How production and cost curves are related, and why it matters for business decisions

The link between production and cost

Cost curves are the mirror image of production curves. Costs depend on how much input is needed for each unit of output, which is what the production function tells us, and on input prices. With input prices given, whenever productivity rises, cost per unit falls, and whenever productivity falls, cost per unit rises.

Short-run relationships

Suppose labour (L) is the only variable input, at a constant wage w, and capital is fixed.

1. Law of variable proportions: the root. As more labour is added to fixed capital, MP_L and AP_L first rise and then fall. Points 2 and 3 turn this into the U-shape of the short-run cost curves: it is the same law seen from the cost side.
2. Marginal product and marginal cost. MC = ΔTVC/ΔQ = w·ΔL/ΔQ = w/MP_L. When the marginal product of labour rises, MC falls; at maximum MP_L, MC is at its minimum; as MP_L falls, MC rises.
3. Average product and average variable cost. AVC = TVC/Q = wL/Q = w/AP_L. When the average product of labour rises, AVC falls; when AP_L reaches its maximum, AVC is at its minimum; when AP_L falls, AVC rises.
4. Total curves. Where the total product curve rises at an increasing rate, the total variable cost curve rises at a decreasing rate, and the other way round.

Diagram. Upper panel: AP_L and MP_L against labour, MP_L peaking first and cutting AP_L at AP_L's maximum. Lower panel: AVC and MC against output, MC reaching its minimum first and cutting AVC at AVC's minimum. The peaks of the upper panel line up with the troughs of the lower one.

Long-run relationship

In the long run all inputs vary, and returns to scale shape the long-run average cost curve: increasing returns give falling LAC (economies of scale), constant returns a flat LAC, and decreasing returns a rising LAC.

Relevance for business decisions

1. Deciding how much to produce. A firm expands output while price or marginal revenue exceeds marginal cost. Since MC rises once marginal product falls, knowing where productivity begins to fall tells the manager where extra output becomes expensive.
2. Hiring and overtime. If adding workers to a fixed set of machines has pushed MP_L down sharply, overtime or extra shifts raise unit costs; the firm may do better by adding machines.
3. Capacity and plant size. Persistently rising short-run costs signal that the plant is too small; the long-run relationship (returns to scale) shows whether a bigger plant would lower unit costs.
4. Pricing and shutting down. The minimum of AVC, which matches maximum AP_L, is the lowest price at which it is worth operating in the short run.
5. Productivity improvement. Training, better layout or new technology raise MP and AP, shifting MC and AVC down, which can be weighed against their cost.
6. Make or buy. Where in-house productivity is low and costs high, buying the component may be cheaper.

Example (our own). A garment exporter in Tiruppur with 50 stitching machines finds that hiring a third shift of tailors raises output by less each week; the rising marginal cost per shirt shows it is cheaper to add machines or outsource than to keep adding labour.

Conclusion

Costs and productivity move inversely: MC = w/MP and AVC = w/AP, so the law of variable proportions produces U-shaped short-run cost curves, and returns to scale shape LAC. Managers use this to choose output, staffing, plant size and prices on the basis of where productivity, and hence cost, is heading.