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The Marginal Rate of Technical Substitution

Learn how to calculate and interpret the marginal rate of technical substitution from production functions and isoquants while keeping output constant.

Intermediate 10 min read

What You'll Learn

  • Define the marginal rate of technical substitution for labor and capital at a fixed output level.
  • Derive its relationship to marginal products and the slope of an isoquant.
  • Calculate the rate from common production functions and verify it using an isoquant equation.
  • Interpret the rate’s units and explain why it is a local, not necessarily finite, trade-off.
  • Identify ratio reversals and other common errors in MRTS calculations.

Why the Marginal Rate of Technical Substitution Matters

A firm can often produce a given output using different combinations of labor and capital. An isoquant represents combinations that produce the same output. The marginal rate of technical substitution (MRTS) describes the local rate at which one input can replace another while output stays fixed. It is a property of the production technology at a particular input combination—not a price ratio or a cost-minimizing rule.

This tutorial uses labor \(L\) and capital \(K\), and defines the MRTS as the amount of capital that can be reduced for a small increase in labor. Stating the order of inputs matters: reversing the inputs gives a reciprocal rate. The MRTS is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: At a differentiable point on an isoquant, the marginal rate of technical substitution of labor for capital is the amount by which capital can be reduced per unit increase in labor, locally, while output remains constant. With this convention, \(\mathrm{MRTS}_{LK}=-dK/dL\) along the isoquant. When both marginal products are positive, \(\mathrm{MRTS}_{LK}=MP_L/MP_K\), where \(MP_L\) and \(MP_K\) are the marginal products of labor and capital.

Marginal Products and the Isoquant Slope

For a production function \(q=f(L,K)\), the marginal product of labor is the additional output from a small increase in labor holding capital fixed. The marginal product of capital is defined in the corresponding way, holding labor fixed. In calculus notation, \(MP_L=\partial q/\partial L\) and \(MP_K=\partial q/\partial K\). These measures describe changes in output, whereas the MRTS describes a trade-off between inputs.

To calculate the MRTS from marginal products, assume the production function is differentiable at the input combination and consider a smooth portion of an isoquant. Along that isoquant, output does not change. The total differential of output is therefore zero:

$$ dq=MP_L\,dL+MP_K\,dK=0. $$

If \(MP_K\ne 0\), rearranging gives the slope of capital with respect to labor along the isoquant. The minus sign in the MRTS definition makes the rate positive when the isoquant slopes downward.

$$ \left.\frac{dK}{dL}\right|_{q}=-\frac{MP_L}{MP_K}, \qquad \mathrm{MRTS}_{LK}=-\left.\frac{dK}{dL}\right|_{q}=\frac{MP_L}{MP_K}. $$

This derivation explains why the marginal products appear in a ratio: an extra small amount of labor raises output by approximately \(MP_L\,dL\), while a small reduction in capital changes output by approximately \(MP_K\,dK\). To keep the two effects offsetting, their sum must be zero. The ratio is local because the marginal products, and thus the slope, can change as the input combination changes.

The formula requires care at boundary or nonsmooth points. If \(MP_K=0\), the ratio is not defined by ordinary division. If marginal products are not positive, the ratio may not have the usual interpretation as a positive amount of one productive input replacing another. At a kink, a single derivative-based slope may not exist. In such cases, inspect the production function or isoquant directly rather than applying the ratio mechanically.

Calculating MRTS from a Production Function

Worked Example: Checking Marginal Products Against an Isoquant

Suppose a workshop’s hypothetical production function is \(q=L^{1/2}K^{1/2}\). At \(L=16\) and \(K=9\), find output, both marginal products, and the MRTS of labor for capital. Verify the answer using the equation of the isoquant. State what the MRTS means locally.

