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My mother and I used to be the same person.
She was fond of saying this, and I would smile indulgently, assuming it was something all mothers said. However, I did not fully grasp her meaning until my grandfather’s funeral.
The news of Grandpa Howard’s death was a shock to everyone. My grandmother had passed away just weeks earlier, so wounds that were not quite healed were ripped open again. Dan, Howard’s son and my stepfather, had flown to Hayden, Idaho to arrange the service. The plan was for my mother and me to drive up that weekend. Afterward, we would all drive home together. Since Dan left, our house had been quiet, neither my mom nor I knowing quite what to say. Part way into the 8-hour drive to Hayden, my mother put on a CD to ease the silence. I lost myself in my thoughts until one phrase caught my attention.
But since it falls unto my lot
That I should rise and you should not
I will gently rise and I’ll softly call,

Goodnight and joy be with you all
I turned to my mother and asked if she would like to sing this with me at Grandpa’s service. She shot me a stunned look, which I understood. My whole family is musical, therefore singing, playing an instrument are as natural as Little League is to other families. I have performed on the piano, the violin and the ukulele, but I am not known to get up and sing in front of people, especially a cappella. My mom, however, said nothing; she just nodded and started the song over.
Again, and again we sang it, trying to perfect harmonies and memorize words. I watched the clouds on the horizon turn from rosy to purple, and from purple to black with our headlights illuminating the lines on the road. Mom and I had some great moments, laughing over errors and hunting for notes. Then there were the low points, such as the 40-mile stretch from Fishtrap to Cheney, Washington spent bickering over wrong lyrics or timing. And yet, by the time we reached Hayden, we had it nailed.
The chapel for Grandpa Howard’s service was beautiful. With no need for a microphone, the testimonies from family and friends echoed down the pews and hung in the air, allowing the emotion to sink in. The final tribute would come from my mother and me; our song. We stood at the front of the church without accompaniment, and instinctively my hand grasped hers right before we opened our mouths for the first note.
The tiny chapel rang with our combined voices and when it ended, my heart was full. Later, back at the house my aunt remarked on what courage and poise I had shown, and how well my voice blended with my mother’s.
“Well,” I replied, catching Mom’s eye over my aunt’s shoulder, “we used to be the same person.”

Exercise in Operation Management

(a) Suppose the manufacturer sells to the retailer at $80/unit. How many ski jackets should the retailer order? How much profit does the retailer expect to make as a result? How much profit will the manufacturer make as a result?
(b) What is the system optimal production quantity and expected profit under global optimization?
(c) Is it possible to find a contract such that both the manufacturer and retailer enjoy a higher expected profit than a)? If so, describe the contract and calculate the expected profit for the manufacturer and the retailer. If not, explain why it is not possible.