Formal example

The Cobb-Douglas anatomic anatomy has connected allotment to calibration back the sum of the exponents adds up to one. The action is:

\ F(K,L)=AK^{b}L^{1-b}

where A > 0 and 0 < b < 1. Thus

\ F(aK,aL)=A(aK)^{b}(aL)^{1-b}=Aa^{b}a^{1-b}K^{b}L^{1-b}=aAK^{b}L^{1-b}=aF(K,L).

But if the Cobb-Douglas assembly action has its accepted form

\ F(K,L)=AK^{b}L^{c}

with 0 < c < 1, again there are accretion allotment if b + c > 1 but abbreviating allotment if b + c < 1, since

\ F(aK,aL)=A(aK)^{b}(aL)^{c}=Aa^{b}a^{c}K^{b}L^{c}=a^{b+c}AK^{b}L^{c}=a^{b+c}F(K,L),

which is greater than or beneath than aF(K,L) as b+c is greater or beneath than one.