Isoquant Curve and Isocost Line Explained (MRTS Formula and Least-Cost Input Mix)

Isoquants and isocosts are the two curves a firm uses to pick its cheapest mix of inputs. An isoquant is a curve showing every combination of labor and capital that produces the same quantity of output, and the marginal rate of technical substitution (MRTS) along it equals MPL ÷ MPK, the slope with its minus sign dropped. An isocost line shows every input combination with the same total cost, wL + rK = C, with slope -w ÷ r, and a firm makes any target output at the lowest cost where the two touch, so MRTS = w ÷ r.

That pair of curves is the firm's version of the consumer's indifference curve and budget line. This guide builds each piece from zero, derives the MRTS formula, and then works one complete cost-minimization problem with every number checked.

What an isoquant shows

"Iso" means equal and "quant" means quantity, so an isoquant is an equal-output curve. Put labor (L) on the horizontal axis and capital (K) on the vertical axis. Every point on one isoquant is an input bundle that produces exactly the same output, say 100 units. A higher isoquant, further from the origin, stands for a larger output, and the whole family of curves is called an isoquant map.

An isoquant describes technology, not prices. It answers the engineering question: which mixes of workers and machines can make 100 units? It says nothing about which mix is cheapest. That second question needs the isocost line.

### Why isoquants slope down and bow toward the origin

Isoquants slope downward because both inputs are productive. If the firm removes some capital, it has to add labor to keep output at 100.

They bow toward the origin because substitution gets harder the further you push it. When a firm has lots of machines and few workers, one more worker is very valuable and can replace many machines. When it already has many workers and few machines, one more worker adds little, so it can replace only a few machines. The rate of substitution falls as you slide down and to the right, which gives the curve its convex shape.

The MRTS formula

The marginal rate of technical substitution is the amount of capital a firm can give up for one more unit of labor while output stays constant. Measured from the curve:

MRTS = -(ΔK ÷ ΔL)

The minus sign turns the negative slope into a positive number. The same quantity can be written with marginal products. Along an isoquant, the output lost from cutting capital must exactly offset the output gained from adding labor:

MPL × ΔL + MPK × ΔK = 0

Rearranging gives -(ΔK ÷ ΔL) = MPL ÷ MPK, so:

MRTS = MPL ÷ MPK

If one more worker adds 12 units of output and one more machine adds 4 units, MRTS = 12 ÷ 4 = 3. One worker can replace 3 machines without changing output. Because MPL falls as labor rises and MPK rises as capital becomes scarce, MRTS falls along the curve. That is the algebra behind the bowed shape. The isoquant MRTS calculator walks through more examples of both methods.

The isocost line

An isocost line shows every combination of labor and capital that costs the same total amount. With wage w, rental rate of capital r and total cost C:

wL + rK = C

Solving for K gives K = C ÷ r - (w ÷ r) × L, so the line has:

A higher cost C shifts the line out in parallel. A change in w or r rotates it. The isocost line calculator and the isocost line glossary entry cover the shifts in more detail. The isoquant vs isocost line comparison sets the two curves side by side.

The cost-minimizing input mix

The firm has a target output and wants to hit it as cheaply as possible. Picture the required isoquant fixed in place and a family of parallel isocost lines. The firm slides toward the lowest isocost line that still touches the isoquant. At that point the line is tangent to the curve, so their slopes are equal:

MRTS = w ÷ r, which is MPL ÷ MPK = w ÷ r

### The least-cost rule

Cross-multiply that condition and you get the per-dollar version:

MPL ÷ w = MPK ÷ r

This is the least-cost rule: the last dollar spent on labor must buy the same extra output as the last dollar spent on capital. Many AP Microeconomics courses teach it in the factor markets unit without the diagram, and the least-cost input combination calculator drills it with numbers. If MPL ÷ w is larger, labor is the better deal per dollar, so the firm should hire more labor and rent less capital until the two ratios meet.

Worked example: finding the least-cost mix

A firm produces with Q = 5 × L^0.5 × K^0.5 and must make Q = 100 units. The wage is w = $20 and capital rents for r = $5.

Step 1: Write the isoquant. Set 5 × √(L × K) = 100. Divide by 5: √(L × K) = 20. Square both sides: L × K = 400. Every bundle whose product is 400 makes 100 units.

