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A manufacturing firm produces and sells a product to a single independent retailer. The product is sold during a single selling season. The manufacturer has production cost is $6.00 per unit and the retailer sells the product to the consumers at a price of $28.00 per unit. Unsold units at the end of the selling season have a salvage value of $2.00. If the retailer cannot meet the demand during the selling season then she has a goodwill penalty cost of $3.00 and the manufacturer has a goodwill penalty cost of $2.5. To procure the product the retailer has a cost of $1.00 per unit. The retailer places an order with the manufacturer at the start of the selling season and the delivery is instantaneous. The consumer demand 𝑋𝑋 follows a Gamma distribution in the interval (0, ∞) with shape parameter 𝛼𝛼 = 1 and scale parameter 𝛽𝛽 = 100 . F) The manufacturer and the retailer have realized that the demand is not fixed. It strongly depends on the retailer’s effort on promoting the product. Therefore, the manufacturer would like to design a contract to coordinate the retailer’s effort on selling the product. Suppose that the retailer’s effort is denoted by 𝑒𝑒 and the cost of the effort is given by 𝑔𝑔(𝑒𝑒) which can be interpreted as the marketing cost when promoting the product. 1. Suppose 𝑔𝑔(0) = 0, 𝑔𝑔′(0) > 0 . Discuss whether 𝑔𝑔′′(𝑒𝑒) > 0 or 𝑔𝑔′′(𝑒𝑒) < 0 and explain why. 2. In this context, the CDF of demand 𝐹𝐹(π‘₯π‘₯|𝑒𝑒) also depends on effort 𝑒𝑒, discuss whether πœ•πœ•πœ•πœ•(π‘₯π‘₯|𝑒𝑒) πœ•πœ•πœ•πœ• > 0 or πœ•πœ•πœ•πœ•(π‘₯π‘₯|𝑒𝑒) πœ•πœ•πœ•πœ• < 0, and explain why. 3. Set up profit functions of retailer, supplier, and the supply chain respectively, i.e. πœ‹πœ‹π‘Ÿπ‘Ÿ , πœ‹πœ‹π‘ π‘ , and πœ‹πœ‹π‘π‘. 4. What is the condition under which the effort 𝑒𝑒 can be coordinated in the supply chain? 5. Consider the buy-back, revenue-sharing, and quantity discount contracts respectively. Discuss which contract can achieve the supply chain coordination in both 𝑒𝑒 and π‘žπ‘ž, and explain the reason.

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