Wolfram Alpha - Business Risk Management - Garven

I would like to call everyone?s attention to Wolfram Alpha, which ?? is an online service that answers factual queries directly by computing the answer from structured data, rather than providing a list of documents or web pages that might contain the answer as a search engine would.??(see http://en.wikipedia.org/wiki/Wolfram_Alpha).? ?I am calling Wolfram Alpha to your attention because I?think that many of you might find it?to be a very useful tool, not only for Finance 4335 but also for other analytically oriented courses which you might be taking.

A good example of how you can put Wolfram Alpha to use is given by the fourth question on Problem Set 1.? There, you were asked to determine the?cost minimizing size of a hospital (in terms of cost per patient-day).? As I point out in the solutions,?this involves determining the number of patient-days which minimizes Y = ?C /X = 4,700,000/X?+ 0.00013X.? Here, X represents the number of patient-days and Y represents the average cost per patient-day.? The answer to this problem is obtained by differentiating Y with respect to X, setting the resulting equation to 0, and?finding the value?for X which solves this equation.? The answer ends up being X = 190,142.

Although it is important that you understand how to solve these kinds of problems on your own, Wolfram Alpha provides a helpful way to validate your analysis.? Furthermore,?Wolfram Alpha?is easy to use because you input commands in plain English; e.g.,?if you want Wolfram Alpha to solve this problem for you, all you have to is type a command like ?differentiate 4,700,000/X + 0.00013X?, or ?dif 4,700,000/X + 0.00013X?, or ?derivative 4,700,000/X + 0.00013X? and then press return.??Let?s use the following command:?derivative 4,700,000/X + 0.00013X.

After you hit the enter key, Wolfram Alpha responds with the following output:

If you click on the ?Show steps? link above, Wolfram Alpha will show you the solution to this problem step-by-step:

Furthermore, Wolfram Alpha identifies two ?roots? for this equation:

These numbers are the values for X that satisfy the first order condition.? Since X = -190,142 is nonsensical, obviously the answer for the problem is X = 190,142 patient days.

You might also find it helpful to?plot the average cost function Y.? Since only positive values for X make any sense, let?s use the following command:

The Plot[] function will produce a nice graph of whatever is in the bracketed terms.? The information located within the {}?brackets simply defines the range of X values that you want plotted.? Wolfram Alpha will then return the following plot:

If you look at this plot, you can clearly see that the minimum point is to the left of X = 200,000; at X = 190,142, you have succeeded in scaling the hospital such that average cost per-patient days is minimized; specifically, Y = ?$4,700,000/X?+ $0.00013X = ?$4,700,000/190,142 + $0.00013(190,142) = $49.44.? Note that if you set X to higher or lower values; say 180,000 or 200,000, that your average cost is higher; at X = 180,000, it is Y = ?$4,700,000/180,000?+ $0.00013(180,000) = $49.51,?and at?X = 200,000, it is Y = ?$4,700,000/200,000 + $0.00013(200,000) = $49.50.

If you would like to learn more about Wolfram Alpha?s features, I recommend watching the following video:

Source: http://risk.garven.com/2012/08/28/wolfram-alpha/

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