Thank you to any one who responds to this in advance. The help is much appreciated. Here is my problem followed by commentary:
The cost of making a part is $3 for labor and $1 for materials. Overhead is fixed at $2000 per week. If more than 5000 units are made each week labor is $4.50 per unit rather than $3, for only those units produced above and beyond 5000. At what level of output will one minimize average cost per unit?
This exercise is giving me a lot of difficulty because I do not know where to start. The general strategy with these type of problems is to set up a function for the quantity one wants minimized and then express this as a function of one variable only (so as not to have to solve by partial differentiation and limit the number of first and second order conditions). Secondly one finds the critical value(s) by the FOC, i.e. setting the first derivative equal to zero and solving. Then to verify that one has indeed found a minimum on a global scale one can use the second or first derivative test perhaps utilizing endpoints if this is appropriate to the domain of the function.
The difficulty I am having in this exercise is that my course materials do not cover mins and maxes of stepwise functions. I have tried many forms to the set up of this problem to no avail. What baffles me is that there are three variables here (units of labor less than 5000, units of labor more than 5000, and units of materials at any output level) which can easily be expressed all as units of labor using the relationship 1/3L=1M.
After doing so however I see no way to set up a single function which makes use of the fact cost of labor=3 for u<5000 and =4.50 for u>5000 and at the same time allows the u>5000 to 'vanish' when u is <5000. As such, I tried variations on another approach...
For this approach I made use of some Calculus III material I got from Math Econ. In this course, I have studied the Lagrange multiplier and as such used various alterations of this knowledge in the hope of making the current word problem a 'constrained' minimization problem.
In doing so I treated 'u' as units less than 5000 and 'v' as units greater than 5000. As such, one can set up the total cost function and the average cost function and then use these two equations to solve for one of the unknowns in terms of the other. However none of these work to any avail as I can not get the 4.50*v (even though it may be expressed in terms of u) to vanish from the equation for output under 5000 units.
So in short I have tried solving this problem both as a single equation all in terms of units and as simultaneous equation in terms of both u and v. Neither of these approaches has worked.
Since a stepwise function does not have a derivative (at all points) it is impossible to set things up and solve things this way even if I knew how.
Please provide some good advice on the set up to this problem and I am confident I will have no troubles taking things from there.
Thanks again and I look forward to correspondence regarding this query soon.
Best Regards,
Khy
The cost of making a part is $3 for labor and $1 for materials. Overhead is fixed at $2000 per week. If more than 5000 units are made each week labor is $4.50 per unit rather than $3, for only those units produced above and beyond 5000. At what level of output will one minimize average cost per unit?
This exercise is giving me a lot of difficulty because I do not know where to start. The general strategy with these type of problems is to set up a function for the quantity one wants minimized and then express this as a function of one variable only (so as not to have to solve by partial differentiation and limit the number of first and second order conditions). Secondly one finds the critical value(s) by the FOC, i.e. setting the first derivative equal to zero and solving. Then to verify that one has indeed found a minimum on a global scale one can use the second or first derivative test perhaps utilizing endpoints if this is appropriate to the domain of the function.
The difficulty I am having in this exercise is that my course materials do not cover mins and maxes of stepwise functions. I have tried many forms to the set up of this problem to no avail. What baffles me is that there are three variables here (units of labor less than 5000, units of labor more than 5000, and units of materials at any output level) which can easily be expressed all as units of labor using the relationship 1/3L=1M.
After doing so however I see no way to set up a single function which makes use of the fact cost of labor=3 for u<5000 and =4.50 for u>5000 and at the same time allows the u>5000 to 'vanish' when u is <5000. As such, I tried variations on another approach...
For this approach I made use of some Calculus III material I got from Math Econ. In this course, I have studied the Lagrange multiplier and as such used various alterations of this knowledge in the hope of making the current word problem a 'constrained' minimization problem.
In doing so I treated 'u' as units less than 5000 and 'v' as units greater than 5000. As such, one can set up the total cost function and the average cost function and then use these two equations to solve for one of the unknowns in terms of the other. However none of these work to any avail as I can not get the 4.50*v (even though it may be expressed in terms of u) to vanish from the equation for output under 5000 units.
So in short I have tried solving this problem both as a single equation all in terms of units and as simultaneous equation in terms of both u and v. Neither of these approaches has worked.
Since a stepwise function does not have a derivative (at all points) it is impossible to set things up and solve things this way even if I knew how.
Please provide some good advice on the set up to this problem and I am confident I will have no troubles taking things from there.
Thanks again and I look forward to correspondence regarding this query soon.
Best Regards,
Khy