Competition with Incomplete Information

Context

• Including uncertainty in market models complicates the analysis. However, some results obtained when ignoring it can be less conducive. For instance, the nature of online marketplaces has many characteristics resembling price competition.

• Yet, many firms make profits in such marketplaces, contrary to the model predictions of deterministic competition in prices.
  • How can uncertainty affect competition in markets?
  • Do the insights and results of competition change when uncertainty is taken into account?
  • Does uncertainty have a uniform impact in all markets?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• Web Search (Case Study)
• Competition in Quantities with Cost Uncertainty
• Competition in Prices with Cost Uncertainty
• Comparison with the Deterministic cases and Conclusions
• Current Field Developments

Learning Objectives

• Introduce cost uncertainty in quantity competition and describe the basic mechanisms.
• Illustrate the impact of uncertainty on the market outcomes of quantity competition.
• Introduce cost uncertainty in price competition and describe the basic mechanisms.
• Illustrate the impact of uncertainty on the market outcomes of price competition.

Web Search

• At the beginning of the 1990s, search engines were not invented yet.

• People had to actually type the exact URL to visit a website.
• So, how did people find websites?

In a Web without Google

Web Crawlers

The Birth of a Tech Giant

• The business model for the PageRank tool was also uncertain.

• With catalog search, Yahoo could place relevant ads in the corresponding catalogs.
• The PageRank tool would take users away from searching the catalog and reduce Yahoo’s advertising revenue.
• Yahoo was not interested in PageRank.
• Soon after, PageRank was patented and became the core algorithm of Google’s search engine.
• Google introduced algorithms associating search keywords with relevant ads.
• Instead of reducing advertising revenue, the alternative approach led to much greater revenues than Yahoo ever achieved.

Competition in Quantities with Cost Uncertainty

• There are two firms in the market.
• The inverse demand function is

\[p(q) = p_{0} + p_{1}q \quad\quad (p_{0}>0,\ p_{1}<0).\]

• Firm \(i=1,2\) has cost function

\[c(q_{i}, s) = c_{s,i} q_{i}.\]

• The random variable \(c_{s,i}\) takes the values \(0 < c_{l} < c_{h} < p_{0}\) with equal probability.
• The value of \(c_{s,i}\) is observed by firm \(i\) but not by firm \(j\) (private information).
• The random variables \(c_{s,i}\) and \(c_{s,j}\) are independent.

The Expected Profit

• The profit of firm \(i\) is stochastic, even after \(c_{s,i}\) is observed, because the cost of firm \(j\) is not observed.
• Given the observed value of \(c_{s,i}\), firm \(i\) maximizes its expected profit.

\[\max_{q_{s,i}} \sum_{r=l,h} \frac{1}{2} \left[ p(q_{s,i} + q_{r,j})q_{s,i} - c(q_{s,i}, s)\right] \quad\quad (s=l,h).\]

Best responses

• The necessary condition for firm \(i\) is

\[p_{0} + 2 p_{1} q_{s,i} + p_{1} \sum_{r=l,h} \frac{1}{2} q_{r,j} = c_{s,i}.\]

• Solving for \(q_{s,i}\) gives the best response of firm \(i\) at state \(c_{s,i}\)

\[q_{s,i} = \frac{c_{s,i} - p_{0} - p_{1} \sum_{r=l,h} \frac{1}{2} q_{r,j}}{2 p_{1}}.\]

Bayesian Nash Equilibrium

• Calculating the expected value of the best responses of \(i\) gives

\[\sum_{s=l,h} \frac{1}{2} q_{s,i} = \frac{\sum_{s=l,h} \frac{1}{2} c_{s,i} - p_{1} \sum_{r=l,h} \frac{1}{2} q_{r,j} - p_{0}}{2 p_{1}}.\]

• Since the game is symmetric, we expect \(q_{s,j}=q_{s,i}\) for both states \(s=l,h\), so

\[\sum_{s=l,h} \frac{1}{2} q_{s,i} = \frac{\sum_{s=l,h} \frac{1}{2} c_{s,i} - p_{0}}{3 p_{1}}.\]

• Substituting into the best response, we obtain

\[q_{s,i} = \frac{3c_{s,i} - 2 p_{0} - \sum_{r=l,h} \frac{1}{2} c_{r,i}}{6 p_{1}} = \frac{3c_{s,i} - 2 p_{0} - \mathbb{E} c_{i}}{6 p_{1}}.\]

What is the Impact of Stochasticity?

