Difficulty: Intermediate | Prerequisites: Derivatives, basic optimisation (Lagrangian helpful but not required).
This material shifts focus from the consumer to the firm. In the short run, at least one input is fixed, so the firm can only adjust output by changing its variable input, and costs are driven by the production function's shape. In the long run, all inputs are variable, and the firm chooses the cheapest combination of inputs to produce any given output level. The tools here (isoquants, isocosts, returns to scale) mirror the consumer tools (indifference curves, budget lines, utility functions) almost exactly, which is intentional. Understanding cost minimisation is a prerequisite for studying perfect competition and monopoly.
In the short run, fixed costs do not change and variable costs rise with output; the shapes of MC, AVC, and ATC are determined by the production function, especially diminishing marginal product. In the long run, the firm picks the cheapest input mix for each output level by setting the ratio of marginal products equal to the ratio of input prices, and whether average cost rises or falls with scale depends on returns to scale.
Production function
A function Q = f(L, K) that maps inputs (labour L, capital K) to the maximum output the firm can produce. Think of it as the firm's recipe.
Marginal product (MP)
The additional output from one more unit of an input, holding the other input fixed. MP_L = ∂Q/∂L. In simple terms, it is how much extra output you get from hiring one more worker.
Diminishing marginal product
The property that each additional unit of a variable input eventually produces less additional output than the previous one. This is the driving force behind the U-shape of cost curves.
Fixed cost (FC)
Costs that do not change with the level of output in the short run. Example: rent on a factory.
Variable cost (VC)
Costs that change with the level of output. Example: wages for hourly workers, raw materials.
Total cost (TC)
TC = FC + VC.
Average fixed cost (AFC)
AFC = FC / Q. Falls continuously as output rises because a fixed amount is spread over more units.
Average variable cost (AVC)
AVC = VC / Q. Typically U-shaped due to diminishing marginal product.
Average total cost (ATC)
ATC = TC / Q = AFC + AVC. Also U-shaped; the gap between ATC and AVC is AFC, which shrinks as Q rises.
Marginal cost (MC)
The additional cost of producing one more unit of output. MC = dTC/dQ = dVC/dQ. MC intersects both AVC and ATC at their minimum points.
Isoquant
A curve showing all input combinations that produce the same level of output. The firm-side equivalent of an indifference curve.
Isocost
A line showing all input combinations the firm can purchase for a given total cost. The equation is wL + rK = C, where w is the wage and r is the rental rate of capital. The firm-side equivalent of the budget constraint.
Returns to scale
How output responds when all inputs are scaled by the same factor t. If f(tL, tK) > tQ, increasing returns. If = tQ, constant returns. If < tQ, decreasing returns.
Long run average cost (LRAC)
The minimum average cost achievable when all inputs are variable. Falls under increasing returns to scale, flat under constant returns, rises under decreasing returns.
In the short run, capital (K) is fixed. The firm can only change output by adjusting labour (L).
Marginal product of labour: MP_L = ∂f(L, K-bar)/∂L. Diminishing MP_L means each additional worker adds less output than the previous one.
Cost structure:
FC = r · K-bar (does not change with Q)
VC = w · L(Q), where L(Q) is the amount of labour needed to produce Q units (the inverse of the production function)
TC = FC + VC
Relationship between MP and MC: MC = w / MP_L. When MP_L is rising, MC is falling. When MP_L is falling, MC is rising. This is why MC is U-shaped.
Key cost curve relationships:
MC intersects AVC at the minimum of AVC.
MC intersects ATC at the minimum of ATC.
ATC is always above AVC; the vertical gap is AFC, which shrinks as Q increases.
When MC < AVC, AVC is falling. When MC > AVC, AVC is rising.
In the long run, both L and K are variable. The firm chooses the input mix that minimises cost for a given output.
Cost minimisation condition: set the ratio of marginal products equal to the ratio of input prices: MP_L / MP_K = w / r. Equivalently, MP_L / w = MP_K / r (the last unit of spending on each input produces the same additional output).
Isoquant: all (L, K) pairs that produce a given Q. Slope = -MP_L / MP_K (the marginal rate of technical substitution, MRTS).
Isocost: all (L, K) pairs costing the same total. Slope = -w / r.
Optimal input mix: the tangency of the isoquant and isocost, where MRTS = w / r.
Deriving the total cost function:
From the tangency condition, express K in terms of L (or vice versa).
Substitute into the production function Q = f(L, K) to get L*(Q) and K*(Q).