Method and output. Substitute the input amounts into the production function:

$$ q=\sqrt{16}\sqrt{9}=4(3)=12. $$

Marginal products. Differentiate the production function with respect to each input:

$$ MP_L=\frac{1}{2}L^{-1/2}K^{1/2} =\frac{1}{2}\frac{\sqrt{9}}{\sqrt{16}} =\frac{1}{2}\frac{3}{4} =\frac{3}{8}, $$
$$ MP_K=\frac{1}{2}L^{1/2}K^{-1/2} =\frac{1}{2}\frac{\sqrt{16}}{\sqrt{9}} =\frac{1}{2}\frac{4}{3} =\frac{2}{3}. $$

MRTS. Divide the marginal product of labor by the marginal product of capital:

$$ \mathrm{MRTS}_{LK} =\frac{MP_L}{MP_K} =\frac{3/8}{2/3} =\frac{3}{8}\frac{3}{2} =\frac{9}{16} =0.5625. $$

Isoquant check. Holding output at 12 means \(\sqrt{LK}=12\), so \(LK=144\) and \(K=144/L\). Differentiating this expression gives \(dK/dL=-144/L^2\). At \(L=16\), the slope is \(-144/256=-9/16\), so the MRTS is \(9/16\), as calculated from the marginal products.

At this point, one additional unit of labor can locally replace about \(0.5625\) units of capital while maintaining output. This is a local approximation, not a promise that the same trade-off applies for a large change in labor. The method also avoids a common mistake: reversing the marginal products would give \(16/9\), which is the reciprocal rate, not the defined \(\mathrm{MRTS}_{LK}\).

Using an Isoquant Equation Directly

When an isoquant can be rearranged to express one input as a function of the other, its derivative gives the slope directly. The negative of that slope is the MRTS under the convention used here. This is also a useful check on a ratio of marginal products: both methods should agree at the same point.

Worked Example: Differentiating a Fixed-Output Relationship

A hypothetical production process has \(q=L^{2/3}K^{1/3}\). At \(L=8\) and \(K=27\), calculate the MRTS of labor for capital in two ways: from marginal products and from the isoquant equation. Include the output level and interpret the result. The method avoids relying on a remembered exponent formula without checking it.

Output at the specified inputs.

$$ q=8^{2/3}27^{1/3}=4(3)=12. $$

Marginal-product method. Differentiate with respect to each input and substitute \(L=8\), \(K=27\):

$$ MP_L=\frac{2}{3}L^{-1/3}K^{1/3} =\frac{2}{3}\left(\frac{1}{2}\right)(3)=1, $$
$$ MP_K=\frac{1}{3}L^{2/3}K^{-2/3} =\frac{1}{3}(4)\left(\frac{1}{9}\right) =\frac{4}{27}. $$

Therefore,

$$ \mathrm{MRTS}_{LK}=\frac{1}{4/27}=\frac{27}{4}=6.75. $$

Isoquant check. On the isoquant for output 12, cube both sides of \(L^{2/3}K^{1/3}=12\), obtaining \(L^2K=1728\). Thus \(K=1728/L^2\), and

$$ \left.\frac{dK}{dL}\right|_q =-\frac{3456}{L^3}, \qquad \left.\frac{dK}{dL}\right|_{L=8} =-\frac{3456}{512} =-\frac{27}{4}. $$

The negative slope confirms an MRTS of \(27/4=6.75\). At this input combination, a small increase of one unit of labor permits a capital reduction of approximately 6.75 units while keeping output at 12. The word “approximately” is important: the derivative describes the slope at the stated point, not a constant rate along the whole isoquant.

A Local Rate Is Not a Finite Input Change

The MRTS is a derivative, so it describes the response to a very small input change. For a finite change, the slope may vary along the isoquant. The exact new input combination must satisfy the production function at the original output level; multiplying the initial MRTS by a large labor change need not give the exact capital reduction.

Worked Example: Comparing a Local Estimate With an Exact Change

A hypothetical production function is \(q=L^2K\). Initially, \(L=4\) and \(K=9\). Find the output and the MRTS of labor for capital at the initial point. Then calculate the exact capital required if labor rises from 4 to 5 while output stays unchanged. Compare the exact reduction with the local estimate. This method avoids treating a point slope as constant over a finite interval.

