Analiza producției

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Pricing Tests and Price Elasticity for one product

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Description of the Price Elasticity model

Thes model helps you solve two common pricing policy questions.

Challenge #1: How do we set the price for a single product to maximize its revenue?

This application uses results of pricing tests to estimate impact of prices on unit sales and revenue. The results can helpyou optimize revenues with limited pricing experiments.

The key to estimating sales from pricing tests is to estimate the price elasticity from test results. The mathematics of price

elasticity is described in the appendix below.

The model computes revenue as price * sales units.

The model includes two types of price elasticity analysis. Simple (or constant) price elasticity. (This is the only elasticity model in the Standard version.) This is the

standard 'textbook' model of price elasticity, based on the relation sales units = constant x price^elasticity,

where elasticity is a constant derived from pricing test data.

Generalize (or non-constant) price elasticity. (Advanced versions offer both elasticity models.) This type of

analysis allows the elasticity to change as a function of price. For a given set of pricing test data, this method

is better at identifying prices that maximize revenue or profits.

Challenge #2: How do we set the price for a single product to maximize profits?

The model computes profit margin as Revenue Costs. Costs can be cost of goods or total costs or whatever you choose,

in order to compute gross margin, operating margin or any other profit margin you choose.

The key step in computing profit margins is to estimate costs as a function of sales levels, as described in the Technical

Notes below.

The model includes Excel charts that provide graphical views of key variables. These charts are part of the model, and

they are included by default in exported Excel workbooks. You can add more charts, import them, and the new charts willbe included in exported Excel workbooks.

As you explore the model, we suggest that you

Read some of the Excel comments that are attached to Analysis Variables throughout the workbook.

These comments also appear in ModelSheet in convenient places.
http://www.modelsheetsoft.com/refer.aspx?s=priceelasticity01.xlsmailto:[email protected]:[email protected]://www.modelsheetsoft.com/refer.aspx?s=priceelasticity01.xls


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Technical Notes

Challenge #1: How do we set the price for a single product to maximize its revenue?

The model assumes that, at a reference price (variable Reference_Price), sales units are known (variable

Reference_Sales_Units).

Assume we have a number of test markets, each of which sells a known number of units (variable Test_Ref_Sales_Units)

at the reference price. The characteristics of test markets that affect the validity of pricing tests are described in the

appendix below.

The model uses the test results (variable Test_Sales_Units) at prices (variable Test_Prices) that you specify for each test

market to estimate the price elasticity (Test_Elasticity) using the regression formula

Test_Sales_Units = Test_Ref_Sales_Units * (Test_Price/Price0) ^ Test_Elasticity

Where does this equation come from?This equation is the exponentiation of the linear regression equation

Ln(Test_Sales_Units) = Ln(Test_Ref_Sales_Units) + Ln(Test_Price/Price0) * Test_Elasticity

This equation is the exponentiated equivalent of a linear approximation, or a first-order Taylor series. Both of these

equations come from accepting the approximation that Test_Ref_Sales_Units and Test_Elasticity are constant over a

range of prices. Their greatest weakness is that, in reality, the elasticity may change as price changes.

Using the results of the regression, the application predicts sales units (Predicted_Sales_Units) for a range of prices

(Prediction_Prices) acording to the formula

Predicted_Sales_Units = Reference_Sales_Units * (Prediction_Price / Reference_Price) ^ Test_Elasticity

Challenge #2: How do we set the price for a single product to maximize profits?

You provide a value for costs at the reference level of sales units, and an estimate of elasticity of costs

with respect to sales units (variable Cost_Elasticity).

This cost model enables you to provide accurate cost information at one sales level, and to model

non-linearities in costs due to economies of scale and capacity limitations. The model estimates costs at each predicted level of unit sales using the formula

Predicted_Cost = Reference_Cost * (Predicted_Sales_Units / Reference_Sales_Units) ^ Cost_Elasticity

Appendix: Introduction to Price Elasticity

Price elasticity is important because it tells how price affects sales levels.

