Is Cobb-Douglas the only output function corresponding to a competitive economy? 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 $\alpha, \beta &gt; 0$. 

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.
Let $a+b&lt;1,\;\; a,b&gt;0$.  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

$$rK +wL = aQ + bQ = (a+b)Q &lt; Q$$

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