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–Either you say “yes”, –Or you say “no” and explain the difference(s) between the contexts.pD(p)Pictures that have boxes representing revenue/profit underneath the demand curve are common in market segmentation and volume discounting. Each box indicates some more revenue.
Page18utdallas.edu/~metinCalculating Differentiated Pricesby Eliminating ArbitrageArbitrage happens when a lower priced product i istransported to be sold as a higher priced product jtransportation cost aijupgraded to be sold as a higher priced product jupgrade cost aijstocked to be sold as a higher priced product jinventory holding cost aijArbitrage can be eliminated with constraints 𝑝?≤ 𝑝?+ 𝑎??and 𝑝?≤ 𝑝?+ 𝑎??on the price decision variables.Constraints are called “arbitrage elimination constraint”. The constraint makes the cost of using arbitrage pi+aijhigher than the price pi.ij𝑎??𝑎??
Page19utdallas.edu/~metinMotivating Example of ChipsIntel chips are sold both in US (country 1) and in Brazil (country 2). The demand functions are given byppbapdppbapd1000048600)(and100000508000)(222111Since the constraint is not satisfied, an arbitrageur can buy chips in Brazil and transport them to US to sell them in US. We need to jointly optimize prices in US and Brazil to eliminate arbitrage rather than separately as we have done above.The transportation cost per chip is $0.08 between these countries.With the constant slope demand curves (lines), the revenue maximizing prices are given as2102012202110151.254.2but 43.22and54.22appbapbap
Page20utdallas.edu/~metinQuadratic Program Formulation to Eliminate ArbitrageWhen we have linear demands di(pi), the profit (pi-ci)di(p) is a quadratic function (it is a polynomial of degree 2). Summing these profits over N regions, we still have a quadratic objective for the joint optimization problem:NipNjiapppdcpiijijNiiiiippN1for 01for subject to)()(max1...1This is a quadratic programming problem and it can be solved efficiently with algorithms similar to those used to solve linear programming problems.
Page21utdallas.edu/~metinSolving Quadratic Program for N=2When there are two regions (US and Brazil), N=2 and we can solve the quadratic program without a software.0,andsubject to)()(max211212212122221111,21ppappapppbappbapppIf the separately found prices satisfy the arbitrage eliminating constraints, they are optimal and we stop. Else let the region with higher price be 1 and the other be 2, we have120102210201andappappIn the optimal solution, the violated constraint must be satisfied as an equality, so we can letnextfindshallthat wesomefor and2211pppapp