Difficulty: Intermediate | Prerequisites: understanding of production functions, isoquants, basic optimisation
Cost minimisation is the producer's core optimisation problem: given input prices, how do you produce a target quantity at the lowest possible cost? The method depends entirely on the shape of the production function. This set of notes covers two special cases, Leontief (fixed proportions) and perfect substitutes, where the standard tangency condition (MRTS = w/r) does not apply. You should already be comfortable with Cobb-Douglas cost minimisation before working through these.
Leontief production functions (fixed proportions) require inputs in a fixed ratio, so you set the components equal, solve for each input in terms of q, and plug into the cost equation. Perfect substitute production functions let you swap one input for another entirely, so the cheapest input (comparing cost per unit of output) handles all the production.
Leontief production function (fixed proportions)
A production function of the form q = c × MIN(aL, bK), where inputs must be used in a fixed ratio. Additional units of one input without additional units of the other produce zero extra output.
Think of it as a recipe: you need exactly 2 eggs and 1 cup of flour per cake. Extra eggs without extra flour do nothing.
Perfect substitutes production function
A production function of the form q = aL + bK, where inputs can fully replace each other. Isoquants are straight lines.
In simple terms, labour and capital are interchangeable, and the firm picks whichever is cheaper per unit of output.
Total cost function, TC(q)
The minimum cost of producing q units of output, expressed as a function of q alone (after solving for the optimal input mix). TC(q) = wL*(q) + rK*(q).
Optimal input mix (L, K)**
The combination of labour and capital that produces a given quantity at the lowest total cost, given input prices w (wage) and r (rental rate of capital).
The MIN function means the firm needs L = 5K for both arguments to bind. Any excess of one input over this ratio is wasted.
Rewriting: q/2 = MIN(L, 5K). At the optimum, L = 5K, so q/2 = L and q/2 = 5K.
Solving for inputs in terms of q:
L* = q/2
K* = q/10
Deriving TC(q):
TC = wL + rK = 8(q/2) + 10(q/10) = 4q + q = 5q
TC(q) = 5q
Numerical check at q = 20:
L* = 20/2 = 10
K* = 20/10 = 2
TC = 8(10) + 10(2) = 80 + 20 = 100, which matches 5(20) = 100
The isoquants are straight lines. The firm compares the cost of producing one unit of output using each input alone:
Cost per unit of output from labour: w/3 = 5/3 per unit of q
Cost per unit of output from capital: r/9 = 12/9 = 4/3 per unit of q
Since 4/3 < 5/3, capital is cheaper per unit of output. The firm uses only capital.
Wait: look at the homework solution. The student wrote L = q/3 and K = q/9 and got TC = 3q. Let us check both corner solutions and the interior.
Corner solution using only labour: L = q/3, K = 0, TC = 5(q/3) = 5q/3
Corner solution using only capital: K = q/9, L = 0, TC = 12(q/9) = 4q/3
Since 4q/3 < 5q/3, the firm uses only capital: K* = q/9, L* = 0, TC(q) = 4q/3
The homework solution appears to use both inputs, arriving at TC = 3q. This would be correct only if the firm were forced to use a specific mix, but with perfect substitutes the cost-minimising firm always goes to a corner. The correct TC(q) = 4q/3.
At q = 15:
L* = 0
K* = 15/9 = 5/3 (approximately 1.67)
TC = 12(5/3) = 20
Note: the student's answer of L* = 5 and K* = 5/3 with TC = 3q uses both inputs, which is not cost-minimising for perfect substitutes. On an exam, always compare corner solutions.
Leontief: q = c × MIN(aL, bK)
TC_{\text{Leontief}} = w \cdot \frac{q}{ca} + r \cdot \frac{q}{cb}where at the optimum aL = bK (set both arguments equal).
Perfect substitutes: q = aL + bK
TC = \min\left(\frac{w}{a}, \frac{r}{b}\right) \cdot qThe firm uses only the input with the lower cost per unit of output (w/a vs r/b).
