Hello this is work in progress
To: Production department
From: Production Analysis Team
Date:
Re: Production Review
You asked us to see give recommendation on how to staff the new 80 guest product line . We have reviewed this and propose that considering we do not have the capabilities to extend capital at this point of time , most cost effective way of staffing this would be to have 2 skilled workers and 10 unskilled workers
WORKOUT:
Based on the production function Q= 1/15 (U^2)(S^2)(D^.5) from your memo, we can differentiate to obtain dQ/dLu ( unskilled ) and dQ/dLs ( skilled ). Also we have D = constant ie D =9. We can simplify the equation to
Q = 1/15 (U^2)(S^2)(9^.5) = 1/5 (U^2)(S^2)
MPLu ( unskilled ) = 1/5 (2)(U)(S^2)
MPLs (skilled ) = 1/5 (2)(S)(U^2)
To minimize cost given the skilled and unskilled wages and rental rate, the marginal product per dollar spent on skilled labor must be equal to the marginal product per dollar spent on unskilled labor. That means that at the ideal input mix,
MPLu / Wu = MPLs / Ws
Wu = $7
Ws = $35
so solving this
MPLu /7 = MPLs/35
We get U = 5S
That solves to unskilled labor to skilled labor ratio as 5:1
Now solving for Q = 80 and substituting in the equation
We have
Q = 1/5 (U^2)(S^2) =80
Or (U*S)^2 = 400
U*S = 20
Substituting U = 5S
5S^2 = 20
S^2 = 4
S = 2 and U = 10