Methodology
Market data ([2], abstract) divided into price tiers:
Club fighter (8), Low (9), Medium (103), High (21 stores...
of 1

Pricing optimization poster version 2 (1)

Published on: Mar 4, 2016
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Transcripts - Pricing optimization poster version 2 (1)

  • 1. Methodology Market data ([2], abstract) divided into price tiers: Club fighter (8), Low (9), Medium (103), High (21 stores) Applied: Method of Least Squares and ANOVA with SPSS Obtained: Demand function constants ๐‘Ž๐‘–, ๐‘๐‘–, c๐‘– > 0 , ๐‘– = 3, 4, 5 at 5% and 10% sig. levels for each price tier (market normals), as well as individual stores Utilized: Normal Curve Shifting [3], to obtain adjusted demand function constants for particular stores within each price tier market Solved: CAS (Maple) :11x11 linear system, testing both adjusted and unadjusted demand function constants, yielding predicted wholesale and retail price i. Functions, in terms of ๐‘ž2, ๐‘Ÿ2, ๐‘˜2 ii. Numerical values, using estimated values for ๐‘ž2, ๐‘Ÿ2, ๐‘˜2 Results For low price tier market, 67% of stores with linear demand constants at 5% or 10% sig. levels. Results: Store #112, 1160 Lake Cook Rd., Buffalo Grove, IL 60090 Demand Function Constants: (Unadjusted) Predicted solutions: (2 lt. cola bottles) Data AnalysisIndustry StructureABSTRACT Simplified Industry Structure References Industry structure: โ€ข Established national manufacturer Coca-Cola Company ยฎ โ€ข Competitor national manufacturer PepsiCo. ยฎ โ€ข Single retailer. Dominickโ€™s Stores ยฎ โ€ข These three players account for an overwhelming majority of the soft drink market Retailer: Buys the two national brands at wholesale and sells them - as well as its own generic brand (private label) - at retail prices to the consumer. Established national manufacturer: Seeks to determine the optimal wholesale price to charge the retailer. Assumes other two players will also maximize. Utilize: โ€ข Analytical Model: Multivariable calculus (optimization with equality constraints / Method of Lagrange Multipliers) [1] โ€ข Empirical Model: โ€ข Regression (Method of Least Squares, ANOVA) โ€ข Normal curve shifting [3] โ€ข Statistical package SPSS โ€ข Large soda market data sets, Dominickโ€™s Stores (141 stores, 400 weeks), University of Chicago Booth School of Business [2]. Let ๐‘Ž๐‘–, ๐‘๐‘–, c๐‘– > 0 , ๐‘– = 3, 4, 5 be constants in the new demand functions, ๐‘„ ๐‘›๐‘ = ๐‘Ž3 โˆ’ ๐‘3 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ + ๐‘3 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ + ๐‘‘3 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘๐‘ ๐‘„ ๐‘๐‘™ = ๐‘Ž4 โˆ’ ๐‘4 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ + ๐‘4 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ + ๐‘‘4 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘๐‘ ๐‘„ ๐‘๐‘ = ๐‘Ž5 โˆ’ ๐‘5 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘๐‘ + ๐‘5 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ + ๐‘‘5 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ to be sub. into Manufacturer, Retailer, and Competitor profit functions: ๐‘š ๐‘ƒ๐‘Ÿ = ( ๐‘ค ๐‘ƒ ๐‘›๐‘ โˆ’ ๐‘š ๐‘‰ ๐‘›๐‘)๐‘„ ๐‘›๐‘ ๐‘Ÿ ๐‘ƒ๐‘Ÿ = ( ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ โˆ’ ๐‘Ÿ ๐‘‰ ๐‘๐‘)๐‘„ ๐‘›๐‘ + ( ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ โˆ’ ๐‘Ÿ ๐‘‰ ๐‘๐‘™)๐‘„ ๐‘๐‘™ + ( ๐‘Ÿ ๐‘ƒ ๐‘๐‘ โˆ’ ๐‘Ÿ ๐‘‰ ๐‘๐‘)๐‘„ ๐‘๐‘ ๐‘ ๐‘ƒ๐‘Ÿ = ( ๐‘ค ๐‘ƒ ๐‘๐‘ โˆ’ ๐‘ ๐‘‰ ๐‘๐‘)๐‘„ ๐‘๐‘ Manufacturer: Max ๐‘š ๐‘ƒ๐‘Ÿ = ๐‘“( ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ , ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ , ๐‘Ÿ ๐‘ƒ ๐‘๐‘ , ๐‘ค ๐‘ƒ ๐‘›๐‘)........objective function Also, Competitor: Max ๐‘ ๐‘ƒ๐‘Ÿ = ๐‘“ ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ , ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ , ๐‘Ÿ ๐‘ƒ ๐‘๐‘ , ๐‘ค ๐‘ƒ ๐‘๐‘ ....objective function For both, then Retailer also Max (Global): โ‡’ Set partial derivatives of ๐‘Ÿ ๐‘ƒ๐‘Ÿ = ๐‘“( ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ , ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ , ๐‘Ÿ ๐‘ƒ ๐‘๐‘) = 0โ€ฆ....constraints Further, assume: ๐‘Ÿ ๐‘‰ ๐‘๐‘™ โ‰ˆ ๐‘ค ๐‘ƒ ๐‘๐‘™ , ๐‘Ÿ ๐‘‰ ๐‘›๐‘ โ‰ˆ ๐‘ค ๐‘ƒ ๐‘›๐‘ , ๐‘Ÿ ๐‘‰ ๐‘๐‘ โ‰ˆ ๐‘ค ๐‘ƒ ๐‘๐‘ For mathematical convenience, let: ๐‘ข1 = ๐‘Ÿ ๐‘ƒ ๐‘›๐‘, ๐‘ข2 = ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ ๐‘ข3 = ๐‘Ÿ ๐‘ƒ ๐‘๐‘ set by the retailer ๐‘ข4 = ๐‘ค ๐‘ƒ ๐‘›๐‘ set by the manufacturer ๐‘ข5 = ๐‘ค ๐‘ƒ ๐‘๐‘ set by the competitor ๐‘ž2 = ๐‘š ๐‘‰ ๐‘›๐‘ incurred by the manufacturer ๐‘Ÿ2 = ๐‘ ๐‘‰ ๐‘๐‘ incurred by the competitor ๐‘˜2 = ๐‘ค ๐‘ƒ ๐‘๐‘™ unknown to man. and comp. COMBINED OPTIMIZATION PROBLEM Objective Functions ๐‘š ๐‘ƒ๐‘Ÿ(๐‘ข1, โ€…๐‘ข2, โ€…๐‘ข3, โ€…๐‘ข4) = (๐‘ข4 โˆ’ ๐‘ž2)(๐‘Ž3 โˆ’ ๐‘3 โ‹… ๐‘ข1 + ๐‘3 โ‹… ๐‘ข2 + ๐‘‘3 โ‹… ๐‘ข3) ๐‘ ๐‘ƒ๐‘Ÿ(๐‘ข1, โ€…๐‘ข2, โ€…๐‘ข3, โ€…๐‘ข5) = (๐‘ข5 โˆ’ ๐‘Ÿ2)(๐‘Ž5 โˆ’ ๐‘5 โ‹… ๐‘ข3 + ๐‘5 โ‹… ๐‘ข1 + ๐‘‘5 โ‹… ๐‘ข2) Constraints ๐ถ1 = ๐‘Ž3 2๐‘3 + ๐‘ข4 2 โ€“ ๐‘4 2๐‘3 ยท ๐‘˜2 โ€“ ๐‘5 2๐‘3 ๐‘ข5 + ๐‘3 + ๐‘4 2๐‘3 ๐‘ข2 + ๐‘5 + ๐‘‘3 2๐‘3 ๐‘ข3 โ€“ ๐‘ข1 = 0 ๐ถ2 = ๐‘Ž4 2๐‘4 + ๐‘˜2 2 โ€“ ๐‘3 2๐‘4 ยท ๐‘ข4 โ€“ ๐‘‘5 2๐‘4 ๐‘ข5 + ๐‘3 + ๐‘4 