<p>The production side is modeled as follows:
There are $m=1,...,j$ identical firms (the number of firms is not necessarily equal to the number of workers of course), that operate in a perfectly competitive environment. This means that firms are price takers, both in the goods market and in the production input market, i.e. they take prices as given when they seek to attain their objective. It also means that markets "clear": in particular, prices adjust without frictions/delay so that all capital and all labor are employed. The firms solve a static (not intertemporal) problem: maximize profits period-by-period separately. Labor is provided totally inelastically, no labor-leisure choice here from the part of workers. Moreover each firm has a constant-returns to scale production function in capital and labor, i.e. the function is homogeneous of degree one.  </p>

<p>Omitting the time subscript, the typical firm's production function is</p>

<p>$$F(K_j, L_j),\;\; j=1,...,m$$</p>

<p>and the objective of the firms is to maximize profits which are defined as the surplus of production over payments to labor $wL_j$, net payments to rented capital $rK_j$, and depreciation $\delta K_j$:</p>

<p>$$\max_{K_j, L_j} \pi = F(K_j, L_j)-wL_j-rK_j-\delta K_j$$</p>

<p>Note that these are "real" magnitudes in the <em>economics</em> sense of the word, i.e. that we have divided throughout by the price of output (we do not show it usually, we just say, "expressed in real terms").</p>

<p>Denote $\kappa_j \equiv K_j/L_j$, the capital-labor ratio <strong>at firm level</strong>. Due to the homogeneity of degree one we can re-write the maximization problem of the firm as</p>

<p>$$\max_{\kappa_j} \pi = L_j\cdot \big[F(\kappa_j, 1)-w-r\kappa_j-\delta \kappa_j\big]$$</p>

<p>Note that $L_j$ has become a multiplicative factor, so we can maximize only the term in brackets, and so only with respect to the capital-labor ratio. We also set $F(\kappa_j, 1) \equiv f(\kappa_j)$ to arrive at</p>

<p>$$\max_{\kappa_j} \pi = \max_{\kappa_j} \big[f(\kappa_j)-w-r\kappa_j-\delta \kappa_j\big]$$</p>

<p>The first order condition for a maximum is for the first derivative to be set equal to zero so</p>

<p>$$f'(\kappa_j) - r - \delta = 0 \implies f'(\kappa_j) - \delta = r$$</p>

<p>This is not yet the funky equation $(6.32)$,although it looks a lot like it, because the latter is expressed in terms of "per capita" capital $k\equiv K/N_1$,  i.e. at the level of individuals/consumers/workers, not at firm level.</p>

<p>How do we arrive at $(6.32)$? Well, since we have assumed that all firms are identical, that labor is provided inellastically, and also that the markets for production inputs clear, we have that</p>

<p>$$mK_j = K \implies K_j = K/m,\;\;\; mL_j = N_1 \implies L_j = N_1/m $$</p>

<p>So</p>

<p>$$\kappa_j = \frac {K_j}{L_j} = \frac {K/m}{N_1/m} = K/N_1 \equiv k$$</p>

<p>and now we have obtained $(6.32)$.</p>

<p>Note how all the assumptions made have been used in order to arrive at this result.</p>

CC BY-SA 3.0
2015-06-25T00:55:51.510Z
2015-06-25T09:36:55.017Z
2015-06-25T09:36:55.017Z
Alecos Papadopoulos
<p>Why Profit maximization implies that rate of return to capital equals the net marginal product of capital. Prove and give intuition why both are equal.</p>

<p>I also wonder that why it doesn't implies rate of return to capital equals the net marginal product of labor?</p>

<p>The highlight part and equation (6.32) is my concern</p>

<p>Please zoom in the web browser to view the image more clearly(press ctrl + turn the scroll wheel of the mouse simultaneously).</p>

<p><img src="https://i.sstatic.net/giKEy.jpg" alt="enter image description here"></p>

<p><img src="https://i.sstatic.net/pqBwG.png" alt="enter image description here"></p>

<p><img src="https://i.sstatic.net/cQ7uL.jpg" alt="enter image description here"></p>