Definition of price elasticity (denoted epsilon)

Let's denote the sales units of a product by Q (for quantity) and the price by P.

Price elasticity relates changes in price to changes in unit sales, according to the formula

Q/Q = epsilon P/P .


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epsilon = ln(Q) / ln(P)

Use of price elasticity to predict sale units at a given price

For price P, the unit sales Q will be, for some constant K,

Q = K * P ^ epsilon

If we have a reference price P0 at which we know that unit sales are Q0, then the relationship can be stated as

Q = Q0 * (P/P0) ^ epsilon

If prices rise by 1%, then unit sales change by epsilon * 1%, and revenues change by (1+epsilon) * 1%.

Different ranges of values for price elasticity indicate different types of reaction to price changes.

epsilon > 0: increasing prices increases unit sales. This occurs for some luxury goods with status appeal.

epsilon = 0: changing price will not affect unit sales, so revenue grows directly with price.

-1 < epsilon < 0: increasing price decreases unit sales, but increases revenues.

epsilon = -1: increasing price decreases unit sales but does not affect revenues at all.

epsilon < -1: increasing prices decreases unit sales so much that revenue declines.

Goods with price elasticities in these ranges often have the following characteristics.

epsilon > 0: luxury goods with status appeal

epsilon = 0: price-inelastic goods

epsilon 0: normal commodities

epsilon < -1: commodities that are overpriced

Using pricing tests to estimate price elasticity

If you can offer different prices in different markets, you can measure the impact of price changes on sales.

Many factors can distort the results of pricing tests.

The different markets may have different distributions of customer behavior, so that combining them in one

test will distort results.

Extraneous factors may distort results unevenly across test markets, such as seasonal effects, amount

of advertising and other promotion, strength of distribution channels.

Price sensitivity may differ over the range of prices tested. To get accurate estimates of price sensitivity, you

must perform separate tests over smaller price ranges.

Customers in one market may find out about lower prices in other markets. This may cause people to buyin markets where you did not expect them to buy, which is likely to distort test results.

Customers may learn that the prices are temporary. If this happens, lower prices will generate artificially

high sales, and higher prices will generate artif icially low sales, as people wait for the test to end.

If your test design minimizes the impact of these problems, you can probably perform a pricing test that yields a good

estimate of price elasticity.

Cost elasticity

We can use price elasticity and cost elasticity to estimate the price that maximizes profits.

Assume we know that, when sales are Q0 units, then costs are Cost0. (This can be cost of good or Cogs pluas opertating

expense, depending what kind of margin you want.) Assume costs as a function of units sold are:

Cost = Cost0 * ((Sales Units) / Q0) ^ Cost_Elasticity


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Challenge #1: How do we set the price for a product line with several products to maximize its revenue?

This application uses results of pricing tests to estimate impact of prices on unit sales and revenue for several related

products. The results can help you optimize revenues with limited pricing experiments.

The key to estimating sales from pricing tests is to estimate the price elasticity for each product, and price cross-elasticities

for each pair of products from test results. The mathematics of price cross-elasticity is described in the appendix below.

The model computes revenue as sum(price[product] * sales_units[product], product in product line)

Challenge #2: How do we set the price for a product line of several products to maximize profits?

The Advanced version of the model computes profit margin as Revenue Costs. Costs can be cost of goods or total costs

or whatever you choose, in order to compute gross margin, operating margin or any other profit margin you choose. This

model assigns costs to each product, but costs can be joint costs of several products.

The key step in computing profit margins is to estimate costs as a function of unit sales levels for the products, asdescribed in the Technical Notes below.

As you explore the model, we suggest that you

Read some of the Excel comments that are attached to Analysis Variables throughout the workbook.

These comments also appear in ModelSheet in convenient places.

