Production and Costs

From Inputs to Outputs to Costs

Production functions map inputs to output: $Q = f(K, L)$. The isoquant (analogous to indifference curves) shows all input combinations producing the same output.

Short run (capital fixed): the total product curve $TP(L)$ shows output as a function of labour alone. Marginal product $MP_L = \Delta Q / \Delta L$ eventually diminishes (law of diminishing returns).

Cost curves in the short run:

$$TC = FC + VC(Q)$$ $$AC = TC/Q = AFC + AVC$$ $$MC = \Delta TC / \Delta Q$$

Key relationships:

• $MC$ intersects $AC$ and $AVC$ at their minimum points (when marginal is below average, average falls; when above, it rises).
• $AFC = FC/Q$ always declines (spreading fixed costs).
• $MC$ first falls (increasing returns from specialisation), then rises (diminishing returns).

Long run (all inputs variable): the firm chooses the cost-minimising input mix. The isocost line $wL + rK = C$ is the firm's budget constraint. Cost minimisation requires tangency: $MRTS = w/r$ (analogous to $MRS = p_x/p_y$ in consumer theory).

$$MRTS = \frac{MP_L}{MP_K} = \frac{w}{r}$$

Returns to scale:

• Constant: doubling inputs doubles output.
• Increasing: doubling inputs more than doubles output (economies of scale).
• Decreasing: doubling inputs less than doubles output (diseconomies).

The long-run average cost (LRAC) curve is U-shaped: economies of scale at low output, constant returns in the middle, diseconomies at high output.

The cost-curve choreography

Tip: The averages-and-marginal logic is universal: when the marginal is below the average, it drags the average down; above, it pulls it up — so $MC$ must cross $AC$ and $AVC$ exactly at their minimum points. Same reason one bad exam drags your GPA down.