Question

Calculate the returns to scale for the following functions:

a) f(x1,x2)= e^(ax1) + bx2^2

b) f(x1, x2,x3)= a*sqrt(x1x2) + x3^b

Answer 1

Answer a) F(X1,X2) = e^(a.X1) + b.(X2^2)

At X1=X2= 1

F(X1,X2) = e^a + b

At X1=X2=2

F(X1,X2) = e^(2a) + b(4)

With the Doubling of inputs X1 and X2 the output F(X1,X2) more than doubled so, it has Increasing returns to scale. [ Double of e^a+b is 2(e^a+b) and e^(2a) +4b is greater than this]

Answer b) F(X1,X2,X3)= a√X1.√X2 + (X3)^b

At X1=X2=X3= 1

F(X1,X2,X3)=a+1

At X1=X2=X3=2

F(X1,X2,X3) = a√4 + 2^b

F(X1,X2,X3) = 2a+ 2b or 2(a+b)

Since with the Doubling of inputs X1,X2 and X3 the output F(X1,X2,X3) exactly doubled so it has constant returns to scale.[ Double of (a+1) is 2(a+1) and that is what we are getting with Doubling of inputs]