An office supply company sells x permanent markers per year at $P per marker. The price- demand equation for these markers is given by P = 10 − 0.001x;

Question

1- Write the Revenues function (10%)

2- What level of production and what price should the company charge for the markers to maximize revenues? (10%) The total cost of manufacturing is: C(x) = 1000 + 2x

3- Write the Company’s Profit function (10%)

4- What level of production and what price should the company charge for the markers to maximize profits? (10%)

5- Draw a graph representing the above-mentioned situation (30%) Now the Government decides to tax the Company in 2€ for each marker produced. Taking into account this additional cost:

6- Write the company’s new Cost function (10%)

7- Write the company’s new Profit function (10%)

8- What level of production and what price should the company charge for the markers to maximize profits (with these new conditions)? (10%)

Answer 1

1) Revenue = Production * Price = (10-.001x)x

2) To Max R, take derivative of R wrt x and set =0. Find x = 5000 units to max R

3) Profit (Income) I= Revenue-Cost = (10-.001x)x-1000-2x=8x-.001x²-1000

4) To Max I, take derivative of I wrt x and set =0. Find x=4000 units to max I

5) Including tax, cost = 1000+4x

6) New I = (10-.001x)x-1000-4x

7) To Max I, take derivative of I wrt x and set=0. Find x= 3000