<p>My professor once said, when doing economics, don't get stuck in mathematics. Math is just a tool.</p>

<p>You know that the price will always be 24 per piece. For (iii), your cost is <span class="math-container">$C(q) = 10q$</span>. What's the cost per piece to produce it? Is it more or less than what you could sell? If it's the former, you're guaranteed to make profit for each piece you make. If it's the latter, you'll make a loss for each piece you make. Given this, what does that say about the quantity you should produce?</p>

<p><em>Edit</em>: well you could get stuck in math... but need to realize that for a function <span class="math-container">$f(x)$</span> to be maximized at <span class="math-container">$x^*$</span> where <span class="math-container">$f'(x^*) = 0$</span> you need the second order condition, namely <span class="math-container">$f''(x^*) &lt; 0$</span>, to hold as well.</p>

CC BY-SA 4.0
2019-10-03T04:26:15.203Z
2019-10-03T15:42:04.137Z
2019-10-03T15:42:04.137Z
Art
<p>You can get some insight by looking at iv). It's a similar problem, but has a very clear and easy to see result.  These questions are helping you remember that "corner solutions" exist, and you should always be on the lookout for them.</p>

<p>Remember that you can't find a unique solution to the optimization problem if your objective function (profits = revenues - costs) has no critical points. This is OK - in the real world, countless frictions exist that are not expressed in your model, but would probably take care of any unintuitive results.</p>

CC BY-SA 4.0
2019-10-03T16:58:31.057Z
Strictly speaking, corner solutions are often unique, but I didn't want to get too nuanced here because there uniqueness usually comes from additional constraints to the problem (e.g., x >= 0).
heh
CC BY-SA 4.0
2019-10-03T16:57:09.033Z
2019-10-03T16:57:09.033Z
heh
<p>I'm having trouble on my homework and I need some help.
A company sells products in a perfectly competitive market, where the price is <span class="math-container">$p = 24.$</span> For each of the following cost functions, write down the company’s profit-maximization problem, and then find <span class="math-container">$q^∗$</span>.</p>
<p>i. <span class="math-container">$C(q) = 2q^3$</span></p>
<p>ii. <span class="math-container">$C(q)=30q+q^2$</span></p>
<p>iii. <span class="math-container">$C(q) = 10q$</span></p>
<p>iv. <span class="math-container">$C(q) = 24q$</span></p>
<p>I finished i and ii on my own, for i, <span class="math-container">$$\max_q \text{Total Revenue}-\text{Total Cost} = \max_q 24q-2q^3,$$</span> then I take the derivative of max, I get <span class="math-container">$24-6(q^*)^2$</span> and set it equal to 0, I get <span class="math-container">$q^*=2$</span> and <span class="math-container">$q^*=-2$</span>, then use the second-order condition find out <span class="math-container">$q^*=2$</span>. For ii, I get <span class="math-container">$q^*=-3$</span>.
But for iii, I get <span class="math-container">$$24q-10q=14q,$$</span> when I take derivative I get <span class="math-container">$14$</span>, how can I set <span class="math-container">$14=0$</span>? What is <span class="math-container">$q^*4$</span> in this situation? Or does <span class="math-container">$q^*$</span> even exist?
Also for iv, the FOC is <span class="math-container">$0=0$</span>, what is <span class="math-container">$q^*$</span> in this situation?
Any help would be appreciated.</p>

<microeconomics><profit-maximization>
profit-maximization
Himekami
# profit-maximization

(Asked by: username_0 on 2019-10-03)

I'm having trouble on my homework and I need some help.
A company sells products in a perfectly competitive market, where the price is $p = 24.$ For each of the following cost functions, write down the company’s profit-maximization problem, and then find $q^∗$.

i. $C(q) = 2q^3$

ii. $C(q)=30q+q^2$

iii. $C(q) = 10q$

iv. $C(q) = 24q$

I finished i and ii on my own, for i, $$\max\_q \text{Total Revenue}-\text{Total Cost} = \max\_q 24q-2q^3,$$ then I take the derivative of max, I get $24-6(q^\*)^2$ and set it equal to 0, I get $q^\*=2$ and $q^\*=-2$, then use the second-order condition find out $q^\*=2$. For ii, I get $q^\*=-3$.
But for iii, I get $$24q-10q=14q,$$ when I take derivative I get $14$, how can I set $14=0$? What is $q^\*4$ in this situation? Or does $q^\*$ even exist?
Also for iv, the FOC is $0=0$, what is $q^\*$ in this situation?
Any help would be appreciated.

**Tags:** microeconomics, profit-maximization

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## 2 answers

### Answer by username_1 on 2019-10-03. Score: 4

My professor once said, when doing economics, don't get stuck in mathematics. Math is just a tool.

You know that the price will always be 24 per piece. For (iii), your cost is $C(q) = 10q$. What's the cost per piece to produce it? Is it more or less than what you could sell? If it's the former, you're guaranteed to make profit for each piece you make. If it's the latter, you'll make a loss for each piece you make. Given this, what does that say about the quantity you should produce?

*Edit*: well you could get stuck in math... but need to realize that for a function $f(x)$ to be maximized at $x^\*$ where $f'(x^\*) = 0$ you need the second order condition, namely $f''(x^\*) < 0$, to hold as well.


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### Answer by username_2 on 2019-10-03. Score: 0

You can get some insight by looking at iv). It's a similar problem, but has a very clear and easy to see result. These questions are helping you remember that "corner solutions" exist, and you should always be on the lookout for them.

Remember that you can't find a unique solution to the optimization problem if your objective function (profits = revenues - costs) has no critical points. This is OK - in the real world, countless frictions exist that are not expressed in your model, but would probably take care of any unintuitive results.

> **Comments:**
> **username_2**: Strictly speaking, corner solutions are often unique, but I didn't want to get too nuanced here because there uniqueness usually comes from additional constraints to the problem (e.g., x >= 0).  

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