Marginal Rate of Technical Substitution
The Marginal Rate of Technical Substitution (MRTS) is a concept that helps to understand how one input can be substituted for another in the production process while keeping the output constant. It measures the rate at which one input (like labor) can be replaced by another input (like capital) without changing the level of output.
Definition:
The MRTS is the rate at which a firm can substitute one factor of production (e.g., labor) for another factor (e.g., capital) while maintaining the same level of output. Mathematically, it is the absolute value of the slope of an isoquant curve (which shows all possible combinations of inputs that produce the same level of output).
Formula:
MRTS = Change in Capital (ΔK) / Change in Labor (ΔL)
Alternatively, it’s the ratio of the marginal products of labor and capital:
MRTS = MPL / MPK
Key Insights:
- Isoquant Curve: Just like an indifference curve in consumer theory, an isoquant shows all combinations of inputs that produce the same level of output. The MRTS is the slope of this curve.
- Substitution Between Inputs: MRTS tells us how much of one input can be substituted for another. For example, if you can reduce the amount of labor (say one worker) and still produce the same amount of output by increasing capital (say adding one machine), MRTS measures the amount of capital you need to increase to offset the decrease in labor.
- Diminishing MRTS: As more and more labor is substituted for capital, the MRTS typically diminishes. This means that to keep output constant, you need increasingly larger amounts of one input (say, labor) to replace a given amount of the other input (capital). This reflects the Law of Diminishing Marginal Returns.
Key Interpretations:
- MRTS = 1: It means that labor and capital can be perfectly substituted for each other. Adding one more unit of labor can replace exactly one unit of capital.
- MRTS > 1: The input being substituted for (e.g., labor) is more productive than the input being added (e.g., capital), so you can replace a large amount of capital with a small increase in labor.
- MRTS < 1: The input being added (e.g., labor) is less productive than the input being substituted for (e.g., capital), so you need more labor to replace less capital.
Understand MRTS using an example
Imagine a factory that produces toys using two inputs:
- Labor (L): workers who manually assemble toys
- Capital (K): machines that automate toy-making
The factory wants to keep toy production constant at 100 units per day. However, it wants flexibility in choosing how much labor and capital to use.
Step-by-Step Input Combinations
Combination A:
- 10 workers
- 5 machines
→ Output = 100 toys
Combination B:
- 12 workers
- 4 machines
→ Output = 100 toys
So, by adding 2 workers, the factory is able to reduce 1 machine and still make the same number of toys.
Calculating MRTS
The Marginal Rate of Technical Substitution (MRTS) tells us how many units of capital (machines) can be reduced when we increase labor (workers) by 1 unit, while keeping output constant.
MRTSL,K=ΔK/ΔL=−1/2=−0.5
This means:
To keep making 100 toys, the factory can give up 0.5 machines for each additional worker hired.
So if they hire 1 extra worker, they need 0.5 fewer machines to do the same job.
Interpretation
- MRTS = 0.5 implies the trade-off between inputs.
- At this point, labor is slightly less productive than capital (you need 2 workers to replace 1 machine).
- It also reflects that machines are efficient, but can be replaced to an extent by manual work.
Diminishing MRTS
Now let’s say the factory keeps adding workers:
Combination C:
- 15 workers
- 2 machines
→ Still 100 toys
This time, it takes 3 extra workers to replace 2 more machines.
MRTSL,K=−2/3≈−0.67
MRTS is decreasing → Each new worker replaces fewer machines than before.
This is called the Law of Diminishing MRTS:
As you keep increasing labor and reducing capital, labor becomes less effective at replacing capital.