Then TC(Q) = w · L*(Q) + r · K*(Q).
Returns to scale: scale all inputs by factor t.
If f(tL, tK) > t · f(L, K): increasing returns to scale (IRS). LRAC is falling.
If f(tL, tK) = t · f(L, K): constant returns to scale (CRS). LRAC is flat.
If f(tL, tK) < t · f(L, K): decreasing returns to scale (DRS). LRAC is rising.
Quick test for Cobb-Douglas f = L^a · K^b: sum the exponents. If a + b > 1, IRS. If a + b = 1, CRS. If a + b < 1, DRS.
\text{MC} = \frac{w}{MP_L} = \frac{dTC}{dQ}\text{ATC} = \frac{TC}{Q} = \text{AFC} + \text{AVC}\text{Cost minimisation: } \frac{MP_L}{MP_K} = \frac{w}{r}\text{Isocost: } wL + rK = C \quad \Rightarrow \quad K = \frac{C}{r} - \frac{w}{r}L\text{Returns to scale test: } f(tL, tK) \; \underset{\text{CRS}}{=} \; t \cdot f(L,K)Students often confuse diminishing marginal product (a short-run concept about one input) with decreasing returns to scale (a long-run concept about scaling all inputs). They are different ideas.
Students sometimes think MC is always rising. MC is U-shaped: it falls while MP is rising, then rises when MP begins to fall.
Students sometimes forget that MC crosses AVC and ATC at their minimum points, not at their intersections with each other.
The slope of the isoquant is not w/r. The slope of the isoquant is MP_L/MP_K (the MRTS). The slope of the isocost is w/r. At the cost-minimising point, they are equal.
⚠️ 2 questions on short-run costs. Know how to derive MC, AVC, ATC from a given production function and identify their relationships.
⚠️ 3 questions on long-run costs. The main task: given a production function and input prices, derive the total cost function by finding the cost-minimising input mix.
⚠️ Returns to scale is a separate exam point. Be able to test any production function (not only Cobb-Douglas) by scaling inputs by t.
⚠️ The relationship between LRAC and returns to scale is listed explicitly. Know that IRS implies falling LRAC, CRS implies flat LRAC, DRS implies rising LRAC.
True or false: AFC eventually rises as output increases. (False: AFC = FC/Q, which falls continuously.)
Fill in the blank: MC = w / ______. (MP_L.)
True or false: if f(2L, 2K) = 3 · f(L, K), the production function exhibits constant returns to scale. (False: output more than doubled, so this is increasing returns to scale.)
True or false: at the cost-minimising input mix, MRTS = w/r. (True.)
Fill in the blank: MC intersects ATC at the ______ of ATC. (minimum.)
Q: A firm has production function Q = 10L^(1/2) with K fixed. The wage is w = 20. What is the marginal cost at Q = 50?
A: From Q = 10L^(1/2), L = Q^2/100. VC = 20 · Q^2/100 = Q^2/5. MC = dVC/dQ = 2Q/5. At Q = 50: MC = 100/5 = 20.
Q: A firm has production function Q = L^(1/2) · K^(1/2), w = 4, r = 1. Derive the total cost function.
A: Cost minimisation: MP_L/MP_K = w/r. MP_L = (1/2)L^(-1/2)K^(1/2), MP_K = (1/2)L^(1/2)K^(-1/2). Ratio = K/L = 4/1, so K = 4L. Substitute into Q = L^(1/2)(4L)^(1/2) = 2L. So L = Q/2, K = 2Q. TC = 4(Q/2) + 1(2Q) = 2Q + 2Q = 4Q.
Q: Does Q = L^(0.3) · K^(0.5) exhibit increasing, constant, or decreasing returns to scale?
A: Sum of exponents = 0.3 + 0.5 = 0.8 < 1. Decreasing returns to scale.
Short-run cost curves feed directly into the perfect competition and monopoly material (Doc 4). The firm's supply curve in perfect competition is the MC curve above AVC. The cost-minimisation problem in the long run mirrors the consumer's utility-maximisation problem: isoquants and isocosts map onto indifference curves and budget lines, with the same tangency condition at the optimum.
Production technology, total product, average product, law of diminishing returns, cost function, cost curves, short-run cost, long-run cost, economies of scale, diseconomies of scale, minimum efficient scale, input demand, factor prices, wage rate, rental rate, MRTS, marginal rate of technical substitution, expansion path, duality of production and cost.