Amonopolist faces a demand curve given by: p = 220 – 3q, where p is the price of the good and q is the quantity demanded. the marginal cost of production is constant and is equal to $40. there are no fixed costs of production. what price should the monopolist charge in order to maximize profit?

Price = $130.00 First, express q in terms of p by solving for q the equation p = 220 -3q. So p = 220 - 3q p - 220 = -3q -p/3 + 220/3 = q Now the total profit will be pq - 40q. Simplify and substitute the equation above for q. So pq - 40q q(p-40) (-p/3 + 220/3)(p-40) -p^2/3 + 40p/3 + 220p/3 - 8800/3 -p^2/3 + 260p/3 + 8800/3 Now since you want the maximum value, that will be where the root(s) of the first derivative of the above expression is 0, so calculate the first derivative. -p^2/3 + 260p/3 + 8800/3 -2p/3 + 260/3 And solve for 0 -2p/3 + 260/3 = 0 260/3 = 2p/3 260 = 2p 130 = p So the best price for maximum profit is 130. Let's verify that. 130 = 220 - 3q 130 + 3q = 220 3q = 90 q = 30 So at that price point, the monopolist will make 30(130-40) = 30(90) = 2700 profit. Let's verify that by checking the price for 29 and 31 units to see if the profit is reduced for both cases. Trying 29 units. p = 220 - 3q p = 220 - 3*29 p = 220 - 87 p = 133 And calculating the new profit. 29(133-40) = 29(93) = 2697 profit. So there's less profit at a higher price. Now try for 31 units. p = 220 - 3q p = 220 - 3*31 p = 220 - 93 p = 127 And calculating the new profit. 31(127-40) = 31(87) = 2697 profit. And there's less profit at a lower price as well.