A foundations note — keeping the building blocks sharp.
On the firm side, the analogue of the utility function is the production function: the maximum output obtainable from given inputs,
Q = f(L, K)with labour L and capital K. “Maximum” matters — it is an efficiency frontier, so the same inputs cannot yield more.
Two derived quantities:
- Marginal product, the extra output from one more unit of an input:
MP_L = ∂Q/∂L,MP_K = ∂Q/∂K. - Average product, output per unit:
AP_L = Q/L.
They are linked by a simple rule: AP_L rises while MP_L is above it and falls once MP_L drops below it, so the MP_L curve cuts AP_L at its maximum.
Diminishing marginal returns. Hold other inputs fixed and keep adding one input; eventually its marginal product falls (∂MP_L/∂L < 0). This is a short-run phenomenon, driven by the fixed factor — more workers spread over the same fixed capital. It is not the same as returns to scale, which vary all inputs together. (MP also often rises first, through specialisation, before it falls.)
The producer’s indifference-curve analogue is the isoquant — input mixes that yield the same output, typically downward-sloping and convex to the origin. Totally differentiating Q along an isoquant (dQ = MP_L·dL + MP_K·dK = 0) gives its slope, the marginal rate of technical substitution:
MRTS_LK = |dK/dL| = MP_L / MP_K— how much capital you can shed per extra unit of labour while holding output constant. Convex isoquants imply a diminishing MRTS: the more labour you already use, the less capital a bit more labour can replace.
Finally, the time horizon. In the short run at least one input is fixed (usually K = K̄), so the firm can only move vertically in L. In the long run everything adjusts, and the firm slides along isoquants to the cost-minimising input mix — the point where MRTS = w/r (the input-price ratio).
Common slips: confusing diminishing returns (one input varies) with diminishing scale returns (all inputs vary together); assuming MP falls from the very first unit; and mixing up the MRTS (a technology-side substitution) with the consumer’s MRS (a taste-side one).