## 7.4Solved Problem

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### SOLVED PROBLEMIs There a Catch?

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Seattle’s Pike Place Fish Market is well known for its freshly caught salmon, halibut, and Alaskan King Crab and for its entertaining staff who hurl a fish into the air on its way to wrapping and checkout.

 Quantity of fishQ Variable costVC Total costTC 30 \$280 \$680 40 320 720 50 440 840 60 600 1,000 70 840 1,240 80 1,160 1,560 90 1,560 1,960 100 2,040 2,440

Consider the following hypothetical daily costs for one of the market’s suppliers, a fisherman who runs a boat that fishes primarily for Chinook Salmon. Each day begins with the decision of whether to take out the boat, given the price he expects to receive at the fish market. Whether he goes out or not, he incurs fixed costs such as dockage, licensing, and mortgage on the boat. In addition to fixed costs, he incurs a variable cost for each fish brought back to port. So, he must also decide how much to catch.

Using the table at right, calculate average variable cost, average total cost, and marginal cost to find the break-even price per fish. If the market price falls to \$14.00 per fish, in the short run, how many fish will the fisherman bring to market?

STEP | 1 Find the average variable cost, average total cost, and marginal cost of a box of fish.Review pages 183–188 in Chapter 6, and Equations 6-3, 6-4, and 6-5.

The average variable cost is equal to the variable cost divided by the quantity (VC/Q), the average total cost is equal to the total cost divided by the quantity (TC/Q), and the marginal cost is the change in the total cost divided by the change in the quantity (ΔTCQ). These costs are calculated for each box in the following table.

 Quantity of fishQ Variable costVC Total costTC Marginal cost MC = ΔTC/ΔQ Average variable cost AVC = VC/Q Average total cost ATC = TQ/Q 30 \$280 \$680 \$9.33 \$22.67 \$4.00 40 320 720 8.00 18.00 12.00 50 440 840 8.80 16.80 16.00 60 600 1,000 10.00 16.67 24.00 70 840 1,240 12.00 17.71 32.00 80 1,160 1,560 14.50 19.50 40.00 90 1,560 1,960 17.33 21.78 48.00 100 2,040 2,440 20.40 24.40

STEP | 2 Find the break-even price per fish.Review pages 209–212 and Figure 7-2.

To find the break-even price, we need to find the minimum average total cost of production. In the table, the minimum average total cost occurs at 60 fish. Thus, the break-even price is \$16.67 per fish.

STEP | 3 If the market price falls to \$14.00 per fish, in the short run how many fish will be brought to market?Review pages 206–209 and the Pitfalls on page 208.

In the case of the price-taking firm, the marginal revenue is equal to the market price. So, to find the optimal quantity, we need to find the point where P = MC. If there is not a point on the table at which P = MC, then the fisherman will want to produce the largest quantity for which P exceeds MC. Going from 40 to 50 fish, the MC is \$12.00, but going from 50 to 60 fish, the MC is \$16.00. Hence, the largest quantity for which P exceeds MC is 50 fish. Although price is less than average total cost, he will still fish because the price is greater than his average variable cost.

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