Consider a cocoa processor who sources cocoa beans from a single supplier using quantity flexibility contract (r,e) where r denotes the unit reservation price and e denotes the unit exercise fee. The processor uses the cocoa beans to produce cocoa butter with selling price of $120 per kg. The processing cost is normalized to zero, so the only cost is the procurement cost of the cocoa beans. Let us assume there is 1-1 conversion between cocoa beans and cocoa butter, i.e., 1 kg of beans yields 1 kg of cocoa butter. The processor can source the cocoa beans from the supplier in advance of the spot market and from a spot market on the day. The processor has a spot price forecast w, $ per kg, for cocoa beans with the following scenarios:

Scenario A: $150 with probability 14

Scenario B: $100 with probability 12

Scenario C: $50 with probability 14

a) A supplier (Supplier1) is offering the contract (51, 38). Is it profitable to order from this supplier? Why (not)?

b) There is another supplier (Supplier2) offering the contract (65, 23). Which supplier should the processor choose to order from? Why?

c) Let us denote as the additional marginal profit of the 1st contract unit with the preferred supplier---that is, the difference between the expected marginal profit of the 1st contracted unit with the preferred supplier (Supplier1 or Supplier2 as you have decided in part b) and the expected marginal profit of the 1st contracted unit with the other supplier. Cocoa processor would like to understand the impact of spot price variability on . What do you expect to happen to when spot price variability increases? Why?

To capture the spot price variability, you can think of a new spot price forecast w, $ per kg, for cocoa beans with the following scenarios:

Scenario A: $180 with probability 14

Scenario B: $100 with probability 12

Scenario C: $20 with probability 14

This new spot price has a higher variability than the original one (they both have the same means but the latter has a larger spread/range).