<p>Since you are interested in labour, let's assume for simplicity that the stock of capital is fixed at $\bar{K}$. Then, the optimal choice of capital and labour is given by:</p>

<p>$$\frac{L^*}{a}=\frac{\bar{K}}{b}$$</p>

<p>Therefore, optimal labour is:</p>

<p>$$L^* = \frac{a}{b}\bar{K}$$</p>

<p>The marginal product of labour depends on how <strong>actual labour</strong> relates to <strong>optimal labour</strong>:</p>

<ul>
<li><p>Case 1: $L = L^*$. In the standard Leontief diagram, with $L$ in the horizontal axis and $K$ in vertical axis, this is any point <strong>on</strong> the optimal path (which function starts at the origin and has slope $\frac{b}{a}$). In this case, $\dfrac{dQ}{dL}=0$.</p></li>
<li><p>Case 2: $L &gt; L^*$. This is when the factors' combination is <strong>below</strong> the $\frac{b}{a}$ path. In this case, $\dfrac{dQ}{dL}=0$.</p></li>
<li><p>Case 3: $L &lt; L^*$. This is when the factors' combination is <strong>above</strong> the $\frac{b}{a}$ path. The solution here depends on how far $L$ is from $L^*$:</p>

<ul>
<li><p>Case 3a: $L^*-L &gt; 1$. This is perhaps the most likely case. Here, the change in labour leaves still with low levels of labour. In this scenario, $MP_L = \dfrac{1}{a}$. This results comes from comparing output before the change in labour: $Q_0=\dfrac{L_0}{a}$ versus after the change: $Q_1=\dfrac{L_0 +\Delta L}{a}$. From here, we conclude that $\dfrac{dQ}{dL}=\dfrac{1}{a}$. Notice that in this case, there is still room for increasing output by increasing labour (i.e. we are still within Case 3).</p></li>
<li><p>Case 3b: $L^*-L = 1$. Here, the change in labour leaves us <strong>on</strong> the optimal path. The change is just as in Case 3a. The differences is that increasing labour further leaves without increases in output. We are then back to case 1.</p></li>
<li><p>Case 3c: $L^*-L &lt; 1$. Here, the change in labour is more than what we actually require. Thus, we move from being in Case 3 to being in Case 2. The change is therefore not $\dfrac{1}{a}$, but equal to $\dfrac{1}{a} \times (L^*-L)$. In other words, the change is proportional to the exact amount of labour input we need to be in the optimal path.</p></li>
</ul></li>
</ul>

<hr>

<p>A graphical example can be seen below:</p>

<p><a href="https://i.sstatic.net/g15af.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/g15af.png" alt="enter image description here"></a></p>

<p>For any labour equal or above $L_0$, further increases in $L$ does not change $q_0$. However, when we are above the optimal path (meaning $K&gt;\bar{K}$), the extra marginal worker does add to production. You can see that depending on how far this point is from the optimal, the three scenarios of Case 3 arise.</p>

<p>If interested (in my personal quest in favour of Open Economics), here is the R code of the graph:</p>

<pre><code>plot(c(0,5), c(0,5), type = "n", xlab = "Labour", ylab = "Capital", yaxt='n', xaxt='n', bty='l', mgp=c(1,1,0))
segments(0, 0, 4.7, 4.7,lwd=2)
text(4.9, 4.8, expression(frac(b,a)),cex = 1.3)
points(2, 2, type="p", pch=19, col="black", bg=NA, cex=1.5)
segments(2, 2, 2, 5,lwd=2.5)
segments(2, 2, 4.7, 2,lwd=2.5)
text(4.9, 2,  expression(q[0]),cex = 1.3)
text(2.4, 1.75, expression(paste("(",L[0],",",bar(K),")")),cex = 1.3)
points(3.5, 2, type="p", pch=19, col="black", bg=NA, cex=1.5)
text(3.5, 2.3, expression(L&gt;L[0]),cex = 1.3)
points(2, 3.5, type="p", pch=19, col="black", bg=NA, cex=1.5)
text(2.4, 3.5, expression(K&gt;bar(K)),cex = 1.3)
</code></pre>