Leontief production is a reasonable model for assembly-line manufacturing where every product requires a fixed number of parts and a fixed amount of labour time: one car needs four tyres and roughly 20 labour-hours, and extra tyres without the labour produce nothing. Perfect substitutes arise in energy generation, where a power plant might switch between natural gas and oil depending on which fuel is cheaper per kilowatt-hour.
Students often try to use the tangency condition (MRTS = w/r) for Leontief or perfect substitute production functions. The tangency condition works only for smooth, convex isoquants (like Cobb-Douglas). Leontief isoquants have a kink; perfect substitute isoquants are linear.
With perfect substitutes, students sometimes use both inputs and call it optimal. Unless the cost per unit of output is identical for both inputs (w/a = r/b), the firm goes to a corner solution.
Students occasionally confuse the coefficients in q = 2 MIN(L, 5K) and set L = K instead of L = 5K. The rule is to set the arguments of MIN equal to each other, not the inputs.
When computing TC for Leontief, students sometimes forget to substitute both L and K back into the cost equation and only account for one input.
⚠️ Expect a question asking you to derive TC(q) for a Leontief production function. The method is mechanical: set the arguments of MIN equal, solve for L and K in terms of q, plug into wL + rK.
⚠️ For perfect substitutes, the exam will likely test whether you recognise the corner solution. Compare w/a to r/b and use the cheaper input only.
⚠️ You may be given a specific q and asked for L* and K*. This is just plugging q into the expressions you derived. Do not skip showing the general TC(q) first.
True or False: For q = 2 MIN(L, 5K), the optimal ratio is L = 5K.
True. Set L = 5K so that both arguments of MIN are equal.
Fill in the blank: With perfect substitutes q = aL + bK, the firm compares ______ to ______ and uses the input with the ______ ratio.
w/a to r/b; lower.
True or False: The tangency condition MRTS = w/r works for Leontief production functions.
False. Leontief isoquants have a kink, so MRTS is undefined at the optimal point.
If q = 3L + 9K with w = 5 and r = 12, which input does the firm use?
Capital only, because r/b = 12/9 = 4/3 < w/a = 5/3.
Q: Given q = 2 MIN(L, 5K) with w = 8 and r = 10, derive TC(q) and find L and K when q = 20.**
A: At the optimum, L = 5K. Then q/2 = L, so L* = q/2 and K* = q/10. TC = 8(q/2) + 10(q/10) = 4q + q = 5q. At q = 20: L* = 10, K* = 2, TC = 100.
Q: Given q = 3L + 9K with w = 5 and r = 12, derive TC(q). Would the firm ever use both inputs?
A: Compare cost per unit of output: w/3 = 5/3 for labour, r/9 = 12/9 = 4/3 for capital. Capital is cheaper. The firm uses only capital: K* = q/9, L* = 0, TC = (4/3)q. The firm would use both inputs only if 5/3 = 4/3, which it does not.
Q: Suppose q = MIN(4L, 2K) with w = 6 and r = 3. Derive TC(q).
A: Set 4L = 2K, so K = 2L. Then q = 4L, giving L* = q/4 and K* = q/2. TC = 6(q/4) + 3(q/2) = 3q/2 + 3q/2 = 3q.
Cost minimisation is the foundation for deriving supply curves. Once you have TC(q), you can compute marginal cost (MC = dTC/dq) and average cost (ATC = TC/q), which are the building blocks of the perfectly competitive firm's output decision covered in the next set of notes. The distinction between Leontief and perfect substitute technologies also matters for understanding factor demand: Leontief production means input demands move in lockstep, while perfect substitutes mean one input can be entirely displaced by a price change.
cost minimisation, Leontief production function, fixed proportions, perfect substitutes production, isoquant, isocost line, total cost function, TC(q), optimal input mix, L*, K*, corner solution, tangency condition, MRTS, wage rate, rental rate of capital, ECON 302, intermediate microeconomics, Penn State, McLeod