Capacity precommitment, communication, and collusive pricing: theoretical benchmark and experimental evidence

Abstract

We use a first-capacity-then-price-setting game as a theoretical benchmark for an experimental study which identifies capacity precommitment, intra-play communication, and prior experience as crucial factors for collusive pricing. The theoretical model determines capacity thresholds above which firms have an incentive to coordinate on higher than equilibrium prices. The experimental data confirm that intra-play communication after capacity but before price choices fosters collusion only for capacity levels exceeding these thresholds. If communication is abandoned, prices nevertheless remain high. Asymmetry in capacities decreases the reliability of price messages as well as the probability to coordinate on equal prices.

Appendix

1.1 A. Mixed-strategy equilibria in the capacity-constrained price game

Let us assume without loss of generality that $\bar{q}_1 \ge \bar{q}_2$. Then mixed-strategy price equilibria exist when capacities are not too small ($\alpha - 2 \bar{q}_1 - \bar{q}_2 < 0$) and not too large ($\bar{q}_2 < \alpha$). Depending on the constellation of installed capacities, one has to distinguish between different types of mixed-strategy equilibria. One of these types covers the relation $2 \bar{q}_1 - \alpha \le \bar{q}_2$. Since we restricted our theoretical and experimental analysis to capacities $\bar{q}_1 \le \alpha /2$, this is the equilibrium type which is generally relevant in our scenario.

As is already known from the analysis of Kreps and Scheinkman (1983), firms 1 and 2 randomize their price decisions according to distribution functions $F_1(p)$ and $F_2(p)$ over a common coincidence interval $[\underline{p}, \bar{p}]$. Since $p_i > \alpha - \bar{q}_1 - \bar{q}_2, i=1,2,$ in the mixed-strategy equilibria and since the probability of a tie proves to be zero, the production of firm i amounts to

$$\begin{aligned} q_i=\left\{ \begin{array}{ll} \min (\bar{q}_i, \alpha - p_i - \bar{q}_j) = \alpha - p_i - \bar{q}_j &{}\quad \text {if } \; p_{i} \; > \; p_j \\ \min (\bar{q}_i, \alpha - p_i) &{}\quad \text {if } \; p_{i} \; < \; p_j, \end{array} \right. \end{aligned}$$

leading to the expected revenues

$$\begin{aligned} E(p_iq_i) = p_i [(1-F_j(p_i)) (\min (\bar{q}_i, \alpha - p_i)) + F_j(p_i) (\alpha - p_i - \bar{q}_j)]\; \end{aligned}$$

(A.1)

for $i,j=1,2,\; i \ne j$. Consider firm 1, charging the upper-bound price $\bar{p}$, at which firm 1 is surely undercut by its rival, i.e., $F_2(p_1=\bar{p})=1$. It maximizes its revenue

$$\begin{aligned} \bar{p} q_1 = \bar{p}(\alpha - \bar{p} - \bar{q}_2) \end{aligned}$$

with respect to $\bar{p}$, thereby determining the upper-bound price

$$\begin{aligned} \bar{p} = (\alpha - \bar{q}_2)/2 \; \end{aligned}$$

(A.2)

and hence the revenue

$$\begin{aligned} \bar{p} q_1 = (\alpha - \bar{q}_2)^2/4. \end{aligned}$$

(A.3)

It follows from (A.2) that $p_1 \le \bar{p} = (\alpha - \bar{q}_2)/2$. Taking into account that our scenario is restricted to constellations where $2 \bar{q}_1 - \alpha \le \bar{q}_2$, this implies that $\bar{q}_1 < \alpha - p_1$ and thus from (A.1)

$$\begin{aligned} E(p_1 q_1) = p_1 [(1-F_2(p_1)) \bar{q}_1 + F_2(p_1) (\alpha - p_1 - \bar{q}_2)]. \end{aligned}$$

(A.4)

