<p>Your production function is basically a Cobb-Douglas function of the form <span class="math-container">$y=A(t_m-a)^\alpha(t_l-b)^\beta$</span>. Therefore, the parameters <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> measure the intensity with which inputs are needed for production. The smaller the value of such parameters, the smaller the marginal productivity of the inputs is. In your context, the company is more labor intense than managerial intense, since <span class="math-container">$\alpha=m&lt;1$</span> and <span class="math-container">$\beta=1$</span>. Finally, the parameters <span class="math-container">$a$</span> and <span class="math-container">$b$</span> measure the minimum level needed by the company of each input in order to operate. In your case, <span class="math-container">$a=0$</span> so the company needs a management to operate. However, <span class="math-container">$b=-n$</span> so the company can produce even if <span class="math-container">$t_l$</span> was equal to zero. Notice that the larger the <span class="math-container">$n$</span>, the less the company needs of labor. </p>

<p>In fact, though you found two equations that characterize equilibrium I think you want to solve for <span class="math-container">$t_m$</span> and <span class="math-container">$t_l$</span> in terms of parameters only (you will see that if  <span class="math-container">$n$</span> is very large, the optimal <span class="math-container">$t_l$</span> is zero. After that, I would plug in the results into the profit function, that will tell you how much profit the manager will get if he decides to enter the industry, and you can compare that to the outside option.</p>

<hr>

<p><strong>Solution:</strong></p>

<p>For the manager to operate the company, the profit function must be larger than the outside option. I will assume that if she operates the company, since she is the manager, she does not need to pay herself, so the costs are only <span class="math-container">$w_lt_l^*$</span>, but her outside option is equal to <span class="math-container">$w_mt_m^*$</span>, the labor income she will earn working for a similar company  that would demand the optimal management labor.  Therefore, she operates the company iff the profit <span class="math-container">$\pi =\rho t_m^m(n+t_l)-w_lt_l-w_mt_m\geq 0$</span>.</p>

<p><strong>Case 1</strong>: (Assume that <span class="math-container">$t_m^*,  t_l&gt;0^*$</span>) </p>

<p>Solving the FOC's, we get <span class="math-container">$$t_m^*=\left(\frac{w_l}{\rho}\right)^{\frac1m} \qquad t_l^*=\frac{w_m}{m \cdot w_l}\left(\frac{w_l}{\rho}\right)^{\frac1m}-n$$</span></p>

<p><span class="math-container">$$\pi^*=\frac{w_m}{m}\left(\frac{w_l}{\rho}\right)^{\frac1m}-w_m\left(\frac{w_l}{\rho}\right)^{\frac1m}-w_l\left(\frac{w_m}{m\cdot w_l}\left(\frac{w_l}{\rho}\right)^{\frac1m}-n\right)=w_ln-w_m\left(\frac{w_l}{\rho}\right)^{\frac1m}$$</span>
Therefore, <span class="math-container">$\pi^*\geq 0$</span> iff  <span class="math-container">$$w_ln\geq w_m\left(\frac{w_l}{\rho}\right)^{\frac1m}$$</span> However, we assumed that <span class="math-container">$t_l^*&gt;0$</span> so 
<span class="math-container">$$\frac{w_m}{m}\left(\frac{w_l}{\rho}\right)^{\frac1m}\geq w_ln$$</span></p>

<p>The two inequalities can hold since <span class="math-container">$1 &gt; m$</span>. These are necessary conditions for the manager to operate the company and demand strictly positive labor. I would interpret them as <span class="math-container">$w_ln$</span> needs to be intermediate.</p>

<p><strong>Case 2:</strong> (<span class="math-container">$t_m^*&gt;0$</span> and <span class="math-container">$t_l^*=0$</span>)</p>

<p>This case occurs if 
<span class="math-container">$$\frac{w_m}{m}\left(\frac{w_l}{\rho}\right)^{\frac1m}\leq w_ln$$</span>
Notice that (after doing some algebra) the optimal profit function <span class="math-container">$\pi^*$</span> is still bigger than zero iff 
<span class="math-container">$$w_ln\geq{w_m}\left(\frac{w_l}{\rho}\right)^{\frac1m}$$</span>
In this case, if the first inequality is satisfied, the second holds too. </p>

<p><strong>Conclusion</strong> Putting the two cases together we conclude that the manager operates her company if and only if 
<span class="math-container">$$w_ln\geq{w_m}\left(\frac{w_l}{\rho}\right)^{\frac1m}$$</span>
She will sometimes hire workers in her company (if <span class="math-container">$w_ln$</span> is not  too large) and sometimes she will not. </p>

