Consider the following problem:

• Demand: q = 100-p
• Retailer: marginal cost of selling r = 10 per unit
• Manufacturer: marginal cost of producing = 40 per unit
• Neither firm faces any competitor
• Suppose now that the retailer and the manufacturer are separate firms.

Write down the profit function of the manufacturer if it sells to the retailer at w. You solved for the quantity the manufacturer will sell a minute ago, so you can use that in the profit function, and then you will have a profit function for the manufacturer that does not have p, but only has w in it. Solve for the optimal w for the manufacturer to charge and calculate the quantity sold under that w.