<p>Let 
<span class="math-container">$$Q = F(K,L)$$</span>
<strong>Assume</strong><br>
a) <span class="math-container">$F(K,L)$</span> exhibits consant returns to scale.  </p>

<p>We need this to aggregate from the individual firms to the total.</p>

<p>b) Price taking behavior  and</p>

<p>c) Profit maximizing behavior  </p>

<p>from the part of the firms.  </p>

<p>Then, throughout the dynamic process,
<span class="math-container">$$w = \frac {\partial F(K,L)}{\partial L} \equiv F_L$$</span></p>

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

<p><span class="math-container">$$\frac {\partial F(K,L)}{\partial L} \equiv h(K,L)$$</span></p>

<p><span class="math-container">$$\dot w = \frac {d}{dt}\frac {\partial F(K,L)}{\partial L} = \frac {d}{dt}h(K,L)= \frac {\partial h(K,L)}{\partial K}\dot K + \frac {\partial h(K,L)}{\partial L}\dot L$$</span></p>

<p><span class="math-container">$$\implies \dot w = F_{KL}\dot K + F_{LL}\dot L$$</span></p>

<p><strong>Assume  further</strong></p>

<p>d) complementarity of inputs, <span class="math-container">$F_{KL}&gt;0$</span>, and  </p>

<p>e) diminishing marginal productss (not in the temporal dimension but in the static dimension, due to diminishing rate of returns), <span class="math-container">$F_{LL}&lt;0, F_{KK}&lt;0$</span>.  </p>

<p>Then, if </p>

<p><span class="math-container">$$\dot K &lt;0, \dot L\geq 0  \implies \dot w &lt;0$$</span></p>

<p>The same treatment wil give us</p>

<p><span class="math-container">$$\dot r = F_{KK}\dot K + F_{KL}\dot L  &gt;0$$</span></p>

CC BY-SA 4.0
2019-04-17T20:53:47.513Z
2019-04-19T13:22:58.487Z
2019-04-19T13:22:58.487Z
Alecos Papadopoulos
<p>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 <span class="math-container">$F(K,L)=K^aL^{1−a} ,0&lt;a&lt;1$</span> , 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 <span class="math-container">$F$</span> with CRS. </p>

CC BY-SA 4.0
2019-04-12T15:37:01.697Z
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).
dismalscience
CC BY-SA 4.0
2019-04-12T15:44:58.353Z
Thanks for your comment. I have edited my question to include as many details about my question as possible.
Vizag
CC BY-SA 4.0
2019-04-15T10:38:56.583Z
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
Vizag
CC BY-SA 4.0
2019-04-16T00:13:30.700Z
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?
AndrewC
CC BY-SA 4.0
2019-04-16T11:32:47.637Z
Yes, general production function in the Solow growth model setting. It satisfies all the required assumptions we have in the Solow growth model
Vizag
<macroeconomics><solow>
Wages in Solow growth model with savings = 0
Vizag
# Wages in Solow growth model with savings = 0

(Asked by: username_0 on 2019-04-12)

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 , 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.

**Tags:** macroeconomics, solow

> **Comments:**
> **username_1**: 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).  
> **username_0**: Thanks for your comment. I have edited my question to include as many details about my question as possible.  
> **username_0**: 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  
> **username_2**: 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?  
> **username_0**: Yes, general production function in the Solow growth model setting. It satisfies all the required assumptions we have in the Solow growth model  

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## 1 answers

### Answer by username_3 on 2019-04-17. Score: 5 | Accepted answer

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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