Question

CHAPTER SUMMARY

647
Using x skilled and y anskilled workers, a manufacturer can produce Q(x, y) = 60x^1/3y^2/3 units per day.
Currently the manufacturer employs 10 skilled
workers and 40 unskilled workers and is planning
to hire 1 additional skilled worker. Use calculus to
estimate the corresponding change that the
manufacturer should make in the level of unskilled
labor so that the total output will remain the
ER SUMMARY
same.

Answer 1

The manufacturer should decrease the level of unskilled labor by 2 for the total output to stay the same.

Explanation:

Q(x,y) = 60 x^⅓ y^⅔

Take the partial derivatives with respect to x and y.

∂Q/∂x = 20 x^-⅔ y^⅔

∂Q/∂y = 40 x^⅓ y^-⅓

So the total differential is:

dQ = ∂Q/∂x dx + ∂Q/∂y dy

dQ = 20 x^-⅔ y^⅔ dx + 40 x^⅓ y^-⅓ dy

If dQ = 0:

0 = 20 x^-⅔ y^⅔ dx + 40 x^⅓ y^-⅓ dy

If x = 10, y = 40, and dx = 1:

0 = 20 (10)^-⅔ (40)^⅔ (1) + 40 (10)^⅓ (40)^-⅓ dy

0 = 20 (4)^⅔ + 40 (1/4)^⅓ dy

-20 (4)^⅔ = 40 (1/4)^⅓ dy

-20 (4)^⅔ (1/4)^⅔ = 40 (1/4)^⅓ (1/4)^⅔ dy

-20 = 40 (1/4) dy

-20 = 10 dy

dy = -2

The manufacturer should decrease the level of unskilled labor by 2 for the total output to stay the same.