<p>Let's do the maths in a simple way:</p>
<p>Take profit function: <span class="math-container">$B = p · f(L,K) - w·L - r·K$</span></p>
<p>Where <span class="math-container">$B$</span> is profit, <span class="math-container">$L$</span> is labour and <span class="math-container">$K$</span> means capital.
<span class="math-container">$f(L,K)$</span> is the product function, <span class="math-container">$p$</span> is price and <span class="math-container">$r$</span> is the rental price of capital and <span class="math-container">$w$</span> is wage.</p>
<p>Take <span class="math-container">$PMg(K)$</span> as marginal product of <span class="math-container">$K$</span> and know that <span class="math-container">$df(K,L)/dK = PMg(K)$</span></p>
<p>In order to optimize <span class="math-container">$B$</span> with <span class="math-container">$K$</span> take the first order derivative:</p>
<p><span class="math-container">$dB/dK = p · PMg(K) - r = 0$</span>, therefore:</p>
<p><span class="math-container">$PMg(K) = r/p$</span>, so the real gains of capital <span class="math-container">$r/p$</span> equal to its marginal product <span class="math-container">$PMg(K)$</span>. This is an equilibrium for capital under perfect competence, if marginal product is greater that its revenue, product would grow that would make marginal product lower (due to the law of diminishing returns).</p>
<p>This is a simplification, I'm not taking in account labour and supposing that capital has a marginal product. Notice that if we not suppose perfect competence, price would be a function of product so we would have different maths.</p>
<p>If you take into account labour and capital which separate marginal products, you have to take the Lagrange optimization and reach:</p>
<p><span class="math-container">$PMg(K)/r = PMg(L)/w$</span></p>
<p>Try to minimize cost fixed to product or maximize product fixed to cost and you will reach that last statement.</p>
<p>Edit: Solve some grammatical issues and other errors</p>

CC BY-SA 4.0
2021-09-28T13:41:01.460Z
There's a typo in the profit function $B$. I also don't find the label $PMg(K)$ particularly mnemonic for "marginal product of capital". Perhaps it's mnemonic in a language other than English?
Herr K.
CC BY-SA 4.0
2021-09-28T13:56:00.333Z
I solved the typo on B (I think that you mean $r · K$ instead of $r · L$.
DPedro
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2021-09-28T13:57:03.603Z
Also, $PMg(.)$ comes from spanish, where means "Producto Marginal". Should be $MgP(.)$ better instead?
DPedro
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2021-09-28T21:10:34.770Z
With this post I'm not saying I trust this theory (depends on the nature of which market are we talking about) what I'm doing is give a response to the question.
DPedro
CC BY-SA 4.0
2021-09-27T20:45:39.843Z
2021-09-28T13:55:24.663Z
2021-09-28T13:55:24.663Z
DPedro
<p>Intuitively it works like this:</p>
<p>There is an infinite number of firms competing for the capital in the economy. If the rental price of capital <span class="math-container">$r$</span> is below its marginal product , then a firm can increase its profit by hiring more capital. In order to do this a <span class="math-container">$firm_1$</span> can offer a slightly higher rate than <span class="math-container">$r$</span> and immediatly all capital in the entire economy would go to <span class="math-container">$firm_1$</span>, leaving all other firms with no production and thus no profit at all. But by that logic another <span class="math-container">$firm_2$</span> would offer an even higher rate that <span class="math-container">$firm_1$</span> and immediately all capital would go to that firm, leaving ...</p>
<p>I hope you get the picture. Firms compete for capital by offering an <span class="math-container">$r$</span> and that <span class="math-container">$r$</span> tends to <span class="math-container">$\partial f(k) / \partial k$</span> in a perfectly competitive market.</p>
<p>Obviously <span class="math-container">$r$</span> cannot exceed <span class="math-container">$\partial f(k) / \partial k$</span>, because a firm would make losses and thus no firm is willing to offer an <span class="math-container">$r$</span> this high.</p>

CC BY-SA 4.0
2021-09-27T20:57:47.847Z
"*Obviously $r$ cannot exceed* $\partial f(k) / \partial k$..." actually, in the Concorde aviation industry $r$ currently exceeds $\partial f(k) / \partial k$, which is why there aren't any Concorde planes in flight.
Giskard
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2021-09-28T11:29:22.137Z
Does capital really work that way? There are different kinds of capital. Newspaper printing presses are no use to a bottling plant, so in case the bottling plant can make more profit from capital, it is still no use for the newspaper printing press to go there, even if it can be teleported for free.
Criticize SE actions means ban
CC BY-SA 4.0
2021-09-28T13:28:11.137Z
@user253751 This question most likely refers to a macro model (Solow, RCK), where there is only one type of output, which can be either consumed or invested in captial.
stollenm
CC BY-SA 4.0
2021-09-28T13:34:39.940Z
@Giskard I was answering in the context of a one-sector macro model. In that case if $r > \partial f(k)/\partial k$, firms wouldn't hire capital and (depending on the assumed production function) no output would be produced at all, giving no utility to consumers. Thus this outcome is most likely no equilibrium state of the economy.
stollenm
CC BY-SA 4.0
2021-09-28T13:45:19.557Z
@stollenm hmm, I hope serious economic decisions are not being made based on such oversimplified models of reality.
Criticize SE actions means ban
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2021-09-27T19:33:16.700Z
2021-09-27T20:57:53.220Z
2021-09-27T20:57:53.220Z
stollenm
<p>Why is it possible to tell that, in the case markets are competitive, capital earns its marginal product? What's the formal explanation and what is the economic intuition behind it?</p>

