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assignment four the monopoly model (I): standard pricing optimal production ………….1 optimal allocation (I) ………….2 optimal allocation (II) ………….5 spring 2016 microeconomi the analytics of cs constrained optimal microeconomics assignment 4 the monopoly model (I): standard pricing the analytics of constrained optimal decisions optimal production / profit maximization 140 130 ► Banana Republic’s customer have the following demand: P = 140 – 6Q ► From the demand function we get immediately the marginal revenue MR = 140 – 12Q monopoly price Pm = 80 MR 110 ► Item are produced at a constant marginal cost: MC = 20 marginal revenue demand 120 100 90 Pm 80 70 monopoly quantity Qm = 10 60 50 40 profit maximization ► Condition: 30 MC = 20 20 MR = MC we get 140 – 12Q = 20 with solution: Qm = 10 , Pm = 140 – 6∙10 = 80 2016 Kellogg School of Management 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Qm 11.67 assignment 4 23.34 page | 1 microeconomics assignment 4 the monopoly model (I): standard pricing the analytics of constrained optimal decisions optimal allocation (I) / profit maximization 140 130 ► Banana Republic has a stock of 12 items already produced. While it faces exactly the same demand as before (P = 140 – 6Q) 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 – 12Q (B): burn units – marginal revenue is in this case MR(B) = 0 “max “ capacity demand 120 110 100 MR(A) 90 Pm 80 70 60 50 40 30 20 10 MR(B) 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 11.67 2016 Kellogg School of Management assignment 4 23.34 page | 2 microeconomics assignment 4 the monopoly model (I): standard pricing the analytics of constrained optimal decisions optimal allocation (I) / profit maximization 140 ► 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. 130 “max “ capacity demand 120 110 100 MR(A) 90 80 allocation to (A) Q(A) = 11.67 70 60 50 40 30 ► It’s easy to see that the marginal revenue from alternative (A) is larger than the marginal revenue from alternative (B) for the first 11.67 units obtained from setting MR(A) = MR(B). 20 10 MR(B) 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 11.67 2016 Kellogg School of Management assignment 4 23.34 page | 3 microeconomics assignment 4 the monopoly model (I): standard pricing the analytics of constrained optimal decisions optimal allocation (I) / profit maximization 140 ► 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) = 0.33 130 120 100 ► No doubt this is puzzling… why burn those units when you can actually still get a fairly high price for them (slightly below P(A) $70/unit)? ► If you choose to still sell these 0.33 units you would actually decrease the price on all previous 11.67 units thus you sell more units at a lower price… 60 ► The marginal revenue measures exactly that: it tells you how the total revenue changes with changes in units … however MR(A) is negative beyond 11.67 burning these 0.33 units gives a higher marginal revenue than from selling through the chain stores 110 90 ► How do you know whether this is better or worse than burning the units? “max “ capacity demand MR(A) 80 allocation to (A) Q(A) = 11.67 70 50 40 30 20 10 MR(B) 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 11.67 2016 Kellogg School of Management assignment 4 23.34 page | 4 microeconomics assignment 4 the monopoly model (I): standard pricing the analytics of constrained optimal decisions optimal allocation (II) / profit maximization 140 ► 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… 130 120 110 100 MR(A) 90 allocation to (A) Q(A) = 10.83 80 70 60 50 ► The marginal revenue MR(A) is largest up to the output Q(A) = 10.83 obtained from the condition 40 MR(A) =MR(C) 20 that is 30 MR(C) 10 140 – 12Q = 10 ► What about the remaining units? “max “ capacity demand MR(B) 0 0 1 2 3 4 5 6 7 8 10.83 2016 Kellogg School of Management assignment 4 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 11.67 23.34 page | 5 microeconomics assignment 4 the monopoly model (I): standard pricing the analytics of constrained optimal decisions optimal allocation (II) / profit maximization 140 ► 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… 130 ► Obviously the 1.17 units are sold under private label at a price of 10 each, thus Q(C) = 1.17 100 “max “ capacity demand 120 selling these 1.17 units gives a higher marginal revenue than from selling through the chain stores or burning them 110 MR(A) 90 allocation to (A) Q(A) = 10.83 80 ► 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 10.83 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 70 60 50 40 30 20 MR(C) 10 MR(B) 0 0 1 2 3 4 5 6 7 8 10.83 2016 Kellogg School of Management assignment 4 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 11.67 23.34 page | 6