Problem involving – Fixed Costs , Variable Costs , Average Fixed and Variable cost
The cost curve for the city water supply is 1/ 4Q2, 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 costs ( FC) and variable costs ( VC). ( Remember that FC are the costs of production when .) Graph these, as well as the cost function, on the same diagram, with Q on the horizontal axis and $ on the vertical axis, from to .
( 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 to .
( c) The formula for marginal cost is . Add this line to the diagram in ( b).
( d) What is the shutdown point— that is, the minimum of AVC? ( Hint: Don’t consider negative quantities. Negative water is not an appealing idea.) 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.)
SAMPLE ANSWER: Answer: Minimum Point for AVC is zero and Marginal Cost intersects AVC at zero only. But if we assume that to be shut down point then it won’t make sense as it will imply that production shouldn’t be started. That won’t appear acceptable option.
( 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 hap-pen 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 hap-pen to the quantity of water supplied?
(j) On the graph for part (b) above now insert the price line for P = $5 per acre ft and P= $1 per acre ft. At each of these prices what is the optimal acre ft supplied to maximize profit? Calculate the profit at each of these prices.
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