Difficulty: Intermediate | Prerequisites: Production functions, marginal product of labour and capital
This material sits at the heart of the theory of the firm. Before tackling this, you should be comfortable with production functions, marginal products, and the idea of diminishing returns. Here you learn how a firm chooses the cheapest combination of inputs to produce a given level of output, which connects directly to the cost curves you will study next.
An isocost line shows every combination of labour and capital a firm can buy for a given budget. The firm minimises cost by choosing the input mix where the isocost line is tangent to the isoquant: at that point, the rate at which the firm can technically swap one input for another (the MRTS) equals the rate at which the market prices those inputs. If the ratios do not match, the firm can always do better by shifting towards the cheaper input.
Isocost line (isocost curve)
A line showing all combinations of two inputs (typically labour and capital) that can be purchased for a given total cost. Its slope equals the negative ratio of input prices: -w/r, where w is the wage rate and r is the rental rate of capital.
Think of it as: a budget line for the firm, just as consumers have budget constraints.
Isoquant
A curve showing all combinations of inputs that produce the same level of output.
In simple terms, this means: the firm's indifference curve, except instead of equal satisfaction, it represents equal output.
Marginal rate of technical substitution (MRTS)
The rate at which a firm can substitute one input for another while keeping output constant. Formally, MRTS = MPL / MPK (when labour is on the horizontal axis and capital is on the vertical axis).
Think of it as: how many units of capital the firm can give up per extra unit of labour without losing any output.
Cost minimisation condition
The firm minimises cost when MRTS = w/r, which is equivalent to MPL/w = MPK/r. This means the last pound spent on each input yields the same additional output.
In simple terms, this means: you are getting equal bang for your buck from every input.
Marginal product of labour (MPL)
The additional output produced by one more unit of labour, holding capital constant.
Marginal product of capital (MPK)
The additional output produced by one more unit of capital, holding labour constant.
Total cost: C = wL + rK, where w = wage, r = rental rate of capital, L = labour, K = capital.
Rearranging for K (on the vertical axis): K = C/r - (w/r)L.
The slope of the isocost line is -w/r.
Worked example: A firm spends $500 on labour (horizontal axis) and capital (vertical axis). Wage rate = $20/hour, rental rate of capital = $25/hour.
Slope = -w/r = -20/25 = -4/5
If the firm spends everything on capital: 500/25 = 20 units of K (vertical intercept).
If the firm spends everything on labour: 500/20 = 25 units of L (horizontal intercept).
At the cost-minimising point, the isocost line is tangent to the isoquant.
This requires: MRTS = w/r, or equivalently, MPL/w = MPK/r.
The expression MRTS = MPL/MPK is the definition of MRTS, not the cost minimisation condition. Exam questions test whether you can distinguish the two.
What MRTS = MPL/MPK is NOT: It is not a cost-minimising expression on its own. It simply defines MRTS. The cost-minimising condition sets MRTS equal to the input price ratio.
If a firm's MRTS is 3 (with capital on the vertical axis, labour on the horizontal axis), this means the marginal product of labour is 3 times the marginal product of capital.
The firm can give up 3 units of capital per additional unit of labour without losing output.
Worked example: A firm employs 100 workers at $10/hour and 50 units of capital at $21/hour. MPL = 3, MPK = 5.
Check: MPL/w = 3/10 = 0.30
Check: MPK/r = 5/21 = 0.238
Since MPL/w > MPK/r, the firm gets more output per pound from labour than from capital.
The firm should employ more labour and less capital to reduce costs at the current output level.
When inputs must be used in fixed proportions (e.g., one pilot per plane), the optimal capital-labour ratio does not change when relative input prices change.
There is no substitution possible, so the MRTS condition does not apply in the usual way.
Worked example: Acme Container Corporation has production function Q = 25L^0.6 K^0.4. Wage = $10/hour, rental rate = $25/hour.
MPL = 0.6 x 25 x L^(-0.4) x K^0.4 = 15 x (K/L)^0.4
MPK = 0.4 x 25 x L^0.6 x K^(-0.6) = 10 x (L/K)^0.6
Cost minimisation: MPL/w = MPK/r
(15(K/L)^0.4) / 10 = (10(L/K)^0.6) / 25
Simplifying: 1.5(K/L)^0.4 = 0.4(L/K)^0.6
Using the shortcut MRTS = w/r: (0.6/0.4)(K/L) = 10/25
1.5(K/L) = 0.4
K/L = 0.4/1.5 = 4/15
The optimal capital-labour ratio is 4/15.