• Firm \(i\) may produce less or more compared to the deterministic case (say \(q\)) depending on the realization of \(c_{s,i}\).
  • If \(c_{s,i}>\mathbb{E} c_{i}\), then production is too costly and \(q_{s,i} < q\).
  • If \(c_{s,i}<\mathbb{E} c_{i}\), then production is cheap and \(q_{s,i}>q\).
  • On average, uncertainty cancels out \(\mathbb{E} q_{i} = q\).
• The market price and quantity can be either greater or less than in the deterministic case, with uncertainty also canceling out on average.
• However, uncertainty reduces the profits of the firms (why?).
  • The more uncertain production gets (i.e., the more volatile is \(c_{s,i}\)), the fewer profits firms make on average.

Competition in Prices with Cost Uncertainty

• There are two firms in the market.
• Firm \(i=1,2\) has cost function

\[c(q_{i}, \theta_{i}) = \theta_{i} q_{i}.\]

• The random variables \(\theta_{i}\) are identically, independently, and uniformly distributed on \([0, 1]\).

• The value of \(\theta_{i}\) is observed by firm \(i\) but not by other firms (private information).

• Let the demand for firm \(i\) be

  \begin{align*} d_{i}(p_{i}, p_{j}) = \left\{\begin{aligned} &1 - p_{i} & p_{i} < p_{j} \\ &\frac{1 - p_{i}}{2} & p_{i} = p_{j} \\ &0 & p_{i} > p_{j} \end{aligned}\right.. \end{align*}

Profit Maximization

• Given the choice \(p_{j}\) of the other firm, firm \(i\) maximizes its expected profit

  \begin{align*} \pi_{i} &= \max_{p_{i}} \mathbb{E} \left[ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(p_{i}(\theta_{i}), p_{j}(\theta_{j}))\right] \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) (1 - p_{i}(\theta_{i})) \mathbb{P}(p_{i}(\theta_{i}) < p_{j}(\theta_{j}) ) \right\} . \end{align*}

Symmetric Bayesian Nash Equilibrium

• With a bit more mathematics and some calculations…
• Firms choose \[p^{\ast}(\theta_{i}) = \frac{1 + 2 \theta_{i}}{3} \quad (i =1, 2).\]
• In contrast with the deterministic case, all firms price above their marginal cost, i.e., \[p^{\ast}(\theta_{i}) > \theta_{i}.\]
• For \(n\) firms the equilibrium prices are \[p^{\ast}(\theta_{i}) = \frac{1 + n \theta_{i}}{n+1} \quad (i =1, \dots, n).\]
• The markup decreases in the number of firms in the market (the market becomes more competitive) \[\frac{\mathrm{d} p^{\ast}(\theta_{i})}{\mathrm{d} n} < 0.\]
• The limiting case gives the perfect competition pricing \[\lim_{n\to\infty} p^{\ast}(\theta_{i}) = \theta_{i}.\]

Current Field Developments

• Models with dynamic, stochastic competition are state of the art in finance and industrial organization.
• Many of them are not analytically solvable, and numerical methods are used to approximate their solutions.
• Such models are also used in estimations and calibrations to
  • examine how well their predictions match observed data, and
  • obtain estimates that help organizations and governments to make educated strategic decisions.

Comprehensive Summary

• The calculations can get complicated even for simple stochastic models.
• However, stochastic components can make some market models more conducive.
• Quantity competition with stochastic marginal costs results to fewer expected profits compared to when uncertainty is absent.
• Price competition with stochastic marginal costs leads to positive markups.
• Firms have positive profits, contrasting the deterministic case.

