<p>The system of FOC of the profit maximization problem when the production function is constant return to scale is a <em>degenerate system</em>. For a good explanation see <a href="https://math.stackexchange.com/questions/3090859/solution-to-a-two-equations-system">this</a>. In short, it means that the system of two equation in two unknown is actually a system of two equation in <strong>one</strong> unknown, which in this case is <span class="math-container">$\frac{K}{L}$</span>.</p>

<p>The solution to this problem needs to satisfy both equations (2) and (3), and this happens only if equation (4) is satisfied. </p>

<p>Hence, to answer the main question, given one price, the other is uniquely identified by equation (4). Given one price, to determine the other it is enough to know the technological parameter, in this case, this is just <span class="math-container">$\alpha$</span>.</p>

CC BY-SA 4.0
2019-07-07T14:52:14.060Z
2019-07-07T14:52:14.060Z
Tecon
<p>This is a subtle issue. First let's present a numerical example to see the head-scratching riddle.
Assume <span class="math-container">$$\alpha =1/2 \implies Y = K^{1/2}L^{1/2}$$</span></p>

<p>and that the exogenously given input prices are </p>

<p><span class="math-container">$$r=1/8,\,\, w=4.$$</span></p>

<p>The f.o.c are</p>

<p><span class="math-container">$$\begin{cases} \frac {Y}{2K} = 1/8 \\
\\
\frac{Y}{2L} = 4 \end{cases} \implies K/4 = 8L \implies \left(L/K\right)^* = 1/32$$</span></p>

<p>This gives us the value of <span class="math-container">$L/K$</span> that satisfies both first-order conditions.</p>

<p>But let's follow the OP's logic, and write/solve the f.o.c. as</p>

<p><span class="math-container">$$\begin{cases} \frac {K^{1/2}L^{1/2}}{2K} = 1/8 \\
\\
\frac{K^{1/2}L^{1/2}}{2L} = 4 \end{cases} \implies \begin{cases} \frac {L}{K} = 1/16 \\
\\
\frac{L}{K} = 1/64 \end{cases}$$</span></p>

<p><strong>Oops.</strong> The system now seems impossible. But it appears that just above we have solved the exact same system. And we didn't do any illegal math operation, like dividing by zero or anything so how is it possible to obtain the above contradictory results?...</p>

<p>But indeed we did something "illegal", although not in terms of mathematical operations in the narrow sense: by solving each f.o.c. <strong><em>separately</em></strong> with respect to <span class="math-container">$L/K$</span> we transformed a problem of optimizing a bivariate function that <em>leads</em> to a system with two unknowns and two equations, into a problem of directly solving a system with <em><strong>one</strong></em> unknown (<span class="math-container">$L/K$</span>), and two equations.<br>
No surprise then that it will have a solution only for specific combinations of the exogenous parameters, which is expressed by eq. <span class="math-container">$(4)$</span> of the OP.  But this is no longer  the profit maximizing problem of the competitive firm. </p>

<p>Another way to say it is that, given that what is treated as exogenous here is the input prices, not all input ratios satisfy the f.o.c. </p>

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2019-01-25T09:17:48.537Z
Thanks for your answer. I don't understand this sentence: "What this relation says is that if we fix a labor/capital ratio, and we want production to be carried out with this ratio, then the combinations (r,w) must obey relation (4)". Why do you say "fix the ratio"? I solve for the ratio, hence, for _any given ratio_, that relationship has to be satisfied. And yes, all price combinations are permissible, however, only the price combinations governed by that relationship allow for a profit-maximizing plan to exist, i.e. there is an interior solution of the PMP.
Tecon
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2019-01-25T11:57:40.150Z
@Tecon In response to your comment I completely re-worked my answer.
Alecos Papadopoulos
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2019-01-25T14:37:20.257Z
Thank you very much @Alecos. Your answer really helped me.
Tecon
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2019-01-25T16:18:10.730Z
Alecos, if you are interested, I posted a follow-up question [here](https://economics.stackexchange.com/questions/26605/solve-for-the-steady-state-with-crs-cobb-douglas-problem-with-the-system-of-equ)
Tecon
CC BY-SA 4.0
2019-01-29T09:41:32.907Z
I am not sure anymore about the correctness of this sentence: "But this is no longer the profit-maximizing problem of the competitive firm." The solution to the PMP of the firm is determined by a system of two equations. This system is "degenerate" in the sense that it can be expressed by a combination of the two unknowns, the ratio $\frac{L}{K}$. **Both** equation need be satisfied, and this does not happen for *any* combination of prices, but only for those combinations that satisfy equation (4).
Tecon
CC BY-SA 4.0
2019-01-29T09:41:37.290Z
In the numerical example you provided, the price ratio, given that technology, cannot support a profit-maximizing plan. To conclude, I believe that equation (4) has to be satisfied in equilibrium, and can, therefore, be used to find, the value of $w$, given the interest rate.
Tecon
CC BY-SA 4.0
2019-01-24T23:23:37.567Z
2019-01-25T11:57:17.003Z
2019-01-25T11:57:17.003Z
Alecos Papadopoulos
<p>A clarification: The values <span class="math-container">$w,r$</span> are not independent of input combinations: you have already solved for <span class="math-container">$K/L$</span>.</p>