View worksheet "Formulas" which shows the named variables and symbolic formulas of the model

in a compact and readable form. The symbolic formulas are not active in this Excel workbook, but they

give you some idea how the model works, and how it looks in ModelSheet.

This Excel workbook was generated by ModelSheet on July 4, 2010, except for this worksheet of comments.

Copyright 2009, 2010 ModelSheet Software, LLC

ModelSheet and the ModelSheet logo are registered trademarks of ModelSheet Software, LLC.


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The predictions in these graphs are based on the Generalized Elasticity method.

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100

$80 $85 $90 $95 $100 $105 $110 $115

Predicted Sales Units versus Price

0102030405060708090

100110120130140150160

$80 $85 $90 $95 $100 $105 $110 $115 $120

Sales Units versus Price in Test Markets


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The predictions in these graphs are based on the Generalized Elasticity method.

$0

$0

$0

$0

$0

$1

$1

$1

$1

$1

$1

$90 $95 $100 $105 $110

Predicted Revenue and Profit versus Price

Predicted Revenue Predicted Margin

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20

Normalized Sales Units versus Price in Test Markets


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Company Name ABC, Inc.

Product Widgets

Name of Test Price Test with Standard Product

Reference Price $100.00

Reference Sales Units 1,000.00

Test Prices

Prices Prices Normalized

Test 1 $95.00 Test 1 0.950

Test 2 $105.00 Test 2 1.050

Test 3 $116.00 Test 3 1.160

Test Sales Units

Test Ref Sales Units Test Sales Units Sales Units Normalized

Test 1 333.33 Test 1 Test 1 0.000

Test 2 333.33 Test 2 Test 2 0.000

Test 3 333.33 Test 3 Test 3 0.000

Total 1,000.00 Total 0.00 Total 0.000

Elasticity #VALUE! Std Error Elasticity #VALUE! R squared #VALUE!

Elasticity Coefficients Std Error Coefficients R squared #VALUE!

1 #VALUE! 1 #VALUE!

2 #VALUE! 2 #VALUE!

Generalized Elasticity

ABC, Inc.WidgetsPrice Test with Standard ProductPrice Test

Shaded cells are input cells. You can enter data in them.Excel formulas in shaded cells are starting suggestions. You can overwrite them.

Reference Information

Regression Statistics

Simple Elasticity (log-linear regression)

Regression equation:

Sales_Units/Ref_Sales_Units = (Price/Ref_Price)^Elasticity

Generalized Elasticity


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Price 1 #VALUE!

Price 2 #VALUE!

Price 3 #VALUE!

Price 4 #VALUE!

Price 5 #VALUE!

Regression equation:

Sales_Units/Ref_Sales_Units = (Price/Ref_Price) Elast1 + (Price/Ref_Price) (2*Elast2)


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Genearalized Elasticity? TRUE

Prediction Prices Predicted Sales Units Predicted Revenue

Price 1 $90 Price 1 #VALUE! Price 1 #VALUE!

Price 2 $95 Price 2 #VALUE! Price 2 #VALUE!Price 3 $100 Price 3 #VALUE! Price 3 #VALUE!



Predicted Costs Predicted Margin Predicted Margin %

Price 1 #VALUE! Price 1 #VALUE! Price 1 #VALUE!Price 2 #VALUE! Price 2 #VALUE! Price 2 #VALUE!

Price 3 #VALUE! Price 3 #VALUE! Price 3 #VALUE!



Reference Sales Units 1,000.00 Reference Cost $72,000.00 Cost Elasticity 0.70

Shaded cells are input cells. You can enter data in them.

ABC, Inc.

Widgets

Price Test with Standard Product

Sales Predictions

Cost = Reference_Cost *(Predicted_Sales_Units/Reference_Sales_Units)^Cost_Elasticity

Cost Parameters

Excel formulas in shaded cells are starting suggestions. You can overwrite them.

