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Unformatted text preview: MS&E 246: Game Theory with Engineering Applications Feryal Erhun Winter, 2010 Problem Set # 5 Due: March 2, 2010, 5:00 PM outside Terman 305 Reading Assignment: Gibbons, Chapter 3; and Mas-Colell/Whinston/Green, Section 8.E. 1. Consider a simple supply chain with one supplier and one retailer. The retailer is selling seasonal goods, and since the suppliers lead-time is longer than the selling season, the goods must be produced before the selling season begins and there is no additional opportunity of replenishment after the demand is observed. Suppose that the probability distribution of the demand F ( x ) is known both by the supplier and the retailer. The suppliers unit cost is c . The selling price P of the retailer is exogenously determined and does not depend on the quantity sold. Note: You may find Leibniz Integral Rule useful in your calculations. Q Z b ( Q ) a ( Q ) f ( x,Q ) dx = Z b ( Q ) a ( Q ) f Q dx + f ( b ( Q ) ,Q ) b Q- f ( a ( Q ) ,Q ) a Q . (a) Consider a centralized supply chain (CSC) where the retailer and the supplier belong to the same company and managed centrally. Write the profit function of the CSC. How many goods will the CSC produce? (b) Now consider a decentralized supply chain (DSC) where the retailer and supplier are independent companies, each trying to maximize profits. The supplier charges a whole- sale price of w per unit. (This is a wholesale price contract.) Write the profit functionper unit....
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