<p>The common usage is probably</p>
<ol>
<li><p><span class="math-container">$\displaystyle \max_{K,L} [F(K,AL) - wL - rK]$</span></p>
</li>
<li><p><span class="math-container">$wL$</span></p>
</li>
<li><p><span class="math-container">$w = \frac{\partial F(K,AL)}{\partial L}$</span></p>
</li>
</ol>
<p>BUT</p>
<p>you could get the other answers just by redefining the unit of measurement of labor. The above answers are valid for &quot;<span class="math-container">$w$</span> is the wage for a unit of labor&quot;, the other answers are valid for &quot;<span class="math-container">$w$</span> is the wage for a unit of <em>effective</em> labor&quot;, so labor unit where technological progress is taken into account. The effective unit is equivalent to <span class="math-container">$1/A$</span> conventional units of labor. It's like saying &quot;using machine X, a worker can now perform the work of 10 normal workers&quot;; this new worker will probably still file 8 hour work days (conventional unit) and not 80 hour work days (effective unit).</p>
<hr />
<p>Another technical aside: there is no difference between
<span class="math-container">$$
\max_L F( A\cdot L)
$$</span>
and
<span class="math-container">$$
\max_{AL} F( A\cdot L)
$$</span>
if <span class="math-container">$A$</span> is constant.</p>

CC BY-SA 4.0
2024-04-12T05:30:47.610Z
I'm not assuming $A$ to be constant. It's a function of time $(t)$, so something like $A(t) = A_0(1+\epsilon)^t$ may also be possible. I think I should have written the notation more clearly.
uninterestedacademic
CC BY-SA 4.0
2024-04-12T05:32:13.140Z
Yes, I understand that, but in your maximization problem it is constant, right? You maximize in each period?
Giskard
CC BY-SA 4.0
2024-04-12T05:33:43.103Z
To put it differently: the value of $A$ is not affected by your choice of the decision parameters $K$ and $L$? (Technological progress is exogenous.)
Giskard
CC BY-SA 4.0
2024-04-12T05:35:53.850Z
Yes, that's correct. It's only a function of time and not labour or capital. I now see what you mean. Thank you.
uninterestedacademic
CC BY-SA 4.0
2024-04-12T05:36:04.593Z
The reason I posted this is because our instructor told us that the growth of capital income and labour income are same in the steady state (assuming constant technological and population growth). While calculating that, he took labour income as $wAL$ and its growth as $\frac{\dot{\overline{wAL}}}{wAL}$. That's what led to my questions. He eventually showed that they're both are equal to $\frac{\dot{A}}{A} + \frac{\dot{L}}{L}$. (Thought I'll write the context.)
uninterestedacademic
CC BY-SA 4.0
2024-04-12T05:41:40.867Z
Now we can't say $\frac{\dot{\overline{wAL}}}{wAL} = \frac{\dot{\overline{wL}}}{wL}$ since $A$ is a function of time and hence can not be pulled out of $\dot{\overline{wAL}}$.
uninterestedacademic
CC BY-SA 4.0
2024-04-12T05:22:01.070Z
2024-04-12T05:22:01.070Z
Giskard
<p>Consider the Solow model with technological progress. Suppose the production function is given by <span class="math-container">$F(K(t),A(t)L(t))$</span>. The variables used are standard.</p>
<p><strong>Questions:</strong></p>
<ol>
<li>Assuming the output price is normalized to <span class="math-container">$1$</span>, is the maximization problem <span class="math-container">$\displaystyle \max_{K,L} [F(K,AL) - wL - rK]$</span> or <span class="math-container">$\displaystyle \max_{K,AL} [F(K,AL) - wAL - rK]$</span>?</li>
<li>Is the labour income <span class="math-container">$wL$</span> or <span class="math-container">$wAL$</span>?</li>
<li>Further, is wage rate <span class="math-container">$w = \frac{\partial F(K,AL)}{\partial L}$</span> or <span class="math-container">$w = \frac{\partial F(K,AL)}{\partial (AL)}$</span>?</li>
</ol>
<p>I believe the rental rate is <span class="math-container">$r = \frac{\partial F(K,AL)}{\partial K}$</span> but if that's not the case, please correct me.</p>

<macroeconomics><wages><solow>
Definition of wage in the Solow Model with Technological Progress
uninterestedacademic
# Definition of wage in the Solow Model with Technological Progress

(Asked by: username_0 on 2024-04-12)

Consider the Solow model with technological progress. Suppose the production function is given by $F(K(t),A(t)L(t))$. The variables used are standard.

**Questions:**

1. Assuming the output price is normalized to $1$, is the maximization problem $\displaystyle \max\_{K,L} [F(K,AL) - wL - rK]$ or $\displaystyle \max\_{K,AL} [F(K,AL) - wAL - rK]$?
2. Is the labour income $wL$ or $wAL$?
3. Further, is wage rate $w = \frac{\partial F(K,AL)}{\partial L}$ or $w = \frac{\partial F(K,AL)}{\partial (AL)}$?

I believe the rental rate is $r = \frac{\partial F(K,AL)}{\partial K}$ but if that's not the case, please correct me.

**Tags:** macroeconomics, wages, solow


---

## 1 answers

### Answer by username_1 on 2024-04-12. Score: 2 | Accepted answer

The common usage is probably

1. $\displaystyle \max\_{K,L} [F(K,AL) - wL - rK]$
2. $wL$
3. $w = \frac{\partial F(K,AL)}{\partial L}$

BUT

you could get the other answers just by redefining the unit of measurement of labor. The above answers are valid for "$w$ is the wage for a unit of labor", the other answers are valid for "$w$ is the wage for a unit of *effective* labor", so labor unit where technological progress is taken into account. The effective unit is equivalent to $1/A$ conventional units of labor. It's like saying "using machine X, a worker can now perform the work of 10 normal workers"; this new worker will probably still file 8 hour work days (conventional unit) and not 80 hour work days (effective unit).

---

Another technical aside: there is no difference between
$$
\max\_L F( A\cdot L)
$$
and
$$
\max\_{AL} F( A\cdot L)
$$
if $A$ is constant.

> **Comments:**
> **username_0**: I'm not assuming $A$ to be constant. It's a function of time $(t)$, so something like $A(t) = A_0(1+\epsilon)^t$ may also be possible. I think I should have written the notation more clearly.  
> **username_1**: Yes, I understand that, but in your maximization problem it is constant, right? You maximize in each period?  
> **username_1**: To put it differently: the value of $A$ is not affected by your choice of the decision parameters $K$ and $L$? (Technological progress is exogenous.)  
> **username_0**: Yes, that's correct. It's only a function of time and not labour or capital. I now see what you mean. Thank you.  
> **username_0**: The reason I posted this is because our instructor told us that the growth of capital income and labour income are same in the steady state (assuming constant technological and population growth). While calculating that, he took labour income as $wAL$ and its growth as $\frac{\dot{\overline{wAL}}}{wAL}$. That's what led to my questions. He eventually showed that they're both are equal to $\frac{\dot{A}}{A} + \frac{\dot{L}}{L}$. (Thought I'll write the context.)  
> **username_0**: Now we can't say $\frac{\dot{\overline{wAL}}}{wAL} = \frac{\dot{\overline{wL}}}{wL}$ since $A$ is a function of time and hence can not be pulled out of $\dot{\overline{wAL}}$.  

---

