6.3 Application I: Deriving the Slope of an Isoquant (MRTS)
Fundamentally, the isoquant illustrates a tradeoff. Suppose Chuck needs to catch 20 fish a day to survive, and he’s currently using some amount of labor and capital $(L,K)$. He might consider fashioning a stronger spear in order to have to spend a bit less time fishing, while keeping his output at 20 fish. To do so, he would need to figure out how much more capital he would need ($\Delta K$) to reduce his time fishing by some amount ($\Delta L$). The rate of additional capital needed per labor reduced, $\Delta K / \Delta L$, is called his marginal rate of technical substitution between labor and capital. (Note: Some textbooks refer to this as the “Technical Rate of Substitution.”)
Visually, the MRTS is represented by the magnitude of the slope of an isoquant:
How do we calculate the MRTS? We can use implicit function theorem to simply assert that the slope along an isoquant is given by \(\left.{dK \over dL}\right|_{f(L,K) = q} = - {\partial f/\partial L \over \partial f/\partial K}\) Since the MRTS is the magnitude of the slope, it’s therefore given by the formula \(MRTS = \left|- {\partial f/\partial L \over \partial f/\partial K}\right| = {MP_L \over MP_K}\) There’s a good economic interpretation of this formula. Let’s think about this in terms of the marginal products of labor and capital. The marginal product of labor says that the change in output due to a small change in labor is given by \(MP_L = {\Delta q \over \Delta L}\) This means that if Chuck uses $\Delta L$ less labor, the drop in his output is approximately \(\Delta q = \Delta L \times MP_L\) Think about the units here: $\Delta q$ is the change in output, so it’s measured in units of output. Likewise, $\Delta L$ is the change in labor, so it’s measured in units of labor. Finally, $MP_L$ is measured in terms of units of output per unit of labor. So if we write this with the units, we have \(\Delta q \text{ units of output} = \Delta L \text{ hours of labor} \times MP_L {\text{units of output} \over \text{hour of labor}}\) Now suppose that at the same time Chuck reduces his labor by $\Delta L$, he increases his capital by $\Delta K$. By the same argument, the amount of additional output he would get is \(\Delta q \text{ units of output} = \Delta K \text{ units of capital} \times MP_K {\text{units of output} \over \text{unit of capital}}\) Finally, let’s say that after both of these changes — the decrease in labor by $\Delta L$ and the increase in capital by $\Delta K$ — that Chuck is producing exactly the same amount of output as he was before: that is, that the net effect of this has been to move along an isoquant. In that case, the loss of quantity due to the change in labor must be exactly offset by the gain in quantity due to the change in capital: \(\Delta K \times MP_K = \Delta L \times MP_L\) Cross-multiplying gives us \({\Delta K \over \Delta L} = {MP_L \over MP_K}\) which is our definition of the MRTS.
Example 1: Linear production function
The first production function we looked at on Monday was the linear function \(q = f(L,K) = 2L + 4K\) where $L$ was hours of labor, and $K$ was the number of nets Chuck used for fishing.
The marginal products of labor in this case are constants: \(\begin{aligned} MP_L &= {df \over dL} = 2 {\text{fish} \over \text{hour}}\\ \\ MP_K &= {df \over dK} = 4 {\text{fish} \over \text{net}} \end{aligned}\) Therefore the MRTS is constant as well: \(MRTS = \frac{MP_L}{MP_K}= \frac{2 \text{ fish/hour}}{4 \text{ fish/net}} = \frac{1 \text{ net}}{2 \text{ hours}}\) Note that this is constant at all levels of $L$ and $K$: regardless of how much labor and capital are used, you can always exchange two hours of labor for one net. Put another way, one net catches four fish; to obtain the same number of fish through labor, you’d need to use two hours of labor.
The graph of this function, seen above, illustrates that the slope of every isoquant is $-1/2$.
Example 2: Cobb-Douglas production function
The second production function we looked at on Monday was the Cobb-Douglas production function \(q = f(L,K) = AL^aK^b\) The marginal products of labor and capital are given by \(\begin{aligned} MP_L &= aAL^{a - 1}K^b\\ MP_K &= bAL^aK^{b-1}\\ \end{aligned}\) This may seem convoluted, but when we take the ratio to form the MRTS it gets much simpler: \(MRTS = {MP_L \over MP_K} = {aAL^{a - 1}K^b \over bAL^aK^{b-1}} = {a \over b}\times{K \over L}\) This has some important qualities:
- The MRTS is increasing in $a$ and decreasing in $b$. Notice that $a$ is the exponent on $L$; the higher $a$ is, the more productive each unit of labor is. If labor is more productive, the MRTS is greater (i.e., the isoquant is steeper), because it would take more capital to make up the production lost by reducing a given labor input. Likewise, $b$ measures the productivity of capital: the more productive capital is, the less capital you would need to make up for a reduction in labor, so the lower the MRTS.
- The MRTS is increasing the capital/labor ratio $(K/L)$. The more capital you use relative to labor, the steeper the isoquant. One way of thinking about this is to realize that labor reinforces capital: the value of $MP_K$ is increasing in $L$. So adding a little bit of labor is extremely valuable if you have lots of capital relative to labor, and less valuable if you have lots of labor already and very little capital. Visually, this means that the isoquants are bowed in toward the origin.
The following diagram allows you to play around with the parameters of $A$, $a$, and $b$, to see how they affect the isoquant map: