A profit maximization problem (whole problem has been solved, I just have question about interpretation)

Question

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<m<1 $ 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.

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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?)

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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 $0<m<1$ 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.

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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.

Answer 1

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.

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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.