1. Chapter 22 Cost Curves
Key Concept: We define average cost and marginal cost curves and see their connections.

2. Chapter 22 Cost Curves
We will talk about cost curves in detail, in particular, average cost and marginal cost.
SR: cs(x2,y)=cv(y)+w2x2= cv(y)+F, suppressing the dependence on x2, we have
cs(y)=cv(y)+F

3. cs(y)=cv(y)+F
ACs(y)
= cs(y)/y
= cv(y)/y+F/y
= AVC(y)+AFC(y)

4. cs(y)=cv(y)+F
MC(y)
= ∆cs(y)/∆y
= [cv(y+∆y)+F-(cv(y)+F)]/∆y
MVC(y)
= ∆cv(y)/∆y
= [cv(y+∆y)-cv(y)]/∆y
Thus MC(y)=MVC(y).

5. Fig. 21.1

6. All the average costs and marginal costs share the same unit, i. e
All the average costs and marginal costs share the same unit, i.e. dollar/output.
MC(0)
= [cv(∆y)-cv(0)]/∆y
= cv(∆y)/∆y
= AVC(0).
We have seen this before (Ch. 15) at MR(0)=AR(0) or MR at x=0 equals p.

7. We now explore the relationship between average cost and marginal cost.
Average grade vs. marginal grade

8. dAVC(y)/dy
= d(cv(y)/y)/dy
= [yd(cv(y)/dy)-cv(y)]/y2
= [MC(y)-AVC(y)]/y
AVC decreasing  MC<AVC
AVC increasing  MC>AVC
AVC flat  MC=AVC

9. dAC(y)/dy
= d[(cv(y)+F)/y]/dy
= [yd((cv(y)+F)/dy)-(cv(y)+F)]/y2
= [MC(y)-AC(y)]/y
AC decreasing  MC<AC
AC increasing  MC>AC
AC flat  MC=AC

10. MC passes through the minimum of both the AVC and AC.
AVC and AC get closer and y becomes larger because AFC (the difference between AC and AVC) is smaller and smaller.

11. Fig. 21.2

12. Since MC(y)=dcv(y)/dy
integrating both sides we get
cv(y)-cv(0)=0yMC(x)dx.
Since cv(0)=0, the area under MC gives you the variable cost.
So there is a connection between the AVC curve and the area under MC curve.

13. Fig. 21.3

14. Fig. 21.2

15. Cost curves. The cost curves for c(y) = y2 + 1.

16. Suppose you have two plants with two different cost functions, what is the cost of producing y units of outputs?
You must use the min cost way. In interior solution, must allocate y=y1+y2 so that MC1(y1)=MC2(y2). In other words, the MC of the firm is the horizontal sum.

17. Fig. 21.5

18. Similarly if a firm sells to two markets, (in interior solution) must sell to the point where two MRs equal.

19. Now we turn to the LR.
LR costs: no fixed costs by definition, but AC curve may still be U-shaped because of the quasi-fixed cost.

20. From above
cs(x2(y),y)=c(y) and
cs(x2,y)c(y) for all x2.
Hence
ACs(x2(y),y)=AC(y) and
ACs(x2,y)AC(y) for all x2.

21. In words, the LR AC is the lower envelope of the SR AC.
This is still true if we have discrete levels of plant size.

22. Fig. 21.6

23. Fig. 21.7

24. Fig. 21.8

25. Regarding MC
since c(y)=cs(x2(y),y), so
MC(y)
=dc(y)/dy
=cs(x2(y),y)/y+[cs(x2,y)/x2]|x2(y)[x2(y)/y]
Note that x2(y) is defined to be the fixed factor which minimizes the cost, in other words, for a given y, cs(x2,y)/x2=0 at x2=x2(y). So LR MC coincides with SR MC.
Mention discrete levels of plant size.

26. Fig

27. Fig. 21.9

28. MC(y)
=cs(x2(y),y)/y
LR MC coincides with SR MC.
Let us work out the Cobb Douglas example to convince ourselves.

29. Chapter 22 Cost Curves
Key Concept: We define average cost and marginal cost curves and see their connections.