CC BY-SA 4.0
2019-10-18T11:41:03.647Z
Thank you so much. Your answer is very explanatory. I understood what I need to do.
studentp
CC BY-SA 4.0
2019-10-18T11:43:43.253Z
For the second question on the determination of conditions, can you please check the Manager's decision problem. Is it completely correct? (Cost function, revenue and profit function are completely correct?) if yes, I will apply what you say, and then I will edit the question. In this way, this question can help everyone. Thus, please check it:) thank you very much.
studentp
CC BY-SA 4.0
2019-10-18T17:56:17.150Z
It looks good to me. The only part that is not clear is the outside option. Your say "the manager wage in the economy", so is that some other parameter $w_0$? or what is it?
Regio
CC BY-SA 4.0
2019-10-18T18:12:43.357Z
Thank you so much for your reply:) No, the question states only that “ suppose that (i) the manager’s outside option is equal to the prevailing manager wage in the economy and (ii) the manager will manage her own company if she establishes one.” there is only the info of $w_m$ and $w_l$. And $p$ is the market price of the good. I guess now everything is clear?
studentp
CC BY-SA 4.0
2019-10-18T22:02:14.183Z
Dear Regio, I solved the question in accordance with what you said. But I cannot get a meaningful result. It doesn’t make sense. Please let me say where is my mistake. I know I disturb you much, but I am happy to get your nice reply. You broaden my mind. Thanks.
studentp
CC BY-SA 4.0
2019-10-19T19:57:23.287Z
There is something I haven't quite figured out, I don't think you are supposed to compare profits with $w_m$, since that assumes the manager will work one hour if she decides no to establish her own company.

As of your solution, do you mind checking it again? There must be something wrong, because the optimal $t_l^*$ should depend on $\rho$. 

Also, since I have answered your original question, can you please mark the answer as accepted?
Regio
CC BY-SA 4.0
2019-10-19T20:02:40.547Z
That’s you mean that I should compare optimal profit $\pi^*$ with $t_m w_m$? ( not only with $w_m$?) yes, I tried to solve it many times,  it I could not find my mistake while solving. Please can you add your solution? I really cannot see my mistake.
studentp
CC BY-SA 4.0
2019-10-19T20:04:01.577Z
And sorry I forgot to put tick. Thank you so much. You are so explanatory and helpful
studentp
CC BY-SA 4.0
2019-10-19T20:14:22.600Z
I corrected $t_l$. But again this result is meaningless? I will be glad if you will show your result :) thank you so much.
studentp
CC BY-SA 4.0
2019-10-22T16:07:59.230Z
$t_l$ was not fully corrected. I just added my solution along with my interpretation of the outside option. Hope this helps.
Regio
CC BY-SA 4.0
2019-10-22T21:47:32.200Z
I don’t know how to thank you. Thanks million times. It is too hard question for my level. But I have understood thanks to you.
studentp
CC BY-SA 4.0
2019-10-23T01:35:42.480Z
You're welcome!
Regio
CC BY-SA 4.0
2019-10-17T22:33:04.670Z
2019-10-22T16:06:56.450Z
2019-10-22T16:06:56.450Z
Regio
<p>I would like to discuss with you about the following production function. </p>

<p><span class="math-container">$$y=f(t_m, t_l)=\rho t_m^m(n+t_l)$$</span></p>

<p>where <span class="math-container">$0&lt;m&lt;1 $</span> and <span class="math-container">$n&gt;0$</span> are fixed parameters.</p>

<p><span class="math-container">$t_m$</span> is manager time.</p>

<p><span class="math-container">$t_l$</span> is labour time. </p>

<p>The manager know the production function freely. </p>

<p>Also let’s define <span class="math-container">$w_m$</span>=manager’s wage and <span class="math-container">$w_l$</span>=labour’s wage for <span class="math-container">$w_m&gt;w_l&gt;0$</span></p>

<p><span class="math-container">$\rho$</span> is the good’s market price. </p>

<p>Extra Notes: </p>

<p>• the manager’s outside option= the  manager wage in the economy  </p>

<p>•the manager is going to  run his own company if he establishes one.</p>

<p>So far, I have defined the production function. </p>

<p>How I have two questions that I would like to discuss with you. </p>

<p>————————</p>

<p><strong>How can I interpret the parameters <span class="math-container">$n$</span> and <span class="math-container">$m$</span>?</strong> It is easy to interpret them mathematically, but I could not decide how to make a comment about these parameters in economic terms?</p>

<p>When I take derivatives </p>

<p><span class="math-container">$\partial y/\partial t_l= t_m ^m&gt;0$</span> which means that as labour’s time increases, production increases, revenue increases and so profit increases. </p>

