The cost curve for the city water supply is C(Q) = 16 + 1/4 Q2, where Q is the amount of water supplied and C(Q) is the cost of providing Q acre-feet of water. (An acre-foot is the amount of water that covers an acre of land to the depth of one foot.)
(a) Provide the formulas for fixed cost (FC) and variable costs (VC). (Remember that FC are the costs of production when Q = 0.) Graph these, as well as the cost function, on the same diagram, with Q on the horizontal axis and $ on the vertical axis, from Q = 0 to Q = 10.
(b) Provide the formulas for average fixed cost (AFC), average variable cost (AVC), and average cost (AC). (Remember that average cost of any kind is the corresponding cost divided by Q.) Graph all these on the same diagram, with Q on the horizontal axis and $/Q on the vertical axis, from Q = 0 to Q = 10.
(c) The formula for marginal cost is MC(Q) = Q/2. Add this line to the diagram in (b).
(d) What is the shutdown point-that is, the minimum of AVC? Does MC intersect AVC at its minimum? (To check, see if AVC and MC have the same value when Q is the amount at the shutdown point.)
(e) What is the minimum value of AC? Does MC intersect AC at its minimum?
(f) Identify the city's supply curve for water.
(g) The price of water is $4/acre-foot. Add this information to the appropriate figure. Is this price above the shutdown point (the minimum value of AVC)? How much water will the city supply, if it is operating to maximize profit? What will its profits be?
(h) The city identifies a new way of treating water that is less expensive than the previous method. What do you expect to happen to the supply curve for water? If the price of water stays constant, what do you expect to happen to the quantity of water supplied?
(i) Providing water to the city takes it out of the river. A biologist discovers that the reduced flows due to urban water use are harming the backstroking twiddler, an endangered species. If the city is required to account for the damage to the twiddler, what do you expect to happen to the supply curve for water? If the price of water stays constant, what do you expect to happen to the quantity of water supplied?