 28. \end{align*}
Group B

Consider the quantity competition with costly entry game. Suppose that firms face market demand \(p(q) = p_{0} + p_{1}q\), have symmetric production costs \(c(q)=c_{1}q\), and entry costs \(\bar c\), where \(p_{0}, c_{1} > 0\) and \(p_{1}<0\). The social welfare as a function of the number of firms is given by \[W(n) = \int_{0}^{n} p(s q(s)) \left(q(s) + sq'(s)\right) \mathrm{d} s  n c(q(n))  n \bar c .\] Calculate the Pareto optimal number of entrants in the market.

By Fermat’s theorem, the welfare is maximized when \(W'(n) = 0\). Using the fundamental theorem of calculus, we get \[p(n q(n))\left(q(n) + n q'(n)\right)  c(q(n))  n c'(q(n))q'(n) = \bar c.\] The optimal quantity produced by each firm when \(n\) firms enter the market is \[q(n) = \frac{p_{0}  c_{1}}{(n + 1) p_{1}},\] hence, one calculates \[\frac{q'(n)}{q(n)} = \frac{1}{n + 1}.\] We will use the last expression to simplify the first order condition. Rearranging the necessary condition and using the linearity of costs gives
\begin{align*} \bar c &= p(n q(n))\left(q(n) + n q'(n)\right)  c(q(n))  n c'(q(n))q'(n) \\ &= p(n q(n))q(n)  c(q(n)) + n \left(p(n q(n))  c'(q(n))\right)q'(n) \\ &= \pi(n) + n \left(p(n q(n))q(n)  c(q(n))\right) \frac{q'(n)}{q(n)} \\ &= \pi(n) \left(1 + n \frac{q'(n)}{q(n)}\right) \\ &= \pi(n) \left(1  n \frac{1}{n + 1}\right) \\ &= \pi(n) \left(\frac{1}{n + 1}\right). \end{align*}
The profit of each firm when \(n\) firms enter the market is \[\pi(n) = \frac{(p_{0}  c_{1})^{2}}{(n + 1)^{2} p_{1}},\] so, we get
\begin{align*} \bar c &= \frac{(p_{0}  c_{1})^{2}}{(n + 1)^{3} p_{1}}. \end{align*}
Therefore, the socially optimal number of firms is given by
\begin{align*} n &= \sqrt[3]{\frac{(p_{0}  c_{1})^{2}}{p_{1} \bar c}}  1. \end{align*}