2๐‘4 ๐‘ข1 + ๐‘‘4 + ๐‘‘5 2๐‘4 ๐‘ข3 โ€“ ๐‘ข2 = 0 ๐ถ3 = ๐‘Ž5 2๐‘5 + ๐‘ข5 2 โ€“ ๐‘‘4 2๐‘5 ยท ๐‘˜2 โ€“ ๐‘‘3 2๐‘5 ๐‘ข4 + ๐‘‘4 + ๐‘‘5 2๐‘5 ๐‘ข2 + ๐‘5 + ๐‘‘3 2๐‘5 ๐‘ข1 โ€“ ๐‘ข3 = 0 ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐ถ๐‘– = ๐‘“๐‘– ๐‘ข1, ๐‘ข2, ๐‘ข3, ๐‘ข4, ๐‘ข5 , ๐‘– = 1,2,3 Method of Lagrange Multipliers, 11x11 system โ‡’ ๐›ป ๐‘š ๐‘ƒ๐‘Ÿ = ๐œ†1 ๐›ป๐ถ1 ๐‘ข1, ๐‘ข2, ๐‘ข3, ๐‘ข4 + ๐œ†2 ๐›ป๐ถ2 ๐‘ข1, ๐‘ข2, ๐‘ข3, ๐‘ข4 + ๐œ†3 ๐›ป๐ถ3(๐‘ข1, ๐‘ข2, ๐‘ข3, ๐‘ข4) ๐›ป ๐‘ ๐‘ƒ๐‘Ÿ = ๐œ‡1 ๐›ป๐ถ1 ๐‘ข1, ๐‘ข2, ๐‘ข3, ๐‘ข5 + ๐œ‡2 ๐›ป๐ถ2 ๐‘ข1, ๐‘ข2, ๐‘ข3, ๐‘ข5 + ๐œ‡3 ๐›ป๐ถ3(๐‘ข1, ๐‘ข2, ๐‘ข3, ๐‘ข5) ๐ถ๐‘– = 0, ๐‘– = 1,2,3 [1] Edwards, C. Henry, David E. Penney, and C. Henry Edwards. Calculus. Upper Saddle River, NJ: Prentice Hall, 2002. Print. [2] Acknowledgements: James M. Kilts Center. University of Chicago. Booth School of Business [3] DeGroot, Morris H., and Mark J. Schervish. Probability and Statistics. Boston: Pearson Education, 2012. Print. [4] Lilien, Gary L., Philip Kotler, and K. Sridhar. Moorthy. Marketing Models. Englewood Cliffs, NJ: Prentice-Hall, 1992. Print. Let ๐‘Ž๐‘–, ๐‘๐‘–, c๐‘– > 0 , ๐‘– = 1, 2, be constants in the demand functions, ๐‘„ ๐‘๐‘™ = ๐‘Ž1 โˆ’ ๐‘1 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ + ๐‘1โ‹… ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ ๐‘„ ๐‘›๐‘ = ๐‘Ž2 โˆ’ ๐‘2 โ‹… ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ + ๐‘2โ‹… ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ to be sub. into Manufacturer and Retailer profit functions: ๐‘š ๐‘ƒ๐‘Ÿ = ( ๐‘ค ๐‘ƒ ๐‘›๐‘ โˆ’ ๐‘š ๐‘‰ ๐‘›๐‘)๐‘„ ๐‘›๐‘ ๐‘Ÿ ๐‘ƒ๐‘Ÿ = ( ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ โˆ’ ๐‘Ÿ ๐‘‰ ๐‘๐‘™)๐‘„ ๐‘๐‘™ + ( ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ โˆ’ ๐‘Ÿ ๐‘‰ ๐‘›๐‘)๐‘„ ๐‘›๐‘ Manufacturer: Max ๐‘š ๐‘ƒ๐‘Ÿ = ๐‘“( ๐‘ค ๐‘ƒ ๐‘›๐‘, ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ , ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ )............objective function But, then Retailer also Max (Global): โ‡’ Set partial derivatives of ๐‘Ÿ ๐‘ƒ๐‘Ÿ = ๐‘“( ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ , ๐‘Ÿ ๐‘ƒ ๐‘๐‘™) to 0. โ€ฆโ€ฆโ€ฆconstraints Further, assume: ๐‘Ÿ ๐‘‰ ๐‘๐‘™ โ‰ˆ ๐‘ค ๐‘ƒ ๐‘๐‘™ , ๐‘Ÿ ๐‘‰ ๐‘›๐‘ โ‰ˆ ๐‘ค ๐‘ƒ ๐‘›๐‘ For mathematical convenience, let: ๐‘ฅ1โ€… = โ€… ๐‘ค ๐‘ƒ ๐‘›๐‘ set by the manufacturer ๐‘ฅ2 โ€…= โ€… ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ ๐‘ฅ3โ€… = โ€… ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ set by the retailer ๐‘˜1โ€… = โ€… ๐‘ค ๐‘ƒ ๐‘๐‘™ unknown to the manufacturer ๐‘ž1โ€… = โ€… ๐‘š ๐‘‰ ๐‘›๐‘ incurred by the