<p><span class="math-container">$\partial y/\partial t_m= m t_m ^{m-1}(n+t_l)&gt;0$</span> which means that as manager’s time increases, production is diminshingly increases (since <span class="math-container">$m-1&lt;0$</span>) and revenue diminishingly increases so profit diminishingly increases. ( I did this comment. But I don’t think it is enough or correct in economic view. How can I say it correctly?)</p>

<p>———————-</p>

<p>My important question is that!</p>

<p><strong>Under Which conditions does the manager establish a company and decide to enter the industry?</strong></p>

<p>For that, I firstly establish the manager’s decision problem as follow:</p>

<p>Revenue = <span class="math-container">$\rho y= \rho t_m^m(n+t_l)$</span></p>

<p>Cost= <span class="math-container">$t_m w_m + t_l w_l$</span></p>

<p>Profit = <span class="math-container">$\pi = \rho t_m^m(n+t_l) -( t_m w_m + t_l w_l)$</span></p>

<p>So his decision problem:</p>

<p><span class="math-container">$$\max \rho t_m^m(n+t_l) -( t_m w_m + t_l w_l)$$</span></p>

<p>Subject to <span class="math-container">$w_m&gt;w_l&gt;0$</span> and <span class="math-container">$0&lt;m&lt;1$</span> and <span class="math-container">$n,p&gt;0$</span> </p>

<p>When we solve it</p>

<p><span class="math-container">$\partial \pi/\partial t_m= \rho m t_m ^{m-1}(n+t_l) -w_m=0 $</span> </p>

<p><span class="math-container">$w_m= \rho m t_m ^{m-1}(n+t_l)$</span></p>

<p><span class="math-container">$\partial \pi/\partial t_l=\rho t_l ^m-w_l=0$</span></p>

<p><span class="math-container">$w_l= \rho t_m ^m$</span></p>

<p>So we know that <span class="math-container">$w_m&gt;w_l&gt;0$</span></p>

<p><span class="math-container">$$ \rho m t_m ^{m-1}(n+t_l)&gt; \rho  t_m^{m}$$</span></p>

<p><span class="math-container">$$n+t_l&gt;{t_m\over m} $$</span></p>

<p>I only find this condition in order to establish a company. ( I am not sure but I hope it is true) But I cannot say something about it in economic view.</p>

<p>I solved everything, just discuss what I did. Is it correct or false? Or how can I interpret them in economic aspect? </p>

<p>Thanks a lot.  </p>

<hr>

<p><strong>EDIT</strong></p>

<p>From the above maximization problem, I obtained the following solutions:</p>

<p>And from <span class="math-container">$w_l= \rho t_m ^m$</span></p>

<p>I get <span class="math-container">$$t^*_m=(w_l/\rho)^{1/m}$$</span></p>

<p>From <span class="math-container">$w_m= \rho m t_m ^{m-1}(n+t_l)$</span></p>

<p>I get <span class="math-container">$$t^*_l= {w_m\over m}p^{1-m/m}w_l^{1-m/m}-n$$</span></p>

<p>When I insert these two <span class="math-container">$t^*_l$</span> and <span class="math-container">$t^*_m$</span> into the profit function, I get </p>

<p><span class="math-container">$$\pi^*=w_l^{1/m}w_m \rho^{1-m/m} {1\over m} (2-mp^{1/m-1})-nw_l$$</span></p>

<p>Then, when I compare this with the outside option <span class="math-container">$w_m$</span>, then I get </p>

<p><span class="math-container">$$\pi^*=w_l^{1/m}w_m \rho^{1-m/m} {1\over m} (2-mp^{1/m-1})-nw_l&gt;= w_m$$</span></p>

<p>But this result doesn’t make sense. I could not make any reasonable interpretation. </p>

<optimization><profit-maximization><dynamic-programming><linear-programming>
A profit maximization problem (whole problem has been solved, I just have question about interpretation)
studentp
# A profit maximization problem (whole problem has been solved, I just have question about interpretation)

(Asked by: username_0 on 2019-10-17)

I would like to discuss with you about the following production function.

$$y=f(t\_m, t\_l)=\rho t\_m^m(n+t\_l)$$

where $0 and $n>0$ are fixed parameters.

$t\_m$ is manager time.

$t\_l$ is labour time.

The manager know the production function freely.

Also let’s define $w\_m$=manager’s wage and $w\_l$=labour’s wage for $w\_m>w\_l>0$

$\rho$ is the good’s market price.

Extra Notes:

• the manager’s outside option= the manager wage in the economy

•the manager is going to run his own company if he establishes one.

