# Subgame perfect equilibrium and expected profit

I am so confused because I cannot set up the monopolist's profit maximization problem.What I did is the following one:

Any help will be appreciated. Thank you.

• Strategy $a$ : Expected payoff from strategy $a$ is $\frac{3}{4} (12000-R)$
• Strategy $b$ : Expected payoff from strategy $b$ is $\frac{1}{4} (16000-R)$
He'll choose strategy $a$ if $\frac{3}{4} (12000-R) \geq \frac{1}{4} (16000-R)$, and strategy $b$ if $\frac{3}{4} (12000-R) < \frac{1}{4} (16000-R)$. Or simply, he'll choose strategy $a$ if $R \leq 10000$, and strategy $b$ if $10000 < R \leq 16000$.
Bank will maximize it's expected profits by choosing $R$ taking as given the entrepreneur's strategy. Therefore, Bank's expected profits equal $\frac{3}{4}R - 1000$ if it chooses $R \leq 10000$, and $\frac{1}{4}R - 1000$ if it chooses $10000 < R \leq 16000$. Clearly, expected profits of the bank are maximum at $R^*=10000$. Observe that entrepreneur is willing to pay more for the loan than $R^*=10000$ since his expected payoff is $\frac{3}{4}(12000 -10000) = 1500 > 0$, but the bank will not charge more than 10000 because that would change the entrepreneur's choice of strategy to $b$ which is a more risky choice and yields lower expected returns to the bank.