8 Oligopoly - Luiscabral.net

Since Q = 13, it follows that b = (2 × 33 − 15 − 12)/(3 × 13) = 1. Therefore, in the initial
equilibrium, Firm 1’s profits are given by
( ) 2 33 + 12 − 2 × 15
̂π 1 =
=
3
( ) 2 15
= 25
3
whereas, with the new technology, profits would be given by
( ) 2 ( ) 2 33 + 12 − 2 × 12 21
̂π 1 =
= = 49
3
3
I conclude that Firm 1 should be willing to pay 49 − 25 = 24 for the new technology.
Given the units I worked with (price in $ per ton and quantity in million tons) this would
be $24 million.
Let us go back to where we started this section. Why is it useful to perform comparative
statics? Take for example the last application considered above. The question we posed
was: How much would Firm 1 (the inefficient firm) be willing to pay for an innovation
that reduces marginal cost to 12 (the efficient firm’s cost level)? Our analysis produced an
answer to this question: Firm 1 would gain 24 from adopting the more efficient technology.
Is it worth going through all the algebraic trouble to get this answer? A simpler estimate
for the gain might be the following: take Firm 1’s initial output as given and compute the
gain from reducing marginal cost. Since initial output is 5 (check), this calculation would
yield the number 5 × (15 − 12) = 15. This number greatly underestimates the true gain of
24. Why? The main reason is that, upon reducing marginal cost, Firm 1 becomes much
more competitive: not only does it increase its margin, it also increases its output.
Let us then consider an alternative simple calculation. Since Firm 1 will have a marginal
cost identical to Firm 2’s, we can estimate the gain by the difference between Firm 2’s and
Firm 1’s initial profits. Firm 2’s output (in the initial equilibrium) is 8 (check), whereas
price is given by 20; its profit is thus 8 × (20 − 12) = 64. Firm 1, in turn, is starting from
a profit of 5 × (20 − 15) = 25. We would thus estimate a gain of 64 − 25 = 39. This time
we are grossly overestimating the true value. Why? The main reason is that, when Firm
1 lowers marginal cost, duopoly competition heats up and price drops. As a result, Firm
2’s profit (which is now equal to Firm 1’s) also drops. In other words, the number π 2 − π 1
over-estimates the gain from technology adoption because it omits the effect of increased
competition.
The advantage of the equilibrium analysis — that is, comparative statics — is that it
takes into account all of the effects that follow from an exogenous change, such as a reduction
in one of the firm’s marginal cost. Taking the initial equilibrium values as constant may
lead to gross misestimation of the impact of an exogenous change, especially if the change
is significant in magnitude (as in the present example).
Exchange rates and sales margin. A French pharmaceutical firm is the sole domestic
producer of a given generic antidepressant. Its marginal cost is e2 per dose. Demand in
France is given by Q = 400−50 p (Q in millions of doses, p in e). There is a second producer
in India whose marginal cost is INR 150 (including transportation cost to the France). The
French regulatory system implies that firms must commit to prices for one year at a time.
Production capacity can be easily adjusted, since the firms produce several other medical