Problem $1.$ Consider the local market demand for fresh smoothies, which is given by: Q $=40020$P where P is price per bottle, and Q is in bottles. Currently, Smoothie$-$King is a monopoly producer with a constant marginal cost of $$10$ per bottle. Suppose there is a new smoothie technology in development that can reduce the marginal cost to $$5$ per bottle.

$($a$)$ What is the Smoothie$-$King$$s current profit$-$maximizing price$/$quantity$?($b$)$ Suppose that Smoothie$-$King could acquire the new technology that reduces its constant marginal cost to $$5.$ How much would the Smoothie$-$King be willing to pay for this technology?

$($c$)$ What if Smoothie$-$King faced a potential entrant, Smoothie$-$Queen that is also considering acquiring the new technology and entering the market? Assume only one firm can acquire the new technology. Assume that upon entry, the two firms will compete via prices in Bertrand competition $($with identical products$).$ How much is Smoothie$-$King willing to pay for this technology now? How much is Smoothie$-$Queen willing to pay for the technology?

$($d$)$ In a different town that faces the same market demand for smoothie, the markets consists of Bertrand$-$Duopoly: Smoothie Store A and Smoothie Store B$.$ Both stores face the same constant marginal cost of $$10/$bottle$.$ What is the current market price and quantity?

$($e$)$ How much would either Smoothie store be willing to pay for the new technology? Again, assume only one store can acquire the new technology. Compare this to Smoothie$-$King$$s WTP in b$)$ and c$).$