Question

The production function for the Roundtree Laser Company is: Q=(10L^.5)(K^.3)(M^.3)

where: Q: number of lasers produced per week L: amount of labor used per week K: the amount of capital used per week M: quantity of raw materials used per week

a) Does the production function exhibit decreasing returns to scale?

b) Does the production function exhibit diminishing marginal returns?

Answer 1

Q=10L^0.5(K^.03)(M^0.3)

(A) since the sum of power of the inputs are =0.5+0.3+0.3

=1.1

Since it is greater than 1,so it shows increasing return to scale.

(B) TV

Q/L,=10*0.5(L^0.5-1)K^0.3*M^0.3

=(5K^0.3*M^0.3)/L^0.5

Marginal return is calculated by taking derivatives with respect to the inputs,

Since the since marginal product of the labor and labor is inversely related, so it shows decreasing return.

Similarly the marginal product of K will be

Q/K=(3L^0.5*M^0.3)/L^0.7

It is also inversely related, so marginal return shows Decreasing return.

Q/M={3L^0.5*M^0.3}\M^0.7

Since the marginal product of M and M is negatively related, so the marginal return shows decreasing return.