Consider a duopoly with firms A and B$,$ which engage in price competition$\mathrm{ (}“$Bertrand$”).$ Both firms offer homogenous products. Total costs are given by

CA$($q$)=10·$ q for firm A and by CB$($q$)=21·$ q for firm B$.$

 Inverse demand is given by

P$($q$)=120-$ q$.$

In this market, firms are only allowed to charge integer values as prices. Moreover, suppose that at a market price equal to their respective marginal cost each firm would rather be active and sell at marginal cost $($yielding zero profits$)$ than not to produce at all $($also yielding zero profits$).$

$($a$)$ Determine the equilibrium profits of the two firms.

$($b$)$ Determine the maximum amount that firm A would be willing to pay firm B to exit the market.

$($c$)$ Determine the minimum amount that firm B would accept to exit the market.