A demand function is a function that maps prices onto quantities. For instance, the following single product, linear demand function: D(p) = 100 - 2p, indicates that setting a price p=10 would lead to a demand quantity of 80 units. The potential market for this product is D(0)=100 units, which corresponds to the total demand when the product is free. In particular, a linear demand function can be inferred from a market potential and a uniform distribution of the willingness-to-pay of the consumers. The reservation price or willingness-to-pay of a consumer is unobservable by the seller and represents the maximum that the individual would pay for a unit of the product. For instance, the previous demand function corresponds to a market potential of 100 units and a consumer willingness-to-pay Unif(0, 50), i.e., the willingness-to-pay is uniformly distributed between $0 and $50 (verify this connection by yourself, recalling that for a Unif(0,50) distribution, P(X>x)=1-x/50 for 0x50).

Exercise Consider a customer basis whose willingness-to-pay for a mechanic pencil is described by a continuous Unif(0,20) distribution. The potential market for this product is 30,000 units, which corresponds to the number of customers that would acquire the product in case its price were zero. Assume that product quantities can take continuous values.

a) Formulate a linear demand function that captures the willingness-to-pay distribution of this market. How many customers would buy the product at p=8? And at p=20? b) Provide a managerial problem definition for the optimization problem that maximizes the seller's revenues. Describe the objective function, decision variables and constraints. Assume that the unit cost is zero, and that the seller can hold infinite inventory. Formulate the problem in Excel. Plot revenues as a function of price, and use Excel Solver to get the optimal solution. Then, answer the questions below: i. Is this a linear program? ii. What is the optimal price? iii. What is the optimal revenue? iv. How many units does the retailer sell when setting the optimal price? v. How much (maximum) should the seller pay to procure a unit of inventory beyond the optimal quantity?

c) Assume now that the seller has 10,000 units of this mechanic pencil in inventory. i. What is the optimal price that the seller has to charge in this case? ii. What is the optimal revenue? iii. How many units would be sold at this new price? iv. How much (maximum) should the seller pay for an extra unit of inventory in this case?

d) How does your answer to (c) change if the seller had 20,000 units of inventory?

e) Going back to the original case, suppose again that the seller has unlimited inventory. Now, after conducting two regional surveys, the demand characterization is fine-tuned as follows: In the West Coast demand has a potential of 20,000 units, with a willingness-to-pay Unif(0,40); and in the East Coast it has a potential of 10,000 units, with a willingness-to-pay Unif(0,10). i. What would be the single nationwide optimal price? Hint: The nationwide demand is a piecewise linear demand function, with a switching point at p=10. ii. Now, assume that the seller could charge a different price for both regions. Should the seller take advantage of this? Justify your answer quantitatively and provide some intuition for the result.

f) Same as (e), but now the seller has 10,000 units to sell across the whole country. Should the seller charge different prices in both regions? Justify your answer quantitatively.