Sales Predictions

Sales Units = Reference_Sales_Units * (Prediction_Price/Reference_Price)^Units_Elasticity

Cost and Margin Predictions


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Variable Display As Dimension Index Formula / Data

Company_Name Company Name

Cost_Elasticity Cost Elasticity

Non_Constant_ElasticityQ Genearalized Elasticity? Global Data: True

Non_Constant_ElasticityQ_plt Genearalized Elasticity? Non_Constant_Terms Data: Non_Constant_ElasticityQ

Predicted_Costs Predicted Costs Predictions Data: var(Reference_Cost*(Predicted_Sales_Units/Reference_Sales_Units)^Cost_Elasticity)

Predicted_Margin Predicted Margin Predictions Data: Predicted_Sales_Units*Prediction_Prices-Predicted_Costs

Predicted_Margin_pct Predicted Margin % Predictions Data: Predicted_Margin/Predicted_Revenue

Predicted_Revenue Predicted Revenue Predictions Data: Prediction_Prices*Predicted_Sales_Units

Predicted_Sales_Units Predicted Sales Units Predictions Data: var(Reference_Sales_Units*if(Non_Constant_ElasticityQ, exp(Prediction_Units_Factor_NonConst), (Predic

Prediction_Prices Prediction Prices Predictions Data: 5*round(minr(ranged("Tests", Test_Prices))/5+(dimitemnum("Predictions")-2)*(maxr(ranged("Tests", Test_P

Prediction_Units_Factor_NonConst Prediction Sales Units Factor Predictions Data: var(ln(Prediction_Prices/Reference_Price)*Test_Elasticity_NonConst12["Non_Constant_Terms.1"]+ln(Predic

Product_Name Product

Reference_Cost Reference Cost

Reference_Price Reference Price

Reference_Sales_Units Reference Sales Units

Test_Elasticity Elasticity Global Data: linest(1, 1, false, ranged("Tests", Test_Ln_Sales_Units_Normed), ranged("Tests", Test_Ln_Prices_Normed

Test_Elasticity_NonConst Generalized Elasticity Predictions Data: var(Test_Elasticity_NonConst12["Non_Constant_Terms.1"]+2*Test_Elasticity_NonConst12["Non_Constant_

Test_Elasticity_NonConst12 Elasticity Coefficients Non_Constant_Terms Data: var(linest(1, dimitemnum("Non_Constant_Terms"), false, ranged("Tests", Test_Ln_Sales_Units_Normed),

Test_Ln_Prices_Normed Ln Prices Tests Data: ln(Test_Prices_Normed)

Test_Ln_Prices_Normed_Sq (Ln Prices)^2 Tests Data: Test_Ln_Prices_Normed^2

Test_Ln_Sales_Units_Normed Ln Sales Units Tests Data: Ln(Test_Sales_Units_Normed)

Test_Name Name of Test

Test_Prices Prices Tests Data: round(1.1*prevde(1.1^(-0.5*length(ranged("Tests", Test_Prices)))*Reference_Price, "Tests"), 0)

ABC, Inc.

Widgets

Price Test with Standard Product

Formulas


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Variable Display Label Comment

Company_Name Company Name Name of the company or organization doing the pricing test

Cost_Elasticity Cost Elasticity The elasticity of cost with respect to changes in units sold; that is, the price sensitivity of costs

to unit sales. The cost used in the model can be cost of goods or operating cost, depending on

what kind of profit margin you want to measure.

Non_Constant_ElasticityQ Genearalized Elasticity? Controls whether simple constant elasticity method or the generalized elasticity method is

used to predict sales based on price. Enter 'True' or 'False'.

The generalized elasticity method builds on constant elasticity by adding nonlinear terms

(quadratic in ln(price)) to the elasticity factor.

Non_Constant_ElasticityQ_plt Genearalized Elasticity? Controls whether simple constant elasticity method or the generalized elasticity method. Used

for graphs.

