Question

1. For each of the production functions below do the following:

1. Determined whether it is a constant, decreasing or increasing returns to scale.
2. Find the expressions for the marginal product of capital and labor. Are they decreasing?

1. Y=A K1/4L3/4                  
2. Y=A K1/3L3/2                 
3. Y=A K1/5L4/6                 
4. Y=K1/3L2/3+K             
5. Y=A K1/3L2/3+K1/2     
6. Y=A K1/3+AL2/3          
7. Y= K13+1/4L1/3                

Answer 1

put K= tK, L=tL in each question

A)

Y`= Ak^1/4 *L^3/4= AK^1/4 *L^3/4(t*^1/4 *t^3/4)

Y'=AY, constant return to scale

B) Y`=Ak^1/3 *L^3/2(t^(1/3+3/2) =

Y`= t^11/6(Y) .. .. Increase return to scale

C) Y`=A K1/5L4/6 (t^(13/15) = t^13/15(Y)

. Decrease return to scale

D) Y`=(K1/3L2/3+K)t= tY. . .Constant return to scale

E) Y`= t(K^1/3 *L^2/3) +t^1/2K^1/2...decresing return to scale

F) Y`=t^1/3A(K^1/3+t^1/3L^2/3) . .

decrease return to scale

g) Y`=t^1/3(K^1/3+1/4L^1/3 ) = t^1/3(Y)

Decrease return to scale

Notes:

1)output changes by Same factor

if all inputs are changed by the same factor.

2) If output changes by less factor b than it is in input ...decrease return to scale,

3)if output changes more than input .. .ncrease return to scale