In order for the risk-neutral firm 1 to be indifferent between all prices $p \in [\underline{p}, \bar{p}]$, it must hold that $E(p_1 q_1) = \bar{p} q_1$. It turns out from (A.3) and (A.4) that, using $F_2(p_1=\underline{p})=0$, the lower-bound price is determined as

$$\begin{aligned} \underline{p} = (\alpha -\bar{q}_2)^2/(4\bar{q}_1). \end{aligned}$$

Furthermore, for all prices $p_1 \in [\underline{p}, \overline{p}]$ the distribution function of firm 2’s price decision must read

$$\begin{aligned} F_2(p) = \frac{\bar{q}_1 - (\alpha - \bar{q}_2)^2/(4p)}{p - \alpha + \bar{q}_1 + \bar{q}_2}, \quad \frac{(\alpha - \bar{q}_2)^2}{4 \bar{q}_1} \le p \le \frac{\alpha - \bar{q}_2}{2}. \end{aligned}$$

Now consider firm 2. Taking into account the upper-bound price (A.2), it holds that $p_2 \le \bar{p} = (\alpha - \bar{q}_2)/2 < (\alpha - \bar{q}_2)$. Then it follows from (A.1) that

$$\begin{aligned} E(p_2 q_2) = p_2 [(1-F_1(p_2)) \bar{q}_2 + F_1(p_2) (\alpha - p_2 - \bar{q}_1)]. \end{aligned}$$

At $p_2=\underline{p}$, implying $F_1(p_2=\underline{p})=0$, the revenue amounts to

$$\begin{aligned} \underline{p} q_2 = (\alpha - \bar{q}_2)^2 \bar{q}_2 /(4 \bar{q}_1). \end{aligned}$$

Therefore, in order for the risk-neutral firm 2 to be indifferent between all prices $p_2 \in [\underline{p}, \bar{p}]$, it must hold that $E(p_2 q_2) = \underline{p} q_2$ so that the distribution function of firm 1’s price decision must read

$$\begin{aligned} F_1(p) = \frac{\bar{q}_2 - (\alpha -\bar{q}_2)^2 \bar{q}_2 /(4\bar{q}_1 p)}{p - \alpha + \bar{q}_1 + \bar{q}_2}, \quad \frac{(\alpha - \bar{q}_2)^2}{4 \bar{q}_1} \le \; p \; < \frac{\alpha - \bar{q}_2}{2}. \end{aligned}$$

At $p=(\alpha - \bar{q}_2)/2$, where $F_1(p_1=\bar{p})=1$, there is a mass point indicating that firm 1 charges this price with positive probability (whereas firm 2 charges this price with density zero). $F_1(p)$ and $F_2(p)$ are the price distribution functions as applied in the main text.

1.2 B. Instructions for the experiment

1.2.1 General information

Thank you for participating in this experiment. Please remain silent and turn off your mobile phones. Please read the instructions carefully and note that they are identical for each participant. During the experiment it is forbidden to talk to other participants. In case you do not follow these rules, we will have to exclude you from the experiment as well as from any payment.

You will receive 15 euros for participating in this experiment. The participation fee and any additional amount of money you will earn during the experiment will be paid out to you privately in cash at the end of the session. No other participant will know how much you earned. All monetary amounts in the experiment will be given in ECU (experimental currency units). At the end, all earned ECUs will be converted into Euro using the following exchange rate:

10.000 ECU = 1 euro

1.2.2 Procedure of the experiment

The experiment consists of 2 parts with 12 rounds each. In each part you will make decisions at three stages. One stage consists of four rounds. You receive the instructions for the second part after finishing the 12 rounds of part 1.

At the end of the experiment, 5 rounds of each part will be randomly selected to determine your payment. You will receive the sum of your payoffs you have earned in 10 rounds. Your total payment will be composed of the participation fee of 15 Euro and the amounts you earned in the 10 randomly selected rounds.