Further Reading

• Belleflamme & Peitz, 2010, sec. 3.1.2
• Watson, 2008, Chapter 26

Mathematical Details

Competition in Quantities with Cost Uncertainty

Consider, again, the Bayesian Nash equilibrium of competition in quantities when firms have symmetric costs, i.e., \[q_{s,i} = \frac{3c_{s,i} - 2 p_{0} - \mathbb{E}c_{i}}{6 p_{1}}.\] We can rewrite the optimal quantity as \[q_{s,i} = \underbrace{\frac{\mathbb{E}c_{i} - p_{0}}{3 p_{1}}}_{:=q} + \underbrace{\frac{c_{s,i} - \mathbb{E}c_{i}}{2 p_{1}}}_{:=u_{s}}.\] The first part on the right hand side of the last expression (i.e., \(q\)) corresponds to the optimal quantity of the model with deterministic costs. From the second term (\(u_{s}\)), we see that if the realized shock (\(c_{s,i}\)) is greater than the average shock, namely \(c_{s,i}-\mathbb{E}c_{i}>0\), then the firm produces less than the in deterministic case (because \(p_{1}<0\)). Analogously, when the realized shock is less than the mean shock, the second part becomes positive, and firm \(i\) produces more compared to the deterministic case. On average, these differences cancel out, as we can see from \[\mathbb{E} q_{i} = \frac{3\mathbb{E} c_{i} - 2 p_{0} - \mathbb{E}c_{i}}{6 p_{1}} = \frac{\mathbb{E}c_{i} - p_{0}}{3 p_{1}}.\] The interpretation is that when the marginal production cost of a firm unexpectedly increases, the firm reacts by reducing the quantity it produces.

The market quantity depends on the realization of both firms’ shock. Specifically, it is \[Q_{s,r} = q_{s,i} + q_{r,j} = 2\frac{\mathbb{E}c_{i} - p_{0}}{3 p_{1}} + \frac{c_{s,i} + c_{r,j} - 2\mathbb{E}c_{i}}{2 p_{1}}.\] Depending on the realizations of \(c_{s,i}\) and \(c_{r,j}\), the market quantity can be greater or less than when costs are deterministic. Nevertheless, on average, we get \[\mathbb{E} Q_{s,r} = 2\frac{\mathbb{E}c_{i} - p_{0}}{3 p_{1}},\] similar to the case without uncertainty. A analogous conclusion is true for the market price, which is given by \[P_{s,r} = p_{0} + p_{1}Q_{s,r} = \frac{2\mathbb{E}c_{i} + p_{0}}{3} + \frac{c_{s,i} + c_{r,j} - 2\mathbb{E}c_{i}}{2}.\] In the stochastic case, prices can be above or below the deterministic price level, but the uncertainty cancels out on average.

The profits of the firm are a bit more tedious to calculate. Specifically,

\begin{align*} \pi_{s,r,i} &= \left(p_{0} + p_{1}Q_{s,r} - c_{s,i}\right) q_{s,i} \\ &= \left(p_{0} + p_{1}\left(2 q + u_{s} + u_{r}\right) - c_{s,i}\right) \left(q + u_{s}\right) \\ &= \left(p_{0} + p_{1}2 q - \mathbb{E}c_{i}\right) q + \left(p_{1}\left(u_{s} + u_{r}\right) + \mathbb{E}c_{i} - c_{s,i}\right) q \\ &\qquad + \left(p_{0} + 2p_{1} q - \mathbb{E}c_{i}\right) u_{s} + \left(p_{1}\left(u_{s} + u_{r}\right) + \mathbb{E}c_{i} - c_{s,i}\right) u_{s}. \end{align*}

Since \(u_{s}\), \(u_{r}\) are independent and \(\mathbb{E}u_{s}=0=\mathbb{E}u_{r}\), expected profits are easily calculated from the last expression as

\begin{align*} \mathbb{E} \pi_{i} &= \left(p_{0} + p_{1}2 q - \mathbb{E}c_{i}\right) q + p_{1}\mathbb{E} u_{s}^2 + \mathbb{E}\left[\left(\mathbb{E}c_{i} - c_{s,i}\right) u_{s}\right] \\ &= \left(p_{0} + p_{1}2 q - \mathbb{E}c_{i}\right) q + p_{1}\mathbb{E} u_{s}^2 -2p_{1} \mathbb{E} u_{s}^2 \\ &= -\frac{(p_{0}-c_{1})^{2}}{9p_{1}} - p_{1}\mathbb{E} u_{s}^2 \\ &= -\frac{(p_{0}-c_{1})^{2}}{9p_{1}} - p_{1}\mathbb{V} u_{s} \\ &= -\frac{(p_{0}-c_{1})^{2}}{9p_{1}} - \frac{1}{4p_{1}}\mathbb{V} c_{i}. \end{align*}

Because profits are quadratic in marginal production costs, they are affected by the uncertainty in production. The more uncertain production gets (i.e., the greater the variance of \(c_{s,i}\) becomes), the fewer profits firms make on average.

