3.7 Appendix B: Deriving the MRT
As we discussed in the math review, the implicit function theorem says that when you have a level set defined by \(f(x,y) = z\) for some constant $z$, the equation for the slope of the level set at any given point may be given by \(\frac{dy}{dx} = -\frac{\partial f/\partial x}{\partial f/\partial y}\) We can use the same approach when developing an expression for the MRT. In general, the equation for the PPF will be something like \(f(x_1,x_2) = k\) for some constant $k$. Therefore, if you’ve derived the equation for the PPF, you can find the MRT (Note: Remember that our convention is to report the MRT as the absolute value of the slope!) as \(MRT = \left|-\frac{\partial f/\partial x_1}{\partial f/\partial x_2}\right|\) For example, in the case of a linear PPF with the equation \(\tfrac{1}{3}x_1 + \tfrac{1}{2}x_2 = 150\) the MRT is \(MRT = \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}\) In the case of the curved PPF with the equation \(\frac{x_1^2}{100} + \frac{x_2^2}{36} = 100\) the MRT is \(MRT = \frac{\frac{x_1}{50}}{\frac{x_2}{18}} = -\frac{9x_1}{25x_2}\)
Relationship to marginal products method
In the text, we said that it was also the case that \(MRT = {MP_{L2} \over MP_{L1}}\) How is this related to the implicit function method above?
Think about how we derived the equation of the PPF when there’s a single input: we started from the resource constraint \(L_1 + L_2 = \overline L\) and then inverted the production functions to find the labor required to produce $x_1$ and $x_2$ — that is, $L_1(x_1)$ and $L_2(x_2)$. We plugged this into the resource constraint to get: \(L_1(x_1) + L_2(x_2) = \overline L\) Now, the left-hand side of this may be thought of as the *labor required to produce the bundle $(x_1,x_2)$: \(L(x_1,x_2) = L_1(x_1) + L_2(x_2)\) so the equation of the PPF is just the level set of this labor requirement function: \(L(x_1,x_2) = \overline L\) Therefore, by the implicit function theorem, the slope of the PPF is \(-\frac{\partial L/\partial x_1}{\partial L/\partial x_2}\) Note that $MP_L = dq/dL$, so in fact the marginal products method yields the same result: \(\frac{MP_{L2}}{MP_{L1}} = \frac{dx_2/dL_2}{dx_1/dL_1} = \frac{dL_1(x_1)/dx_1}{dL_2(x_2)/dx_2} \equiv \frac{\partial L(x_1,x_2)/\partial x_1}{\partial L(x_1,x_2)/\partial x_2}\)
The MRT with more than one input
Astute readers will have noticed that this might seem like a special case, and ask what the MRT is when there are more than one input (and therefore more than one relevant marginal product). The answer is that it’s a bit more complicated, because we would need to solve an optimization problem to determine the best combination of inputs to produce at each point along the PPF, and we haven’t yet introduced the math to do that. However, we can note that it would remain true that for each input, the MRT would be equal to the ratio of the marginal products: for example, if we had $x_1 = f_1(L_1,K_1)$ and $x_2 = f_2(L_2,K_2)$, then we would have both \(MRT = \frac{MP_{L2}}{MP_{L1}}\) and \(MRT = \frac{MP_{K2}}{MP_{K1}}\) Importantly, note that this implies that \(\frac{MP_{L1}}{MP_{K1}} = \frac{MP_{L2}}{MP_{K2}}\) which means that at each point along the PPF, the marginal rates of technical substitution must be equal across goods: \(MRTS_1(L_1,K_1) = MRTS_2(L_2,K_2)\) To see why this must hold, go through the following thought experiment:
- Suppose Chuck was producing some amount $x_1$ of fish, using $L_1$ labor and $K_1$ capital.
- Now suppose he decided to produce the same amount of fish using one more unit of labor — that is, move along his isoquant for $x_1$ a bit to the right.
- Because of the definition of his MRTS, this would mean he would use approximately $MRTS_1$ less unit of capital.
- Because he can only produce two goods, this means he is using 1 unit less labor and $MRTS_1$ more unit of capital to produce coconuts.
- Since we are along the PPF, we know that he cannot produce more coconuts by doing this. Therefore he must be staying along his isoquant for coconuts, meaning we must have $MRTS_1 = MRTS_2$ This is a very hand-wave-y “proof,” but hopefully it suffices for this level of analysis. If you continue on in economics you’ll see a much more rigorous analysis of this!