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:


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.
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.