Profit-Maximizing Utilization of an Input

[Bedee2704]

Hello everybody,
would like to know how to go onto the logic of the following task re the subject mentioned above.
You got a production process with two inputs prices and values for MP, that is P (Labor) = $75/ MP (Labor) = 5 units as well as P (Capital) = $600/ MP (Capital) = 30 units.
next step is they divide 75 over 5 and 600 over 30 to arrive at the price per unit of output and building the reciprocal of the result to arrive at the output/ additional dollar spent.
Then they conclude by reducing capital by one unit would reduce output by 30 units and reduce costs by $600 and spending $450 on 6 additional units of labor would increase output by 6 x 5 units, thereby restoring output to its previous level and reduce the cost of production by $150.
I don’t get a grip of the calculus/ logic of reducing capital by one untit would reduce output by 30 units and reduce costs by $600 and how I reduced the the cost of production by $150.
Can somebody explain the path of the calculation as a step by step approach?
Reference is page 84 Kaplan Schweser 2014 Economics Chapter
I thank you
PW

 

[Oscar]

The cost-minimizing factor input utilization is given when:
MPLabor/PLabor = MPCapital/PCapital
That means that by re-allocating the inputs between labor and capital no additional cost reduction at a constant output can be achieved.
In the example you have:
MPLabor/PLabor = 5/75 = 0.067
MPCapital/PCapital = 30/600 = 0.05
Obviously the additional output per $ spent for labor is greater than for capital. Hence, it is rational to employ more labor at the expense for capital. Recall that that we have diminishing marginal productivities for each input factor. If you increase labor and simultaneously decrease capital this will lead MPLabor/PLabor to decrease and MPCapital/PCapital to increase and at a certain point both will match.
In order to make both terms equal you must ask yourself the following question: If I reduce capital by one unit (as described above) the output will go down by 30 units (MPCapital); in order to produce this 30 units with additional labor how much more units of labor do I need to employ?
The answer is 6 additional units of labor (=30 units/5 MPLabor). By doing so, you have saved total costs of $150 (+$600 savings from one unit of capital minus $450 costs of 6 additional units of labor.
Since you are still producing the same output as before this input allocation is preferable to the initial one.
Regards,
Oscar