Marginal Rate of Technical Transformation

The Marginal Rate of Technical Transformation is the quantity of capital (K) to give up to get an extra unit of labor (L). $$MRT = -\frac{w}{r}$$

Example

Ziggy pays a grandma (L) $100000 to bake cookies for a year, and the cost of a oven (K) for the year is $10000.

A grandma is worth `\frac{100000}{10000} = 10` ovens.

The marginal rate of technical transformation is

$$ MRTT = - 10 $$

the minus sign indicates that, to hire (+) one grandma, Ziggy can forego (-) 10 ovens.

The marginal rate of technical transformation is the slope of the cost curves:

$$ \begin{align*} C &= rK + wL \\ C &= 10000K + 100000L \\ -10000K &= 100000L - C \\ K &= - \frac{100000}{10000}L + \frac{C}{100000} \\ K &= - 10L + \frac{C}{100000} \end{align*} $$

Question

Now hiring a grandma (L) to bake cookies costs is `w = $58` and ovens cost `r = $870`.

How many grandmas can Ziggy hire if he avoids installing a new oven?

Hiring a grandma costs `\frac{58}{870} = 1 / 15` oven(s).