<p>1) Unless you assume that the price of the output is equal to 1, there is a minor mistake, as output prices affect factor prices.</p>

<p>2) If there is a profit maximizing pair <span class="math-container">$(K,L)$</span>, then for all <span class="math-container">$\alpha \in \mathbb{R}_+$</span> <span class="math-container">$(\alpha K,\alpha L)$</span> will also be profit maximizing. This is because the production function you are looking at has constant returns to scale (CRS). </p>

<p>A consequence of CRS is that if the function is differentiable then both marginal products will be homogeneous of degree zero, hence <span class="math-container">$r$</span> and <span class="math-container">$w$</span> will only determine the ratios of <span class="math-container">$K$</span> and <span class="math-container">$L$</span>, not their level.</p>

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2019-01-25T09:31:47.477Z
Thanks, @denesp. What I find interesting is that, for instance, given $r$, I can get the value of $w$ that allows for a profit-maximizing plan to exist, _independently_ of $\frac{K}{L}$. This can be useful, I guess. when we compute the steady-state allocation in an economy with CRS Cobb-Douglas; the interest rate is fixed by the consumption side and, thanks to that equation, we can easily get the wage. Do you think this argument is correct?
Tecon
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2019-01-25T14:03:03.273Z
"given $r$, I can get the value of $w$ that allows for a profit-maximizing plan to exist" This is true. "*independently* of $\frac{K}{L}$". Again, I don't know what you mean by this. $K$ and $L$ are not parameters, they are endogenous variables and will take values as functions of $r$ and $w$.
Giskard
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2019-01-25T14:03:45.327Z
On the macro argument: Sure, but I have always been wary of the CRS assumption in macroeconomics models.
Giskard
CC BY-SA 4.0
2019-01-24T17:31:29.327Z
2019-01-24T17:31:29.327Z
Giskard
<p>There is a market economy with technology given by:</p>

<p><span class="math-container">$$Y = K^\alpha L^{1-\alpha} \tag{1}$$</span></p>

<p>Firms behave competitively and input prices are:</p>

<p><span class="math-container">$$r = \alpha K^{\alpha-1}L^{1-\alpha} = \alpha(\frac{L}{K})^{1-\alpha} \tag{2} $$</span></p>

<p><span class="math-container">$$w = (1-\alpha) K^\alpha L^{-\alpha} = (1-\alpha) (\frac{L}{K})^{-\alpha} \tag{3}$$</span></p>

<p>I can solve for <span class="math-container">$\frac{L}{K}$</span>, and obtain the following expression:</p>

<p><span class="math-container">$$r = \alpha (1-\alpha)^{\frac{1-\alpha}{\alpha}}w^{\frac{\alpha-1}{\alpha}} \tag{4}$$</span></p>

<p>Which is a function that links the prices independently of the inputs combination. </p>

<p>1) Is this correct?</p>

<p>2) What is the economic interpretation of the last result?</p>

<macroeconomics><labor-economics><production-function><technology><capital>
Cobb-Douglas production function, given $w$ get $r$ regardless of input levels. Why?
Tecon
# Cobb-Douglas production function, given $w$ get $r$ regardless of input levels. Why?

(Asked by: username_0 on 2019-01-24)

There is a market economy with technology given by:

$$Y = K^\alpha L^{1-\alpha} \tag{1}$$

Firms behave competitively and input prices are:

$$r = \alpha K^{\alpha-1}L^{1-\alpha} = \alpha(\frac{L}{K})^{1-\alpha} \tag{2} $$

$$w = (1-\alpha) K^\alpha L^{-\alpha} = (1-\alpha) (\frac{L}{K})^{-\alpha} \tag{3}$$

I can solve for $\frac{L}{K}$, and obtain the following expression:

$$r = \alpha (1-\alpha)^{\frac{1-\alpha}{\alpha}}w^{\frac{\alpha-1}{\alpha}} \tag{4}$$

Which is a function that links the prices independently of the inputs combination.

1) Is this correct?

2) What is the economic interpretation of the last result?

**Tags:** macroeconomics, labor-economics, production-function, technology, capital

---

## 3 answers

### Answer by username_0 on 2019-07-07. Score: 0 | Accepted answer

The system of FOC of the profit maximization problem when the production function is constant return to scale is a *degenerate system*. For a good explanation see [this](https://math.stackexchange.com/questions/3090859/solution-to-a-two-equations-system). In short, it means that the system of two equation in two unknown is actually a system of two equation in **one** unknown, which in this case is $\frac{K}{L}$.

The solution to this problem needs to satisfy both equations (2) and (3), and this happens only if equation (4) is satisfied.

Hence, to answer the main question, given one price, the other is uniquely identified by equation (4). Given one price, to determine the other it is enough to know the technological parameter, in this case, this is just $\alpha$.


