<p>The fact that the wage rate is strictly positive depends on the assumption of <strong>competitive markets</strong> and on the assumption of <strong>strictly positive marginal</strong> <strong>products</strong> (and, of course, on the assumption that <span class="math-container">$F$</span>  is differentiable)<span class="math-container">$^1$</span>.  In particular, the marginal product of labour is (<strong>Assumption 1</strong>):</p>
<p><span class="math-container">$$ F_L(K,L, A)\equiv \frac {\partial F(K,L, A)}{\partial L} &gt;0$$</span></p>
<p>We know that in a competitive market, in equilibrium, the wage rate is equal to the marginal productivity of labour<span class="math-container">$^2$</span>:</p>
<p><span class="math-container">$$w=F_L(K,L, A)\equiv \frac {\partial F(K,L, A)}{\partial L} &gt;0\;\;\;\; \;\;(1)$$</span></p>
<p>so <span class="math-container">$w$</span> is strictly positive.</p>
<p>As <span class="math-container">$w&gt;0$</span>, so <span class="math-container">$w\neq 0$</span>, we have that <span class="math-container">$(2.4)$</span> implies <span class="math-container">$(2.3)$</span>. Indeed, <span class="math-container">$(2.4)$</span> states:</p>
<p><span class="math-container">$$(L(t))-\bar L(t)) w(t) =0.^3\;\;\;\;\; (2.4)$$</span></p>
<p>As <span class="math-container">$w\neq 0$</span>,  to satisfy this equality, must be</p>
<p><span class="math-container">$$L(t))-\bar L(t) =0, $$</span></p>
<p>that is</p>
<p><span class="math-container">$$L(t))=\bar L(t).\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2.3) $$</span></p>
<hr />
<p><span class="math-container">$^1$</span><sup> Constant returns to scale play no role here. They  will be important later in the model, because they allow to write the <em>intensive production function</em></sup></p>
<p><span class="math-container">$^2$</span> <sup>You can remember from microeconomic theory that, in a competitive market, the profit maximizing firms equate the value of the marginal product of a factor input to its price, in particular equate marginal productivity of labour to the wage rate. See also Acemoglu, <em>cit.</em> p.33, where you find the formula <span class="math-container">$(1)$</span> I wrote above. </sup></p>
<p><span class="math-container">$^3$</span> <sup> <span class="math-container">$\bar L(t)$</span> stands for the (anelastic with respect to wage) supply of labour.</sup></p>

CC BY-SA 4.0
2022-12-09T16:38:03.867Z
2022-12-09T21:34:07.453Z
2022-12-09T21:34:07.453Z
BakerStreet
<p>I would like to know if you could help me to solve this exercise from <em>Acemoglu's book &quot;Introduction to Modern Economic Growth&quot;</em></p>
<p><strong>Exercise 2.1 : Show that competitive labor markets and Assumption 1 imply that the wage rate must be strictly positive and thus (2.4) implies (2.3)</strong></p>
<p>Assumption 1:
<a href="https://i.sstatic.net/5ufLR.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/5ufLR.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/CuAhX.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/CuAhX.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$$L(t)=\bar L(t)\;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; (2.3)$$</span></p>
<p>I've been thinking about it for a few days, but I really don't know where to start. I would be very grateful for your help</p>

<macroeconomics><economic-growth>
How to show that in Solow's model competitive labour markets and the assumption of constant returns to scale imply that the wage is positive?
Juan Esteban Amaris Rojas
# How to show that in Solow's model competitive labour markets and the assumption of constant returns to scale imply that the wage is positive?

(Asked by: username_0 on 2022-12-08)

I would like to know if you could help me to solve this exercise from *Acemoglu's book "Introduction to Modern Economic Growth"*

**Exercise 2.1 : Show that competitive labor markets and Assumption 1 imply that the wage rate must be strictly positive and thus (2.4) implies (2.3)**

Assumption 1:
[![enter image description here](https://i.sstatic.net/5ufLR.png)](https://i.sstatic.net/5ufLR.png)

[![enter image description here](https://i.sstatic.net/CuAhX.png)](https://i.sstatic.net/CuAhX.png)

$$L(t)=\bar L(t)\;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; (2.3)$$

I've been thinking about it for a few days, but I really don't know where to start. I would be very grateful for your help

**Tags:** macroeconomics, economic-growth

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## 1 answers

### Answer by username_1 on 2022-12-09. Score: 3

The fact that the wage rate is strictly positive depends on the assumption of **competitive markets** and on the assumption of **strictly positive marginal** **products** (and, of course, on the assumption that $F$ is differentiable)$^1$. In particular, the marginal product of labour is (**Assumption 1**):

$$ F\_L(K,L, A)\equiv \frac {\partial F(K,L, A)}{\partial L} >0$$

We know that in a competitive market, in equilibrium, the wage rate is equal to the marginal productivity of labour$^2$:

$$w=F\_L(K,L, A)\equiv \frac {\partial F(K,L, A)}{\partial L} >0\;\;\;\; \;\;(1)$$

so $w$ is strictly positive.

As $w>0$, so $w\neq 0$, we have that $(2.4)$ implies $(2.3)$. Indeed, $(2.4)$ states:

$$(L(t))-\bar L(t)) w(t) =0.^3\;\;\;\;\; (2.4)$$

As $w\neq 0$, to satisfy this equality, must be

$$L(t))-\bar L(t) =0, $$

that is

$$L(t))=\bar L(t).\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2.3) $$

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$^1$ Constant returns to scale play no role here. They will be important later in the model, because they allow to write the *intensive production function*

$^2$ You can remember from microeconomic theory that, in a competitive market, the profit maximizing firms equate the value of the marginal product of a factor input to its price, in particular equate marginal productivity of labour to the wage rate. See also Acemoglu, *cit.* p.33, where you find the formula $(1)$ I wrote above.

$^3$  $\bar L(t)$ stands for the (anelastic with respect to wage) supply of labour.


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