Short-run and long-run expansion paths

The optimal input combination

A firm using labour (L) and capital (K) produces a given output at least cost where an isoquant is tangent to an isocost line, that is, where the marginal rate of technical substitution equals the ratio of input prices: MRTS_LK = MP_L/MP_K = w/r, with w the wage and r the price of capital. An expansion path joins such points as output rises, showing how the firm's input mix changes with its scale.

Long-run expansion path

In the long run both inputs are variable. Holding w and r constant, the firm faces a set of parallel isocost lines, higher lines meaning larger outlays.

Diagram. L on the horizontal axis, K on the vertical axis. Isoquants Q₁ = 100, Q₂ = 200 and Q₃ = 300 are tangent to isocost lines C₁, C₂ and C₃ at E₁, E₂ and E₃. The curve OE₁E₂E₃ is the long-run expansion path.

• Every point on it is a least-cost combination: at each, MRTS = w/r.
• For a homogeneous production function the path is a straight line from the origin: the firm keeps the same K/L ratio as it grows.
• Its shape shows the bias of growth: if it bends towards the L-axis, expansion is labour-using; towards the K-axis, capital-using.
• It lies within the ridge lines, where both marginal products are positive.

Short-run expansion path

In the short run capital is fixed at K̄ (the plant is given) and only labour can be varied.

Diagram. A horizontal line at height K̄ cuts the isoquants Q₁, Q₂, Q₃ at S₁, S₂, S₃. To raise output the firm can only move rightward along this line, adding labour. This line is the short-run expansion path.

• Given the fixed plant K̄, each point on the line is the cheapest way to produce that output in the short run: the firm can only vary labour, so the labour needed for the output is its best short-run choice. The short-run path thus shows the optimal combination for each output under the constraint of fixed capital.
• Only one point on it, where it crosses the long-run path (say S₂ = E₂), is also a least-cost combination when all inputs can vary. There the fixed plant happens to be the right size for that output.
• At other outputs the isoquant cuts the line instead of touching an isocost line: MRTS ≠ w/r. To the left of E₂ the firm has too much capital for its output; to the right, too little capital and too much labour. Either way the output costs more than it would with the best plant, so short-run costs exceed long-run costs except at that one output.

Distinction between the two

Conclusion

The long-run expansion path shows the cheapest combination of all inputs for each level of output, so it guides decisions on plant size and technique. The short-run path shows how output can be raised with a given plant: every point on it is the least-cost combination given that fixed capital, but only the output the plant was built for, where it crosses the long-run path, is also the least-cost choice when all inputs can vary, which is why firms plan in the long run and operate in the short run.