The Relation between Inventory Investment and Price Dynamics in a Distributive Firm Summer Workshop of Economic Theory in Hokkaido University August, 213 Akiyuki Tonogi, Institute of Innovation Research Hitotsubashi University Hitotsubashi University 1
Table of Contents 1. Introduction 2. Empirical Facts 3. The Model 4. Numerical Experiment 5. Conclusion Hitotsubashi University 2
Introduction Hitotsubashi University 3
Background Rigidity of prices is important concept in macroeconomics, especially in the New Keynesian framework Many economists have engaged in the empirical research on micro data of prices to examine the concept of price rigidity In the course of studies, we rediscover the importance of transitory discount sale in the pricing behavior Hitotsubashi University 4
Background: Discount sale Share and Frequency of Discount Sale (%) 45 4 35 3 25 2 15 1 5 Bargain Sales Ratio Daily Frequency of Bargain Sales Note: (1) The bargain sale period is defined as the period that an item is sold at a lower price than its regular price by the difference of more or equal to2 JPY. The regular price is defined by the weekly mode price. (2) The ratio of bargain sales is the share of the amount sold during bargain sale periods. Hitotsubashi University 5
Background Kehoe and Midrigan (212) Prices are sticky after all Consistent with the New Keynesian Framework Abe and Tonogi (21) Finding extreme higher frequency of daily price changes than the previous studies Importance of discount sale in pricing Sudo, Ueda, and Watanabe (213) Point out the correlation of discount sale and business cycle Hitotsubashi University 6
Purpose: Various Pricing Strategies Quantities sold are concentrated on discount sales days Heterogeneity in pricing behavior is large among stores (B) (yen) 2 quantity: RHS 18 price: LHS 16 14 12 1 8 6 4 2 28/1/1 28/11/1 28/12/1 (unit) 6 5 4 3 2 1 (A) (yen) (unit) (C) (yen) (unit) 2 18 quantity: RHS price: LHS 16 14 12 1 8 6 4 2 28/1/1 28/11/1 28/12/1 1 9 8 7 6 5 4 3 2 1 2 18 quantity: RHS price: LHS 16 14 12 1 8 6 4 2 28/1/1 28/11/1 28/12/1 3 25 2 15 1 5 Hitotsubashi University 7
Purpose: Previous Works Review of studies on inventory investment and Pricing behavior Blinder (1982) Theoretical analysis of pricing behavior with quadratic inventory storage cost in a production firm No discount sale Aguirregabiria (1999) Theoretical and empirical analysis of pricing behavior with linear storage cost and fixed order cost in a distributive firm Success to generate discount sale Target of modeling is monthly data We should regard the discount sales as phenomena in short term, which can be observed in daily data Hitotsubashi University 8
Purposes of the Paper Investigating the empirical relations among a firm s probability of price change, business scale, frequency of discount sales, and price elasticity of demand using daily scanner data Constructing a model with an (S, s) inventory policy in order to match daily pricing behavior of distributive firms Examining the dynamic nature of price and quantity behavior in numerical experiments Hitotsubashi University 9
Empirical Facts Hitotsubashi University 1
Daily Scanner Data Nikkei-POS Data from Nikkei Digital Media Item: a Cup-Noodle Period: from January 1,28 to December 31, 28 Number of Store: 253 Summary of Statistics (Pooled) Variable Obs Mean Std. Dev. Min Max storecode 92,598 69. 379.9 1 1152 date 92,598-15.7 28/1/1 28/12/31 quantity 85,27 2.8 49. 1 3,529 price 85,27 128.7 16.3 16 178 visitor 92,163 4,64.8 2,748.7 169 29,149 Hitotsubashi University 11
Fact 1: Discount Sale is a Source of Price flexibility price change probability.8.7.6.5.4.3.2 y =.2184x +.2373 (.67) R² =.4.1...1.2.3.4.5.6.7 frequency of discount sale Note: Figures in parentheses are standard deviation of the coefficient Hitotsubashi University 12
Fact 2: Capacity of Storage Brings Price Changes price change probability.8.7.6.5.4.3.2 y =.564x +.224 (.1) R² =.175.1.. 1. 2. 3. 4. 5. 6. 7. coefficient of variation for quantity Note: Figures in parentheses are standard deviation of the coefficient Hitotsubashi University 13
Fact 3: High Price Store Intend to Set Prices Sticky price change probability.8.7.6.5.4.3.2 y =.35x +.7495 (.1) R² =.588.1. 1 11 12 13 14 15 16 17 average price (yen) Note: Figures in parentheses are standard deviation of the coefficient Hitotsubashi University 14