An isocost line reveals the input combinations that can be purchased for a given total cost.
Shifting the isocost outward (parallel) means higher total cost. Changing the slope means a change in relative input prices.
In the long run, all costs are variable. There are no fixed costs.
In the short run, at least one input is fixed (typically capital).
Isocost line: C = wL + rK
Slope of isocost: -w/r
MRTS (definition): MRTS = MPL / MPK
Cost minimisation condition: MPL/w = MPK/r (equivalently, MRTS = w/r)
Optimal K/L ratio (Cobb-Douglas): For Q = A x L^a x K^b, the optimal ratio is K/L = (b/a)(w/r)
Cost minimisation is why firms relocate factories to countries with lower wages (changing w shifts the optimal input mix toward more labour). It also explains automation decisions: when wages rise relative to the cost of machinery, firms substitute capital for labour. Airlines, which operate with fixed proportions (one cockpit crew per aircraft), cannot substitute in this way, which is why pilot wage increases hit their costs directly.
Students often confuse the definition of MRTS (MPL/MPK) with the cost minimisation condition (MRTS = w/r). The definition is always true by construction. The condition holds only at the optimum.
Students assume that if one input is more productive, the firm should use more of it. Not necessarily. What matters is productivity per pound spent (MPL/w vs. MPK/r).
Students forget that with fixed-proportions technology, changes in relative input prices do not alter the input ratio.
Students think "all costs are variable in the long run" means the long run is always cheaper. It means the firm has more flexibility, but total cost depends on the output level.
Know the difference between the MRTS definition and the cost minimisation condition. This is a common trick question.
Be able to calculate the slope of an isocost line from input prices.
Be able to determine which direction a firm should adjust inputs when MPL/w does not equal MPK/r.
Know that tangency of isocost and isoquant means output is produced at minimum cost.
For Cobb-Douglas production functions, be ready to derive the optimal K/L ratio using the shortcut.
True or False: MRTS = MPL/MPK is the cost minimisation condition.
Fill in the blank: The slope of the isocost line equals ______.
True or False: In the long run, all costs are fixed.
Fill in the blank: A firm minimises cost when the marginal product per pound spent is ______ across all inputs.
True or False: If a firm uses fixed-proportions technology, a change in relative input prices changes the optimal input ratio.
Answers:
False. That is the definition of MRTS. The cost minimisation condition is MRTS = w/r.
-w/r (negative wage rate divided by rental rate of capital).
False. In the long run, all costs are variable.
Equal.
False. With fixed proportions, the ratio stays the same regardless of prices.
Q: A firm spends $500 on labour and capital. Wage = $20/hr, rental rate = $25/hr. What is the slope of the isocost curve?
A: -w/r = -20/25 = -4/5.
Q: A firm's MRTS is 3 (capital on vertical axis, labour on horizontal). What does this tell you about marginal products?
A: The marginal product of labour is 3 times the marginal product of capital.
Q: A firm has MPL = 3, MPK = 5, w = $10, r = $21. Should it adjust its input mix?
A: Yes. MPL/w = 0.30 > MPK/r = 0.238. The firm should use more labour and less capital.
Q: When an isocost line is tangent to an isoquant, what do we know?
A: Output is being produced at minimum cost. The firm cannot rearrange inputs to produce the same output more cheaply.
Q: With the production function Q = 25L^0.6 K^0.4, w = $10, r = $25, what is the optimal K/L ratio?
A: K/L = (0.4/0.6)(10/25) = (2/3)(2/5) = 4/15.
This material connects directly to cost curves (the next topic): the minimum-cost input combination at each output level traces out the firm's total cost curve. It also connects to the theory of consumer choice, where the analogous condition is MRS = price ratio. The logic is identical, applied to firms instead of households.
The Cobb-Douglas production function reappears in growth theory and labour economics, where the exponents represent income shares going to labour and capital.
Related Terms / Search Tags
isocost line, isocost curve, isoquant, MRTS, marginal rate of technical substitution, cost minimisation, cost minimization, MPL/w = MPK/r, input substitution, Cobb-Douglas production function, optimal input mix, capital-labour ratio, fixed proportions, perfect complements, long-run costs, variable costs, wage rate, rental rate of capital, tangency condition, producer equilibrium