scale property of multi-variable functions

Question

Match the multi-variable functions on the left with the scale property

Q(K,L)=2K^0.5L¹

Q(K,L)=7K^0.9L^0.1

Q(K,L)=5K^0.2L^0.6

Increasing returns to scale

Constant returns to scale

Decreasing returns to scale

Answer 1

All three functions are of the form

Q(K,L) = cK^aL^b (1)

for some constants a, b, c.

So we can analyze all the three given functions at once by analyzing (1).

To determine the scale property of (1), consider some constant t, and evaluate

Q(tK, tL) = c(tK)^a(tL)^b = t^a+bcK^aL^b = t^a+bQ(K,L)

So

if a+b > 1 then Q(K,L) exhibits increasing returns to scale,

if a+b < 1 then Q(K,L) exhibits decreasing returns to scale,

if a+b = 1 then Q(K,L) exhibits constant returns to scale.