A firm imports the product it sells from Indonesia and it sells it in the domesticmarket in Australia. The price it pays to import the product is given by PI = 4 + 16 Q. The pricethe firm receives when it sells the product in Australia is given by P = 12 − 13 Q. Furthermore itcosts 2 dollars per unit to ship the product from the Indonesia to Australia.a) (4 points) What is the quantity that maximizes the profit of the firm? What is the max-imum profit?b) (4 points) Suppose that the Indonesian government imposes an export tax of 3 dollars perunit. Write down the profit function after the tax. What is the quantity that maximizesthe profit after the tax, and what is the maximum profit after the tax?c) (2 points) In the situation described in part c), if the firm chooses to import the quantitythat maximizes its profit then what is the export tax revenue received from the firm?
Question
A firm imports the product it sells from Indonesia and it sells it in the domesticmarket in Australia. The price it pays to import the product is given by PI = 4 + 16 Q. The pricethe firm receives when it sells the product in Australia is given by P = 12 − 13 Q. Furthermore itcosts 2 dollars per unit to ship the product from the Indonesia to Australia.a) (4 points) What is the quantity that maximizes the profit of the firm? What is the max-imum profit?b) (4 points) Suppose that the Indonesian government imposes an export tax of 3 dollars perunit. Write down the profit function after the tax. What is the quantity that maximizesthe profit after the tax, and what is the maximum profit after the tax?c) (2 points) In the situation described in part c), if the firm chooses to import the quantitythat maximizes its profit then what is the export tax revenue received from the firm?
Solution
a) The profit function of the firm is given by the difference between the revenue and the cost. The revenue is the price the firm receives times the quantity sold, P*Q, and the cost is the price the firm pays to import the product plus the shipping cost, (PI + 2)*Q.
So, the profit function is: Profit = P*Q - (PI + 2)*Q = (12 - 13Q)Q - (4 + 16Q + 2)Q = -29Q^2 + 12Q - 6Q = -29Q^2 + 6Q
To find the quantity that maximizes the profit, we take the derivative of the profit function with respect to Q and set it equal to zero: d(Profit)/dQ = -58Q + 6 = 0 Solving for Q gives Q = 6/58 = 0.1034
Substituting Q = 0.1034 into the profit function gives the maximum profit: Profit = -29*(0.1034)^2 + 6*(0.1034) = 0.317
b) If the Indonesian government imposes an export tax of 3 dollars per unit, the cost to import the product increases by 3 dollars per unit. So, the new profit function is: Profit = P*Q - (PI + 2 + 3)*Q = (12 - 13Q)Q - (4 + 16Q + 2 + 3)Q = -32Q^2 + 12Q - 9Q = -32Q^2 + 3Q
Taking the derivative of the new profit function with respect to Q and setting it equal to zero gives: d(Profit)/dQ = -64Q + 3 = 0 Solving for Q gives Q = 3/64 = 0.0469
Substituting Q = 0.0469 into the new profit function gives the maximum profit after the tax: Profit = -32*(0.0469)^2 + 3*(0.0469) = 0.069
c) The export tax revenue received from the firm is the tax per unit times the quantity imported, which is 3Q = 30.0469 = 0.1407.
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