1. Costs
10-1

2. Drawing on Chapter 10
Original graphics & text copyright © The McGraw-Hill Companies, Inc. All rights reserved.

3. Outline Costs in the short run
Allocating production between two processes
The relationship among MP, AP, MC, and AVC
Costs in the long run
Long-run costs and the structure of industry
The relationship between long-run and short-run cost curves
10-3

4. Costs In The Short Run Suppose Output Q=F(K,L) K the fixed input
L the variable input
Their respective prices r and w
Then
Fixed cost: FC = rK0
Variable cost (VC): VCQ = wL
Total cost:
TC = FC+VCQ = rK0+wL
FC: does not vary with the level of output in the short run. (The cost of all fixed factors of production.)
VC: cost that varies with the level of output in the short run. (The cost of all variable factors of production.)
TC) all costs of production: the sum of variable cost and fixed cost.
10-4

5. Figure 10.1: Output as a Function of One Variable Input
Total product curve for Kelly’s cleaners.
10-5

6. Figure 10.2: The Total, Variable, and Fixed Cost Curves
K= 120 machine hours
r = $0.25/machine hour
Hence FC
w = $10/hour
Show calculation of one point on VC curve
Go to experiment results.
10-6

7. Figure 10.3: The Production Function Q = 3KL, with K = 4
Return from experiment results.
We can mimic this real process with relatively simple algebraic functions. This is very simple, but part of some famous models, such as the Ricardian model of international trade that you studied in principles.
10-7

8. Figure 10.4: The Total, Variable, and Fixed Cost Curves for the Production Function Q-3KL
After reviewing this ...
A somewhat more realistic case would be based on a Cobb-Douglass production function (sketch production function, VC curve, each curving in a single direction, each the opposite direction of the other).
10-8

9. Unit Costs In The Short Run
Average costs
Average fixed cost:
𝑨𝑭𝑪 𝑸 = 𝑭𝑪 𝑸
Average variable cost:
𝑨𝑽𝑪 𝑸 = 𝑽𝑪 𝑸 𝑸
Average total cost: 𝑨𝑻𝑪 𝑸 = 𝑻𝑪 𝑸 𝑸
= 𝑨𝑭𝑪 𝑸 + 𝑨𝑽𝑪 𝑸
Marginal cost
𝑴𝑪 𝑸 = ∆ 𝑽𝑪 𝑸 ∆𝑸 = ∆ 𝑻𝑪 𝑸 ∆𝑸
Average fixed cost (AFC): fixed cost divided by the quantity of output.
Average variable cost (AVC): variable cost divided by the quantity of output.
Average total cost (ATC): total cost divided by the quantity of output.
Marginal cost (MC): change in total cost that results from a 1-unit change in output.
Then back to the experiment results.
10-9

10. Figure 10.5: The Marginal, Average Total, Average Variable, and Average Fixed Cost Curves
Graphing the Short-run Average and Marginal Cost Curves
Geometrically, average variable cost at any level of output Q may be interpreted as the slope of a ray to the variable cost curve at Q.
Geometrically, MC is the slope of the TC curve and VC curve at any given level of Q.
Draw a VC curve and mark MC as its slope at two dissimilar places, with corresponding plot below of MC.
When MC is less than average cost (either ATC or AVC), the average cost curve must be decreasing with output; and when MC is greater than average cost, average cost must be increasing with output. Why?
10-10

11. Figure 10.6: Quantity vs. Average Costs
10-11

12. Figure 10.7: Cost Curves for a Specific Production Process
Again, for the simplest algebraic production function worth using, we can derive these curves. We will encounter constant MC functions in the simplest models of monopoly, etc.
10-12

13. Figure 10.8: The Minimum Cost Production Allocation Among Processes
We can also use constant MC production functions immediately.
Let QT be the total amount to be produced, and let Q1 and Q2 be the amounts produced in the first and second processes, respectively. And suppose the marginal cost in either process at very low levels of output is lower than the marginal cost at QT units of output in the other (which ensures that both processes will be used).
What will the firm do if, given current output choices, MCA<MCB? Repeat that change in allocation between processes until there is no more reason to change. How will that lead toward the minimum cost allocation?
The values of Q1and Q2 that solve this problem will then be the ones that result in equal marginal costs for the two processes.
10-13

14. Figure 10.9: The Relationship Between MP, AP, MC, and AVC
MP pulls up AP as long as MP > AP.
Higher labor productivity (output per unit of labor) means more output can be created with less labor, and therefore lower additional cost per unit (MC).
As MC falls below AVC, MC pulls down AVC.
The opposite happens in each case as MP falls below AP (MP equalling AP at maximum AP) and MC rises above AVC (equal at maximum AVC).
10-14

15. Costs In The Long Run Let C measure the cost of production.
We assume that each firm minimizes its cost, C, of producing a given amount of output, say Q0.
Why is this reasonable?
An isocost line shows the set of input bundles with a given cost, say C0.
In the long run, all costs are variable, so we need not distinguish TC, FC, and VC.
Each input bundle on a given isocost line costs the same amount.
10-15

16. Figure 10.10: The Isocost Line
This is just like a consumer’s budget line, with C instead of M, and each price and quantity named a bit differently.
10-16

