the analytics of constrained optimal decisions microeco nomics spring 2016 the monopoly model (I): standard pricing ………….1optimal production ………….2 optimal allocation (I) assignment four ………….5optimal allocation (II)

microeconomic s the analytics of constrained optimal decisions assignment 4 the monopoly model (I): standard pricing  2016 Kellogg School of Management assignment 4 page |1 optimal production / profit maximization ► Banana Republic’s customer have the following demand: P = 140 – 6 Q ► Item are produced at a constant marginal cost: MC = 20 demand monopoly quantity Q m = 10 PmPm monopoly price P m = 80 MR MC = 20 QmQm  marginal revenue ► From the demand function we get immediately the marginal revenue MR = 140 – 12 Q  profit maximization ► Condition: MR = MC we get 140 – 12 Q = 20 with solution: Q m = 10, P m = 140 – 6∙10 = 80

microeconomic s the analytics of constrained optimal decisions assignment 4 the monopoly model (I): standard pricing  2016 Kellogg School of Management assignment 4 page |2 optimal allocation (I) / profit maximization ► Banana Republic has a stock of 12 items already produced. While it faces exactly the same demand as before ( P = 140 – 6 Q ) the marginal cost is now zero (up to maximum capacity) because any production cost is sunk. ► The problem is now one of allocation of available units (since they are already produced) rather then how many units to produce. ► Another important issue here is to clearly understand what are the allocation alternatives and their corresponding payoffs. ► The two alternatives are: (A): sell through the store – marginal revenue is MR (A) = 140 – 12 Q (B): burn units – marginal revenue is in this case MR (B) = 0 demand PmPm MR (A) MR (B) “max “ capacity

microeconomic s the analytics of constrained optimal decisions assignment 4 the monopoly model (I): standard pricing  2016 Kellogg School of Management assignment 4 page |3 optimal allocation (I) / profit maximization ► The problem can be restated as: … we have 12 units to allocate between two alternatives that give different payoffs … ► We are given the marginal revenue that we obtain for each unit allocated to a certain alternative... ► … which gives a very easy way to decide on how to allocate each unit... ► Start with the first unit…allocate it to the alternative that gives the highest marginal revenue, then the second … and so on … until the last unit is allocated. ► It’s easy to see that the marginal revenue from alternative (A) is larger than the marginal revenue from alternative (B) for the first units obtained from setting MR (A) = MR (B). demand MR (A) “max “ capacity allocation to (A) Q (A) = MR (B)

microeconomic s the analytics of constrained optimal decisions assignment 4 the monopoly model (I): standard pricing  2016 Kellogg School of Management assignment 4 page |4 optimal allocation (I) / profit maximization ► Since 12 units are sold through the stores the price will be P (A) = 140 – 6∙11.67 = 70 ► The remaining 0.33 units are burned… thus Q (B) = 0.33 ► No doubt this is puzzling… why burn those units when you can actually still get a fairly high price for them (slightly below $70/unit)? ► If you choose to still sell these 0.33 units you would actually decrease the price on all previous units thus you sell more units at a lower price… ► How do you know whether this is better or worse than burning the units? ► The marginal revenue measures exactly that: it tells you how the total revenue changes with changes in units … however MR (A) is negative beyond demand P (A) MR (A) “max “ capacity allocation to (A) Q (A) = burning these 0.33 units gives a higher marginal revenue than from selling through the chain stores MR (B)

microeconomic s the analytics of constrained optimal decisions assignment 4 the monopoly model (I): standard pricing  2016 Kellogg School of Management assignment 4 page |5 optimal allocation (II) / profit maximization ► There are three alternatives now: (A) sell through chain stores MR (A) (B) burn MR (B) = 0 (C) sell as private label MR (C) = 10 ► The problem essentially remains the same: how should we allocate the 12 units among the three alternatives given their payoffs. ► Same logic applies: each unit should be allocated to the alternative that offers the highest marginal revenue… ► The marginal revenue MR (A) is largest up to the output Q (A) = obtained from the condition MR (A) = MR (C) that is 140 – 12 Q = 10 ► What about the remaining units? demand MR (A) “max “ capacity allocation to (A) Q (A) = MR (B) MR (C) 10.83

microeconomic s the analytics of constrained optimal decisions assignment 4 the monopoly model (I): standard pricing  2016 Kellogg School of Management assignment 4 page |6 optimal allocation (II) / profit maximization ► The remaining 1.17 units can be burned, for a marginal revenue of 0, or sold under private label for a marginal revenue of 10… ► Obviously the 1.17 units are sold under private label at a price of 10 each, thus Q (C) = 1.17 ► Again, it might be counterintuitive to sell 1.17 units at $10 per unit under private label when those units could be sold at a price (slightly below) $70 through the chain stores… ► Adding the extra units to the chain stores sales would decrease the price on all previous units which would result in an overall lower revenue than selling them at $10 under the private label … ► No unit is burned, thus Q (B) = 0 demand MR (A) “max “ capacity allocation to (A) Q (A) = MR (B) MR (C) selling these 1.17 units gives a higher marginal revenue than from selling through the chain stores or burning them