Competition in Prices with Cost Uncertainty

Suppose that there are \(n\) firms in the market. Firm \(i=1,2,\dots,n\) has cost function \[c(q_{i}, \theta_{i}) = \theta_{i} q_{i},\] where the random variables \(\theta_{i}\) are identically, independently, and uniformly distributed on \([0, 1]\). The value of \(\theta_{i}\) is observed by firm \(i\) but not by other firms (private information).

Let the demand for firm \(i\) be

\begin{align*} d_{i}(p_{1}, \dots, p_{n}) = \left\{\begin{aligned} &1 - p_{i} & p_{i} < \min_{j\neq i} p_{j} \\ &\frac{1 - p_{i}}{m} & p_{i} = \min_{j\neq i} p_{j},\ m=\left|\operatorname{arg}\min_{j\neq i} p_{j}\right| \\ &0 & p_{i} > \min_{j\neq i} p_{j} \end{aligned}\right.. \end{align*}

How do Symmetric Bayesian Nash Equilibria Look Like? The Bayesian Nash Equilibria of the game specify best responses (prices) for each state (\(\theta\)). Thus, we look for functions \(p_{i}^{\ast}\) giving a price for each \(\theta\in[0,1]\). Due to symmetry, we expect \(p^{\ast} = p_{i}^{\ast}\) for every firm \(i\). The probability of two or more firms having the same costs is zero. So, we can focus on firms with different costs. Additionally, We expect \(p^{\ast}(1)=1\) because prices below cost (\(p^{\ast}(1)<\theta=1\)) or above the market size (\(p^{\ast}(1)>d_{i}(0)\)) give negative profit. From our intuition in the deterministic case, we expect \(p^{\ast}\) to be increasing in the marginal cost \(\theta\).

We focus on smooth best response functions \(p^{\ast}\). Given the choices \(p_{j}\) of other firms, firm \(i\) maximizes expected profits

\begin{align*} \pi_{i} &= \max_{p_{i}} \mathbb{E} \left[ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(p_{1}(\theta_{1}), \dots, p_{i}(\theta_{i}), \dots, p_{n}(\theta_{n}))\right] \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \mathbb{P}(p_{i}(\theta_{i}) < \min_{j\neq i} p_{j}(\theta_{j}) ) \right\} \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \prod_{j\neq i} \mathbb{P}(p_{i}(\theta_{i}) < p_{j}(\theta_{j}) ) \right\} \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \prod_{j\neq i} \mathbb{P}(p_{j}^{-1}(p_{i}(\theta_{i})) < \theta_{j} ) \right\} \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \prod_{j\neq i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) \right\} . \end{align*}

To find the necessary conditions, by Pontryagin’s maximum principle, we can maximize pointwise. We then get

\begin{align*} & (1 - 2p_{i}(\theta_{i}) + \theta_{i}) \prod_{j\neq i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) - \\ & \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \sum_{k \neq i} \frac{\mathrm{d} p_{j}^{-1}(p_{i}(\theta_{i}))}{\mathrm{d} p_{i}(\theta_{i})} \prod_{j\neq k, i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) = 0 . \end{align*}

By the inverse mapping theorem

\begin{align*} & (1 - 2p_{i}(\theta_{i}) + \theta_{i}) \prod_{j\neq i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) - \\ & \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \sum_{k \neq i} \frac{1}{p_{j}’(p_{j}^{-1}(p_{i}(\theta_{i})))} \prod_{j\neq k, i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) = 0. \end{align*}

By symmetry, we expect \(p_{i} = p_{j} = p^{\ast}\), so we can simplify to

\begin{align*} & (1 - 2p^{\ast}(\theta_{i}) + \theta_{i}) (1 - \theta_{i})^{n-1} - \\ & \left( p^{\ast}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) (n - 1) \frac{1}{(p^{\ast})’(\theta_{i})} (1 - \theta_{i})^{n-2} = 0. \end{align*}

Thus, the necessary condition specifies a first order differential equation that \(p^{\ast}\) needs to satisfy.