If you suffer a loss in the 10 selected rounds, you can pay it in cash or balance it by completing additional tasks at the end of the experiment. Please note that these additional tasks can only be used to compensate for possible losses, but not to increase your earnings. Additionally, you will receive a payment for one task from the questionnaire part. Hence, you will receive the participation fee and payment for the questionnaire part in any case.

1.2.3 Introduction

In this experiment you take the role of a manager in a company. You decide how many units of a good your company should produce. This amount specifies the capacity of your company. Afterwards, you choose the selling price for the produced good. Your company has a competing company which produces the same good. You compete against the other company in four rounds. Afterwards, another competitor will be randomly assigned to you. You will not be informed about the other manager’s identity.

In each round of the experiment, you will first make decisions about the capacity, followed by decisions about the price. At the beginning of each round, you and the company you are competing with will decide about your capacity simultaneously and independently of each other. The capacity corresponds to the amount of goods that your company is planning to produce in this round. Every capacity unit costs a fixed amount. After your capacity decision, you will be informed about the capacity decision of your competing company. Afterwards, both companies choose their selling price for their good at the same time. The company with the lower price gets the chance to sell its produced amount first. The company with the higher price can sell something only if the preferred company has produced too little to sell something to every interested costumer. If the prices are equal, the customers are distributed to both companies equally (if the number of customers is odd, it will be rounded to the next higher even number). In any case, both companies have to pay their production costs. This holds even if they have not sold anything.

1.2.4 Definition of the experiment—part 1

The experiment consists of two parts which are divided in three stages. At the beginning of each stage, the groups of two companies are assigned by chance and anew.

One stage consists of four rounds in which you interact with the same competing company. The procedure is as follows:

• 1. Both companies choose a capacity between 0 and 100 at the same time. The costs are 80 ECU for each capacity unit.

• 2. The capacity decisions are announced.

• 3. Both companies choose a price between 0 and 200 for the produced good at the same time.

• 4. The chosen prices are announced, the produced goods are sold and both companies come to know their earnings or losses and the ones of the other company.

The demand of your produced good complies with your chosen price: The more expensive the good, the less it is bought.

The number of customers for the produced good is computed by the software and depends on price p. It equals the amount 200-p. This means that the number of customers declines with a rising price.

Your payment is composed of your revenues minus your production costs. For every sold good you receive the chosen price and pay production costs for every produced unit. The amount you sell depends on whether your price is lower, higher or equal to the price of the other company in your group:

• (a) Your price is lower.

In this case you first get the chance to sell your produced amount. For every unit sold you receive your price p. If an amount M is requested, you earn p × M. From this you have to subtract the production costs of 80 ECU per produced unit. If the requested amount M exceeds your capacity, the other company can sell something at its higher price. This will work if interested customers remain that are ready to pay the higher price.

• (b) Your price is higher.

In this case you only sell something if the other company has produced too little. If there are remaining customers who are ready to buy at the higher price, they will buy from you. For every unit sold you receive the chosen price. Even if you sell nothing or less than you have produced you have to pay all the production costs at the amount of 80 × your capacity.

• (c) Both prices are equal.

In this case both companies can each sell to one half of the customers. Even if you sell less than you have produced you have to pay all the production costs.

Therefore, your payment can be summarized as follows:

Your price × sold amount − 80 × your capacity

Please note that the production costs of 80 × your capacity also arise even if you sell nothing or less than you produced in one round.

You will receive instructions for part 2 at the end of stage 1.

Before part 1 of the experiment begins, we ask you to answer a few control questions to help you understand the rules of the experiment. This is followed by one practice round, so that you can become familiar with the structure of the experiment. If you have any questions, please raise your hand.

1.2.5 Instructions for part 2

Part 2 also consists of three stages. At the beginning of each stage the groups of two companies are chosen randomly.

Afterwards, there will be four rounds in each stage. The process of these rounds only differs from the rounds in part 1 insofar that the companies can tell the other company the price they will determine before choosing a price. This statement is not binding, e.g., the actual decision can be different from the price which was told.

If you have any questions about part 2, please raise your hand.