Solution
To Deter or Not to Deter
Show the problem again
Market demand is $P = 100 - Q$, where $Q$ is total quantity, and producing costs nothing. An incumbent first commits to a quantity $q_1 \ge 0$ (any real number). A potential entrant observes $q_1$ and then decides whether to enter. Entering costs a fixed amount of $49$, after which the entrant chooses his own quantity $q_2$ and the two firms sell at the resulting price. An entrant who is indifferent between entering and staying out stays out. The incumbent maximises his own profit and both are rational. What is the incumbent's profit in the subgame-perfect equilibrium?
Worked solution
The answer is 1250. If the entrant enters, he best-responds with $q_2 = (100-q_1)/2$ and earns $((100-q_1)/2)^2 - 49$. He stays out iff $(100-q_1)/2 \le 7$, that is $q_1 \ge 86$. Deterrence therefore costs the incumbent a monopoly profit of $86\cdot 14 = 1204$ at the cheapest deterring quantity (more capacity only lowers profit). The alternative is to accept entry. Then the incumbent is a Stackelberg leader, choosing $q_1$ to maximise $q_1(100-q_1)/2$, so $q_1=50$, $q_2=25$, and his profit is $50\cdot 25 = 1250$ (the entrant earns $625-49>0$, so he does enter). Since $1250 > 1204$, deterrence does not pay: the incumbent lets the entrant in. With a larger fixed entry cost, such as 225, the comparison reverses and deterrence wins.
Source: Entry deterrence by commitment, after Spence (1977) and Dixit (1980), in a linear-demand Cournot setting. Statement written for AxiomIQ.