<p>The profit of the firm by definition is:</p>
<p><span class="math-container">$$\Pi= TR-TC$$</span></p>
<p>Where <span class="math-container">$\Pi$</span> is profit, <span class="math-container">$TR$</span> total revenue and <span class="math-container">$TC$</span> total cost. By definition total revenue is price times quantity <span class="math-container">$TR=P \cdot Q$</span> (in your case quantity is <span class="math-container">$x$</span>). Price could also be function of quantity, the problem does not mention whether firm is price taker or not. However, reading between the lines it looks like firm is assumed to be price taker otherwise it would be weird not to provide demand or inverse demand function (unless this is supposed to be nasty trick question I would say it’s fair to assume firm is price taker).</p>
<p>Hence plugging in your parameters we have:</p>
<p><span class="math-container">$$\Pi = 400x - 2x^2 -900$$</span></p>
<p>Taking derivative the above wrt <span class="math-container">$x$</span> and equating it to 0 we get conditions for profit maximization:</p>
<p><span class="math-container">$$400-4x=0 \implies 100=x^*$$</span></p>
<p>With <span class="math-container">$x=100$</span> the profit is:</p>
<p><span class="math-container">$$\Pi = 400 \cdot 100 - 2 \cdot 100^2 -900=19100$$</span></p>
<p>The caveat is that the above would hold only for firm that is price taker (perfect competition). If it would be monopoly/monopolistically competitive firm the equilibrium price would not stay 400 when firm would change output (except in some corner cases like monopoly with perfectly elastic demand).  But I think perfect competition is unstated assumption here.</p>

CC BY-SA 4.0
2022-05-31T11:55:40.967Z
2022-05-31T11:55:40.967Z
1muflon1
<p>For a given item, if we know that the equilibrium price is 400 and the cost of manufacturing quantity <span class="math-container">$x$</span> of the item is <span class="math-container">$2x^3+900$</span>, then how can I find the profit?</p>
<p>My initial thoughts are that this is insufficient information to work out profit, but I am curious to know whether or not there is a way.</p>

CC BY-SA 4.0
2022-05-31T09:58:51.647Z
You're right. There is insufficient information to answer this question. However, if we know that this firm is a price taker and choose quantity such that it maximises its profit then we can use the information to solve the problem and determine the optimal profit.
Amit
CC BY-SA 4.0
2022-05-31T10:30:17.420Z
Thanks for your help. How would I do that for this particular problem? @Amit
FD_bfa
CC BY-SA 4.0
2022-05-31T11:18:06.897Z
Just try and solve this maximisation problem: $\max_{x\geq 0} \ 400x - 2x^3 - 900$. Update your answer with your work.
Amit
<macroeconomics><microeconomics><stock-market><price><equilibrium-price>
Calculating profit from price and cost
FD_bfa
# Calculating profit from price and cost

(Asked by: username_0 on 2022-05-31)

For a given item, if we know that the equilibrium price is 400 and the cost of manufacturing quantity $x$ of the item is $2x^3+900$, then how can I find the profit?

My initial thoughts are that this is insufficient information to work out profit, but I am curious to know whether or not there is a way.

**Tags:** macroeconomics, microeconomics, stock-market, price, equilibrium-price

> **Comments:**
> **username_1**: You're right. There is insufficient information to answer this question. However, if we know that this firm is a price taker and choose quantity such that it maximises its profit then we can use the information to solve the problem and determine the optimal profit.  
> **username_0**: Thanks for your help. How would I do that for this particular problem? @username_1  
> **username_1**: Just try and solve this maximisation problem: $\max_{x\geq 0} \ 400x - 2x^3 - 900$. Update your answer with your work.  

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## 1 answers

### Answer by username_2 on 2022-05-31. Score: 1 | Accepted answer

The profit of the firm by definition is:

$$\Pi= TR-TC$$

Where $\Pi$ is profit, $TR$ total revenue and $TC$ total cost. By definition total revenue is price times quantity $TR=P \cdot Q$ (in your case quantity is $x$). Price could also be function of quantity, the problem does not mention whether firm is price taker or not. However, reading between the lines it looks like firm is assumed to be price taker otherwise it would be weird not to provide demand or inverse demand function (unless this is supposed to be nasty trick question I would say it’s fair to assume firm is price taker).

Hence plugging in your parameters we have:

$$\Pi = 400x - 2x^2 -900$$

Taking derivative the above wrt $x$ and equating it to 0 we get conditions for profit maximization:

$$400-4x=0 \implies 100=x^\*$$

With $x=100$ the profit is:

$$\Pi = 400 \cdot 100 - 2 \cdot 100^2 -900=19100$$

The caveat is that the above would hold only for firm that is price taker (perfect competition). If it would be monopoly/monopolistically competitive firm the equilibrium price would not stay 400 when firm would change output (except in some corner cases like monopoly with perfectly elastic demand). But I think perfect competition is unstated assumption here.


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