Cost-minimising choice of factor inputs

Types of Cost, Revenue & Profit

Exam Tip: Remember the key relationships:

  • Profit = Total Revenue – Total Cost

  • Average Cost (AC) = Total Cost / Quantity

  • Marginal Cost (MC) = ΔTC / ΔQ

Fixed Costs (FC) 💰

Costs that do not change with the level of output. Think of the rent for a factory or the salary of a manager.

Variable Costs (VC) 🔧

Costs that vary directly with output. For example, raw materials or hourly wages.

Total Cost (TC) 📊

\$TC = FC + VC\$

Average Cost (AC) 📈

\$AC = \frac{TC}{Q}\$

Marginal Cost (MC) ➕

\$MC = \frac{\Delta TC}{\Delta Q}\$

Total Revenue (TR) 💵

\$TR = P \times Q\$

Profit (π) 🏆

\$\pi = TR - TC\$

Short‑Run vs Long‑Run Production

AspectShort RunLong Run
Fixed InputsSome inputs are fixed (e.g., factory size)All inputs can vary
Cost BehaviourTC = FC + VC (FC cannot change)FC can change with scale
Production Function\$Q = f(L, K_{\text{fixed}})\$\$Q = f(L, K)\$

Analogy: Imagine baking a cake. In the short run you can add more eggs (variable input) but the oven size (fixed input) stays the same. In the long run you can buy a bigger oven if you want to bake more cakes.

Cost‑Minimising Choice of Factor Inputs

When a firm wants to produce a given quantity at the lowest possible cost, it must choose the right mix of labour (L) and capital (K). The rule of thumb:

Use the ratio of marginal products equal to the ratio of input prices:

\$\frac{MPL}{MPK} = \frac{w}{r}\$

  • \$MP_L\$ = marginal product of labour (additional output from one more worker)

  • \$MP_K\$ = marginal product of capital (additional output from one more unit of capital)

  • \$w\$ = wage rate (price of labour)

  • \$r\$ = rental rate of capital (price of capital)

Step‑by‑Step Example 🍕

  1. Suppose a pizza shop wants to produce 100 pizzas.

  2. Current wage is £10 per hour and rental cost of ovens is £5 per hour.

  3. Calculate \$MPL\$ and \$MPK\$ from the production function (e.g., \$Q = 10L^{0.5}K^{0.5}\$).

  4. Set up the ratio: \$\frac{MPL}{MPK} = \frac{10}{5} = 2\$

  5. Solve for the optimal L and K that satisfy this ratio while producing 100 pizzas.

  6. Check that total cost \$TC = wL + rK\$ is minimised.

Exam Tip: When given a production function, always compute the marginal products first. Then use the price ratio to find the optimal input mix. Remember to verify that the chosen combination actually produces the required output.

Short‑Run Cost Minimisation

In the short run, one input (usually capital) is fixed. The firm minimises cost by choosing the optimal amount of the variable input (labour) such that:

\$\frac{MPL}{w} = \frac{1}{ACL}\$

or simply: keep hiring workers until the marginal cost of hiring an extra worker equals the marginal revenue product of that worker.

Long‑Run Cost Minimisation

All inputs are variable. The firm chooses L and K to minimise total cost for a given output level, satisfying the ratio condition above.

Analogy: Think of a school project where you can choose how many classmates (labour) and how many calculators (capital) to use. The goal is to finish the project for the least amount of effort and money.

Key Take‑Away Points for the Exam

  • Always distinguish between fixed and variable costs.

  • Know how to calculate AC, MC, TR, and profit.

  • Understand the difference between short‑run and long‑run production.

  • Use the marginal‑product to price ratio for cost minimisation.

  • Practice deriving marginal products from common production functions.

  • Check that the chosen input mix actually meets the output target.

Final Exam Tip: In multiple‑choice questions, look for the answer that matches the condition \$\frac{MPL}{MPK} = \frac{w}{r}\$ or its equivalent. In short‑answer questions, show the steps: compute marginal products, set up the ratio, solve for inputs, and confirm cost minimisation. Good luck! 🎓