Chapter 21 Cost Curves

1. Chapter 21 Cost Curves • SR: cs(x2,y)=cv(y)+w2x2= cv(y)+F, suppressing the dependence on x2, we have cs(y)=cv(y)+F and ACs(y)=cv(y)/y+F/y=AVC(y)+AFC(y). • SR: MC(y)=∆cs(y)/∆y=[cv(y+∆y)+F-(cv(y)+F)]/∆y • MVC(y)= ∆cv(y)/∆y= [cv(y+∆y)-cv(y)]/∆y, so MC(y)=MVC(y).

2. Fig. 21.1

3. MC(0)= [cv(∆y)-cv(0)]/∆y= cv(∆y)/∆y=AVC(0). • The units for MC and AVC are both dollar/output. • 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

4. dAC(y)/dy=d[(cv(y)+F)/y]/dy=[yd((cv(y)+F)/dy)-(cv(y)+F)]/y2=[MC(y)-AC(y)]/ydAC(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 • MC passes through the minimum of both the AVC and AC. AVC and AC get closer and y becomes larger.

5. Fig. 21.2

6. 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. • 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 MC(y1)=MC(y2). In other words, the MC of the firm is the horizontal sum.

7. Fig. 21.3

8. Fig. 21.5

9. Similarly if a firm sells to two markets, (in interior solution) must sell to the point where two MRs equal. • LR costs: no fixed costs by definition, but AC curve may still be U-shaped because of the quasi-fixed cost. • 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.

10. Fig. 21.6

11. Fig. 21.7

12. 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. • 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.

13. Fig. 21.8

14. Fig. 21.9

15. Fig. 21.10