Step 2: Find MRTS. For this function MPL = 0.5 × Q ÷ L and MPK = 0.5 × Q ÷ K (the Cobb-Douglas guide derives these). So MRTS = MPL ÷ MPK = (0.5Q ÷ L) ÷ (0.5Q ÷ K) = K ÷ L.

Step 3: Set MRTS = w ÷ r. w ÷ r = 20 ÷ 5 = 4, so K ÷ L = 4, or K = 4L.

Step 4: Substitute into the isoquant. L × 4L = 400, so 4L² = 400, L² = 100 and L = 10. Then K = 4 × 10 = 40.

Step 5: Compute the cost. C = 20 × 10 + 5 × 40 = 200 + 200 = $400.

Check 1, output: 5 × √(10 × 40) = 5 × √400 = 5 × 20 = 100. Correct.

Check 2, the least-cost rule: MPL = 0.5 × 100 ÷ 10 = 5 units and MPK = 0.5 × 100 ÷ 40 = 1.25 units. MPL ÷ w = 5 ÷ 20 = 0.25 and MPK ÷ r = 1.25 ÷ 5 = 0.25. Equal, so no dollar can be moved to raise output. MRTS = 5 ÷ 1.25 = 4 = w ÷ r.

Now compare other bundles on the same 100-unit isoquant:

Every row makes 100 units, but only (10, 40) costs $400. The MRTS column falls down the table, which is the diminishing MRTS that bows the isoquant. Moving from (8, 50) to (10, 40) gives an arc MRTS of -(-10 ÷ 2) = 5, which sits between the point values 6.25 and 4, as it should.

The isocost line through the optimum is 20L + 5K = 400. Its intercepts are 400 ÷ 20 = 20 workers and 400 ÷ 5 = 80 machines, and its slope is -4. Bundles like (20, 20) lie on the higher $500 line.

### Reading a bundle that is not optimal

At (20, 20), MPL = 0.5 × 100 ÷ 20 = 2.5 and MPK = 0.5 × 100 ÷ 20 = 2.5. Per dollar, labor gives 2.5 ÷ 20 = 0.125 units while capital gives 2.5 ÷ 5 = 0.5 units. Capital is four times the better deal, so the firm should swap workers for machines, moving up the isoquant toward (10, 40).

What happens when an input price changes

Suppose the wage falls to $5 while r stays at $5. Now w ÷ r = 1, so K = L. Substituting: L × L = 400, so L = 20 and K = 20. Cost = 5 × 20 + 5 × 20 = 100 + 100 = $200. Check output: 5 × √(20 × 20) = 5 × 20 = 100. Holding output at 100, the cheaper input gets used more (labor rises from 10 to 20) and the firm substitutes away from capital (40 down to 20). This is the substitution effect at a fixed output. In practice the lower wage also cuts costs, so the firm may produce more, and that output effect raises demand for both inputs. Which effect wins decides whether capital demand falls or rises, a link explored in factor markets.

Special isoquant shapes

• Perfect substitutes: the isoquant is a straight line with a constant MRTS. The firm usually ends at a corner, using only the input with more output per dollar.
• Perfect complements: the isoquant is L-shaped, as with one driver per truck. Extra units of one input alone add nothing, and the firm buys the inputs in fixed proportions whatever the prices.

The tangency rule MRTS = w ÷ r only applies to smooth, bowed isoquants like the one in the example.

Isoquant vs indifference curve

The diagrams look alike, so exams like to test the difference. The isoquant vs indifference curve comparison goes deeper; here is the summary.

The biggest difference is the label. An isoquant marked 200 really is twice the output of one marked 100. Indifference curve labels only rank bundles, so "twice as happy" means nothing.

Common mistakes

• Flipping the ratio. MRTS is MPL ÷ MPK when labor is on the horizontal axis, not MPK ÷ MPL. Check by asking which input is being added.
• Treating the isoquant as affordable. An isoquant shows what is technically possible. Only the isocost line involves prices.
• Forgetting the sign. The isoquant slope is negative, while MRTS is reported as a positive number.
• Comparing raw marginal products. A higher MPL does not mean "hire more labor". Divide by the price first, as in MPL ÷ w versus MPK ÷ r.
• Applying tangency to every shape. With perfect substitutes or complements the answer is a corner or a kink, not a tangency.

Once a firm knows its cheapest input mix for every output level, those minimum costs trace out its cost curves, and the production and costs module covers those curves for the short run and the long run.