CC BY-SA 3.0
2017-03-16T07:36:35.700Z
I assume that the $\frac{b}{a}$ path to which you are referring is the line through the origin on a graph with the L on the horizontal and K on the vertical that connects the "vertex" points of the L-shaped Leotief isoquants?
user278039
CC BY-SA 3.0
2017-03-16T09:11:33.093Z
Yes, precisely. Once I have more time I will draw a graph to make the point clearer.
luchonacho
CC BY-SA 3.0
2017-03-16T17:33:23.477Z
Since the OP stated the problem in discrete changes (and specifically of a one-unit change), perhaps it would be beneficial to enhance/clarify your answer with the needed additional condition that when $L_0< L^*$ _and_ $(L_0 + \Delta L)/a < \bar K/b$, etc... because a discrete change may push labor _above_ its optimal level.
Alecos Papadopoulos
CC BY-SA 3.0
2017-03-17T09:53:50.763Z
@AlecosPapadopoulos You are right. There could be different possibilities. Will add as soon as possible. Thanks.
luchonacho
CC BY-SA 3.0
2017-03-15T13:54:58.490Z
2017-03-17T11:41:19.503Z
2017-03-17T11:41:19.503Z
luchonacho
<p>Find marginal product of labor of a Leontief production function.  For example,
$$f(L, K) = min\{\frac{L}{a}, \frac{K}{b}\}$$</p>

<p><strong>MY ATTEMPT</strong></p>

<p>Now as I understand it the marginal product of labor, $MP_L$ of this function cannot be found in the "normal" way because this function is not differentiable.  However, by definition, marginal product of labor means the extra output when one puts in an additional unit of labor.  We assume capital stays constant.</p>

<p>Say, we produce originally $q$ units of output.</p>

<p>Thus, if $\frac{L}{a}+\Delta L &lt; \frac{K}{b}$ where we are increasing the quantity of labor, then we are on the vertical portion of the isoquant (Labor on the horizontal axis, Capital on vertical).  Thus, we would increase our output by $q+\Delta L$.</p>

<p>If $\frac{L}{a}+\Delta L&gt;\frac{K}{b}$, we would still be producing $q$ units.</p>

CC BY-SA 3.0
2017-03-15T13:24:21.427Z
This looks good. However, you are not increasing output by $q+\Delta L$. Rather, that is total production. Your MPL is just $\Delta L$
123
<production-function><leontief>
Leontief function marginal product of labor/capital
user278039
# Leontief function marginal product of labor/capital

(Asked by: username_0 on 2017-03-15)

Find marginal product of labor of a Leontief production function. For example,
$$f(L, K) = min\{\frac{L}{a}, \frac{K}{b}\}$$

**MY ATTEMPT**

Now as I understand it the marginal product of labor, $MP\_L$ of this function cannot be found in the "normal" way because this function is not differentiable. However, by definition, marginal product of labor means the extra output when one puts in an additional unit of labor. We assume capital stays constant.

Say, we produce originally $q$ units of output.

Thus, if $\frac{L}{a}+\Delta L < \frac{K}{b}$ where we are increasing the quantity of labor, then we are on the vertical portion of the isoquant (Labor on the horizontal axis, Capital on vertical). Thus, we would increase our output by $q+\Delta L$.

If $\frac{L}{a}+\Delta L>\frac{K}{b}$, we would still be producing $q$ units.

**Tags:** production-function, leontief

> **Comments:**
> **username_1**: This looks good. However, you are not increasing output by $q+\Delta L$. Rather, that is total production. Your MPL is just $\Delta L$  

---

## 1 answers

### Answer by username_2 on 2017-03-15. Score: 5 | Accepted answer

Since you are interested in labour, let's assume for simplicity that the stock of capital is fixed at $\bar{K}$. Then, the optimal choice of capital and labour is given by:

$$\frac{L^\*}{a}=\frac{\bar{K}}{b}$$

Therefore, optimal labour is:

$$L^\* = \frac{a}{b}\bar{K}$$

The marginal product of labour depends on how **actual labour** relates to **optimal labour**:

* Case 1: $L = L^\*$. In the standard Leontief diagram, with $L$ in the horizontal axis and $K$ in vertical axis, this is any point **on** the optimal path (which function starts at the origin and has slope $\frac{b}{a}$). In this case, $\dfrac{dQ}{dL}=0$.
* Case 2: $L > L^\*$. This is when the factors' combination is **below** the $\frac{b}{a}$ path. In this case, $\dfrac{dQ}{dL}=0$.
* Case 3: $L < L^\*$. This is when the factors' combination is **above** the $\frac{b}{a}$ path. The solution here depends on how far $L$ is from $L^\*$:

  + Case 3a: $L^\*-L > 1$. This is perhaps the most likely case. Here, the change in labour leaves still with low levels of labour. In this scenario, $MP\_L = \dfrac{1}{a}$. This results comes from comparing output before the change in labour: $Q\_0=\dfrac{L\_0}{a}$ versus after the change: $Q\_1=\dfrac{L\_0 +\Delta L}{a}$. From here, we conclude that $\dfrac{dQ}{dL}=\dfrac{1}{a}$. Notice that in this case, there is still room for increasing output by increasing labour (i.e. we are still within Case 3).
  + Case 3b: $L^\*-L = 1$. Here, the change in labour leaves us **on** the optimal path. The change is just as in Case 3a. The differences is that increasing labour further leaves without increases in output. We are then back to case 1.
  + Case 3c: $L^\*-L < 1$. Here, the change in labour is more than what we actually require. Thus, we move from being in Case 3 to being in Case 2. The change is therefore not $\dfrac{1}{a}$, but equal to $\dfrac{1}{a} \times (L^\*-L)$. In other words, the change is proportional to the exact amount of labour input we need to be in the optimal path.

---

A graphical example can be seen below:

[![enter image description here](https://i.sstatic.net/g15af.png)](https://i.sstatic.net/g15af.png)

For any labour equal or above $L\_0$, further increases in $L$ does not change $q\_0$. However, when we are above the optimal path (meaning $K>\bar{K}$), the extra marginal worker does add to production. You can see that depending on how far this point is from the optimal, the three scenarios of Case 3 arise.

If interested (in my personal quest in favour of Open Economics), here is the R code of the graph:

```
plot(c(0,5), c(0,5), type = "n", xlab = "Labour", ylab = "Capital", yaxt='n', xaxt='n', bty='l', mgp=c(1,1,0))
segments(0, 0, 4.7, 4.7,lwd=2)
text(4.9, 4.8, expression(frac(b,a)),cex = 1.3)
points(2, 2, type="p", pch=19, col="black", bg=NA, cex=1.5)
segments(2, 2, 2, 5,lwd=2.5)
segments(2, 2, 4.7, 2,lwd=2.5)
text(4.9, 2,  expression(q[0]),cex = 1.3)
text(2.4, 1.75, expression(paste("(",L[0],",",bar(K),")")),cex = 1.3)
points(3.5, 2, type="p", pch=19, col="black", bg=NA, cex=1.5)
text(3.5, 2.3, expression(L>L[0]),cex = 1.3)
points(2, 3.5, type="p", pch=19, col="black", bg=NA, cex=1.5)
text(2.4, 3.5, expression(K>bar(K)),cex = 1.3)

```

> **Comments:**
> **username_0**: I assume that the $\frac{b}{a}$ path to which you are referring is the line through the origin on a graph with the L on the horizontal and K on the vertical that connects the "vertex" points of the L-shaped Leotief isoquants?  
> **username_2**: Yes, precisely. Once I have more time I will draw a graph to make the point clearer.  
> **username_3**: Since the OP stated the problem in discrete changes (and specifically of a one-unit change), perhaps it would be beneficial to enhance/clarify your answer with the needed additional condition that when $L_0< L^*$ _and_ $(L_0 + \Delta L)/a < \bar K/b$, etc... because a discrete change may push labor _above_ its optimal level.  
> **username_2**: @username_3 You are right. There could be different possibilities. Will add as soon as possible. Thanks.  

---