CC BY-SA 3.0
2015-06-24T22:44:40.410Z
Could you please define mathematically what you mean by rate of return to capital? I think what you state only holds w.r.t. the marginal rate of return.
Giskard
CC BY-SA 3.0
2015-06-24T23:36:12.307Z
Have you written down the profit maximization problem, go over maximizing it (in order to "prove"), and then maybe wonder "what an abstract mathematical operation like maximizing a function has to do with the economic criterion on how much capital to rent in order to have the biggest possible profits?" (for the "intuition" part). How would _you_ go about determining in a practical, trial-and-error approach,  the level of capital that would give you the highest profits? This is a practical attempt to "involve you so that you understand".
Alecos Papadopoulos
CC BY-SA 3.0
2015-06-24T23:56:56.033Z
@AlecosPapadopoulos - Math is my strength, please use math intense approach.
Victor
CC BY-SA 3.0
2015-06-25T00:02:24.247Z
I honestly don't get it: you are studying the Overlapping Generations model, and you have never encountered the standard profit maximization problem of a competitive firm at some introductory microeconomics course?
Alecos Papadopoulos
CC BY-SA 3.0
2015-06-25T00:10:26.217Z
@AlecosPapadopoulos Honestly, i am almost graduating  with math major from college and want to prepare for graduate economics course for MS of economics, but haven't get to study these two terminology in both introductory and intermediate course... Also, equation (6.32) look funky to me.
Victor
CC BY-SA 3.0
2015-06-25T00:14:17.173Z
Oh, I see. Well I'll be happy to oblige and prepare an answer -but before preparing for macroeconomics, prepare for microeconomics, it would be my advice.
Alecos Papadopoulos
CC BY-SA 3.0
2015-06-25T04:52:29.793Z
I'm voting to close this question as a low-quality question, as it contains pictures which are not there to illustrate your problem but as part of your question. I really like your question, however pictures are (a) not searchable, which defeats the whole point of a this SE, (b) are harder to read and (usually, but no in this case) (c) show a lack of effort. Could you add the equation to your question and just leave the chapter of the book as supplemental material?
The Almighty Bob
CC BY-SA 3.0
2015-06-25T14:28:53.373Z
@Victor Please edit your question to remove the pictures so the material is searchable. Alecos Papadopoulos already answered your question, but it would be beneficial to the community to have this question formatted in an appropriate way.
cc7768
<macroeconomics><microeconomics>
Why does profit maximization implies that both of the following are equal?
Victor
# Why does profit maximization implies that both of the following are equal?

(Asked by: username_0 on 2015-06-24)

Why Profit maximization implies that rate of return to capital equals the net marginal product of capital. Prove and give intuition why both are equal.

I also wonder that why it doesn't implies rate of return to capital equals the net marginal product of labor?

The highlight part and equation (6.32) is my concern

Please zoom in the web browser to view the image more clearly(press ctrl + turn the scroll wheel of the mouse simultaneously).

![enter image description here](https://i.sstatic.net/giKEy.jpg)

![enter image description here](https://i.sstatic.net/pqBwG.png)

![enter image description here](https://i.sstatic.net/cQ7uL.jpg)

