Business Calculus

 
Discussion Question 2[Module 4(x410)]
[After you have completed Module 4, provide brief responses to parts “a” and “b” of this item.]
     In previous Modules the concept of price elasticity of demand was examined in situations where a product’s Demand model was linear.  In some instances, this linear Demand model provided the basis to develop the product’s Revenue model.  Not surprisingly, a 2nd order Revenue model would result from a 1storder Demand model.
     Now that 3rd order models have been examined, it should be possible to deal with price elasticity of demand situations where a product’s Demand model is quadratic, and where the development of a Revenue model from this 2nd order Demand model results in a 3rd order model.  For example, given the quadraticDemand model x = 187,500 – 25(p)2, the Revenue model would be R = (unit price)(units sold)= (p)(x) = [(p)][187,500 – 25(p)2] ==> 187,500(p)1 – 25(p)3.
     Using techniques covered in Module 4, it should be possible to find the unit price “p” associated with maximum Revenue (assuming no limitations from restricted domain considerations on values of “p”).  Thus…for Revenue models in the form R = f(p), “Rmax” occurs at the unit price “p” associated with [d(R)/d(p)] = (R)’ = 0.
R = 187,500(p)1 – 25(p)3 ==> (R)’ = 187,500 – 75(p)2
            @ 0 = (R)’ = 187,500 – 75(p)2 ==> 75(p)2 = 187,500 ==>
                1(p)2 = $2,500.00 ==> 1(p) = $50.00   [ignore, of course, the negative root –$50.00]
            @ p = $50…Rmax = 187,500(50)1 – 25(50)3 = $6,250,000.
     In HW08 from Module 3B, it was demonstrated that the price “p” where maximum Revenue occurs (assuming no limitations from restricted domain considerations on values of “p”) is the same as the price “p” where unit elasticity (ɳ = –1) occurs.  Given that the general form of any product’s elasticity expression may be written as…
ɳ =  [(p) / (x)] [d(x) / d(p)] or  [(p) / (x)] [(x)′]
…(a) find the specific elasticity expression for this product whose Demand model is x = 187,500 – 25(p)2and then (b) show the steps you would use to verify that the price “p” associated with unit elasticity is, in fact, p = $50.