17. Figure 10.11: The Maximum Output for a Given Expenditure
This is not the way we would expect a firm to make its production choices – as we will see – but it illustrates parallels with what we’ve already studied.
Finding the maximum output for a given expenditure is the same process as for maximizing a consumer’s utility given her/his budget constraint.
The input combination chosen is on the highest isoquant touched by the isocost line in question. In this case, that is found where MRTS = w/r: the benefit in capital saved by employing another unit of labor is equal its cost in foregone capital (the rate at which capital must be given up in order to employ more labor, at the relative price or opportunity cost of labor). [Translate and graphically illustrate this in utility/budget terms to see the parallel.]
The only difference is that here output definitely has a cardinal interpretation (twice as much is twice as much), while we try to maintain an ordinal interpretation of utility (more is more, but we don’t care how much more).
10-17

18. Figure 10.12: The Minimum Cost for a Given Level of Output
To find the minimum cost bundle we
- begin with a specific isoquant,
- then superimpose a map of isocost lines, each corresponding to a different cost level.
Assuming
strict convexity of the isoquants (neither perfect substitutes nor perfect complements in production),
smooth isoquants (continuously diminishing MRTS), and
production that requires a positive amount of both inputs,
The least-cost input bundle corresponds to the point of tangency between an isocost line and the specified isoquant (there are no corner solutions).
The input combination chosen is on the lowest isocost line touched by the isoquant in question. In this case, that is found where MRTS = w/r: the benefit in capital saved by employing another unit of labor is equal its cost in foregone capital (the rate at which capital must be given up in order to employ more labor, at the relative price or opportunity cost of labor).
Let’s do a numerical example.
With perfect substitutes in production, we can have corner solutions without tangency. Draw an example.
With perfect complements in production (Leontieff production function), we can have an interior solution without tangency.
Let’s draw pictures to illustrate.
10-18

19. Figure 10.13: Different Ways of Producing 1 Ton of Gravel
Differences in the prices faced by different firms with access to the same production technology explain differences in their use of inputs.
(As usual, note that the slope should be marked as a ratio of vertical to horizontal changes, not an angle.)
10-19

20. Figure 10.14: The Effect of a Minimum Wage Law on Unemployment of Skilled Labor
Correspondingly, if a price, say of labor, changes, we would expect a firm to substitute the now cheaper input for the now more expensive input.
(As usual, note that the slope should be marked as a ratio of vertical to horizontal changes, not an angle.)
10-20

21. The Relationship Between Optimal Input Choice And Long-run Costs
How do costs (total, average, and marginal) change across possible output levels for a particular production process?
The output expansion path traces out minimum cost input combinations in the isoquant map with isocost lines of given slope and increasing cost and output.
Again, for minimum cost input combinations to occur we are assuming that the production technology is such that positive amounts
10-21

22. Figure 10.15: The Long-Run Expansion Path
Let’s do a numerical example.
Then let’s try to derive this from our experimental data.
10-22

23. Figure 10.16: The Long-Run Total, Average, and Marginal Cost Curves
10-23

24. The Relationship Between Optimal Input Choice And Long-run Costs
With _____ returns to scale, as output grows, inputs and LR total cost grow _____ proportionally, so LR average and marginal cost are _____.
Constant
exactly constant
Decreasing
more than increasing
Increasing
less than decreasing
10-24

25. Figure 10.17: The LTC, LMC and LAC Curves with Constant Returns to Scale
Re-use the examples of constant, decreasing, and increasing returns to scale hand-drawn in connection with production (from Besanko & Braeutigam, p. 210). Suppose w=1 and r=2. Calculate in a table the LTC, LAC, and LMC schedules for each and draw the corresponding curves.
10-25

26. Figure 10.18: The LTC, LAC and LMC Curves for a Production Process with Decreasing Returns to Scale
10-26

27. Figure 10.19: The LTC, LAC and LMC Curves for a Production Process with Increasing Returns to Scale
10-27

28. Long-run Costs And The Structure Of Industry
The number of firms in an industry depends on the minimum efficient scale of production: the minimum output level at which LAC is minimized.
Natural monopoly: an industry whose market output is produced at the lowest cost when production is concentrated in the hands of a single firm.
10-28

29. Figure 10.20: LAC Curves Characteristic of Highly Concentrated Industrial Structures
10-29

30. Figure 10.21: LAC Curves Characteristic of Unconcentrated Industry Structures
10-30

31. Figure 10.22: The Family of Cost Curves Associated with a U-Shaped LAC
Each subscript reflects a different level of fixed input(s) (“plant size”) and associated fixed cost, numbered in ascending order.
What is the plant size that minimizes the (SR)ATC of producing Q1? 1. What is that minimum ATC? That’s also the LAC at Q1. What is the MC of producing one more unit of output? It’s the SMC1 at Q1, and that’s also the LMC at Q1.
Repeat that question and answer for Q2, then for Q3.
We’ve thus defined the LAC and LMC curves. The LAC curve is the “lower envelope of the (SR)ATC curves.
Let’s look at the experiment’s results for a numerical example. (We can go back to the cleaned 2010 results for a clearer tradeoff of ATC-minimizing choices of plant sizes for different levels of output.)
10-31

32. Figure A10.1: The Short-run and Long-Run Expansion Paths
We’ll go into these topics from the appendix only if we have extra time.
If so,
Note that this example has constant returns to scale, and can be explored using the “3-D” graphs to which we earlier followed a hyperlink, with the sum of the exponents in the Cobb-Douglass production function equal to 1.
Let’s do a numerical example on the board.
10-32

33. Figure A10.2: The LTC and STC Curves Associated with the Isoquant Map in Figure A.10.1
The LTC which is a straight ray from the origin of the graph reflects the constant returns to scale.
10-33

34. Figure A10.3: The LAC, LMC, and Two ATC Curves Associated with the Cost Curves from Figure A.10.2
And the constant LAC=LMC reflect the constant returns to scale.
10-34