manufacturer OPTIMIZATION PROBLEM Objective Function ๐‘š ๐‘ƒ๐‘Ÿ ๐‘ฅ1, ๐‘ฅ2, ๐‘ฅ3 = = ๐‘ฅ1(๐‘Ž2 โˆ’ ๐‘2 ๐‘ฅ2 + ๐‘2 ๐‘ฅ3) + ๐‘ฅ2(โˆ’โ€…๐‘2 ๐‘ฅ1 โˆ’ ๐‘2 ๐‘ž1) + ๐‘ฅ3(๐‘2 ๐‘ฅ1โ€… โˆ’ ๐‘2 ๐‘ž1) Constraints โ„Ž1 ๐‘ฅ1, ๐‘ฅ2, ๐‘ฅ3 = ๐‘Ž1 2๐‘1 + 1 2 ๐‘˜1 โˆ’ ๐‘2 2๐‘1 ๐‘ฅ1 + ๐‘1 + ๐‘2 2๐‘1 ๐‘ฅ2 โˆ’ ๐‘ฅ3 = 0 โ„Ž2 ๐‘ฅ1, ๐‘ฅ2, ๐‘ฅ3 = ๐‘Ž2 2๐‘2 + 1 2 ๐‘ฅ1 โˆ’ ๐‘1 2๐‘2 ๐‘˜1 + ๐‘1 + ๐‘2 2๐‘2 ๐‘ฅ3 โˆ’ ๐‘ฅ2 = 0 Method of Lagrange Multipliers, 5x5 system โ‡’ ๐›ป ๐‘š ๐‘ƒ๐‘Ÿ(๐‘ฅ1, โ€…๐‘ฆ1, โ€…๐‘ง1) = โ€…๐œ‡ โ‹… ๐›ปโ„Ž1(๐‘ฅ1, โ€…๐‘ฆ1, โ€…๐‘ง1)โ€…+ โ€…๐œ† โ‹… ๐›ป๐‘”1(๐‘ฅ1, โ€…๐‘ฆ1, โ€…๐‘ง1) โ„Ž1(๐‘ฅ1, โ€…๐‘ฆ1, โ€…๐‘ง1) = 0 โ„Ž2(๐‘ฅ1, โ€…๐‘ฆ1, โ€…๐‘ง1) = 0 ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐‘ = ๐‘1+๐‘2 2 For ๐‘Ž1 = ๐‘Ž2, ๐‘1 = ๐‘2, ๐‘1 = ๐‘2 the analytical solutions are: ๐‘ฅ1 = ๐‘ค ๐‘ƒ ๐‘›๐‘ = ๐‘Ž+๐‘๐‘ž+๐‘๐‘˜ 2๐‘ ; ๐‘ฅ2 โ€…= โ€… ๐‘Ÿ ๐‘ƒ ๐‘›๐‘ = 3๐‘Ž+๐‘๐‘ž+๐‘๐‘˜ 4๐‘ ; ๐‘ฅ3โ€… = โ€… ๐‘Ÿ ๐‘ƒ ๐‘๐‘™ = ๐‘Žโˆ’๐‘˜(๐‘โˆ’๐‘) 2(๐‘โˆ’๐‘) Price Optimization of an Established National Brand in the Presence of two Competitors, Part 1 Stavros Christofi, PhD; Cameron Sakurai; Alex Potocki Western Connecticut State University, Department of Mathematics Assumptions โ€ข Wholesale prices are set first, and then the retailer sets retail prices. โ€ข Wholesale prices are known only by those who set them, as well as the retailer. โ€ข Excluded promotional offers from data. โ€ข Standard practice for private labels: Retailer buys private label at wholesale price from some unknown manufacturer. โ€ข Demand functions are linear (others may be considered) Profit = Revenueโˆ’Costs = (Price)(Quantity)โˆ’[(Fixed Costs)+(Var. Cost/Qu.)(Quantity)] = [(Price)โˆ’(Var. Cost/Quantity)](Quantity)โˆ’(Fixed Costs) ๐‘ƒ๐‘Ÿ = ๐‘ƒ โˆ’ ๐‘‰ ๐‘„ โˆ’ ๐น ๐‘ค.๐‘™.๐‘”, ๐น=0 ๐‘ƒ๐‘Ÿ = ๐‘ƒ โˆ’ ๐‘‰ ๐‘„ Where, ๐‘„ = f(P) = demand function Formulation i. Unadjusted Constants i. Adjusted Constants ii. Adjusted Constants, Conclusion Individual stores within the club fighter and high price tier stores displayed similar results. Observe that, for above store, ๐‘คโ„Ž๐‘œ๐‘™๐‘’๐‘ ๐‘Ž๐‘™๐‘’ < ๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘–๐‘™ providing retailer with 69% markup for the national brand and 61% markup for the competitors brand. Our work, based on undergraduate mathematics, should provide some basis for pricing behavior in the marketplace. Future Research Explore demand functions other than linear to better fit the middle price tier for the soft drink market. Investigate markets other than soft drinks.

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