<p>It is generally true that a profit-maximizing firm with a constant-returns to scale technology can produce a positive output only if the profit is zero. Output prices are pinned down by the zero-profit condition.</p>
<p>It follows that the profit-maximizing output level is completely indeterminate at equilibrium prices. However, equilibrium output is not indeterminate. Supply has to equal demand; such a firm will just produce enough to satisfy the market demand.</p>

CC BY-SA 4.0
2021-08-31T10:31:06.130Z
Do you not also need that the firm is a taker of output prices?
BrsG
CC BY-SA 4.0
2021-08-31T14:37:48.740Z
My question was related more to the input choice. What is the point of taking FOC wrt inputs if there is not solution to that problem? I would understand this if the firm takes $Y$ as given at the beginning and optimizes constrained on the production function. Then we would have profits as function of $Y$, and then one should pin down $Y$ and input prices by equating their supplies and demands. Is this correct? I would still not get the point of taking FOC anyway, and it happens everywhere in macro.
Jsck
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2021-08-31T22:26:29.510Z
@Jsck But then you would solve the usual problem of minimizing $WL+RK$ subject to $AK^{\alpha}L^{1-\alpha}=y$, which is well defined.
Michael Greinecker
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2021-08-31T22:27:12.227Z
@BrsG That is part of the definition of a competitive equilibrium. If the firm is not a price taker things are very different.
Michael Greinecker
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2021-09-01T15:50:02.990Z
@MichaelGreinecker I know, the thing that bugs me about the uncostrained problem is that it does not have a solution, even keeping $Y$ indeterminate. What is the point of taking FOC if there is no solution even taking prices as given?
Jsck
CC BY-SA 4.0
2021-08-30T22:49:28.143Z
2021-08-30T22:49:28.143Z
Michael Greinecker
<p>In most macroeconomic papers it is taken as given that the aggregate prodution function is <span class="math-container">$Y=AK^{\alpha}L^{1-\alpha}$</span>, and that the optimality conditions for inputs determine input demands:
<span class="math-container">$$
\max_{K,L} AK^{\alpha}L^{1-\alpha}-WL-RK \\
\alpha A K^{\alpha-1}L^{1-\alpha}=R \\
(1-\alpha) A K^{\alpha}L^{-\alpha}=W
$$</span>
However, it looks like the profit function does not have a global maximum given input prices, hence it is impossible to determine the optimal choice of inputs.</p>
<p>In fact, even trying to reduce the dimensionality of the problem by using the tangency condition one gets
<span class="math-container">$$
K=\frac{\alpha}{1-\alpha}\frac{W}{R}L \\
\implies Y=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L\\
\max_{L}\left[A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}-\frac{W}{1-\alpha}\right]L
$$</span>
which does not have a finite solution if the term in square brackets is positive.</p>
<p>What am I getting wrong? Why these conditions are widely used in macro even though they don't look correct from a mathematical point of view?</p>

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2021-08-30T19:20:08.753Z
Where are the power $\alpha$ and $1-\alpha$ in the expression of $Y=...$?
Bertrand
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2021-08-30T19:25:37.417Z
Sorry for the typo, I put them.
Jsck
CC BY-SA 4.0
2021-08-30T19:29:56.403Z
The term $L^{1-\alpha}$ is missing in the last eq.
Bertrand
CC BY-SA 4.0
2021-08-30T19:34:45.167Z
No it's not: $AK^{\alpha}L^{1-\alpha}=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}L\right)^{\alpha}L^{1-\alpha}=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L^{\alpha+1-\alpha}=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L$
Jsck
<macroeconomics><production-function><optimization><cobb-douglas>
Does global maximum of CRS Cobb-Douglas profit exist
Jsck
# Does global maximum of CRS Cobb-Douglas profit exist

(Asked by: username_0 on 2021-08-30)

In most macroeconomic papers it is taken as given that the aggregate prodution function is $Y=AK^{\alpha}L^{1-\alpha}$, and that the optimality conditions for inputs determine input demands:
$$
\max\_{K,L} AK^{\alpha}L^{1-\alpha}-WL-RK \\
\alpha A K^{\alpha-1}L^{1-\alpha}=R \\
(1-\alpha) A K^{\alpha}L^{-\alpha}=W
$$
However, it looks like the profit function does not have a global maximum given input prices, hence it is impossible to determine the optimal choice of inputs.

In fact, even trying to reduce the dimensionality of the problem by using the tangency condition one gets
$$
K=\frac{\alpha}{1-\alpha}\frac{W}{R}L \\
\implies Y=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L\\
\max\_{L}\left[A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}-\frac{W}{1-\alpha}\right]L
$$
which does not have a finite solution if the term in square brackets is positive.

What am I getting wrong? Why these conditions are widely used in macro even though they don't look correct from a mathematical point of view?

**Tags:** macroeconomics, production-function, optimization, cobb-douglas
> **Comments:**
> **username_1**: Where are the power $\alpha$ and $1-\alpha$ in the expression of $Y=...$?  
> **username_0**: Sorry for the typo, I put them.  
> **username_1**: The term $L^{1-\alpha}$ is missing in the last eq.  
> **username_0**: No it's not: $AK^{\alpha}L^{1-\alpha}=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}L\right)^{\alpha}L^{1-\alpha}=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L^{\alpha+1-\alpha}=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L$  

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## 1 answers

### Answer by username_2 on 2021-08-30. Score: 4

It is generally true that a profit-maximizing firm with a constant-returns to scale technology can produce a positive output only if the profit is zero. Output prices are pinned down by the zero-profit condition.

It follows that the profit-maximizing output level is completely indeterminate at equilibrium prices. However, equilibrium output is not indeterminate. Supply has to equal demand; such a firm will just produce enough to satisfy the market demand.

> **Comments:**
> **username_3**: Do you not also need that the firm is a taker of output prices?  
> **username_0**: My question was related more to the input choice. What is the point of taking FOC wrt inputs if there is not solution to that problem? I would understand this if the firm takes $Y$ as given at the beginning and optimizes constrained on the production function. Then we would have profits as function of $Y$, and then one should pin down $Y$ and input prices by equating their supplies and demands. Is this correct? I would still not get the point of taking FOC anyway, and it happens everywhere in macro.  
> **username_2**: @Jsck But then you would solve the usual problem of minimizing $WL+RK$ subject to $AK^{\alpha}L^{1-\alpha}=y$, which is well defined.  
> **username_2**: @BrsG That is part of the definition of a competitive equilibrium. If the firm is not a price taker things are very different.  
> **username_0**: @MichaelGreinecker I know, the thing that bugs me about the uncostrained problem is that it does not have a solution, even keeping $Y$ indeterminate. What is the point of taking FOC if there is no solution even taking prices as given?  

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