<p>Let $a+b&lt;1,\;\; a,b&gt;0$.  Pay the factors of production their marginal product:</p>

<p>$$rK = \frac {aQ}{K} K= aQ,\;\;\ wL=\frac {bQ}{L} L=bQ$$</p>

<p>So total payments to factors of production will be</p>

<p>$$rK +wL = aQ + bQ = (a+b)Q &lt; Q$$</p>

<p>And the question is: <strong>who gets the rest of the output that has been produced?</strong></p>

CC BY-SA 4.0
2018-09-06T21:35:20.167Z
2018-09-06T22:03:19.330Z
2018-09-06T22:03:19.330Z
Alecos Papadopoulos
<p>Any constant returns to scale function is compatible witha competitive economy. Cobb-Douglas is not the only one. Google CES production function.</p>

<p>Also, product can be wasted or be an externality, so even in non CRS competitive economy can exist.</p>

CC BY-SA 4.0
2018-09-07T10:48:28.893Z
2018-09-07T10:48:28.893Z
chatGPT
<p>Apologies if this is a rather simple question, I appreciate any guidance. </p>

<blockquote>
  <p>$$ Q(K,L) = AK^\alpha L^{\beta} $$</p>
  
  <p>where A is a constant. Identify the conditions on $\alpha$ and $\beta$
  for output function Q to correspond to a competitive economy.</p>
</blockquote>

<p>So a competitive economy is one where the factors of production are paid their marginal product. I am aware that the Cobb-Douglas production function alignes with this, so I would have: $ \alpha + \beta = 1$ and $\alpha, \beta &gt; 0$. </p>

<p><strong>But is this really the only and most general answer?</strong> I cannot see why we couldn't have different exponents and still pay the factors of production their marginal product.</p>

<macroeconomics><competitive-equilibrium><cobb-douglas>
Is Cobb-Douglas the only output function corresponding to a competitive economy?
Jhonny
# Is Cobb-Douglas the only output function corresponding to a competitive economy?

(Asked by: username_0 on 2018-09-05)

Apologies if this is a rather simple question, I appreciate any guidance.

> $$ Q(K,L) = AK^\alpha L^{\beta} $$
>
> where A is a constant. Identify the conditions on $\alpha$ and $\beta$
> for output function Q to correspond to a competitive economy.

So a competitive economy is one where the factors of production are paid their marginal product. I am aware that the Cobb-Douglas production function alignes with this, so I would have: $ \alpha + \beta = 1$ and $\alpha, \beta > 0$.

**But is this really the only and most general answer?** I cannot see why we couldn't have different exponents and still pay the factors of production their marginal product.

**Tags:** macroeconomics, competitive-equilibrium, cobb-douglas

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## 2 answers

### Answer by username_2 on 2018-09-07. Score: 4 | Accepted answer

Any constant returns to scale function is compatible witha competitive economy. Cobb-Douglas is not the only one. Google CES production function.

Also, product can be wasted or be an externality, so even in non CRS competitive economy can exist.


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### Answer by username_1 on 2018-09-06. Score: 5

Let $a+b<1,\;\; a,b>0$. Pay the factors of production their marginal product:

$$rK = \frac {aQ}{K} K= aQ,\;\;\ wL=\frac {bQ}{L} L=bQ$$

So total payments to factors of production will be

$$rK +wL = aQ + bQ = (a+b)Q < Q$$

And the question is: **who gets the rest of the output that has been produced?**


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