Consider a monopolist in a market with linear inverse demand p(q) = 4 − q/2

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Consider a monopolist in a market with linear inverse demand p(q) = 4 − q/2. The monopolist's cost function is c(q) = 2q.

(1) Write down the monopolist's profit function. Compute the profit-maximizing quantity and the corresponding price.

(2) Assume that a 2% tax is levied on the monopolist's profits. Does this have any effect on its choices of output level and output price?

(3) Consider now a quantity tax of $1 per output unit sold. Compute the optimal output level and the corresponding output price. How does this tax affect the monopolist's choices of output and price, and its profits? 

(Hint: Note that a quantity tax of of $1 per output unit sold is equivalent to raising the marginal cost by $1. Why?)

(4) We say that the monopolist passes on the tax to the consumer if it raises the price by more than the tax ($1 here). Is this the case with the quantity tax in (2)?