Two variable inputs: isoquants, isocosts and long-run equilibrium
An isoquant shows input mixes giving the same output. The least-cost mix is where it touches the isocost line, so MRTS = w/r.
Isoquants and MRTS
In the long run, all inputs are variable. An isoquant (equal-product curve) joins all combinations of labour (L) and capital (K) that produce the same output. Its slope is the marginal rate of technical substitution (MRTS): the amount of capital that can be replaced by one more unit of labour with output unchanged. MRTS = MPL ÷ MPK, and it diminishes along the isoquant. Like indifference curves, isoquants slope downwards, are convex to the origin, never intersect, and higher isoquants show more output. Unlike indifference curves, they carry a measurable output number (e.g. 1,000 shirts).
Worked example: the isocost line and the least-cost check
A garment firm has Rs. 100,000 a day for inputs. A worker costs w = Rs. 2,000 a day and a machine costs r = Rs. 5,000 a day. (a) Find the isocost line. (b) At its current mix, MPL = 40 units and MPK = 120 units. Is it using the least-cost mix?
- (a) Isocost: 2,000L + 5,000K = 100,000.
- Maximum L (K = 0) = 100,000 ÷ 2,000 = 50 workers. Maximum K (L = 0) = 100,000 ÷ 5,000 = 20 machines.
- Slope = −w ÷ r = −2,000 ÷ 5,000 = −0.4. One mix on the line: 25 workers and 10 machines (50,000 + 50,000 = 100,000). ✓
- (b) MPL ÷ w = 40 ÷ 2,000 = 0.02 units per rupee. MPK ÷ r = 120 ÷ 5,000 = 0.024 units per rupee.
- A rupee spent on machines adds more output (0.024 > 0.02), so the mix is not least-cost.
Isocost runs from 50 workers to 20 machines, slope −0.4. The firm should shift spending from labour to machines until MPL/w = MPK/r.
The producer's long-run equilibrium
The firm is in equilibrium where an isoquant is tangent to an isocost line. There, MRTS = w ÷ r, which is the same as MPL ÷ w = MPK ÷ r: the last rupee spent on each input adds the same output. This can be read two ways: the firm produces the maximum output for a given cost, or the minimum cost for a given output. The isoquant must also be convex at that point. If MPL/w is greater than MPK/r, the firm should use more labour; if smaller, more capital.
Expansion path and ridge lines
Returns to scale (all inputs changed together)
| Type | If all inputs double, output… | Example |
|---|---|---|
| Increasing returns to scale | More than doubles (e.g. × 2.5) | Specialisation and larger machines in a bigger factory |
| Constant returns to scale | Exactly doubles (× 2) | Building a second identical production line |
| Decreasing returns to scale | Less than doubles (e.g. × 1.5) | Management becomes hard to coordinate in a very large firm |
The firm's long-run input decision
Sri Lankan apparel makers such as MAS and Brandix must weigh labour against machines. When wages rise relative to machine costs (w/r up), the isocost line becomes steeper and the least-cost mix moves towards automation, such as automated cutting.
Short answer: “What is an isoquant?” A curve joining all combinations of two inputs (labour and capital) that produce the same level of output.
Structured question: “A firm has Rs. 100,000, w = Rs. 2,000 and r = Rs. 5,000; MPL = 40 and MPK = 120. Is it minimising cost? Explain.” MPL/w = 0.02 and MPK/r = 0.024. Not equal, so it should use more capital and less labour until the ratios are equal.
Essay: “Explain the producer's equilibrium with two variable inputs using isoquants and isocost lines, and explain the expansion path and ridge lines.” Define isoquant, MRTS, isocost; draw the tangency; state MRTS = w/r and MPL/w = MPK/r; show the expansion path and ridge lines bounding the economic region.
Isoquant = same quantity; isocost = same cost. Least cost where they kiss: MPL/w = MPK/r. Ridge lines fence the economic region.