---

### Answer by username_1 on 2019-01-24. Score: 4

This is a subtle issue. First let's present a numerical example to see the head-scratching riddle.
Assume $$\alpha =1/2 \implies Y = K^{1/2}L^{1/2}$$

and that the exogenously given input prices are

$$r=1/8,\,\, w=4.$$

The f.o.c are

$$\begin{cases} \frac {Y}{2K} = 1/8 \\
\\
\frac{Y}{2L} = 4 \end{cases} \implies K/4 = 8L \implies \left(L/K\right)^\* = 1/32$$

This gives us the value of $L/K$ that satisfies both first-order conditions.

But let's follow the OP's logic, and write/solve the f.o.c. as

$$\begin{cases} \frac {K^{1/2}L^{1/2}}{2K} = 1/8 \\
\\
\frac{K^{1/2}L^{1/2}}{2L} = 4 \end{cases} \implies \begin{cases} \frac {L}{K} = 1/16 \\
\\
\frac{L}{K} = 1/64 \end{cases}$$

**Oops.** The system now seems impossible. But it appears that just above we have solved the exact same system. And we didn't do any illegal math operation, like dividing by zero or anything so how is it possible to obtain the above contradictory results?...

But indeed we did something "illegal", although not in terms of mathematical operations in the narrow sense: by solving each f.o.c. ***separately*** with respect to $L/K$ we transformed a problem of optimizing a bivariate function that *leads* to a system with two unknowns and two equations, into a problem of directly solving a system with ***one*** unknown ($L/K$), and two equations.  
No surprise then that it will have a solution only for specific combinations of the exogenous parameters, which is expressed by eq. $(4)$ of the OP. But this is no longer the profit maximizing problem of the competitive firm.

Another way to say it is that, given that what is treated as exogenous here is the input prices, not all input ratios satisfy the f.o.c.

> **Comments:**
> **username_0**: Thanks for your answer. I don't understand this sentence: "What this relation says is that if we fix a labor/capital ratio, and we want production to be carried out with this ratio, then the combinations (r,w) must obey relation (4)". Why do you say "fix the ratio"? I solve for the ratio, hence, for _any given ratio_, that relationship has to be satisfied. And yes, all price combinations are permissible, however, only the price combinations governed by that relationship allow for a profit-maximizing plan to exist, i.e. there is an interior solution of the PMP.  
> **username_1**: @Tecon In response to your comment I completely re-worked my answer.  
> **username_0**: Thank you very much @Alecos. Your answer really helped me.  
> **username_0**: Alecos, if you are interested, I posted a follow-up question [here](https://economics.stackexchange.com/questions/26605/solve-for-the-steady-state-with-crs-cobb-douglas-problem-with-the-system-of-equ)  
> **username_0**: I am not sure anymore about the correctness of this sentence: "But this is no longer the profit-maximizing problem of the competitive firm." The solution to the PMP of the firm is determined by a system of two equations. This system is "degenerate" in the sense that it can be expressed by a combination of the two unknowns, the ratio $\frac{L}{K}$. **Both** equation need be satisfied, and this does not happen for *any* combination of prices, but only for those combinations that satisfy equation (4).  
> **username_0**: In the numerical example you provided, the price ratio, given that technology, cannot support a profit-maximizing plan. To conclude, I believe that equation (4) has to be satisfied in equilibrium, and can, therefore, be used to find, the value of $w$, given the interest rate.  

---

### Answer by username_2 on 2019-01-24. Score: 3

A clarification: The values $w,r$ are not independent of input combinations: you have already solved for $K/L$.

1) Unless you assume that the price of the output is equal to 1, there is a minor mistake, as output prices affect factor prices.

2) If there is a profit maximizing pair $(K,L)$, then for all $\alpha \in \mathbb{R}\_+$ $(\alpha K,\alpha L)$ will also be profit maximizing. This is because the production function you are looking at has constant returns to scale (CRS).

A consequence of CRS is that if the function is differentiable then both marginal products will be homogeneous of degree zero, hence $r$ and $w$ will only determine the ratios of $K$ and $L$, not their level.

> **Comments:**
> **username_0**: Thanks, @denesp. What I find interesting is that, for instance, given $r$, I can get the value of $w$ that allows for a profit-maximizing plan to exist, _independently_ of $\frac{K}{L}$. This can be useful, I guess. when we compute the steady-state allocation in an economy with CRS Cobb-Douglas; the interest rate is fixed by the consumption side and, thanks to that equation, we can easily get the wage. Do you think this argument is correct?  
> **username_2**: "given $r$, I can get the value of $w$ that allows for a profit-maximizing plan to exist" This is true. "*independently* of $\frac{K}{L}$". Again, I don't know what you mean by this. $K$ and $L$ are not parameters, they are endogenous variables and will take values as functions of $r$ and $w$.  
> **username_2**: On the macro argument: Sure, but I have always been wary of the CRS assumption in macroeconomics models.  

---