Fact 4: More Monopolistic Power Discounts Prices More 35. 3. y = 93.357x + 9.2225 (11.92) R² =.1963 markdown rate 25. 2. 15. 1. 5...1.5..5.1.15.2.25 1/estimated demand elasticity for price Note: Figures in parentheses are standard deviation of the coefficient Hitotsubashi University 15
Fact 5: No-relation between Markdown Rate and Discount Sales Frequency frequency of discount sales.7.6.5.4.3.2.1 y =.3x +.279 (.2) R² =.1. 35. 3. 25. 2. 15. 1. 5.. average of markdown rate (%) Note: Figures in parentheses are standard deviation of the coefficient Hitotsubashi University 16
Daily Scanner Data Summary of Statistics (Between) Within M oment Obs M ean Std. Dev. M in M ax # of Days 253 364 2 344 366 Visitors/ day 253 4,62 2,499 3 11,68 Coefficient of Variance (Visitors/Day) 253.196.74.77.426 Average Price 253 129.421 11.436 13.922 159.266 Mode Price 253 133.269 13.8 11. 16. Median Price 253 133.269 13.8 11. 16. Standard Deviation of Price Level 253 11.475 4.13.194 27.874 Skewness of Price 253-1.18 1.46-7.858 2.541 Kurtosis of Price 253 6.957 1.42 1.25 14.189 Probability of Price Changes 253.299.164.3.726 Average Rate of Price Changes 253.273.669 -.358 7.522 Standard Deviation of Price Change Rate 253 5.857 5.569.76 64.85 Probability of Bargain Sales 253.283.15..662 Average Discount Rate 252-14.455 5.584-31.685-3.77 Standard Deviation of Discount Rate 252 6.439 2.999. 2.6 Sales Quantity/Day 253 2 16 2 89 Sales Amount/Day 253 2,27 1,749 22 9,853 Coefficient of Variance (Sales Quantity/Day) 253 1.72.95.53 5.72 Coefficient of Variance (Sales Amount/Day) 253 1.48.71.53 4.24 Price Elastisity of Demand 252 27. 4.75 5.1 471.7 Hitotsubashi University 17
Five Facts on the Discount Sale Fact 1: Price movements in discount sales is an important cause of price changes Fact 2: Probability of price changes has a positive correlation to capacity of goods storage in a retail store Fact 3: Probability of price changes has a negative correlation to the average price level of a retail store Fact 4: Average markdown rates has a negative correlation with price elasticity of demand Fact 5: Frequency of discount sales has no correlation to average markdown rates Hitotsubashi University 18
The Model Hitotsubashi University 19
Environment of a Distributive Firm Facing demand function in selling market Price-taker in buying market: Linear operation cost function: Power inventory storage cost function: Hitotsubashi University 2
Environment of a Distributive Firm Profit function of the distributive firm π Fixed ordering cost: Fixed menu cost: Fixed stock-out penalty cost: Indicator Functions: 1, if, if, 1, if, if, 1, if, if Hitotsubashi University 21
Environment of a Distributive Firm The firm cannot sell the goods in quantity over stored goods quantity The firm cannot buy the goods over in amount over operating fund Transition equations Inventory stocks: 1 Operating funds: Hitotsubashi University 22
Problem for the Distributive Firm max 1, 1,,,, s.t.,,,,, 1, if, if, 1, if, if, 1, if, if, 1, given,, given,,,,,,, where p,ρ, 1,, and 1,, Hitotsubashi University 23
List of Variables and Parameters Variables Parameters Selling price Price elasticity of demand Selling quantity Curvature of inventory cost Demand state Depreciation rate of inventory Purchasing price Discount rate Purchasing quantity Inventory cost technology Inventory stock Order cost Operating fund Operating cost Indicator of order Stock-out cost Indicator of price change Indicator of stock-out Current profit Hitotsubashi University 24
Discount Sales in This Model Euler equation of the model Where,,. Selling Quantity in Period t Selling Quantity in Period t+1 ここに数式を入力します * * Hitotsubashi University 25
Parameterization in Benchmark Case Parameters Value Price elasticity of demand 18. Curvature of inventory cost 2. Depreciation rate of inventory. Discount rate.1 Inventory cost technology.1 Order cost 2. Operating cost.5 Stock-out cost 5. Menu cost.3 Hitotsubashi University 26
A Output of Solved Model Hitotsubashi University 27
A Output of Solved Model Hitotsubashi University 28
Numerical Experiment Hitotsubashi University 29
Numerical Experiment: Menu Cost (A). (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (B).1 (selling price) 1.5 (sellingquantity) 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (C).3 Benchmark Case (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (D).1 (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Rise in menu cost reduces the frequency of discount sales and goods order The higher menu cost leads the higher average price and lower frequency of price changes Note: line denotes selling price (LHS) and bar denotes selling quantity (RHS) Hitotsubashi University 3
