1. “Zoras” is a commercial fishery whose costs are estimated as c(y) = 25 + 5y + y^2, where y represents the number of pounds of fish caught/sold.
(a) Compute Zoras’ average variable cost function, AVC(y).
(b) Compute Zoras’ average cost function, AC(y).
(c) Zoras’ marginal costs are given by MC(y) = 5 + 2y. Draw by hand Zoras’ average and marginal cost curves.
(d) Let p be the price per pound of fish, and assume that this price can never fall below 5 dollars per pound, so that p ≥ 5. Find the formula for y, the amount of fish sales that maximizes Zoras’ profits, as a function of p.
(e) Would Zoras ever want to shut-down production?
(f) Imagine that, unless they (at least) break even, Zoras can sell their fleet and avoid paying any fixed costs. At what price would Zoras choose to exit the market?
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The costs of “Zoras”, a commercial fishery, are estimated as c(y) = 25 + 5y + y^2, where y represents the number of pounds of fish caught/sold.
The fixed costs are 25 and the variable costs are given by VC(y) = 5y+y^2.
The average variable cost function AVC(y) = (5y+y^2)/y = 5 + y
The average cost function, AC(y) = (25 + 5y + y^2)/y = 25/y + 5 + y
Zoras’ marginal costs are given by MC(y) = 5 + 2y.
If p is taken to be the price per pound of fish, and it is assumed that this price can never fall below $5 per pound, so that p ≥ 5.
Profits are maximized at the point where marginal cost is equal to average total cost:
5+2y = 25/y + 5 + y
5y + 2y^2 = 25 + 5y + y^2
25 = y^2
y = 5
The fishery makes maximum profits when 5 fish are sold.
To break even, the price at which the fish should be sold is $15.
The fishery would have to shut down when the price of fish falls below $15.
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