EXPF50 Cobb-Douglas employee-budget

A certain assembly facility follows the Cobb–Douglas production formula

where P is the production, currently at 1000 assembled items per year, x is the number of employees and y is the daily operating budget.

Find dy/dx and evaluate the function at 70 employees.

It is easier to see what dy/dx represents and interpret the function in the context of the solution to the problem.

To determine dy/dx first take a total derivative of the function.

A little bit of convenient algebra on the derivative produces

The specification of 70 employees means that x is 70. This number, along with the production number of 1000, will produce y.

The most convenient way to find y is to raise both sides of the equation to the 1/0.7 power.

With the values of x and y we can determine dy/dx for 70 employees.

The quantity y is the daily budget so this number means that at this level of employees and daily budget you decrease the daily budget by $19.30 for each additional employee. Conversely the number tells you that another employee is worth a $19.30 decrease in daily budget.

This information along with employee cost information tells you whether to increase daily budget and reduce the number of employees or increase the number of employees and decrease daily budget.