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Monopolist profit maximization after tax and with price ceiling

Se the attached file.

Consider a monopolist facing a market demand curve given by q = 186- p(q). The monopolist's fixed costs and variable costs are equal to Cf = 2400 and C(q) = q2 /10 +10q, respectively ........(in the latter equation its actually q squared over ten plus ten q)

Calculate:

a) the monopolist's price-quantity combination that maximizes profits and the level of profits obtained;
b) the monopolist's price-quantity combination that maximizes profits in case a fixed tax T = 1000 is introduced and the level of profits obtained;
c) the monopolist's price-quantity combination that maximizes profits in case a tax for each unit of product sold tq = 11 is introduced, the amount paid in taxes and the level of profits obtained after taxes;
d) the monopolist's price-quantity combination that maximizes profits in case a tax of the tπ = 50% on the profit is introduced; the amount paid in taxes and the level of profits obtained after taxes;
e) the monopolist's quantity and profits in case the government fixes a p = 90.

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a)
Profit is maximized at the quantity where Marginal Revenue (MR) = Marginal Cost (MC)

From the Demand curve:
Q = 186 - P
Rearranging:
P = 186 - Q

Total Revenue (TR) = PQ
TR = (186 - Q)Q
TR = 186Q - Q^2

MR is the derivative of TR:
MR = 186 - 2Q

Total Cost (TC) = Variable Cost (VC) + Fixed Cost (FC)
TC = 0.1Q^2 + 10Q + 2400

MC is the derivative of TC:
MC = 0.2Q + 10

To maximize profit, let MR = MC
186 - 2Q = 0.2Q + 10
186 - 10 = 2Q + 0.2Q
176 = 2.2Q
Q = 80

From the Demand curve:
P ...

Solution Summary

This solution shows how to calculate the profit-maximizing price and quantity for a monopolist, including under scenarios where it must pay:
- a fixed tax of 1000
- a variable tax of 11 per unit
- a tax of 50% of total profit

and also where the government has imposed a price ceiling.

$2.19