What Is the Marginal Rate of Technical Substitution (MRTS)?
The marginal rate of technical substitution (MRTS) measures how much of one production input (usually capital, K) a firm can give up if it adds one more unit of another input (usually labor, L), while keeping the same level of output. In short: it quantifies the trade‑off between inputs along an isoquant (a curve of constant output).
Key takeaways
– MRTS = the absolute slope of an isoquant (rate of trade between inputs holding output constant).
– Algebraically: MRTS(L,K) = −ΔK/ΔL (for a small change) = MP_L / MP_K, where MP_L and MP_K are the marginal products of labor and capital.
– In cost minimization, the optimal input mix satisfies MRTS = w / r (w = wage, r = rental price of capital).
– Most isoquants are convex and MRTS typically falls as L rises (diminishing MRTS). Special cases: perfect substitutes (constant MRTS) and perfect complements (MRTS not useful except at kink).
Formula for the MRTS
– Definition: MRTS(L, K) = −ΔK / ΔL (change in capital allowed per unit change in labor, holding output constant).
– Using marginal products: MRTS(L, K) = MP_L / MP_K.
– In differential form (continuous case): MRTS = −(dK/dL)|Q = MP_L / MP_K.
How to calculate MRTS — practical steps
1. Specify the production function Q = f(L, K) and fix a target output Q0.
2. Compute marginal products:
– MP_L = ∂f/∂L
– MP_K = ∂f/∂K
3. Compute MRTS = MP_L / MP_K. Interpret as the amount of K forgone per one additional unit of L (note sign convention: ΔK is negative when L increases).
4. For cost minimization, compare MRTS to the price ratio w/r:
– If MRTS > w/r, marginal product per dollar is higher for labor → hire more labor and reduce capital.
– If MRTS w/r → increase labor, reduce capital.
– If MP_L/MP_K < w/r → increase capital, reduce labor.
6. Check feasibility constraints (discrete inputs, capacity, adjustment costs).
7. Recompute after changes (MRTS generally changes as L and K change).
Limitations and caveats
– MRTS assumes divisibility and continuous adjustment of inputs — not always realistic.
– Measurement: estimating marginal products can be noisy; functional form assumptions matter.
– Adjustment costs, contracts, and short‑run irreversibility can prevent immediate movement to the MRTS = w/r condition.
– Technology changes, scale economies, and complementarities can alter MRTS across ranges.
The bottom line
MRTS is a central concept in production theory: it links technology (marginal products) to substitution possibilities between inputs and, combined with input prices, yields the cost‑minimizing input mix. Practically, firms use MRTS to assess how flexible their production is to input price changes and to guide input allocation — recognizing real‑world frictions and measurement challenges.
Source
– Investopedia, “Marginal Rate of Technical Substitution (MRTS)” — https://www.investopedia.com/terms/m/marginal-rate-technical-substitution.asp (article summary and definitions used here).