An oyster farmer produces two types of oysters - Crassostrea-Gigas (Pacific Oysters) and Crassostrea-Sikamea (Kumamoto Oysters)....

 

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An oyster farmer produces two types of oysters - Crassostrea-Gigas (Pacific Oysters) and Crassostrea-Sikamea (Kumamoto Oysters). The farmer's total cost function is: TC = 0.45 P+P XK + 0.76 K where: P is the quantity of Pacific Oysters (in kgs) and K is the quantity of the Kumamoto Oysters (in kgs). The market prices that can be charged are determined by supply and demand forces and are influenced by the quantity of each type of oyster according to the following equations: Pp=32-P+3K Pk 13+ 2.2P-2K for the price (in dollars per kg) of Pacific Oysters and for the price (in dollars per kg) of Kumamoto Oysters. The total revenue is given by the equation: TR = Pp P + Pk K and the profit given by the equation Profit TR-TC (a) Use a substitution of the price variables to express the profit in terms of P and K only. (b) Use the method of Lagrange Multipliers to find the maximum profit when total production (quantity) is restricted to 2023 kgs. Note P or K need not be whole numbers. (c) Be sure to show that your solution is a maximum point. An oyster farmer produces two types of oysters - Crassostrea-Gigas (Pacific Oysters) and Crassostrea-Sikamea (Kumamoto Oysters). The farmer's total cost function is: TC = 0.45 P+P XK + 0.76 K where: P is the quantity of Pacific Oysters (in kgs) and K is the quantity of the Kumamoto Oysters (in kgs). The market prices that can be charged are determined by supply and demand forces and are influenced by the quantity of each type of oyster according to the following equations: Pp=32-P+3K Pk 13+ 2.2P-2K for the price (in dollars per kg) of Pacific Oysters and for the price (in dollars per kg) of Kumamoto Oysters. The total revenue is given by the equation: TR = Pp P + Pk K and the profit given by the equation Profit TR-TC (a) Use a substitution of the price variables to express the profit in terms of P and K only. (b) Use the method of Lagrange Multipliers to find the maximum profit when total production (quantity) is restricted to 2023 kgs. Note P or K need not be whole numbers. (c) Be sure to show that your solution is a maximum point.