Economic Help.

Question

Suppose it costs $30 for each lobster trap set. Lobsters sell for $15. If X traps are set, the harvest rate of lobsters, L, as a function of the number of traps, is given by: L = 45 X –X2. With no restrictions on the number of traps, and open access to the waters, how many traps will be set? How many lobsters will be harvested? How much profit will be realized from lobstering?

Answer 1

This is a profit maximization question. Firms maximize profits by setting marginal revenue equal to marginal cost. To solve this question we first need to solve for total revenue and total cost then solve for marginal revenue and marginal cost to calculate the profit maximizing quantity then solve for the profit which is defined as Total revenue minus total cost.

 

First: Total Revenue is given as Price*Quantity. We have the price per lobster as $15 so Price*Quantity is given as $15*(45x-x^2) or TR=675x-15x^2.

 

Second: We need to calculate total cost. Total Cost is given as the formula TC=C*Q. We have Q again given as Q=45x-x^2 and Cost or C is given as 30. So Total Cost, TC is TC=$30(45x-x^2). Distributing the 30 we get the total cost equation as, TC=1350x-30x^2.

 

Third: Solve for marginal revenue and marginal cost. To do this we need to take the partial derivative of both the total revenue and total cost formula. @TR/@Q=675-30x=0 and the marginal cost is @TC/@Q=1350-60x=0 since these are both equal to zero we can now set marginal revenue equal to marginal cost which is, 675-30x=1350-60x and solve for x. Collecting terms and solving we have 30x=675 so X or the Quantity that maximizes profit is x=675/30 or x=22.5.

 

Fourth: We have the harvest value of 22.5 as computed in the last step so now we need to compute the profit, remember that profit is defined as Total Revenue - Total Cost so Pr=TR-TC. To solve for p we plug the 22.5 into both the total revenue and total cost equation and calculate P. So Pr=(675x-15x^2)-(1350x-30x^2) so substituting in 22.5 we need to solve the following equation with arithmetic, Pr=(675*(22.5)-15*(22.5^2))-(1350*(22.5)-30*(22.5^2)) which gives us the equation Pr=7593.75-15187.5 or Pr=-7593.75 or a loss of $7593.75 for the industry, perhaps they should consider switching to crabs instead of lobster. Hope this has been of some help.