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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 $a_K<1$ 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 $a_K<1$ 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.

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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 $f(0)=0$, $f’(k)>0, f’’(k)<0$ ... 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 $a_k<1$ - and hence the model should always converge.

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  • $\begingroup$ The formal argument is that $f''(k)<0$, in particular tangent line is above the graph. Since graph is $>0$ everywhere for $k\geq 0$, the intercept of tangent line is $>0$. Thus $a_K<1$ by intercept positive. That is the formal argument. What is the intuitive argument why this is true? $\endgroup$ – user45765 May 27 at 17:00
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    $\begingroup$ @user45765 the intuition is that the more capital you use per worker the lower will be the marginal product, eg with 1 monitor I might be able to code 100 lines per hour but with two 150 with 3 175 and so forth because having extra screens allows me to save time that would normally be consumed during the switching between script and output screen/graphs etc but each new screen adds less than last and at some point adding extra screens will not make me any more productive. This would imply that when you increase the amount of capital by 1 percent the output should increase by less than that. $\endgroup$ – 1muflon1 May 27 at 17:07
  • $\begingroup$ I get the main point. However, there are industry that will get increasing return to scale. And it seems that your argument is an uniform principle asserting eventual decrease of marginal return is true in all cases. If a worker is replaced by a machine, you will still experience decreasing of marginal return eventually by your argument as every action takes time to respond. In other words, your statement is more or less asserting eventual decreasing marginal return rather than decreasing of marginal return all the time. $\endgroup$ – user45765 May 27 at 18:06
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    $\begingroup$ @user45765 1.The diminishing marginal returns does not necessarily depend on the returns to scale production function has. Cobb-Douglas function has constant returns to scale but still capital itself within it exhibits diminishing marginal returns and you get constant returns only when both capital and labor increase proportionally, not if only K increases. 2. I purposely written in answer that this is realistic assumption on macroeconomic scale - from macro perspective it’s not realistic for capital having increasing marginal returns due to underlaying scarcity $\endgroup$ – 1muflon1 May 27 at 18:12
  • $\begingroup$ I see. Thanks a lot for clarification. $\endgroup$ – user45765 May 27 at 18:19
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In a one line exercice, it is possible to show that in perfect competition, an output elasticity wrt any input greater than one, implies negative profits.

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