Long Run function

Let’s break down long-run cost functions clearly and simply. In the long run, all inputs are variable (e.g., labor, capital), unlike the short run where at least one input is fixed. Long-run cost functions help firms decide the most efficient scale of production when they have full flexibility to adjust resources.

We’ll explain:

  1. LTC (Long-Run Total Cost)

  2. LAC (Long-Run Average Cost)

  3. LMC (Long-Run Marginal Cost)

1. LTC – Long-Run Total Cost

Definition: LTC is the total cost a firm incurs to produce any level of output in the long run when it can change all input levels (labor, machines, land, etc.).

Key Points:

  • It shows the minimum cost of producing each output level with the most efficient input combination.

  • It’s influenced by the production technology, input prices, and scale of operations.

Example: Imagine a coffee shop wants to produce 1,000 cups per day.

  • In the short run: It may have a small machine and few workers—costly and inefficient.

  • In the long run: It can upgrade machines, hire the right number of workers, rent a bigger space, and optimize everything.
    → The LTC will show the lowest possible cost of producing those 1,000 cups with full flexibility.

2. LAC – Long-Run Average Cost

Definition: LAC is the cost per unit of output in the long run. It is found by dividing LTC by output (Q):


LAC = \frac{LTC}{Q}

Key Points:

  • LAC is a U-shaped curve due to:

    • Economies of scale (costs decrease as output increases).

    • Constant returns to scale (costs stabilize).

    • Diseconomies of scale (costs increase as output gets too large).

  • It shows the optimal size of the firm.

Real Story Example: A small bakery grows over time:

  • At first, producing 100 loaves/day costs $2/loaf.

  • At 500 loaves/day, bulk buying reduces cost to $1/loaf.

  • At 2,000 loaves/day, managing so many employees increases cost to $1.50/loaf.

The lowest point on the LAC curve shows the most efficient scale—maybe 500 loaves/day.

3. LMC – Long-Run Marginal Cost

Definition: LMC is the additional cost of producing one more unit of output in the long run.


LMC = \frac{ΔLTC}{ΔQ}

Key Points:

  • It tells whether increasing output will increase or reduce cost per unit.

  • When LMC < LAC → LAC is falling.

  • When LMC > LAC → LAC is rising.

  • When LMC = LAC → LAC is at its minimum point (most efficient).

Practical Example:

  • A tech firm adds new production lines to produce more gadgets.

  • If LMC is $90 and current LAC is $100 → each new gadget reduces average cost → good to expand.

  • If LMC is $110 while LAC is $100 → expansion raises costs → not efficient.

Summary Table:

Concept

Formula

Meaning

Shape

Role

LTC

Total cost to produce Q units

Shows total cost with optimal input mix

Increasing

Total cost planning

LAC

LTC ÷ Q

Cost per unit in long run

U-shaped

Shows efficiency scale

LMC

ΔLTC ÷ ΔQ

Cost of one more unit in long run

V or upward sloping

Decision to expand

Final Insight:

In real business decisions, firms use LAC and LMC to choose:

  • What scale to operate at (small, medium, large plant)

  • When to expand or downsize

  • What mix of technology and labor to use

Analogy: Think of a farmer choosing between using donkeys, tractors, or drones. In the short run, he’s stuck with one. In the long run, he chooses the most efficient one for his scale—that’s what long-run cost functions help with.

Would you like a graph or diagram for these concepts?