Mathematics for Economist 1: Profit maximization

Mathematics for Economist 1: Profit maximization

Postby econmas » Fri Feb 19, 2021 3:47 am

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.
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Re: Mathematics for Economist 1: Profit maximization

Postby Baltuilhe » Fri Feb 19, 2021 7:28 pm

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)

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