1. LONG-RUN COSTS
In the long-run there are no fixed inputs, and therefore no fixed costs. All costs are variable.
Another way to look at the long-run is that in the long-run a firm can choose any amount of fixed costs it wants for making short-run decisions.

2. Cost Minimization
Suppose a firm has a production function with two variable inputs, labor (L), and capital (K).
Q = f(L, K)
This production function can be represented by an isoquant map.

3. Some isoquants: Q1 < Q2 < Q3
capital
(K) Q2 Q1
labor (L)

4. Marginal Rate of Technical Substitution
The Marginal Rate of Technical Substitution of L or K is the amount of K it takes to make up for the loss of one unit of L, output constant.
MRTS (L for K) = -KL, output constant.
MRTS is (minus) the slope of an isoquant.

5. Problem
The firm is told it has to produce a particular output level, say, Q*.
The firm can buy L and K at fixed known prices,
PL and PK.
How should the firm choose L and K if it must produce Q* at minimum cost?

6. The firm's total cost is its expenditure on all of its inputs:
C = PL L + PKK
For a given value of C, the above expression is known as an "isocost" line, or "cost constraint".

7. The firm's cost contraint can be rewritten as
K = (C/ PK ) - (PL /PK ) L
and drawn on the same set of axes as the firm's isoquants.

8. The slope of the isocost line is (PL /PK )
capital
(K)
The slope of the isocost line is (PL /PK )
C"/ PK
C"/ PL
labor (L)

9. If Q* is to be produced, what's the lowest possible cost of production?
capital
(K)
C"/ PK Q*
C"/ PL
labor (L)

10. Exactly C* is the minimum cost of producing Q*.
What's the rule? How much L and K are used?
capital
(K)
C*/ PK
C"/ PK Q*
C*/ PL
C"/ PL
labor (L)

11. L* and K* are the best amounts of the inputs.
MRTS (L for K) = (PL /PK )
L* and K* are the best amounts of the inputs.
capital
(K)
C*/ PK K* Q*
C*/ PL L*
labor (L)

12. Finding the Long-run Total Cost Curve
The Long-Run Total Cost Curve shows the minimum cost of producing any output when all inputs are variable.
In this case the number of inputs is two.
We show how to find a couple of points on the firm's LRTC curve.

13. Q* and C* are one point of the firm's LRTC curve.
What's the cost of producing Q' (>Q*)?
capital
(K)
C*/ PK K* Q' Q*
C*/ PL L*
labor (L)

14. Use these points to start plotting the LRTC curve.
capital
(K)
C'/ PK
C*/ PK K* Q' Q*
C*/ PL L*
labor (L)

15. Sketch in the LRTC curve.Q* Q' Q

16. From this LRTC curve you can find the corresponding average and marginal cost curves.

17. The Long-run Average Cost Curve
The long-run average cost curve shows the minimum average cost at each output level when all inputs are variable, that is, when the firm can have any plant size it wants.
There is a relationship between the LRAC curve and the firm's set of short-run average cost curves.

18. SR and LR Average Costs
Economists use the term “plant size” to talk about having a particular amount of fixed inputs. Choosing a different amount of plant and equipment (plant size) amounts to choosing an amount of fixed costs.
Economists want you to think of fixed costs as being associated with plant and equipment. Bigger plants have larger fixed costs.

19. If each plant size is associated with a different amount of fixed costs, then each plant size for a firm will give us a different set of short-run cost curves.
Choosing a different plant size (a long-run decision) then means moving from one short-run cost curve to another.

20. Economists usually assume that plant size is infinitely divisible (variable). In the case of finely divisible plant size, the LRAC curve might look like this:
Each small U-shaped
curve is a SAC curve.
$/Q
LRAC
The LRAC
curve. Q
Average costs for a
typical firm.

21. In the preceding graph, each short-run cost curve corresponds to a particular amount of fixed inputs.
As the fixed input amount increases in the long run, you move to different SR cost curves, each one corresponding to a particular plant size.

22. Notice in the graphs of LRAC curves presented so far that the curves have been drawn to be U-shaped. That is, when output is increasing LRAC at first falls, and then eventually rises.
The overall shape of the long-run average cost curve depends on the technology of production.

23. For example, advantages implicit in large scale production (with large plants) may allow firms to produce large outputs at lower cost per unit.
On the other hand, firms may get so big that ever increasing managerial and monitoring costs may cause unit costs to rise.

24. ECONOMIES OF SCALE: When output increases, long-run average costs decline.
LRAC shows
economies of
scale here.
$/Q
LRAC Q
Average costs for a
typical pizza firm.

25. DISECONOMIES OF SCALE: When output increases, long-run average costs increase.
LRAC shows
diseconomies of
scale here.
$/Q
LRAC Q
Average costs for a
typical pizza firm.

26. For the U-shaped long-run average cost curve, there are economies of scale over small outputs, and diseconomies of scale at larger outputs.

27. Not all firms necessarily suffer from diseconomies of scale at large outputs.
When a firm has economies of scale over a range of outputs big enough to supply the total market demand, that firm is called a natural monopoly.

28. Electric power generation
Naturally monopolies have long-run average cost curves that look like this:
$/Q
LRAC Q
Electric power generation
in a local market

29. As we will see, firms in perfect competition must have U-shaped long-run average cost curves.
One conclusion from this is that only certain industries can be expected to be perfectly competitive. And a crucial factor is the technology of production, since that is what determines the shape of the long-run average cost curve.