Difficulty: Intermediate | Prerequisites: Comfort with exponents and basic differentiation. You should have read the cost functions notes (Part 1) and be familiar with Cobb-Douglas production functions from Ch. 6.
In the previous notes you worked with cost functions that were handed to you. Here, you derive the cost function yourself, starting from the firm's production technology. This is the core mechanical skill of intermediate micro: given a production function and input prices, build the cost curve.
In the short run, at least one input (typically capital) is fixed. The firm can only adjust labour. That constraint means the firm may be stuck using an inefficient input mix, and the resulting cost function reflects that inflexibility. Understanding this derivation also sets up the contrast with long-run costs, where everything is adjustable.
To get a short-run cost function, fix capital, write the production function in terms of labour alone, invert it to express labour as a function of output, and then multiply by the wage (plus fixed capital cost). Because of diminishing marginal product of labour, short-run costs rise faster than proportionally with output, producing diseconomies of scale.
Short-run production function
The production function with one or more inputs held constant. In the standard setup, capital K is fixed and only labour L can vary: q = f(L, K-bar).
Think of it as: your factory is a fixed size, and all you can do is hire more (or fewer) workers.
Diminishing marginal product of labour (diminishing MPL)
The property that each additional worker adds less output than the one before. Formally, the second derivative of production with respect to L is negative.
In simple terms, cramming more workers into the same factory eventually means they get in each other's way.
Short-run total cost (SRTC)
Total cost when at least one input is fixed: SRTC = wL + rK-bar. The rK-bar portion is fixed cost; the wL portion is variable.
Short-run average cost (SRAC or SRATC)
SRTC divided by quantity: SRAC = SRTC / q.
Short-run marginal cost (SRMC)
The derivative of SRTC with respect to q. It tells you what one more unit of output costs when capital cannot adjust.
Diseconomies of scale
When average cost rises as output increases. In the short run, this happens because the fixed input constrains production, so the firm needs disproportionately more labour (and hence cost) for each additional unit.
Production function: q = L^0.75 × K^0.25
Input prices: w = $16 (wage), r = $4 (rental rate of capital)
Short-run constraint: K is fixed at 16
Total cost = labour cost + capital cost = wL + rK = 16L + 4(16) = 16L + 64.
The 64 is fixed cost (capital cannot change). The 16L is variable cost.
Substitute K = 16 into the production function.
q = L^0.75 × 16^0.25 = L^0.75 × 2 = 2L^0.75.
This is the short-run production function: output depends only on labour.
From q = 2L^0.75, divide both sides by 2: q/2 = L^0.75.
Raise both sides to the power 4/3 (the reciprocal of 0.75): L = (q/2)^(4/3).
This tells you the minimum labour needed to produce any given q.
MPL = dq/dL = 2 × 0.75 × L^(-0.25) = 1.5L^(-0.25).
As L increases, L^(-0.25) decreases, so MPL falls.
This confirms diminishing marginal product of labour.
Intuitively: with a fixed amount of capital, each additional worker has less equipment to work with, so their contribution to output shrinks.
Substitute L = (q/2)^(4/3) into TC = 16L + 64.
SRTC = 16(q/2)^(4/3) + 64.
Simplifying: (q/2)^(4/3) = q^(4/3) / 2^(4/3). And 16 / 2^(4/3) = 16 / 2.52 (approx.) = 2^4 / 2^(4/3) = 2^(8/3).
So SRTC = 2^(8/3) × q^(4/3) + 64. (Exact form.)
The variable cost term has q raised to the power 4/3, which is greater than 1.
That means variable cost grows faster than proportionally with output.
Average cost rises with output: this is diseconomies of scale in the short run.
The source is diminishing MPL. You need disproportionately more labour for each additional unit.
SRMC = dSRTC/dq = 16 × (4/3) × (1/2)^(4/3) × q^(1/3) = (4/3) × 2^(8/3) × q^(1/3).
SRAC = SRTC / q = 2^(8/3) × q^(1/3) + 64/q.
Set SRMC = SRAC and solve for q.
The algebra is messy, but the principle is the same as in Part 1: MC crosses ATC at the minimum of ATC.
Alternatively, take the derivative of SRAC with respect to q, set it equal to zero, and solve.