Numerical Experiment: Storage Cost (A). (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (B).5 (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (C).1 Benchmark Case (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (D).2 (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 With a low storage cost, the retail shop makes discount sales frequently With a high storage cost, the retail shop does not make discount sales frequently in order to save inventory holding costs It is complicated in substitution and income effect Note: line denotes selling price (LHS) and bar denotes selling quantity (RHS) Hitotsubashi University 31
Numerical Experiment: Order Cost (A) 1. (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (B) 1.5 (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (C) 2. Benchmark Case (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (D) 3. (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 With a very small order cost, retail shop has no bunching motivation of inventory investment The higher order cost leads to the lower frequency of inventory investments and lower frequency of price changes At the same time, the higher order cost leads to higher average price Note: line denotes selling price (LHS) and bar denotes selling quantity (RHS) Hitotsubashi University 32
Numerical Experiment: Price Elasticity of Demand (A) 14 (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (B) 18 Benchmark Case (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2 The price elasticity of demand has no effect on discount sale s frequency It only affects the markdown ratio in the discount sale.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 (C) 26 (selling price) (sellingquantity) 1.5 1 1. 8 6.95 4.9 2.85 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Note: line denotes selling price (LHS) and bar denotes selling quantity (RHS) Hitotsubashi University 33
Summary Table of Numerical Experiments elasticity of demand order cost storing cost menu cost selling price selling quantity price change bargain sale average SD max min mode average SD max min mode frequency average average rate frequency rate of discount. 1..5 1.4.91 1.4 1.67 2.1 6..5.5.5 9.2.33 1.75.1 1.1.5 1.4.91 1.4 1.43 1.88 6..5.5.43 9.21.43 7.17 18 2..1.3 1.2.5 1.4.91 1.4 1.28 1.92 6..5.5.29 13.81.14 13.81.6 1.2.5 1.4.91 1.4 1.28 1.92 6..5.5.29 13.81.14 13.81.1 1.2.4 1.4.91 1.4 1.11 1.73 6..5.5.22 13.81.11 13.81..88..89.88.89 9.5.5 1. 9. 1..5.59. NaN.5.92.3.94.88.94 5.33 3.3 1. 3. 3..67 6.69.33 6.69 18 2..1.3 1.2.5 1.4.91 1.4 1.28 1.92 6..5.5.29 13.81.14 13.81.15 1.2.5 1.4.91 1.4 1.28 1.92 6..5.5.29 13.81.14 13.81.2 1.1.5 1.4.91 1.4 1.33 1.65 5..5.5.5 8.53.33 8.33 1..89..89.89.89 9.. 9. 9. 9.. NaN. NaN 1.5 1..5 1.4.9 1.4 2. 2.51 7..5 1..6 9.77.6 7.45 18 2..1.3 1.2.5 1.4.91 1.4 1.28 1.92 6..5.5.29 13.81.14 13.81 2.5 1.2.4 1.4.91 1.4 1.11 1.73 6..5.5.22 13.81.11 13.81 3. 1.2.4 1.4.91 1.4 1.11 1.73 6..5.5.22 13.81.11 13.81 14 1.3.6 1.5.88 1.5 1.29 1.93 6..5.5.29 17.75.14 17.75 18 2..1.3 1.2.5 1.4.91 1.4 1.28 1.92 6..5.5.29 13.81.14 13.81 26 1.1.4 1.3.92 1.3 1.57 2.62 8..5.5.29 1.66.14 1.66 Hitotsubashi University 34
Conclusion Hitotsubashi University 35
Conclusion:Summary of the Work Empirical facts Variable Covariance Numerical Experiments Fact 1 Frequency of price changes Frequency of discount sales Positive Discount sale is a major reason of price changes Fact 2 Frequency of price changes Capacity of retail shops Positive Storage cost making the relation Fact 3 Frequency of price changes Average price of selling Negative Menu cost and Order cost making the relation Fact 4 Average markdown rate Price elasticity of demand Negative Price elasticity of demand affects the markdown rate in the model Fact 5 Frequency of price changes Average markdown rate Nothing High elasticity of demand has no effect on discount sale Frequency Hitotsubashi University 36
1 Conclusion Established the five empirical facts using daily POS data of a storable processed food 2 Constructed a distributive firm s model which generates an endogenous discount sales behavior without demand state or purchasing price movements 3 Succeeded to replicate the daily behavior of retail shops which are described in the five empirical facts in the numerical experiments Hitotsubashi University 37
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Thank you for listening! Hitotsubashi University 4