Essay About 0P And Linear Price-Response Function D

Decision Science Price Differentiation Homework
KhalilLara AkkadDCSN 211: Report on Chapter 3 & 4Work in groups of two. Prepare a paper report including answers to the following questions. Submit one report per group at the beginning of the class on the due date. Have a second copy of your report with you to correct mistakes during class time.Consider a linear price-response function d (p)=D-m*p. In Chapter 3 (Basic Price Optimization), we established that the total revenue is maximized when p=D/(2*m). At what price is the total contribution maximized, assuming incremental cost c? Answer: The price that maximizes revenue is always lower than the price that maximizes total contribution (unless incremental costs are 0). Revenue maximizing price = P’ = D/2*m where m is the slope of the linear price response function.Contribution maximizing price = P* Max m(p*) = Max (p*- c) d(p) To maximize total contribution, we take the derivative of m(p) and set it as = to 0 to find p*.m’(p*) = d’(p*)(p*- c) + d(p*)d’(p*)(p*- c) + d(p*) = 0p* – c = – d(p*)/d’(p*)p* = – d(p*)/d’(p*) + c                    Contribution maximizing price (p*)[pic 1]Find a general formula for any values of D, p, and c. Show all steps of your analysis. Test your formula on one example for which you know the correct solution.Given: linear price response function d (p) = D – m*p & revenue maximizing price = p’ = D/2*mm(p) = (p – c) d(p)m(p) = (p – c)(D – m*p)m(p) = (p – c)(D – m*(D/2*m)m (p) = (p – c)(D – D/2)Testing of general formula:Using an example from the power point presentation on chp. 3:what price maximizes the total revenue: p = 6.25$ for d (p) = 10,000-800pgiven incremental costs = $5.00In the example, m(p) = $6,250Testing our function with the same parameters:m(p) = (6.25 – 5)(10,000 – 10,000/2) = ($1.25)(5000) = $6,250Consider a logit price-response function d (p) = C*exp(-(a+b*p))/(1+exp(-(a+b*p))) with parameters C=100, a=-10, and b=0.1. Determine the price at which the total contribution is maximized, assuming incremental cost c=50. Report the optimal price and the total contribution obtained at the optimal price. You may want to use an optimization solver to solve this question.First step: find ^ = – (a/b) = – (-10/0.1) = $100[pic 2]Second step: d(p) =  =  =  = 50 [pic 3][pic 4][pic 5]Third step: d’(p) =  =  =  = -2.5 [pic 6][pic 7][pic 8]Fourth step: m(p) = (p – c) d(p)To maximize it, use the derivative:m’(p*) = d’(p*)(p*- c) + d(p*) = (-2.5) (p* – 50) + 50 (-2.5) (p* – 50) + 50 = 0                  contribution maximizing price (p*) = $70[pic 9]Replacing p* = optimal price in m(p):m(p) = (p – c) d(p) = (70 – 50) 50 = $1,000 (total contribution at optimal price)Solve Exercise 1 (Arbitrage) from the end of Chapter 4 in the Phillips textbook. Note that some questions may be solved analytically, while others may require an optimization solver.For Pleasantville: m(p1) = (p1 – c) d(p1) = (p1 – 5) (10,000 – 800p1) = 14,000p1 – 800p12 – 50,000