So far, I have defined the production function.

How I have two questions that I would like to discuss with you.

————————

**How can I interpret the parameters $n$ and $m$?** It is easy to interpret them mathematically, but I could not decide how to make a comment about these parameters in economic terms?

When I take derivatives

$\partial y/\partial t\_l= t\_m ^m>0$ which means that as labour’s time increases, production increases, revenue increases and so profit increases.

$\partial y/\partial t\_m= m t\_m ^{m-1}(n+t\_l)>0$ which means that as manager’s time increases, production is diminshingly increases (since $m-1<0$) and revenue diminishingly increases so profit diminishingly increases. ( I did this comment. But I don’t think it is enough or correct in economic view. How can I say it correctly?)

———————-

My important question is that!

**Under Which conditions does the manager establish a company and decide to enter the industry?**

For that, I firstly establish the manager’s decision problem as follow:

Revenue = $\rho y= \rho t\_m^m(n+t\_l)$

Cost= $t\_m w\_m + t\_l w\_l$

Profit = $\pi = \rho t\_m^m(n+t\_l) -( t\_m w\_m + t\_l w\_l)$

So his decision problem:

$$\max \rho t\_m^m(n+t\_l) -( t\_m w\_m + t\_l w\_l)$$

Subject to $w\_m>w\_l>0$ and $0 and $n,p>0$

When we solve it

$\partial \pi/\partial t\_m= \rho m t\_m ^{m-1}(n+t\_l) -w\_m=0 $

$w\_m= \rho m t\_m ^{m-1}(n+t\_l)$

$\partial \pi/\partial t\_l=\rho t\_l ^m-w\_l=0$

$w\_l= \rho t\_m ^m$

So we know that $w\_m>w\_l>0$

$$ \rho m t\_m ^{m-1}(n+t\_l)> \rho t\_m^{m}$$

$$n+t\_l>{t\_m\over m} $$

I only find this condition in order to establish a company. ( I am not sure but I hope it is true) But I cannot say something about it in economic view.

I solved everything, just discuss what I did. Is it correct or false? Or how can I interpret them in economic aspect?

Thanks a lot.

---

**EDIT**

From the above maximization problem, I obtained the following solutions:

And from $w\_l= \rho t\_m ^m$

I get $$t^\*\_m=(w\_l/\rho)^{1/m}$$

From $w\_m= \rho m t\_m ^{m-1}(n+t\_l)$

I get $$t^\*\_l= {w\_m\over m}p^{1-m/m}w\_l^{1-m/m}-n$$

When I insert these two $t^\*\_l$ and $t^\*\_m$ into the profit function, I get

$$\pi^\*=w\_l^{1/m}w\_m \rho^{1-m/m} {1\over m} (2-mp^{1/m-1})-nw\_l$$

Then, when I compare this with the outside option $w\_m$, then I get

$$\pi^\*=w\_l^{1/m}w\_m \rho^{1-m/m} {1\over m} (2-mp^{1/m-1})-nw\_l>= w\_m$$

But this result doesn’t make sense. I could not make any reasonable interpretation.

**Tags:** optimization, profit-maximization, dynamic-programming, linear-programming

---

## 1 answers

### Answer by username_1 on 2019-10-17. Score: 1 | Accepted answer

Your production function is basically a Cobb-Douglas function of the form $y=A(t\_m-a)^\alpha(t\_l-b)^\beta$. Therefore, the parameters $\alpha$ and $\beta$ measure the intensity with which inputs are needed for production. The smaller the value of such parameters, the smaller the marginal productivity of the inputs is. In your context, the company is more labor intense than managerial intense, since $\alpha=m<1$ and $\beta=1$. Finally, the parameters $a$ and $b$ measure the minimum level needed by the company of each input in order to operate. In your case, $a=0$ so the company needs a management to operate. However, $b=-n$ so the company can produce even if $t\_l$ was equal to zero. Notice that the larger the $n$, the less the company needs of labor.

In fact, though you found two equations that characterize equilibrium I think you want to solve for $t\_m$ and $t\_l$ in terms of parameters only (you will see that if $n$ is very large, the optimal $t\_l$ is zero. After that, I would plug in the results into the profit function, that will tell you how much profit the manager will get if he decides to enter the industry, and you can compare that to the outside option.

---

**Solution:**

For the manager to operate the company, the profit function must be larger than the outside option. I will assume that if she operates the company, since she is the manager, she does not need to pay herself, so the costs are only $w\_lt\_l^\*$, but her outside option is equal to $w\_mt\_m^\*$, the labor income she will earn working for a similar company that would demand the optimal management labor. Therefore, she operates the company iff the profit $\pi =\rho t\_m^m(n+t\_l)-w\_lt\_l-w\_mt\_m\geq 0$.