The best responses are obtained by solving the differential equation. Rearranging gives

\begin{align*} (p^{\ast})’(\theta_{i}) &= \frac{\left( p^{\ast}(\theta_{i}) - \theta_{i} \right) (1 - p_{i}(\theta_{i})) (n - 1)}{(1 - 2p^{\ast}(\theta_{i}) + \theta_{i}) (1 - \theta_{i})}. \end{align*}

We solve using the method of undetermined coefficients for \(p^{\ast}(\theta_{i}) = p_{0} + p_{1}\theta_{i}\). By \(1=p(1)\), this simplifies to \(p^{\ast}(\theta_{i}) = 1 - p_{1} + p_{1}\theta_{i}\). Substituting gives

\begin{align*} p_{1} &= \frac{(1 - p_{1}) p_{1} ( 1 - \theta_{i})^{2} (n - 1)}{(2 p_{1} - 1)(1 - \theta_{i})^{2}}. \end{align*}

Thus,

\begin{align*} p_{1} = \frac{n}{n+1} \quad\text{and}\quad p_{0} = \frac{1}{n+1}. \end{align*}

In the symmetric Bayesian Nash equilibrium firms choose \[p^{\ast}(\theta_{i}) = \frac{1 + n \theta_{i}}{n+1} \quad (i =1, \dots, n).\] In contrast with the deterministic case, all firms price above their marginal cost, i.e., \[p^{\ast}(\theta_{i}) > \theta_{i}.\] The markup decreases in the number of firms in the market (the market becomes more competitive) \[\frac{\mathrm{d} p^{\ast}(\theta_{i})}{\mathrm{d} n} < 0.\] The limiting case gives the perfect competition pricing \[\lim_{n\to\infty} p^{\ast}(\theta_{i}) = \theta_{i}.\]

Exercises

Group A

1. Consider a Cournot duopoly with information asymmetry. The market inverse demand is \(p(q) = 10 - q\). The cost function of firm \(1\) is \(c(q)=q\), which is common knowledge. The cost of firm two is \(c(q)=0\) with probability \(1/2\) and \(c(q)=2 q\) with probability \(1/2\). The cost of firm \(2\) is drawn by nature, but it is only observed by firm \(2\). The firms simultaneously choose their quantities.

   1. Calculate the Bayesian Nash equilibrium of the game.
   2. Calculate the market quantity and price for both states of the game.
   3. Calculate the firms’ expected profits. That is, calculate the profits the firms expect before nature draws the marginal cost of firm \(2\). Which firm achieves greater profit? Why?
   1. Firm \(2\) observes its marginal cost and solves \[\max_{q_{s,2}} \left\{ p(q_{1} + q_{s,2})q_{s,2} - c_{s,2}q_{s,2}\right\} \quad\quad (c_{s,2} = 0,2).\] From this, we can calculate the two first order conditions (one for each state) as \[q_{s,2} = \frac{-c_{s,2} + 10 - q_{1}}{2} \quad\quad (c_{s,2} = 0,2).\] Further, we find that the expected quantity produced by firm \(2\) is \[\mathbb{E}q_{2} = \frac{9 - q_{1}}{2}.\] Firm \(1\) does not observe the marginal cost of firm \(2\), so it maximizes the expected profit \[\max_{q_{1}} \left\{ \mathbb{E} p(q_{1} + q_{2})q_{1} - q_{1}\right\},\] which gives the best response \[q_{1} = \frac{9 - \mathbb{E}q_{2}}{2}.\] Substituting the expected quantity produced by firm \(2\) gives \(q_{1} = 3\). Finally, substituting in the best response of firm \(2\), we get \[q_{0,2} = \frac{21}{6},\] when nature draws the low cost and \[q_{2,2} = \frac{15}{6},\] when nature draws the high cost.
   2. When the marginal cost of firm \(2\) is zero, the market quantity is \[q_{0} = q_{1} + q_{0,2} = 3 + \frac{21}{6} = \frac{39}{6},\] and the market price is \[p(q_{0}) = 10 - q_{0} = \frac{21}{6}.\] When the marginal cost of firm \(2\) is equal to \(2\), the market quantity is \[q_{2} = q_{1} + q_{2,2} = 3 + \frac{15}{6} = \frac{33}{6},\] and the market price is \[p(q_{2}) = 10 - q_{2} = \frac{27}{6}.\]
   3. The expected profit of firm \(2\) is \[\mathbb{E}\pi_{2} = \frac{1}{2} \frac{21}{6} \frac{21}{6} + \frac{1}{2} \left(\frac{27}{6} - 2\right)\frac{15}{6} = \frac{37}{4}.\] The profit of firm \(1\) is \[\mathbb{E}\pi_{1} = \frac{1}{2} \left(\frac{21}{6} - 1\right)3 + \frac{1}{2} \left(\frac{27}{6} - 1\right) 3 = 9.\] Firm \(2\) makes, on average, more profit than firm \(1\). Firm \(2\) has more information about the state of nature and can adjust its decisions better than firm \(1\).

