Question

Given F(K,L) = AKa Lb, show that a and b determine whether the function is contact, increasing or decreasing return to scale (show one example in each case).

Answer 1

Using the properties of homogeneity

Multiply each input by a constant, say, $k$ , and factor

F($k$ K,$k$ L) = $A(kK)^a(kL)^b$

= $Ak^aK^ak^bL^b$

= $k^{a+b}(AK^aL^b)$

= $k^{a+b}(F(K,L))$

For a strict Cobb-Douglas production function, where $a+b=1$ , exhibits a constant returns to scale. Where $a+b≠1$ (a generalized production function), if $a+b>1$ , the function exhibits an increasing returns to scale and a decreasing returns to scale if $a+b<1$