Document Sample

Price elasticities and linear regression A short review Price elasticities • We are interested to measure the effect of price changes on quantity sold. • The price elasticity can be computed as follows: % Change in unit sales %Q Q P Price Elasticity % Change in price %P P Q • An example: – Suppose we found out that if we raise price by 5%, sales will drop by 10% – What is the demand elasticity with respect to price? Solution • Fill in the parts of the formula: – %ΔQ= - 0.1 (-10%) – %ΔP= 0.05 (5%) • So the elasticity equals -0.1 / 0.05 = -2 • This elasticity is called point elasticity. Its value depends on the price-point where it is calculated Another example • Suppose we observe the following data: – In period 1, sales equals 100 units, at a price of $ 8.00. – In period 2, sales equals 140 units, at a price of $ 6.00. • What is the price elasticity? Answer • Compute the percentage changes: – Price was decreased from $8 to $6, which is 25% – Sales increases from 100 to 140, which is 40%. • Fill this in in the formula: – The elasticity equals 0.4 / -0.25 = -1.6 • Next, we introduce arc-elasticity. Why arc-elasticity? • Suppose price changes from $3.00 to $4.00 • This equals an increase of 33%. • However, if we have a price changes from $4.00 to $3.00, we have a reduction of 25%. • Thus, there are two ways of viewing what happens between $3.00 and $4.00. Solution: arc-elasticity • Use average price and quantity Q / Q Price Arc - Elasticity P / P • where the bar above the symbol means that we take the average of the observed values Example of arc-elasticity • Company X has done the following marketing research for product Z • When they price their product at $100, the demand equals 1000 units. • If they price the product at $90, the demand equals 1200 units. • What is the arc-elasticity for product Z? Answer • Use the formula: – The numerator equals (1200-1000) / ( average[1200,1000] ) – The denominator equals (90-100) / ( average[90,100] ) • Combined, this yields an arc-elasticity of 0.1818 / -0.1053 = -1.7273 Arc versus point-elasticity • We could also have computed the point elasticities for this product, for the two price points given • Changing from $100 to $90, the point- elasticity would be -2 • Changing from $90 to $100, the point- elasticity would be -1.5 • As the two points get closer together, arc- elasticity approaches the point elasticities Regression Use it to determine price sensitivity Regression • Another way to obtain price sensitivity estimates is through regression • Regression is a mathematical technique that we will use to relate sales to price levels • This enables: – predictions of future sales levels – price optimization Simple linear demand model Input data for regression: • A standard place to begin is: p = price q = quantity qi = a + b pi + ei Results from the estimation: • Elasticity is a function of b: a = intercept b = slope q p p Elasticity b p q q • We can also use the model to predict profits Profit = q (p - c) • Using calculus we can find the optimal price: profit bc a c a 0 p * p 2b 2 2b REMEMBER: Input data for regression: p = price Example q = quantity Results from the estimation: a = intercept b = slope • Suppose we apply regression to sales data, and find the following demand function: • q = a + b p = 100 – 4p • Unit cost equals $5 • What is the optimal price? • p* = [(-4 * 5) – 100] / [2*-4] = 15 • What is the price elasticity when p = 12? • When p = 12 then q = 52 • So the elasticity = b * (p/q) = -0.92 Is this really optimal?? • Use sensitivity analysis: using the regression result, compute q and profit for various levels of p. This gives the following table... and graph p q profit Profit for various prices $9 64 $256 $450 $10 60 $300 $400 $11 56 $336 $350 $12 52 $364 $300 $13 48 $384 Profit $250 $14 44 $396 $200 $15 40 $400 $150 $16 36 $396 $100 $17 32 $384 $50 $18 28 $364 $0 $19 24 $336 $10 $12 $14 $16 $18 $20 price (p) $20 20 $300 Log demand model • If the relationship between price and sales is non-linear, it is better to use a log model: AGAIN: ln(qi )= a + b ln(pi ) + ei Input data for regression: • Elasticity is a function of b: p = price q = quantity Results from the estimation: q p Elasticity b a = intercept p q b = slope • We can also use the model to predict profits Profit = q (p - c) • Using calculus we can find the optimal price: profit c 0 p * p 1 1 b Relationship with cost-plus • Note that this optimal price c p * 1 1 b resembles the cost-plus markup rule: p = c / (1 - margin) • In cost-based pricing, this margin is often determined beforehand, which is generally not a good pricing approach. • However, using regression, it is the customer’s price sensitivity which enters the cost-plus markup rule, by setting margin = -1/b • This margin maximizes profit Example • Suppose we find that there is a non-linear relationship between sales and price… – Then we estimate the log-demand model. • Suppose we find the following parameters: – ln(q) = 1 - 2 ln(p) • What is the optimal price, using unit cost=5? • Answer: p* = 5 / [1+ 1/-2 ] = 10 • What is the price elasticity when p = 12? • Answer: The elasticity is equal to b, so it’s equal to -2 When to use what? • Price elasticity alone can only be used to determine an optimal price if we are dealing with a constant elasticity across prices (for example: log-demand model). Each price-quantity point on the function has the same elasticity, but there is only one price that maximizes profit. • For linear price-quantity schedules (for example: linear regression), we saw that we need both the slope and intercept to determine optimal price. Such demand functions have an infinite number of point elasticities (it is different for each price), but only one point maximizes profits. • Using the slope from a finite interval as an approximation of point elasticity will yield a different estimate, depending on whether the movement is "up" or "down". Arc elasticity averages these movements to produce a single value for elasticity. However, arc elasticity, nor point elasticity alone, based on these intervals, can be used to find optimal prices. • When in doubt, use a table of values for prices, quantity, and profits to confirm optimal price calculations. These tables have the added value of providing insight into the sensitivity of prices, quantities, and profits. General tip for completing this quiz.. • It makes sense to create a spreadsheet with important formulas, while making the quiz. • You can find the important formulas in this PPT • This will save you a lot of time ! Excel and regression • Excel contains a Regression Add-In – (See Tools | Data Analysis | select regression) • If Data Analysis doesn’t show up: – you may need to add an add-in, through Tools | Add-ins. Choose Analysis ToolPak • On the next slides, we have a step-by-step description of how to perform regression in Excel How to perform regression in Excel • Save the data to an Excel-file (there’s a link under the table) • Open the Excel-file • The regression add-in accepts only variables in columns, so you may need to transpose the data: • Copy the data • Select a destination cell (e.g. A7) • Select PasteSpecial and check ‘transpose’ • Click okay • Run the regression Add-in (see next slide) Run the regression add-in • Select Tools | Data Analysis | Regression • Set Input Y Range: – click next to the textbox – Select the range • Do same for X range • Check Labels if your ranges contain the variable labels Results Regressing sales on price will produce the following result. The most important numbers are in bold Interpretation of output: 1. The R-Square describes how well the model fits the observed data. Closer to 1 means better fit. 2. Intercept displays the value for a in the regression equation qi = a + b pi + ei 3. Price displays the value for b 4. The standard errors tells us how certain we are about these values. The lower the standard error, relative to the coefficient value, the more certain we are. Log-demand model • If you need to use regression for the log- demand model ln(qi )= a + b ln(pi ) + ei • then – You will need to ln-transform the data first – Use extra columns for these data (e.g. 2 slides back: columns D and E) – In these cells, use the formula LN(…) – Apply regression to these cells, in stead of the original ones. Finally… • You can use the output to compute optimal prices and predict sales. • Use the formulas in this PPT • ... and make that Excel-sheet with useful formulas, and use it for all your future pricing questions!

DOCUMENT INFO

Shared By:

Categories:

Tags:
price elasticities, price elasticity, demand elasticities, percentage change, income elasticity, quantity demanded, price changes, price elasticity of demand, income elasticities, Natural Gas

Stats:

views: | 5 |

posted: | 3/26/2011 |

language: | English |

pages: | 26 |

OTHER DOCS BY nikeborome

How are you planning on using Docstoc?
BUSINESS
PERSONAL

Feel free to Contact Us with any questions you might have.