• Chapter 22 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).

• 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

• 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

• MC passes through the minimum of both the AVC and AC. AVC and AC get closer and y becomes larger.

• Since MC(y)=dcv(y)/dy, integrating both sides we get cv(y)-cv(0)=0

yMC(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.

• 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.

• 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.