Ratio of Marginal Products (MPL / MPK)
Substitution of Inputs (ΔK / ΔL)
The additional output generated by adding one unit of labor.
The additional output generated by adding one unit of capital.
How much capital is reduced while keeping output constant?
How much labor is added to compensate for the lost capital?
Marginal Rate of Technical Substitution:
0.00
How to Calculate the Marginal Rate of Technical Substitution
The Marginal Rate of Technical Substitution (MRTS) is a fundamental concept in microeconomics that describes the rate at which a firm can substitute one factor of production (usually capital) for another (usually labor) while maintaining the same level of output. It represents the absolute value of the slope of an isoquant curve.
Understanding the Formula
There are two primary ways to calculate MRTS, depending on the data available. The calculator above supports both methods.
MPL (Marginal Product of Labor): The extra output produced by one additional unit of labor.
MPK (Marginal Product of Capital): The extra output produced by one additional unit of capital.
This formula assumes that the gain in output from extra labor matches exactly the loss in output from reduced capital to stay on the same isoquant.
Method 2: Using Changes in Inputs
MRTS(L, K) = – (ΔK / ΔL)
Where:
ΔK (Change in Capital): The amount of capital reduced (usually a negative number).
ΔL (Change in Labor): The amount of labor increased (usually a positive number).
Note: MRTS is typically expressed as a positive number, which is why we take the absolute value or negate the ratio.
Step-by-Step Calculation Example
Imagine a furniture factory producing wooden chairs. The production manager wants to know how many machines can be replaced by workers without dropping the total daily production of chairs.
Scenario:
Marginal Product of Labor (MPL): 10 chairs per worker.
Marginal Product of Capital (MPK): 20 chairs per machine.
Calculation:
Using the Marginal Product formula:
MRTS = MPL / MPK = 10 / 20 = 0.5
Interpretation:
An MRTS of 0.5 means that 1 unit of labor can substitute for 0.5 units of capital. In simpler terms, labor is half as productive as capital at this specific point of production. To replace 1 machine (Capital), you would need 2 workers (Labor) to maintain the same output level.
Why is MRTS Important?
Calculating the MRTS helps firms minimize costs. By comparing the MRTS to the ratio of input prices (Wage Rate / Rental Rate of Capital), a firm can determine the optimal combination of labor and capital. If the MRTS is higher than the price ratio, the firm should use more labor. If it is lower, the firm should use more capital to maximize efficiency.
Diminishing Marginal Rate of Technical Substitution
As you move down an isoquant (substituting capital for labor), the MRTS typically diminishes. This means that as you add more and more labor, it becomes increasingly difficult to replace capital, requiring more workers to replace the same amount of machinery. This reflects the law of diminishing marginal returns.
function toggleInputs() {
var method = document.getElementById('calcMethod').value;
var mpInputs = document.getElementById('mpInputs');
var deltaInputs = document.getElementById('deltaInputs');
if (method === 'mp_ratio') {
mpInputs.style.display = 'block';
deltaInputs.style.display = 'none';
} else {
mpInputs.style.display = 'none';
deltaInputs.style.display = 'block';
}
// Hide results when switching modes until recalculated
document.getElementById('resultOutput').style.display = 'none';
}
function calculateMRTS() {
var method = document.getElementById('calcMethod').value;
var mrts = 0;
var explanation = "";
var error = false;
if (method === 'mp_ratio') {
// Get inputs
var mpl = parseFloat(document.getElementById('mpl').value);
var mpk = parseFloat(document.getElementById('mpk').value);
// Validation
if (isNaN(mpl) || isNaN(mpk) || mpl < 0 || mpk < 0) {
alert("Please enter valid positive numbers for Marginal Products.");
return;
}
if (mpk === 0) {
alert("Marginal Product of Capital cannot be zero (division by zero).");
return;
}
// Calculation: MRTS = MPL / MPK
mrts = mpl / mpk;
explanation = "At this point, 1 unit of Labor can replace " + mrts.toFixed(2) + " units of Capital while keeping output constant. Formula: MPL (" + mpl + ") / MPK (" + mpk + ")";
} else {
// Get inputs
var deltaK = parseFloat(document.getElementById('deltaK').value);
var deltaL = parseFloat(document.getElementById('deltaL').value);
// Validation
if (isNaN(deltaK) || isNaN(deltaL)) {
alert("Please enter valid numbers for the changes in inputs.");
return;
}
if (deltaL === 0) {
alert("Change in Labor cannot be zero.");
return;
}
// Calculation: MRTS = |Delta K / Delta L|
// We use Math.abs because users might enter Delta K as negative or positive
// but MRTS is expressed as the magnitude of the trade-off.
mrts = Math.abs(deltaK / deltaL);
explanation = "You are exchanging " + Math.abs(deltaK) + " units of Capital for " + Math.abs(deltaL) + " units of Labor. The rate of substitution is " + mrts.toFixed(2) + ".";
}
// Display Results
document.getElementById('mrtsValue').innerHTML = mrts.toFixed(4);
document.getElementById('mrtsText').innerHTML = explanation;
document.getElementById('resultOutput').style.display = 'block';
}