How to Calculate MRTS Along an Isoquant
MRTS along an isoquant equals minus the slope, −(ΔK ÷ ΔL), and also equals MPL ÷ MPK, the capital given up for one more unit of labor.
The Isoquant MRTS formula
Calculator
Enter two points on one isoquant, the marginal products and the input prices to get MRTS and the least-cost check.
Workers used in the starting input bundle.
Machines used alongside those workers.
Workers after sliding along the curve. Output must be unchanged.
Machines the firm can drop to and still make the same output.
Extra output from one more worker at the starting point.
Extra output from one more machine at the starting point.
Price of one unit of labor.
Price of one unit of capital.
Moving along this isoquant, each extra worker replaces 3 machines with output unchanged.
- MRTS from marginal products
- 3
- Input price ratio (w ÷ r)
- 2
- Least-cost check
- Use more labor, less capital
MPL ÷ MPK gives 3, the same trade read off the marginal products instead of the slope.
The market lets the firm swap 2 machines for one worker at the same total cost.
MRTS is 3 against a price ratio of 2, and the cheapest way to make this output is where those two are equal.
How to calculate Isoquant MRTS, step by step
- 1Pick two points on the same isoquant. Both input bundles have to produce the same output, since MRTS only means something while output is held fixed.
- 2Take the change in each input. ΔL is the labor added and ΔK is the capital given up as you slide down the curve, so ΔK comes out negative.
- 3Divide and flip the sign. MRTS = −(ΔK ÷ ΔL), which turns the negative slope into a positive rate of substitution.
- 4Cross-check with the marginal products. MRTS also equals MPL ÷ MPK, so the ratio of marginal products should match the slope you measured over a small step.
- 5Compare MRTS with the input price ratio. The cheapest way to make that output has MRTS = w ÷ r. A larger MRTS means labor swaps for capital faster than it costs, so shift toward labor.
Worked example: Isoquant MRTS
A firm producing 500 units uses 10 workers and 40 machines. Sliding along the same isoquant to 12 workers, it still makes 500 units with only 34 machines. ΔL = 12 − 10 = 2 and ΔK = 34 − 40 = −6, so MRTS = −(−6 ÷ 2) = 3, meaning each extra worker replaces 3 machines. At the starting point MPL = 30 units and MPK = 10 units, so MPL ÷ MPK = 30 ÷ 10 = 3 as well. The wage is $20 and a machine rents for $10, giving w ÷ r = 20 ÷ 10 = 2. Since 3 is greater than 2, labor replaces capital faster than it costs, so the firm should use more labor and less capital to make those 500 units.
Isoquant MRTS questions
What is the difference between MRTS and MRS?
MRTS is the production version, the capital a firm gives up for one more unit of labor while output stays fixed. MRS is the consumption version, the units of one good a shopper gives up for one more of another while utility stays fixed.
Why do isoquants bow toward the origin?
MRTS falls as labor replaces capital. Each added worker has less capital to work with, so MPL falls while MPK rises, and the ratio MPL ÷ MPK shrinks as you move right along the curve.
Can two isoquants cross?
No. Each isoquant carries one output level, so a crossing point would have to produce two different quantities from the same input bundle, which is impossible.
What does a straight-line isoquant mean?
Constant MRTS, so the two inputs are perfect substitutes and the firm will use whichever one is cheaper per unit of output. A right-angled isoquant is the opposite case, perfect complements used in fixed proportions.
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