Group B

1. Consider a Cournot duopoly with information asymmetry. The market inverse demand is \(p(q) = p_{0} + p_{1}q\). The cost function of firm \(1\) is \(c(q)=c_{1}q\), which is common knowledge. The cost of firm two is \(c(q)=c_{\theta,2}q\), where \(c_{\theta,2}\) is a positive random variable. The cost \(c_{\theta,2}\) is drawn by nature, but it is only observed by firm \(2\). The firms simultaneously choose their quantities.

   1. Calculate the Bayesian Nash equilibrium of the game.
   2. Calculate the market quantity and price.
   3. Calculate the firms’ profits.
   4. Suppose that \(c_{1}>1\) and \(c_{\theta,2}=c_{1} + \theta\), where \(\theta=1\) with probability \(p\) and \(\theta=-1\) with probability \(1-p\). Which firm is producing more on expectation (Before nature draws the marginal cost of firm \(2\))? Why?

   1. Firm \(2\) observes its marginal cost and solves \[\max_{q_{\theta,2}} \left\{ p(q_{1} + q_{\theta,2})q_{\theta,2} - c_{\theta,2}q_{\theta,2}\right\}.\] From this, we can calculate the state dependent first order conditions (one for each state) as \[q_{\theta,2} = \frac{c_{\theta,2} - p_0 - p_{1}q_{1}}{2 p_{1}}.\] Further, we calculate the expected quantity produced by firm \(2\) as \[\mathbb{E}q_{2} = \frac{\mathbb{E}c_{2} - p_0 - p_{1}q_{1}}{2 p_{1}}.\] Firm \(1\) does not observe the marginal cost of firm \(2\), so it maximizes the expected profit \[\max_{q_{1}} \left\{ \mathbb{E} p(q_{1} + q_{2})q_{1} - c_{1}q_{1}\right\},\] which gives the best response \[q_{1} = \frac{c_{1} - p_0 - p_{1}\mathbb{E}q_{2}}{2 p_{1}}.\] Substituting the expected quantity produced by firm \(2\) gives \[q_{1} = \frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}}.\] Finally, substituting in the best response of firm \(2\), we get \[q_{\theta,2} = \frac{3 c_{\theta,2} + \mathbb{E} c_{2} - 2c_{1} - 2p_0}{6 p_{1}}.\]

   2. The total market quantity is

      \begin{align*} q_{\theta} &= q_{1} + q_{\theta,2} \\ &= \frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}} + \frac{3 c_{\theta,2} + \mathbb{E} c_{2} - 2c_{1} - 2p_0}{6 p_{1}} \\ &= \frac{2c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} - 4p_0}{6 p_{1}}, \end{align*}

      and the market price is

      \begin{align*} p(q_{\theta}) &= p_{0} + p_{1}q_{\theta} = \frac{2c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0}{6}. \end{align*}

   3. The profit of firm \(2\) is

      \begin{align*} \pi_{\theta,2} &= \left( \frac{2c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0}{6} - c_{\theta,2} \right) \frac{3 c_{\theta,2} + \mathbb{E} c_{2} - 2c_{1} - 2p_0}{6 p_{1}} \\ &= - \frac{\left(2c_{1} - 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0\right)^{2}}{36 p_{1}}. \end{align*}

      The profit of firm \(1\) is

      \begin{align*} \pi_{\theta,1} &= \left( \frac{2c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0}{6} - c_{1} \right) \frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}} \\ &= \frac{-4c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0}{6} \frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}} . \end{align*}

   4. We have \(q_{1} > \mathbb{E} q_{2}\) if and only if \[\frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}} > \frac{3 \mathbb{E} c_{2} + \mathbb{E} c_{2} - 2c_{1} - 2p_0}{6 p_{1}},\] which simplifies to \[c_{1} < \mathbb{E} c_{2}.\] By the distributional assumptions of \(c_{2}\), the last condition is equivalent to \[0 < \mathbb{E} \theta = 2p - 1.\] If the average shock to the marginal cost of firm \(2\) is positive (or equivalently \(p>1/2\)), then firm \(2\)’s production cost is on average greater than that of firm \(1\), and the expected production of firm \(2\) is less than that of firm \(1\).

References

References