Mathematics for Economist 1: Profit maximization

[econmas]

Given demand and cost function as;
price=25-3Q
Cost=[tex]Q^2+6Q[/tex]
1.Find the profit function.
2.Find the price and output that maximizes the profit.
3. Find the maximum profit.

[Baltuilhe]

Good night!

[tex]p(q)=25-3q\\
C(q)=q^2+6q[/tex]

1) Profit:
Let's discover, first, the revenue:
[tex]R(q)=q.p(q)=q(25-3q)=25q-3q^2[/tex]

Profit is Revenue minus Cost. So:
[tex]P(q)=R(q)-C(q)=25q-3q^2-\left(q^2+6q\right)\\
P(q)=25q-3q^2-q^2-6q\\
P(q)=19q-4q^2[/tex]

2)
The price and output that maximizes the profit:
[tex]\\P'(q)=0\\
19-8q=0\\
8q=19\\
q=\frac{19}{8}\\
q=2,375\\
q=2[/tex] (output)

price:
[tex]p(q)=25-3q\\
p(2)=25-3(2)=25-6=19[/tex] (price)

3) Maximum profit
[tex]P(2)=19(2)-4(2)^2\\
P(2)=38-16=22\\[/tex]

Hope to Have Helped! (3H)

[Guest]

I find it peculiar that while Price is, of course, the "price per item" which is why "revenue" is price times quantity, cost is total cost. (Which it clearly is because it is increasing while "cost per item made" typically decreases due to "economies of scale.)