Question

Consider the following hypothetical market for solar panels. Demand for solar panels is given by p(Q) = 33 – Q and marginal cost is initially 15. However, an investment in R&D can lower the marginal cost to 11.

1. Suppose the market is a monopoly. How much would a monopoly be willing to invest to reduce its marginal cost from 15 to 11?

2. Now assume the market for solar panels is dominated by two firms who compete via Bertrand competition. What is the new Nash equilibrium if firm #1’s costs fall from 15 to 11? What are each firm’s quantity and profit?

3. In question 2, how much would firm #1 be willing to pay to develop its idea and reduce cost from 15 to 11?

4. Now suppose the market for solar panels is a monopoly (with constant marginal cost 15). Firm #2 is a producer of windmill energy that is looking to get into the solar panel business. Suppose a start-up invents a new way to produce solar panels with constant marginal cost 11. The startup is willing to sell exclusive ownership of this technology to either firm #1 (the monopoly producer of solar panels), or firm #2 (the windmill producer looking to begin selling solar panels). If firm #2 purchases the technology, the market switches to Bertrand competition, where firm #1 has constant marginal cost 15 and firm #2 has constant marginal cost 11. How much is firm #2 willing to spend to buy the technology?

5. In question 4, how much is firm #1 willing to spend to buy the technology?

6. Now suppose firm #1 and firm #2 are already competing via Bertrand competition, with constant marginal cost 15. Suppose it is impossible to obtain exclusive ownership of the idea. As soon as one firm invents the innovation (or purchases the innovation from a startup), both firms have their costs fall from 15 to 12, since the other can copy the idea. How much would firms be willing to spend invest to lower their costs from 15 to 12?

Answer 1

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1. If the market is a monopoly, without investing

the revenue of the monopolist will be PQ and costs will be 15Q

Profit=PQ-15Q=(33-Q)Q-15Q= (18-Q)Q

For maximum profit, we differentiate and equate to 0, we get

18-2Q=0

Q=9

So profits= (18-9)*9=81

After investing, since the profits will become

PQ-15Q=(33-Q)Q-11Q= (22-Q)Q

For maximum profit, we differentiate and equate to 0, we get

22-2Q=0

Q=11

So profits= (22-11)*11=121

Since the difference in profit in both cases is 121-81=40, the monopolist will be willing to invest upto 40 or less for R&D to reduce the marginal cost from 15 to 11.

2. Under bertrand competition

If the first firm's MC is 11 and the second's is 15

The second will have 0 profit since Firm 1 can always undercut its price and take away the entire market. (if it charges 16, firm 1 can charge 15, if firm 2 charges 15, firm 1 will charge just under 15 and still take away the entire market)

Similarly we can assume the Firm 1 will charge lesss than or just equal to 15 to take away the entire market.

Thus for nash equilibrium, we can assume firm 1 will charge 15 and get the entire market and firm 2 has 0 quantity and 0 profits produced.

For firm 1, P=15=33-Q according to demand curve

Q=33-16=18

Therefore profits= 15Q-11Q=72

3. Since in bertrand competition, if both had a marginal cost of 15, both will keep producing at competitive price=15 and thus both will have 0 profits, however if the mc of firm 1 drops to 11, it will have a profit of 72 as calculated above. Therefore firm 1 will be willing to pay upto 72 for reducing its cost from 15 to 11.

4. In this case the marginal cost of firm 1 is 15, and the marginal cost of firm 2 will become 11 if it purchases the technology. As we calculated in this bertrand duopoly with 1 firm having mc 15 and the other 11, the profit of the one with 11 will be 72 and the one with mc=15 will be 0, so firm 2 will be willing to pay upto 72 to get this technology. (Firm 1's initial profit was 81 with monopoly and MC=15 and it now becomes 0, so it should be willing to pay 81)

Hope it helps. Do ask for any clarifications required.