CC BY-SA 4.0
2021-09-28T14:23:03.480Z
Even if perfect competition is a useful model for some kinds of markets, it is rather implausible that it could be so for the market for a capital good.  One of the assumptions of perfect competition is perfect information, but for a capital good that would require perfect information over the whole life of the good.  The practicality is that investment in capital goods always involves risk and uncertainty (about future demand for  what the goods can produce, if not about their productive capacity).
Adam Bailey
<macroeconomics>
Market competitiveness → Capital earns its marginal product. But why?
shannontesla
# Market competitiveness → Capital earns its marginal product. But why?

(Asked by: username_0 on 2020-01-01)

Why is it possible to tell that, in the case markets are competitive, capital earns its marginal product? What's the formal explanation and what is the economic intuition behind it?

**Tags:** macroeconomics
> **Comments:**
> **username_1**: Even if perfect competition is a useful model for some kinds of markets, it is rather implausible that it could be so for the market for a capital good.  One of the assumptions of perfect competition is perfect information, but for a capital good that would require perfect information over the whole life of the good.  The practicality is that investment in capital goods always involves risk and uncertainty (about future demand for  what the goods can produce, if not about their productive capacity).  

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## 2 answers

### Answer by username_2 on 2021-09-27. Score: 2

Let's do the maths in a simple way:

Take profit function: $B = p · f(L,K) - w·L - r·K$

Where $B$ is profit, $L$ is labour and $K$ means capital.
$f(L,K)$ is the product function, $p$ is price and $r$ is the rental price of capital and $w$ is wage.

Take $PMg(K)$ as marginal product of $K$ and know that $df(K,L)/dK = PMg(K)$

In order to optimize $B$ with $K$ take the first order derivative:

$dB/dK = p · PMg(K) - r = 0$, therefore:

$PMg(K) = r/p$, so the real gains of capital $r/p$ equal to its marginal product $PMg(K)$. This is an equilibrium for capital under perfect competence, if marginal product is greater that its revenue, product would grow that would make marginal product lower (due to the law of diminishing returns).

This is a simplification, I'm not taking in account labour and supposing that capital has a marginal product. Notice that if we not suppose perfect competence, price would be a function of product so we would have different maths.

If you take into account labour and capital which separate marginal products, you have to take the Lagrange optimization and reach:

$PMg(K)/r = PMg(L)/w$

Try to minimize cost fixed to product or maximize product fixed to cost and you will reach that last statement.

Edit: Solve some grammatical issues and other errors

> **Comments:**
> **username_3**: There's a typo in the profit function $B$. I also don't find the label $PMg(K)$ particularly mnemonic for "marginal product of capital". Perhaps it's mnemonic in a language other than English?  
> **username_2**: I solved the typo on B (I think that you mean $r · K$ instead of $r · L$.  
> **username_2**: Also, $PMg(.)$ comes from spanish, where means "Producto Marginal". Should be $MgP(.)$ better instead?  
> **username_2**: With this post I'm not saying I trust this theory (depends on the nature of which market are we talking about) what I'm doing is give a response to the question.  

---

### Answer by username_4 on 2021-09-27. Score: 1

Intuitively it works like this:

There is an infinite number of firms competing for the capital in the economy. If the rental price of capital $r$ is below its marginal product , then a firm can increase its profit by hiring more capital. In order to do this a $firm\_1$ can offer a slightly higher rate than $r$ and immediatly all capital in the entire economy would go to $firm\_1$, leaving all other firms with no production and thus no profit at all. But by that logic another $firm\_2$ would offer an even higher rate that $firm\_1$ and immediately all capital would go to that firm, leaving ...

I hope you get the picture. Firms compete for capital by offering an $r$ and that $r$ tends to $\partial f(k) / \partial k$ in a perfectly competitive market.

Obviously $r$ cannot exceed $\partial f(k) / \partial k$, because a firm would make losses and thus no firm is willing to offer an $r$ this high.

> **Comments:**
> **username_5**: "*Obviously $r$ cannot exceed* $\partial f(k) / \partial k$..." actually, in the Concorde aviation industry $r$ currently exceeds $\partial f(k) / \partial k$, which is why there aren't any Concorde planes in flight.  
> **username_6**: Does capital really work that way? There are different kinds of capital. Newspaper printing presses are no use to a bottling plant, so in case the bottling plant can make more profit from capital, it is still no use for the newspaper printing press to go there, even if it can be teleported for free.  
> **username_4**: @user253751 This question most likely refers to a macro model (Solow, RCK), where there is only one type of output, which can be either consumed or invested in captial.  
> **username_4**: @Giskard I was answering in the context of a one-sector macro model. In that case if $r > \partial f(k)/\partial k$, firms wouldn't hire capital and (depending on the assumed production function) no output would be produced at all, giving no utility to consumers. Thus this outcome is most likely no equilibrium state of the economy.  
> **username_6**: @stollenm hmm, I hope serious economic decisions are not being made based on such oversimplified models of reality.  

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