How to Calculate Technical Rate of Substitution

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Technical Rate of Substitution Calculator .trs-calculator-wrapper { max-width: 800px; margin: 0 auto; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; line-height: 1.6; color: #333; } .trs-calc-box { background-color: #f8f9fa; border: 1px solid #e9ecef; border-radius: 8px; padding: 30px; margin-bottom: 40px; box-shadow: 0 4px 6px rgba(0,0,0,0.05); } .trs-calc-title { text-align: center; margin-bottom: 25px; color: #2c3e50; font-size: 24px; font-weight: 700; } .trs-input-group { margin-bottom: 20px; } .trs-input-label { display: block; margin-bottom: 8px; font-weight: 600; color: #495057; } .trs-input-field { width: 100%; padding: 12px; border: 1px solid #ced4da; border-radius: 4px; font-size: 16px; box-sizing: border-box; transition: border-color 0.15s ease-in-out; } .trs-input-field:focus { border-color: #007bff; outline: 0; box-shadow: 0 0 0 0.2rem rgba(0,123,255,.25); } .trs-btn { display: block; width: 100%; padding: 14px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 18px; font-weight: 600; cursor: pointer; transition: background-color 0.2s; margin-top: 20px; } .trs-btn:hover { background-color: #0056b3; } .trs-result-box { margin-top: 25px; padding: 20px; background-color: #ffffff; border-left: 5px solid #28a745; border-radius: 4px; display: none; box-shadow: 0 2px 4px rgba(0,0,0,0.05); } .trs-result-value { font-size: 32px; font-weight: 700; color: #28a745; margin-bottom: 10px; } .trs-result-text { font-size: 16px; color: #666; } .trs-article { margin-top: 50px; } .trs-article h2 { color: #2c3e50; border-bottom: 2px solid #eee; padding-bottom: 10px; margin-top: 30px; } .trs-article p { margin-bottom: 15px; font-size: 17px; } .trs-article ul { margin-bottom: 20px; padding-left: 20px; } .trs-article li { margin-bottom: 8px; } .trs-formula-box { background: #eef2f5; padding: 15px; border-radius: 4px; font-family: 'Courier New', monospace; text-align: center; font-weight: bold; margin: 20px 0; } .error-msg { color: #dc3545; font-size: 14px; margin-top: 5px; display: none; }
Technical Rate of Substitution (TRS) Calculator
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How to Calculate Technical Rate of Substitution (TRS)

The Technical Rate of Substitution (TRS), also frequently referred to as the Marginal Rate of Technical Substitution (MRTS), is a fundamental concept in microeconomics and production theory. It measures the rate at which a firm can substitute one input (such as capital) for another (such as labor) while keeping the total level of output constant.

Understanding the TRS is crucial for businesses aiming to optimize production efficiency and minimize costs. It represents the absolute value of the slope of an isoquant curve at any given point.

The TRS Formula

The technical rate of substitution is calculated by taking the ratio of the marginal products of the two inputs. In a standard production function involving Labor (L) and Capital (K), the formula is:

TRS = MPL / MPK

Where:

  • MPL (Marginal Product of Labor): The additional output generated by adding one more unit of labor, holding capital constant.
  • MPK (Marginal Product of Capital): The additional output generated by adding one more unit of capital, holding labor constant.

Alternatively, if you are looking at discrete changes along an isoquant, the formula can be expressed as the negative change in capital divided by the change in labor:

TRS = – (ΔK / ΔL)

Step-by-Step Calculation Guide

To calculate the TRS for your production process, follow these steps:

  1. Determine the Marginal Product of Labor (MPL): Calculate how many additional units of product are created when you add one worker (or hour of labor).
  2. Determine the Marginal Product of Capital (MPK): Calculate how many additional units of product are created when you add one machine (or unit of capital).
  3. Divide MPL by MPK: Perform the division to find the ratio.

Real-World Example

Imagine a furniture factory producing wooden chairs. The inputs are carpenters (Labor) and automated saws (Capital).

  • If hiring one additional carpenter allows the factory to produce 20 more chairs per day (MPL = 20).
  • If installing one additional automated saw allows the factory to produce 10 more chairs per day (MPK = 10).

Using the calculator above:

TRS = 20 / 10 = 2.0

Interpretation: The firm can substitute 2 units of Capital (saws) for every 1 unit of Labor (carpenter) added, without changing the total output of chairs. This ratio helps managers decide the most cost-effective combination of workers and machines based on their respective costs.

Why TRS Matters

The Technical Rate of Substitution is vital for achieving allocative efficiency. To minimize costs, a firm should adjust its inputs until the TRS equals the ratio of the input prices (Wage Rate / Rental Rate of Capital). If the TRS is higher than the price ratio, the firm should use more labor and less capital, and vice versa.

function calculateTRS() { // 1. Get input elements var mplInput = document.getElementById('mplInput'); var mpkInput = document.getElementById('mpkInput'); var resultBox = document.getElementById('resultBox'); var trsResult = document.getElementById('trsResult'); var trsInterpretation = document.getElementById('trsInterpretation'); var mplError = document.getElementById('mplError'); var mpkError = document.getElementById('mpkError'); // 2. Parse values var mpl = parseFloat(mplInput.value); var mpk = parseFloat(mpkInput.value); // 3. Reset error states mplError.style.display = 'none'; mpkError.style.display = 'none'; resultBox.style.display = 'none'; mplInput.style.borderColor = '#ced4da'; mpkInput.style.borderColor = '#ced4da'; var hasError = false; // 4. Validate inputs if (isNaN(mpl)) { mplError.style.display = 'block'; mplInput.style.borderColor = '#dc3545'; hasError = true; } if (isNaN(mpk) || mpk === 0) { mpkError.style.display = 'block'; mpkInput.style.borderColor = '#dc3545'; hasError = true; } if (hasError) { return; } // 5. Calculate TRS formula: TRS = MPL / MPK var trsValue = mpl / mpk; // 6. Round to 2 decimal places for display var displayValue = trsValue.toFixed(2); // 7. Update DOM trsResult.innerHTML = displayValue; // Generate dynamic interpretation text var interpretation = "To maintain the same output level, you can give up " + displayValue + " units of Capital for every 1 unit of Labor added."; trsInterpretation.innerHTML = interpretation; resultBox.style.display = 'block'; }

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