Benjamin C. answered • 06/21/20
Economics Grad Student; Former TA; Math, Writing, Physics
This is a pretty extensive problem with 8 different questions, so I am not going to answer all of them here. Instead, I will give some insight into how to set up the questions so that, hopefully, you can solve the problem on your own.
1) Revenue is defined as the total money that a firm intakes. This differs from profit, which is the total money that a firm intakes after accounting for all of its costs. Revenue is defined as the total quantity multiplied by the total price. If a firm sells 10 apples at a price of $1 per apple, the revenue for that firm would be $10. In the problem, the price is given as P = 7 - .002*x, and the quantity is defined as x. So the revenue function would simply be 7*x - .002*x^2.
2) The cost function is irrelevant when maximizing revenue. To maximize revenue with respect to the quantity produced, simply take the first order derivative of the revenue function with respect to x. Then, substitute the resulting value of x into the price function given in the problem to find the corresponding price.
3) See the definition of profit mentioned above.
4) To maximize profit, follow the same algorithm as in question 2, but with the cost function subtracted from the revenue function.
5) It is difficult to explain how to draw a graph in text, and this question does not specify which function(s) to graph. You will have a price, revenue, cost, and profit function. Each of these relates y to x. They can all be graphed for any number of points by hand, using a calculator, or using a software.
6, 7, 8) A tax of $3 for each marker produced can be expressed mathematically by " - 3*x ". Add this tax term into each equation and follow the methods outlined above.
