Suppose you are the sole provider of water in a small town. The demand function
ID: 1107473 • Letter: S
Question
Suppose you are the sole provider of water in a small town. The demand function for the town's water supply is given by the function:
Q = 650 - 4P.
Your cost function is:
TC = 200 + 2Q2
.
A. Calculate P, Q, consumer surplus and .
B. Given the above values, the city government steps in and sets the price so that deadweight loss is minimized. What is the new P, Q, consumer surplus and ?
C. Calculate the deadweight loss before and after government regulation.
D. Suppose after one year, your costs of providing water increase (due to some regulation recently passed that required you reduce the level of nickel in the drinking water to 90 ppb - parts per billion).
Your new cost function is:
TC = 900 + 3Q2
.
What is the new P, Q, and consumer surplus given this change to the cost function?
E. If the government had not changed the price, what would be the firm’s profit?
Explanation / Answer
Q = 650 - 4P
4P = 650 - Q
P = 162.5 - 0.25Q
TC = 200 + 2Q2
Marginal cost (MC) = dTC / dQ = 4Q
(A) Profit is maximized by equating Marginal revenue (MR) with MC.
Total revenue (TR) = P x Q = 162.5Q - 0.25Q2
MR = dTR / dQ = 162.5 - 0.5Q
Equating with MC,
162.5 - 0.5Q = 4Q
4.5Q = 162.5
Q = 36 (Considering interger value for quantity)
P = 162.5 - (0.25 x 36) = 162.5 - 9 = 153.5
TR = P x Q = 153.5 x 36 = 5,526
TC = 200 + (2 x 36 x 36) = 200 + 2,592 = 2,792
Profit = TR - TC = 5,526 - 2,792 = 2,734
From demand function, When Q = 0, P = 162.5 (Reservation price)
Consumer surplus (CS) = Area between demand curve and market price = (1/2) x (162.5 - 153.5) x 36 = 18 x 9
= 162
(B) Deadweight loss is minimized when Price equals MC.
162.5 - 0.25Q = 4Q
4.25Q = 162.5
Q = 38 (Considering integer value for quantity)
P = MC = 4Q = 4 x 38 = 152
TR = 152 x 38 = 5,776
TC = 200 + (2 x 38 x 38) = 200 + 2,888 = 3,088
Proft = 5,776 - 3,088 = 2,688
CS = (1/2) x (162.5 - 152) x 38 = 19 x 10.5 = 199.5
(C)
Deadweight loss before regulation = (1/2) x Difference in price x Difference in quantity
= (1/2) x (153.5 - 152) x (38 - 36) = (1/2) x 1.5 x 2 = 1.5
After regulation, deadweight loss is zero since Price being equal to MC leads to maximized total surplus and zero deadweight loss.
(D) TC = 900 + 3Q2
MC = dTC / dQ = 6Q
Equating MR & MC,
162.5 - 0.5Q = 6Q
6.5Q = 162.5
Q = 25
P = 162.5 - (25 x 0.25) = 162.5 - 6.25 = 156.25
TR = 156.25 x 25 = 3,906.25
TC = 900 + (3 x 25 x 25) = 900 + 1,875 = 2,775
Profit = 3,906.25 - 2,775 = 1,131.25
CS = (1/2) x (162.5 - 156.25) x 25 = (1/2) x 6.25 x 25 = 78.125
NOTE: First 4 parts are answered.
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