As a product, Oranges are notoriously difficult to produce. The yield varies acc

Question

As a product, Oranges are notoriously difficult to produce. The yield varies according to the weather, insects, lack of insects, worker productivity, etc.. Assume that the cost of producing oranges varies according to the equation C(q) = (X3)/10000000 - (3X2)/4000 + (87X)/42 + 510 where x is lbs of oranges. The market price for oranges is $1.10 per pound with an unanswered demand of 3,900 lbs of oranges (these are obtained from the intersection of the aggregate supply and demand curves.)

a.) How many oranges should you produce in order to maximize profit?

b.) What is that profit?

c.) Suppose the price of oranges suddenly rises to $1.25 without changing the total quantity demanded

(Yes, I know this requires multiple shifts in the market equilibrium or extremely inelastic demand, assume they happen.) How many should you produce now? What will your profit be?

Explanation / Answer

(a) cost: (X3)/10000000 - (3X2)/4000 + (87X)/42 + 510

market price: 1.1X

so profit = market price for X pounds - cost of X pounds

P(X) = 1.1X - (X3)/10000000 + (3X2)/4000 - (87X)/42 - 510

for maximum profit, P'(X) = 0

so P'(X) = 1.1 - 3X2/107 + 3X/2000 - 87/42 = 0

solving, X ~ 4137 pounds Ans

(b) Using this value of X, P(4137) = $14063 pounds Ans

(c) repeat (a) and (b) by replacing market price = $1.1 by $1.25.

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