**Tags:** macroeconomics, microeconomics
> **Comments:**
> **username_1**: Could you please define mathematically what you mean by rate of return to capital? I think what you state only holds w.r.t. the marginal rate of return.  
> **username_2**: Have you written down the profit maximization problem, go over maximizing it (in order to "prove"), and then maybe wonder "what an abstract mathematical operation like maximizing a function has to do with the economic criterion on how much capital to rent in order to have the biggest possible profits?" (for the "intuition" part). How would _you_ go about determining in a practical, trial-and-error approach,  the level of capital that would give you the highest profits? This is a practical attempt to "involve you so that you understand".  
> **username_0**: @AlecosPapadopoulos - Math is my strength, please use math intense approach.  
> **username_2**: I honestly don't get it: you are studying the Overlapping Generations model, and you have never encountered the standard profit maximization problem of a competitive firm at some introductory microeconomics course?  
> **username_0**: @AlecosPapadopoulos Honestly, i am almost graduating  with math major from college and want to prepare for graduate economics course for MS of economics, but haven't get to study these two terminology in both introductory and intermediate course... Also, equation (6.32) look funky to me.  
> **username_2**: Oh, I see. Well I'll be happy to oblige and prepare an answer -but before preparing for macroeconomics, prepare for microeconomics, it would be my advice.  
> **username_3**: I'm voting to close this question as a low-quality question, as it contains pictures which are not there to illustrate your problem but as part of your question. I really like your question, however pictures are (a) not searchable, which defeats the whole point of a this SE, (b) are harder to read and (usually, but no in this case) (c) show a lack of effort. Could you add the equation to your question and just leave the chapter of the book as supplemental material?  
> **username_4**: @Victor Please edit your question to remove the pictures so the material is searchable. Alecos Papadopoulos already answered your question, but it would be beneficial to the community to have this question formatted in an appropriate way.  

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## 1 answers

### Answer by username_2 on 2015-06-25. Score: 6 | Accepted answer

The production side is modeled as follows:
There are $m=1,...,j$ identical firms (the number of firms is not necessarily equal to the number of workers of course), that operate in a perfectly competitive environment. This means that firms are price takers, both in the goods market and in the production input market, i.e. they take prices as given when they seek to attain their objective. It also means that markets "clear": in particular, prices adjust without frictions/delay so that all capital and all labor are employed. The firms solve a static (not intertemporal) problem: maximize profits period-by-period separately. Labor is provided totally inelastically, no labor-leisure choice here from the part of workers. Moreover each firm has a constant-returns to scale production function in capital and labor, i.e. the function is homogeneous of degree one.

Omitting the time subscript, the typical firm's production function is

$$F(K\_j, L\_j),\;\; j=1,...,m$$

and the objective of the firms is to maximize profits which are defined as the surplus of production over payments to labor $wL\_j$, net payments to rented capital $rK\_j$, and depreciation $\delta K\_j$:

$$\max\_{K\_j, L\_j} \pi = F(K\_j, L\_j)-wL\_j-rK\_j-\delta K\_j$$

Note that these are "real" magnitudes in the *economics* sense of the word, i.e. that we have divided throughout by the price of output (we do not show it usually, we just say, "expressed in real terms").

Denote $\kappa\_j \equiv K\_j/L\_j$, the capital-labor ratio **at firm level**. Due to the homogeneity of degree one we can re-write the maximization problem of the firm as

$$\max\_{\kappa\_j} \pi = L\_j\cdot \big[F(\kappa\_j, 1)-w-r\kappa\_j-\delta \kappa\_j\big]$$

Note that $L\_j$ has become a multiplicative factor, so we can maximize only the term in brackets, and so only with respect to the capital-labor ratio. We also set $F(\kappa\_j, 1) \equiv f(\kappa\_j)$ to arrive at

$$\max\_{\kappa\_j} \pi = \max\_{\kappa\_j} \big[f(\kappa\_j)-w-r\kappa\_j-\delta \kappa\_j\big]$$

The first order condition for a maximum is for the first derivative to be set equal to zero so

$$f'(\kappa\_j) - r - \delta = 0 \implies f'(\kappa\_j) - \delta = r$$

This is not yet the funky equation $(6.32)$,although it looks a lot like it, because the latter is expressed in terms of "per capita" capital $k\equiv K/N\_1$, i.e. at the level of individuals/consumers/workers, not at firm level.

How do we arrive at $(6.32)$? Well, since we have assumed that all firms are identical, that labor is provided inellastically, and also that the markets for production inputs clear, we have that

$$mK\_j = K \implies K\_j = K/m,\;\;\; mL\_j = N\_1 \implies L\_j = N\_1/m $$

So

$$\kappa\_j = \frac {K\_j}{L\_j} = \frac {K/m}{N\_1/m} = K/N\_1 \equiv k$$

and now we have obtained $(6.32)$.

Note how all the assumptions made have been used in order to arrive at this result.


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