Exercise: maximizing profit

Assume that you are the owner of a small business that produces T-shirts. Your the total revenue for your business can be modeled by the following equation:

$TR= -(0.2Q-1)^2+15$

and your total cost corresponds to this function:

$TC= (0.1Q-2)^2+5$

Find the point at which your firm maximizes its profit. Then, find how much profit the firm if able to earn at that point.

Using the total cost and total revenue functions we can set up the profit function:

$\Pi= -(.02Q-1)^2-(0.1Q-2)^2+10$

Then, realize that if you want to find the maximum profit, take the derivative of the function and set it up equal to zero, and solve for Q. This is equivalent of taking the derivative of the total cost and total revenue functions and setting them equal to each other. In this problem, I chose the latter option as it was explained in the previous lessons. Notice that profit is often denoted by a capital pi.

We are assuming that the TR>TC for some positive value, you can check for yourself. However, if we did not know that, we would first take the first derivative of profit, solve for Q (let's call this value Q1) and then take its second derivative, if the latter is negative at Q1, you have a local maximum, and check for multiple local maxima.

Find the MR:

$MR:-0.4(0.2Q-1)$

Find the MC:

$MC:0.2(0.1Q-2)$

Set them equal to each other, and solve for Q:

$\\ 1). -0.4(.2Q-1)=0.2(0.1Q-2) \\ 2). \frac{Q}{2}=4\rightarrow Q=8$

Therefore, the firm maximizes its profit when it produces 8 T-shirts. At this level of production, the firm is able to earn a profit of $8.2.

$\Pi = -(0.2(8)-1)^2-(0.1(8)-2)^2+10= 8.2$

Let graph the TC and TR functions and the points at which the profit is optimized:

You can see that the vertical distance is maximized when Q=8. Also, notice that the vertical distance represents profit.

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