A foundations note — keeping the building blocks sharp.
With a fixed factor in the short run, a firm’s total cost splits in two:
TC = FC + VC(Q)fixed cost FC (independent of output — the rent on the plant) and variable cost VC(Q). At zero output TC = FC. From these comes the whole cost family:
AFC = FC/Q, AVC = VC/Q, AC = TC/Q = AFC + AVC, MC = dTC/dQ = dVC/dQTwo facts do most of the work:
- Marginal cost depends only on variable cost. Fixed (and sunk) costs never enter a marginal decision — a point people forget when they let overhead sway a per-unit choice.
- AFC falls monotonically, so the gap between
ACandAVCisAFC, and it narrows as output grows.
Why MC cuts the averages at their minima. A marginal below the average pulls the average down; a marginal above it pushes the average up. So the MC curve passes exactly through the minimum of AVC and the minimum of AC — meeting AVC’s minimum first (to the left), since AVC bottoms out before AC.
The long run is an envelope. With every input variable, there is no fixed cost, and for each level of output the firm can pick the best plant size. So the long-run average cost LRAC is the lower envelope of the family of short-run AC curves — at each output it takes whichever short-run curve is cheapest, tangent to it there. (Note: it is a curve of tangencies, not a line through the short-run minima — a classic error.)
Scale. A falling LRAC is economies of scale — specialisation, spreading fixed costs, bulk buying; a rising LRAC is diseconomies, from the cost of coordinating a larger operation. The bottom of a U-shaped LRAC is the minimum efficient scale. This concerns cost, and is related to but not the same as returns to scale, which concerns output.
Common slips: confusing MC (a derivative) with AVC (a ratio); letting fixed or sunk costs enter the marginal calculation; and drawing LRAC through the short-run cost minima instead of as their tangency envelope.