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Consider a natural monopoly with market demand
P = 100 10Q
and whose total cost curve and marginal cost curve can be expressed as
TC = 100 + 20Q + Q2
MC = 20 + 2Q
a. Write an equation for this natural monopolys average total cost (ATC) curve?
b. Using the equation you wrote in (a) fill in the following table.

Q

ATC

1

2

4

5

10

20

c. Draw a graph of this firms demand curve, ATC curve, and MC curve. At what (Q, P) do MC and ATC intersect one another? Show how you found there coordinates. Is this (Q, P) pair a likely equilibrium in this market? Explain your answer fully.
d. Suppose this monopolist acts as a single price profit maximizing monopolist. Determine the monopolists price, quantity, and profit with this scenario. Round your answers to the nearest hundredth.
e. Suppose this monopolist is regulated to produce the socially optimal amount of output (where P = MC for the last unit produced and sold). Given this regulatory decision, determine the monopolists price, quantity, and profit. Round your answers to the nearest hundredth.
f. Suppose this monopolist is regulated to produce that quantity where profit is equal to zero. Given this regulatory decision, determine the monopolists price, quantity, and profit. Round your answers to the nearest hundredth.
Answers: a. ATC = (100/Q) + 20 + Q b. Q ATC 1 $121 per unit 2 72 4 49 5 45 10 40 20 45 c. ATC = MC (100/Q) + 20 + Q = 20 + 2Q (100/Q) = Q Q = 10 units of output MC = 20 + 2Q MC = 20 + 2(10) = 40 P = $40 per unit of output (Q, P) = (10 units of output, $40 per unit of output) This point lies to the right of the demand curve and will therefore not be a likely (Q, P) equilibrium in this market since at a price of $40 per unit, there is only demand for 6 units of output. Or, at a quantity of 10 units, demanders willingness to pay can be calculated as $0 per unit of output. d. MR = 100 20Q MC = 20 + 2Q MR = MC determines the single price monopolists optimal quantity. So, 100 20Q = 20 + 2Q or Q = 3.64 units of output Use the demand curve to find the price the monopolist should charge for this level of output: P = 100 10Q = 100 10(3.64) = $63.60 per unit of output Profit = TR TC TR = P*Q = ($63.60 per unit of output)(3.64 units of output) = $231.50 TC = 100 + 20Q + Q2 TC = 100 + 20(3.64) + (3.64)(3.64) TC = $186.05 Profit = $231.50 – $186.05 Profit = $45.45 e. The socially optimal amount of good in this market is that quantity where MC intersects the demand curve. So, MC = 20 + 2Q and demand can be written as P = 100 10Q. Thus, 20 + 2Q = 100 10Q or Q socially optimal = 6.67 units of output. (This is an increase in the level of output you found in (d): remember that the monopolist left to their own devices will constrict…

output.) Use either the demand curve or the MC curve to find the price: P = 100 10Q = 100 10(6.67) = $33.30 per unit of output. (Again, compare the price you found in (d) to this price: remember that the monopolist left to their own devices will charge a higher price than would a market that operated as a perfectly competitive market.) Profit = TR TC. TR = P*Q = ($33.30 per unit of output)(6.67 units of output = $222.11. TC = 100 + 20Q + Q2 . TC = 100 + 20(6.67) + (6.67)(6.67) = $277.89. Profit = $222.11 – $277.89 = -$55.78. f. If the monopolist is regulated to produce where P = ATC, then we know that the firm will earn zero economic profits (but we will confirm this once we get underway). Use the demand curve and the ATC curve to solve for the quantity the firm will produce. Thus, 100 10Q = (100/Q) + 20 + Q or 0 = 11Q2 + (-80)Q + 100. Now, you have the quadratic equation and will need to use the quadratic formula to solve for a solution. So, Q = 80 [(80)(80) (4)(11)(100)]1/2/[(2)(11)] Q = (80 + 44.8)/22 or Q = (80 44.8)/22 (this second solution is the one that you need to discard-look at your graph and see if you can find which (Q, P) this second solution corresponds to). Q = 5.67 units of output To find P, use the demand curve and this quantity: P = 100 10(Q) = 100 10(5.67) = $43.30 per unit of output. Now, check that the firm earns zero economic profit when it produces 5.67 units of output and sells this output at $43.30 per unit of output. TR = P*Q = ($43.30 per unit of output)(5.67 units of output) = $245.51. TC = 100 + 20Q + Q2 . TC = 100 + 20(5.67) + (5.67)(5.67) = $245.55. We have TC slightly greater than TR and this is due to rounding error.

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