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I am trying to understand the change of wage rate and rental rate in the Solow growth model with s = 0. It is clear that capital per capita will approach 0 (due to depreciation of capital). Also is clear to me the fact that the rental rate will rise because of decreasing marginal returns assumed on the production function. Since factors are paid their marginal products in case of Constant Returns to Scale, rental rates rise. I am confused about the marginal product of labor overtime in a general setting. For example if I take $F(K,L)=K^aL^{1−a} ,0<a<1$ , then I can see that marginal product of labor will fall if we assume constant or increasing labor force. But I am unable to show this for a general $F$ with CRS.

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    $\begingroup$ Hi, I think this question has potential but you’ll get a better response to it if you both take the time to write the text that you took a picture of, and include the relevant context (I see that the question seems to refer to another Solow model in a previous question). $\endgroup$ – dismalscience Apr 12 at 15:37
  • $\begingroup$ Thanks for your comment. I have edited my question to include as many details about my question as possible. $\endgroup$ – Vizag Apr 12 at 15:44
  • $\begingroup$ I see a wall of text with many sentences, but I do not know what the question is. $\endgroup$ – Kenny LJ Apr 15 at 10:36
  • $\begingroup$ Hi Kenny. Thanks for your comment. My question is to show the evolution of Marginal Product of Labor with time as capital per capita decreases. Will it increase or decrease? Basically what the labor wage rate will be with time as the capital per capita decreases $\endgroup$ – Vizag Apr 15 at 10:38
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    $\begingroup$ My confusion stems from what you're interested in when you say "general" production function. Usually, Solow models only consider production functions that satisfy the Inada conditions. Of the CES functions, only the Cobb Douglass production function satisfies this (which what Solow used himself). So are you interested in generalizing away from the Inada conditions? (If so, then it's pretty easy to come up with an example with a constant wage rate and CRS production.) Or is there another specific generalization you'd like to make? $\endgroup$ – AndrewC Apr 16 at 0:13
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Let $$Q = F(K,L)$$ Assume
a) $F(K,L)$ exhibits consant returns to scale.

We need this to aggregate from the individual firms to the total.

b) Price taking behavior and

c) Profit maximizing behavior

from the part of the firms.

Then, throughout the dynamic process, $$w = \frac {\partial F(K,L)}{\partial L} \equiv F_L$$

i.e. the wage is equal to the marginal product of labor. For clarity, write

$$\frac {\partial F(K,L)}{\partial L} \equiv h(K,L)$$

$$\dot w = \frac {d}{dt}\frac {\partial F(K,L)}{\partial L} = \frac {d}{dt}h(K,L)= \frac {\partial h(K,L)}{\partial K}\dot K + \frac {\partial h(K,L)}{\partial L}\dot L$$

$$\implies \dot w = F_{KL}\dot K + F_{LL}\dot L$$

Assume further

d) complementarity of inputs, $F_{KL}>0$, and

e) diminishing marginal productss (not in the temporal dimension but in the static dimension, due to diminishing rate of returns), $F_{LL}<0, F_{KK}<0$.

Then, if

$$\dot K <0, \dot L\geq 0 \implies \dot w <0$$

The same treatment wil give us

$$\dot r = F_{KK}\dot K + F_{KL}\dot L >0$$

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  • $\begingroup$ This is beautiful. Thank you! $\endgroup$ – Vizag Apr 17 at 21:11

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