Initial output.

$$ q=4^2(9)=16(9)=144. $$

Marginal products and local MRTS. Differentiating gives \(MP_L=2LK\) and \(MP_K=L^2\). At the initial input combination,

$$ MP_L=2(4)(9)=72, \qquad MP_K=4^2=16, \qquad \mathrm{MRTS}_{LK}=\frac{72}{16}=\frac{9}{2}=4.5. $$

Equivalently, on the isoquant \(L^2K=144\), \(K=144/L^2\), so \(dK/dL=-288/L^3\). At \(L=4\), this slope is \(-288/64=-4.5\), agreeing with the marginal-product calculation.

Exact adjustment. After labor rises to 5, keep output at 144 and solve for capital:

$$ 5^2K_{\text{new}}=144, \qquad K_{\text{new}}=\frac{144}{25}=5.76. $$

Capital falls from 9 to 5.76, an exact reduction of \(9-5.76=3.24\) units. The initial MRTS gives a local estimate of \(4.5(5-4)=4.5\) units of capital. That estimate differs from the exact reduction because the slope changes as the input combination moves along the isoquant.

The exact check is that \(5^2(5.76)=25(5.76)=144\), so output is unchanged. The local estimate is useful for a small change, but it must not be presented as the exact answer for a finite change.

Interpreting and Checking the Rate

Under the stated convention, \(\mathrm{MRTS}_{LK}\) is measured in units of capital per unit of labor. Its value depends on both the production function and the input combination. It does not, by itself, say which input combination a firm should choose. That choice requires additional information, such as input prices and a cost objective.

A reliable calculation can be checked in three ways:

1
Fix the output level.
Confirm that the input change being considered stays on the same isoquant.
2
Use the stated input order.
For labor replacing capital, calculate \(MP_L/MP_K\) and take the negative of the slope \(dK/dL\).
3
Check the sign and units.
A downward-sloping isoquant has a negative slope, while the defined MRTS is positive when both marginal products are positive. Its units are capital per labor.

A common exam trap is to report the absolute isoquant slope as though it were the slope itself, or to use \(MP_K/MP_L\) for \(\mathrm{MRTS}_{LK}\). For example, if \(MP_L=6\) and \(MP_K=2\), the defined MRTS is \(6/2=3\), and the isoquant slope is \(-3\). The reversed ratio \(2/6=1/3\) answers a different question: the reciprocal trade-off. Always name which input is changing and which is being reduced.

Key takeaway: For a differentiable production function with positive marginal products, the MRTS of labor for capital is \(MP_L/MP_K\), equal to the negative slope of the isoquant. Interpret it as a local input trade-off at a specified combination, and verify its direction, sign, and units before using it.

Check Your Understanding

For each problem, state the input order clearly and distinguish local rates from exact finite changes where relevant.

  1. A production function is \(q=L^{1/2}K^{1/2}\). At \(L=25\), \(K=4\), calculate \(MP_L\), \(MP_K\), and \(\mathrm{MRTS}_{LK}\). (Answer: \(MP_L=0.2\), \(MP_K=1.25\), and \(\mathrm{MRTS}_{LK}=0.16\) units of capital per unit of labor.)
  2. A process has \(q=L^{1/3}K^{2/3}\). At \(L=8\), \(K=27\), find output and the MRTS of labor for capital. (Answer: \(q=18\) and \(\mathrm{MRTS}_{LK}=27/16=1.6875\).)
  3. For \(q=3L+K\), derive the isoquant slope and \(\mathrm{MRTS}_{LK}\). If labor rises by 2 units, how much capital must fall to maintain output? (Answer: slope \(dK/dL=-3\), \(\mathrm{MRTS}_{LK}=3\), and capital must fall by exactly 6 units.)
  4. A production function is \(q=LK\), with \(L=6\) and \(K=10\). Find the local MRTS of labor for capital and estimate the capital change for an increase of 0.5 units of labor. (Answer: \(\mathrm{MRTS}_{LK}=10/6=5/3\); the local estimate is a capital reduction of \(5/6\) unit.)
  5. For \(q=L^2K\), the initial combination is \(L=3\), \(K=8\). Calculate the initial output and local MRTS. Then find the exact capital reduction if labor increases from 3 to 4 at unchanged output. (Answer: initial output \(72\), local \(\mathrm{MRTS}_{LK}=16/3\), and exact capital reduction \(8-72/16=3.5\) units.)

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