Price setting firms with cost reduction technology

As part of my undergraduate studies in Industrial Economics, I am trying to solve the following question:

Two price-setting firms are competing in a market for a homogeneous product. There are 10,000 people in the population, each of whom is willing to pay at most 10 for one unit of the good. Initially, both firms have a marginal cost of 5. Assume that the firms are not capacity constrained and cannot collude. What is the equilibrium in this market and what are the firms’ profits?

Suppose now that a new technology becomes available that lowers the marginal cost to 3. The cost to a firm of purchasing this technology is 10,000. The firms must now simultaneously decide whether to adopt the new technology or not, and following this decision, they simultaneously set prices. Each firm can observe whether its rival acquired the new technology or not before setting its price. What is (are) the equilibrium (equilibria) in the market now?

I am struggling with both parts of the question but I think I got the answer to the first part more or less correct. It comes down to a Bertrand equilibrium where both firms get a fraction of the total market share, charge price equal to marginal cost (i.e. 5) and make 0 profit. Thoughts?

Now the second part is more tricky since the per unit cost will depend on the total output of the firm so it can spread the 10,000 investment over the number of units manufactured. Some loose thoughts:

• I can see that if both firms decide to invest in the cost reduction technology and they split the market in equal parts, i.e. each produce 5,000 units, then they will need to charge 5 to not incur loses, i.e. there is no point in investing.
• I also get that if a firm were to supply all the market, its average cost would be 4.

The above two points make me think that the possible equilibria are:

1. Both firms invest in the cost reduction technology and upon seeing that the other firm has also invested, decide to split the market into equal parts so as not to incur losses.
2. One firm decides to invest and the other doesn't, upon seeing that the other has invested, the non-investing firms decides to not compete since it knows the other firm will supply all the market at a cost slightly below its own marginal cost of 5. The investing firm supplies all the market and makes monopoly profit.

Option 2 doesn't look like a Nash equilibrium to me since both firms know they can get monopoly profit if they invest and the other doesn't and 0 profit if they both invest and split the market so they both decide to invest and the only Nash equilibrium is option 1.

Do I make any sense? Help appreciated.

Your reasoning for the 1st part is correct.

The 2nd part is a 2-period game. You should try to solve it by backward induction. First you go through all the 2nd period subgames. Then you can use the payoffs of the subgames and write up the investment decision subgame as a bimatrix game. However, the question is poorly written and as stated this game has no Nash equilibria in pure continous strategies. Let's go through the reasoning as to why.

First, let's find the NE for all possible cases in the 2nd period.

• Both firms have adopted the technology and have marginal cost 3. They will engage in a Bertrand game, set their prices to 3 and make 0 profit.
• No firm has invested, both charge a price of 5 and make zero profit.
• Now on to tricky case. What if one firm 1 has invested and firm 2 hasn't? It is crucial to know how firms split the market if they both charge the same price in this case. Since this is not given in the question, it is not really possible give a definite answer. I will go through two cases.

First, let's assume there's a 50:50 split with identical prices. Any equilibrium price above 5 can be ruled out by the usual Bertrand game argeuments. Firm 2 will never charge a price below 5, since it will make a loss. Firm 1 will never charge a price below 5, because given $p_2 \geq 5$ it could always increase it's profits by raising the price. So really the only candidate for an equilibrium is that both firms charge 5. However, if we assume a 50:50 split in this case, then firm 1 could increase it's profit by slightly decreasing the price and gaining the whole market. As a result, there is no equilibrium in this subgame, and hence no equilibrium for the overall game.

Second, let's assume arbitrarily that if both firms charge the same price, then all customers will buy from firm 1. In this case $p_1 = p_2 = 5$ is a Nash equilibrium.

You could then go on to find the NE of the whole game for different assumptions on the payoff split.

• Thanks for your answer. I understand that the 10,000 investment is a sunk cost but surely it should signal a commitment in the case where one firm invest and the other doesn't. The one that invests can potentially charge 3, its marginal cost, where the other one only can go as low as 5. Shouldn't the non-investment firm get this and realise the other firm will supply the whole market at a lower cost? Also, besides being seen as an investment, couldn't the 10,000 be seen as a fixed cost of entry? Feb 3, 2017 at 5:55
• I still don't see your "tricky case" reasoning as valid, I am inclined to agree with @Bee Dev on this part. If firm 1 has invested and firm 2 hasn't then 1 can undercut $p_2$ ever so slightly and supply the whole market almost doubling its profit. To me the only NE for this game is where both invest in the cost reducing technology. I've thought more about it and this looks like a prisoner's dilemma where innocent is investing and guilty is not investing. Feb 4, 2017 at 16:18
• I agree with your reasoning, but per definition of NE, there can't be a strategy that is even slightly better, and with the limit argument there always exists some incrementally higher price that increases profits. Both investing can't be an NE because both firms would incur an overall loss and could deviate to an overall payoff of zero. Feb 4, 2017 at 22:10
• Ok, I see your point about the limit argument. I also understand the argument about sunk costs and that they don't matter when setting the price in the second period. But we are considering the investment costs when determining the NE by saying that both investing can't be a NE because both would incur a loss. Now, my understanding of what a NE is that both firms can't increase their expected payoff by making another move, and it tells me the best they can do is invest, split the market 50:50 and charge 5 which would allow them to recover the investment amount. They don't have an... Feb 5, 2017 at 10:56
• ...incentive to undercut one another because in doing so, they would trigger a price war that would end up with both charging 3, still splitting the market 50:50 and incurring a loss. The reason both not investing or one investing and the other not are not NE, in my opinion, is that in those cases, they can always invest and expect the other one not to do so. Feb 5, 2017 at 10:58

Your answer to part one is correct. You should respond to the second part in the context of game theory by analyzing possible outcomes for each participants' choice of action.

Possible outcome #1: Both firms adopt the new technology. Both companies will incur a loss of 10,000 (cost of adopting ew technology). Equilibrium price is 3

Possible outcome #2: Only Company A adopts new technology: In this case, company A will maximize its profit by charging a price that infinitly approaches 5 but never reaches actaully reaches it which can be demonstrated by $\lim_{n\rightarrow&space;5}&space;(n)&space;=5$

Thus Company A has a price "indefinitely approaching" 5. A market share of 100% of population i.e., 10000, a cost per product of 3 and new technology cost of 10000. Thus profit is $(5-3)\times&space;10000-10000&space;=&space;10000$

Company B will have 0 gain and 0 output

Possible outcome #3: Only company B adopts new technology: Reverse outcome #2

Possible outcome #4: Neither company utilizes new technology: Same as answer to part one.

• Hi, I realized that my answer is wrong and I will edit it. $p \to 5$ cannot be a Nash equilibrium either since there is always a price that's epsilon higher that will increase your profit. Feb 3, 2017 at 19:47
• I agree with the outcomes, I got to that point too. But which ones constitute a NE? To me, only outcome #1 as I replied to @Tobias in his answer. Feb 4, 2017 at 16:20
• Case #1 is certainly not an equilibrium, because both firms would incur a loss of 10000 when they could always not invest and make 0 profit. Feb 4, 2017 at 22:07