Predicted_Costs Predicted Costs The predicted costs for each level of unit sales predicted by the price elasticity and cost

elasticity analyses

Predicted_Margin Predicted Margin The predicted profit margin (e.g. contribution margin or operating margin) at each price level

using the predictions for sales units and costs

Predicted_Margin_pct Predicted Margin % The predicted profit margin percentage, defined as (profit margin) / revenue, for each

prediction price level

Predicted_Revenue Predicted Revenue The predicted revenue at each price level using the predictions for sales units

Predicted_Sales_Units Predicted Sales Units The predicted sales units at each price level

Prediction_Prices Prediction Prices The prices at which sales units of the product are predicted

Prediction_Units_Factor_NonConst Prediction Sales Units Factor The prediction factor for sales units at each price level

Product_Name Product

Reference_Cost Reference Cost The cost incurred when sales units equal the reference amount (Reference_Sales_Units)

Reference_Price Reference Price The reference price at which we know that sales units equal Reference_Sales_Units

Reference_Sales_Units Reference Sales Units The number of units sold when the price is the reference price (Reference_Price)

Test_Elasticity Elasticity Elasticity of sales units with respect to changes in price, estimated from pricing tests

Test_Elasticity_NonConst Generalized Elasticity Elasticity of sales units with respect to changes in price, as a function of price level. Estimated

from pricing tests.

Note on definitions: Price elasticity it defined by the local relationship of sales units and price:

elasticity = price/units * d(units)/d(price). Only in the case of constant elasticity does this imply

the well-known relationship units = constant * price^elasticity over a range of prices.

Test_Elasticity_NonConst12 Elasticity Coefficients Coefficients that define non-constant price elasticity of sales units with respect to changes in

prices, estimated from pricing tests

Test_Ln_Prices_Normed Ln Prices Ln(Test_Prices / Reference_Price), for use in the regression equationTest_Ln_Prices_Normed_Sq (Ln Prices)^2 Ln(Test_Prices / Reference_Price)^2, for use as quadratic term in nonlinear regression

equation for price elasticity

Test_Ln_Sales_Units_Normed Ln Sales Units Ln(Test_Sales_Units / Reference_Sales_Units), for use in the regression equation

Test_Name Name of Test A name that uniquely identifies this pricing test

Test_Prices Prices The prices used in the various pricing tests

Test_Prices_Normed Prices Normalized The ratio (test price)/(reference price) for each test market. The log of this variable is used in

the regression equation to compute elasticity.

Test_R_squared R squared The coefficient of determination (commonly known as r-squared) for the regression of log units

versus log prices in test markets

Test_R_squared_NonConst R squared The coefficient of determination (commonly known as r-squared) for the regression of log units

versus log prices in test markets

ABC, Inc.WidgetsPrice Test with Standard ProductLabels


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Test_Ref_Sales_Units Test Ref Sales Units The reference sales units that can be sold in each test market when the price is the reference

price (Reference_Price)

Test_Sales_Units Test Sales Units The sales units measured at the test price in each test market in the pricing test

Test_Sales_Units_Normed Sales Units Normalized The ratio (test sales units)/(test reference sales units) for each test market. The log of this

variable is used in the regression equation to compute elasticity.

Test_StdError_Elasticity Std Error Elasticity The standard error of the estimate of the elasticity from the market testTest_StdError_Elasticity_NonConst Std Error Coefficients The standard error of the estimate of the elasticity from the market test

Dimension (item) Display Item As Total As Level As Comment

Non_Constant_Terms Non-Constant Terms Total Non-Constant_Terms

1 1 Nonlinear_Order

2 2

Predictions Predictions Total Predictions A list of the price levels at which sales units are predicted from the market test data

Price_1 Pr ice 1 Predic t ions

Price_2 Price 2

Price_3 Price 3

Price_4 Price 4

Price_5 Price 5

Tests Tests Total Tests A list of the test markets where the pricing test is performed

Test_1 Test 1 Tests

Test_2 Test 2Test_3 Test 3

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