Production function:
q = L^{0.75} K^{0.25}Short-run production function (K = 16):
q = 2L^{0.75}Labour demand (inverted):
L = \left(\frac{q}{2}\right)^{4/3}Short-run total cost:
SRTC = 16 \left(\frac{q}{2}\right)^{4/3} + 64 = 2^{8/3} q^{4/3} + 64Marginal product of labour:
MPL = 1.5 L^{-0.25}Short-run marginal cost:
SRMC = \frac{4}{3} \cdot 2^{8/3} \cdot q^{1/3}Diminishing marginal product is visible in nearly every business that tries to ramp up quickly without expanding its facilities. A restaurant with a fixed kitchen can hire more cooks, but eventually they are tripping over each other and the kitchen produces meals more slowly per cook. The short-run cost function captures exactly this: the more you try to squeeze out of a fixed setup, the more expensive each additional unit becomes.
This is also why firms invest in expanding capacity (buying more capital) rather than endlessly hiring. Removing the short-run constraint is the transition to the long run.
Students often confuse diminishing marginal product with negative marginal product. Diminishing MPL means each extra worker adds less than the previous one, but output is still increasing. Negative MPL (output actually falls) is a separate, more extreme condition.
A common error is forgetting to include the fixed capital cost when writing SRTC. The full cost is wL + rK, not just wL.
Students sometimes invert the production function incorrectly by forgetting to handle the exponent properly. If q = 2L^0.75, you must divide by 2 first, then raise to the power 4/3. The order matters.
Some students think diseconomies of scale in the short run means the firm is doing something wrong. It is a consequence of fixed inputs, not mismanagement.
⚠️ The full derivation (production function to cost function) is a classic exam problem. Practice the steps until you can do them without looking at notes.
⚠️ Be ready to explain diminishing MPL in plain English and connect it to rising marginal cost.
⚠️ You may be asked whether the short-run cost function exhibits economies or diseconomies of scale. Look at the exponent on q in the variable cost term: greater than 1 means diseconomies.
⚠️ Know how to find the minimum of SRAC by setting SRMC = SRAC.
True or false: In the short run, all inputs can be adjusted. (False. At least one input, typically capital, is fixed.)
Fill in the blank: Diminishing MPL means the production function is ______ in L. (concave)
If q = 2L^0.75, how many units of labour are needed to produce q = 16? (L = (16/2)^(4/3) = 8^(4/3) = 16.)
True or false: Short-run diseconomies of scale arise because the firm is poorly managed. (False. They arise from fixed inputs.)
True or false: SRTC includes both variable and fixed costs. (True.)
Q: A firm has q = L^0.75 × K^0.25 with w = $16, r = $4, and K fixed at 16. Write the short-run total cost as a function of labour.
A: SRTC = wL + rK = 16L + 4(16) = 16L + 64.
Q: What is the short-run production function when K = 16?
A: q = L^0.75 × 16^0.25 = L^0.75 × 2 = 2L^0.75.
Q: Invert the short-run production function to express L as a function of q.
A: q = 2L^0.75, so L^0.75 = q/2, so L = (q/2)^(4/3).
Q: Does this production function exhibit diminishing marginal product of labour? Explain intuitively.
A: Yes. MPL = 1.5L^(-0.25), which decreases as L rises. Intuitively, with capital fixed, each extra worker has less equipment to work with and contributes less additional output.
Q: Write the short-run total cost function as C(q).
A: Substitute L = (q/2)^(4/3) into SRTC = 16L + 64: SRTC(q) = 16(q/2)^(4/3) + 64 = 2^(8/3) × q^(4/3) + 64.
Q: Does the short-run total cost function exhibit economies or diseconomies of scale? How do you know?
A: Diseconomies of scale. The variable cost term has q raised to the 4/3 power, which is greater than 1. Average cost rises as output grows.
This derivation is the bridge between production theory (Ch. 6) and cost theory (Ch. 7). The short-run cost function you derive here is one of many possible short-run curves, each corresponding to a different level of fixed capital. The envelope of all those short-run curves gives the long-run average cost curve, which is covered in the next set of notes.
The concept of diminishing MPL will reappear when you study labour markets, where firms hire workers up to the point where the value of MPL equals the wage.
Short-run cost, short-run total cost, SRTC, SRAC, SRMC, production function, Cobb-Douglas, diminishing marginal product, MPL, marginal product of labour, labour demand, fixed capital, variable inputs, diseconomies of scale, cost derivation, invert production function, ECON 300, intermediate microeconomics, Chapter 7, University of Washington