**Case 1**: (Assume that $t\_m^\*, t\_l>0^\*$)

Solving the FOC's, we get $$t\_m^\*=\left(\frac{w\_l}{\rho}\right)^{\frac1m} \qquad t\_l^\*=\frac{w\_m}{m \cdot w\_l}\left(\frac{w\_l}{\rho}\right)^{\frac1m}-n$$

$$\pi^\*=\frac{w\_m}{m}\left(\frac{w\_l}{\rho}\right)^{\frac1m}-w\_m\left(\frac{w\_l}{\rho}\right)^{\frac1m}-w\_l\left(\frac{w\_m}{m\cdot w\_l}\left(\frac{w\_l}{\rho}\right)^{\frac1m}-n\right)=w\_ln-w\_m\left(\frac{w\_l}{\rho}\right)^{\frac1m}$$
Therefore, $\pi^\*\geq 0$ iff $$w\_ln\geq w\_m\left(\frac{w\_l}{\rho}\right)^{\frac1m}$$ However, we assumed that $t\_l^\*>0$ so
$$\frac{w\_m}{m}\left(\frac{w\_l}{\rho}\right)^{\frac1m}\geq w\_ln$$

The two inequalities can hold since $1 > m$. These are necessary conditions for the manager to operate the company and demand strictly positive labor. I would interpret them as $w\_ln$ needs to be intermediate.

**Case 2:** ($t\_m^\*>0$ and $t\_l^\*=0$)

This case occurs if
$$\frac{w\_m}{m}\left(\frac{w\_l}{\rho}\right)^{\frac1m}\leq w\_ln$$
Notice that (after doing some algebra) the optimal profit function $\pi^\*$ is still bigger than zero iff
$$w\_ln\geq{w\_m}\left(\frac{w\_l}{\rho}\right)^{\frac1m}$$
In this case, if the first inequality is satisfied, the second holds too.

**Conclusion** Putting the two cases together we conclude that the manager operates her company if and only if
$$w\_ln\geq{w\_m}\left(\frac{w\_l}{\rho}\right)^{\frac1m}$$
She will sometimes hire workers in her company (if $w\_ln$ is not too large) and sometimes she will not.

> **Comments:**
> **username_0**: Thank you so much. Your answer is very explanatory. I understood what I need to do.  
> **username_0**: For the second question on the determination of conditions, can you please check the Manager's decision problem. Is it completely correct? (Cost function, revenue and profit function are completely correct?) if yes, I will apply what you say, and then I will edit the question. In this way, this question can help everyone. Thus, please check it:) thank you very much.  
> **username_1**: It looks good to me. The only part that is not clear is the outside option. Your say "the manager wage in the economy", so is that some other parameter $w_0$? or what is it?  
> **username_0**: Thank you so much for your reply:) No, the question states only that “ suppose that (i) the manager’s outside option is equal to the prevailing manager wage in the economy and (ii) the manager will manage her own company if she establishes one.” there is only the info of $w_m$ and $w_l$. And $p$ is the market price of the good. I guess now everything is clear?  
> **username_0**: Dear Regio, I solved the question in accordance with what you said. But I cannot get a meaningful result. It doesn’t make sense. Please let me say where is my mistake. I know I disturb you much, but I am happy to get your nice reply. You broaden my mind. Thanks.  
> **username_1**: There is something I haven't quite figured out, I don't think you are supposed to compare profits with $w_m$, since that assumes the manager will work one hour if she decides no to establish her own company.  As of your solution, do you mind checking it again? There must be something wrong, because the optimal $t_l^*$ should depend on $\rho$.   Also, since I have answered your original question, can you please mark the answer as accepted?  
> **username_0**: That’s you mean that I should compare optimal profit $\pi^*$ with $t_m w_m$? ( not only with $w_m$?) yes, I tried to solve it many times,  it I could not find my mistake while solving. Please can you add your solution? I really cannot see my mistake.  
> **username_0**: And sorry I forgot to put tick. Thank you so much. You are so explanatory and helpful  
> **username_0**: I corrected $t_l$. But again this result is meaningless? I will be glad if you will show your result :) thank you so much.  
> **username_1**: $t_l$ was not fully corrected. I just added my solution along with my interpretation of the outside option. Hope this helps.  
> **username_0**: I don’t know how to thank you. Thanks million times. It is too hard question for my level. But I have understood thanks to you.  
> **username_1**: You're welcome!  

---

