1. the analytics of constrained optimal decisions microeco nomics spring 2016 dynamic pricing (I) ………….1setup ………….2 uniform pricing assignment eight ………….4 dynamic pricing

2. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |1 airline industry  We observe an increase in airline ticket price as the date of take-off approaches  Clearly the static/uniform pricing monopoly model is too simple to capture and explain this behavior  We have to develop a dynamic model that would capture: - the market participants in each time interval - the optimal behavior of the monopolist  The market : consists of many buyers (more than the capacity of the plane) with different willingness to buy the airline ticket, each buyer will buy at most one ticket when the price for the ticket is below buyer’s willingness to pay  Time frame : there are two periods considered such that - in the first period a certain fraction of the total buyers are searching for an airline ticket and these buyers are those with the lowest willingness to pay - in the second period the remaining fraction of the buyers (that either did not get a ticket in the first period or did not participate in the first period) are searching for an airline ticket  No arbitrage : buyers with low valuation that buy a ticket in the first period are not able to re-sell the ticket in the second period. setup assumptions

3. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |2 airline industry 500 250 demand MR 250 ► Demand: P ( Q ) = 500 – Q ► Marginal revenue is obtained as MR ( Q ) = 500 – 2 Q ► Marginal cost is MC ( Q ) = 0 ► Optimal output (A) and price (B) Q m = 250 and P m = 250  With uniform pricing tickets are sold at the same price and the whole market is served at once.  Notice that 50 seats will be empty (airplane capacity is 300).  Total revenue (and profit since costs are assumed to be zero) is in this case: TR = P m ∙ Q m = 250∙250 = 62,500 capacity = 300 300 (A) (B) key points uniform pricing

4. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |3 airline industry uniform pricing - model ► How would you formulate the problem of finding the optimal solution for the uniform pricing case? ► “Ingredients”: demand P = 500 – Q objective maximize profit  = P  Q = (500 – Q )  Q for 0  Q  300 ► From an algebraic perspective, just take the first derivative of the objective function (with respect to Q ), set that expression to zero and solve for the Q ; this will give you 500 – 2 Q = 0 with solution Q = 250. ► From a modelling perspective the analyst is considering a “one-shot” market even though the market may be functioning several periods. In other words, say the market is open for two days, this formulation of the problem implies that the same price is offered in both periods.

5. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |4 airline industry dynamic approach - introduction 500 demand high value 500 – M low value first period market M second period market 500 – M ► Let’s assume that there are M buyers that are searching for a ticket in the first period (and these are the buyers with the lowest valuation) ► The remaining 500 – M are entering the market only in the second period. ► The buyers that entered the market in the first period but could not buy a ticket could potentially re-enter the market in the second period, however they will not get a ticket in the second period for sure (since they’ll have a valuation far below the valuation of all others present in the market)

6. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |5 airline industry myopic pricing 500 M demand 0 ► First period market ► We can think of the first period market as a “stand-alone” market with a demand P 1 ( Q ) = M – Q and a corresponding marginal revenue MR 1 ( Q ) = M – 2 Q ► Optimal output (C) and price (D) Q 1 = M/ 2 and P 1 = M /2  Only M /2 tickets are sold in the first period which means there are another 300 – M /2 seats available for the second period market.  Of course this means that there are M /2 persons ( M total trying to buy a ticket in the first period, but only M /2 got a ticket) still being in the market in the second period M M /2 tickets sold first period MR 1 first period market (C) (D) ticket move to second period key points

7. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |4 airline industry myopic pricing 500 demand 250 ► Second period market ► We know there are 300 – M /2 tickets left for the second period market. Notice that in the second period the airline will not sell more than 250 tickets. ► Thus:  case (i) if 300 – M /2  250 ( M  100) the airliner will limit the second period ticket sale to 250  case (ii) if 300 – M /2 100) the airliner will have a positive marginal revenue for the last ticket available for the second period market thus it will sell all the remaining tickets ► There are 500 – M “high value” buyers entering the second period market.  case (ii) if 300 – M /2 < 500 – M ( M < 400) the airliner will not be able to serve all “high value” buyers in the second period; otherwise all “high value” buyers will be serve MR 2

8. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |7 airline industry myopic pricing 500 demand 500 – M residual capacity = 300 – M /2 250 ► Second period market M  100 ► We can think again of the second period market as a “stand-alone” market with a demand P 2 ( Q ) = 500 – Q for Q ≤ 500 – M and a corresponding marginal revenue MR 2 ( Q ) = 500 – 2 Q for Q ≤ 500 – M ► Optimal output (E) and price (F) Q 2 = 250 and P 2 = 250  Only 250 tickets are sold in the second period since there are too many tickets left from first period market.  The buyers moved from first period have their willingness to pay too low to count in the second period. (E) (F) 250 tickets sold second period 0 MR 2 125 300 – M /2 key points second period market

9. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |8 airline industry myopic pricing 500 demand 500 – M residual capacity = 300 – M /2 300 – M /2 ► Second period market 100 < M  400 ► We can think again of the second period market as a “stand-alone” market with a demand P 2 ( Q ) = 500 – Q for Q ≤ 500 – M and a corresponding marginal revenue MR 2 ( Q ) = 500 – 2 Q for Q ≤ 500 – M ► Optimal output (E) and price (F) Q 2 = 300 – M /2 and P 2 = 200 + M /2  All remaining 300 – M /2 tickets are sold in the second period since the profit maximization is constrained by the available capacity.  The buyers moved from first period have their willingness to pay too low to count in the second period. (E) (F) 200 + M /2 tickets sold second period 0 MR 2 key points second period market

10. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |9 airline industry myopic pricing 500 demand 300 – M /2 residual capacity = 300 – M /2 500 – M second period market ► Second period market 400 < M  500 ► We can think again of the second period market as a “stand-alone” market with a demand P 2 ( Q ) = 500 – Q for Q ≤ 500 – M and a corresponding marginal revenue MR 2 ( Q ) = 500 – 2 Q for Q ≤ 500 – M ► Optimal output (E) and price (F) Q 2 = 500 – M and P 2 = M  Not all remaining 300 – M /2 tickets are sold in the second period since the profit maximization is constrained by the number of “high value” buyers.  The buyers moved from first period have their willingness to pay too low to count in the second period. (E) (F) M tickets sold second period 0 MR 2 key points

11. microeconomic s the analytics of constrained optimal decisions assignment 8 dynamic pricing (I)  2016 Kellogg School of Management assignment 8 page |10 airline industry myopic pricing prices tickets P1P1 P2P2 Q2Q2 Q1Q1 ► First period market ► Optimal output and price: Q 1 = M/ 2 and P 1 = M /2 ► Second period market ► Optimal output and price: 0 < M  100: Q 2 = 250 and P 2 = 250 100 < M  400: Q 2 = 300 – M /2 and P 2 = 200 + M /2 400 < M  500: Q 2 = 500 – M and P 2 = M