Is there any fundamental reason that output elasticity of capital $<1$ in general? In Solow growth model, if one perturb the saving, there is convergence of old equilibrium capital to new equilibrium capital as investment breaking point capital is attractor. 

However, in derivation of convergence, one needs to invoke $a_K$ elasticity of output to capital. Assuming growth in knowledge, growth of population and decay of capital. Then I need $a_K&lt;1$ to guarantee convergence of old equilibrium to new equilibrium. I am assuming Inada condition on output per effective labor and knowledge is only labor augmenting in the model.

Is $a_K&lt;1$ predetermined by Solow growth model or some condition on the model? This discussion does not assume any particular form of production function, in particular, Cobb-Douglas function.

Ref. Romer, Advanced Macroeconomics, Chpt 1, Sec 5 on speed of convergence.
The elasticity of output with respect to capital will be less than 1 due to the diminishing marginal returns of capital - this is both realistic on macroeconomic scale and also one of the central assumptions of the model.

According to Romer’s advanced macroeconomics, pp 12 section 1.2 assumptions:


  “The intensive-form production function, $f(k)$, is assumed to satisfy $f(0)=0$, $f’(k)&gt;0, f’’(k)&lt;0$ ... Thus the assumptions that $f’(k)$ is positive and $f’’(k)$ negative imply that the marginal product of capital is positive, but that it declines as capital (per unit of effective labor) rises.”


This is one of the standard assumptions of Solow-Swan models so $a_K$ (or in Romers book $\alpha_K$) will be less than 1 just by assumptions of the model.

Side note: this is actually also implied by the Inada conditions themselves (which as Romer points out are stronger than needed for the model's central result) since $\lim_{k\rightarrow 0 } f’(k)=\infty$ and $\lim_{k\rightarrow \infty } f’(k)=0$ together with other assumptions of the model imply that $a_k&lt;1$ - and hence the model should always converge.