# Marginal rate of technical substitution

I am a student in an intermediate microeconomics class and am having a little trouble understanding the marginal rate of technical substitution. I understand that it represents the amount that labor (capital) has to be decreased for capital (labor) to be increased and stay on the same isoquant, but I am having trouble understanding it in practice.

For instance when the MRTS = K/L and capital is 4 and labor is 1 than the MRTS is 4/1. Does that mean that mean if capital is increased by 4 units then labor has to be decreased by 1 unit to stay on the same isoquant? Or does it mean that mean if capital is decreased by 4 units then labor has to be increased by 1 unit to stay on the isoquant? Or am I way off in my understanding?

We start with a production function: $f(k,l)$, which tells us what quantity will be produced if we use $k$ units of capital and $l$ units of labour. Suppose that we cange that quantity of capital by $dk$ units and change the quantity of labour by $dl$ units. If this change is chosen so that we stay on the same isoquant then it must be that the total production doesn't change:

$$\frac{\partial f(k,l)}{\partial k}dk+\frac{\partial f(k,l)}{\partial l}dl=0.$$

Rearranging, we get

$$\frac{dk}{dl}=-\frac{\frac{\partial f(k,l)}{\partial l}}{\frac{\partial f(k,l)}{\partial k}}.$$

This is the marginal rate of technical substitution, the slope of the isoquant. It has the same interpretation as any other slope. It means that if I increase labour by one unit then I can decrease capital by $$\frac{\frac{\partial f(k,l)}{\partial l}}{\frac{\partial f(k,l)}{\partial k}}$$ units.

Example Suppose we have a Cobb-Doublas production function: $f(k,l)=k^a l^{1-a}$. We have

$$\frac{\partial f(k,l)}{\partial k}=ak^{a-1}l^{1-a}$$

$$\frac{\partial f(k,l)}{\partial l}=(1-a)k^{a}l^{-a}$$

$$MRTS=-\frac{(1-a)k^{a}l^{-a}}{ak^{a-1}l^{1-a}}=-\frac{1-a}{a}\frac{k}{l}.$$

Now we can plug some numbers in to this example to see how it works. Suppose $a=1/2$, and we currently have $k=4$ and $l=1$. Then, we have

$$MRTS=-\frac{1/2}{1/2}\frac{4}{1}=-4.$$

So, for each unit increase in the quantity of labour employed, we can decrease the quantity of capital employed by 4 units.

This works backwards too, so every unit decrease in $l$ would demand a 4 unit increase in $k$ to keep us on the same isoquant.