Estimating Demand Curves and Using Solver to Optimize Price - Pricing - Marketing Analytics: Data-Driven Techniques with Microsoft Excel (2014) 

Marketing Analytics: Data-Driven Techniques with Microsoft Excel (2014)

Part II. Pricing

Chapter 4: Estimating Demand Curves and Using Solver to Optimize Price

Chapter 5: Price Bundling

Chapter 6: Nonlinear Pricing

Chapter 7: Price Skimming and Sales

Chapter 8: Revenue Management

Chapter 4. Estimating Demand Curves and Using Solver to Optimize Price

Understanding how pricing impacts revenues and profitability is one of the most important issues faced by managers. To do so, managers need to understand how consumers' willingness to purchase changes at different price levels and how these changes impact profitability. (That is, managers need to understand the demand curve.) To do so effectively, this chapter covers the following topics:

·        Using the Excel Trend Curve and Goal Seek to obtain back-of-the-envelope estimation of a demand curve.

·        Using the Excel Solver to determine the profit maximizing price.

·        Examining the effect of product tie-ins on the optimal product price.

·        Using the SolverTable add-in to quickly price thousands of products!

Estimating Linear and Power Demand Curves

In this section you learn how to fit the two most frequently used demand curves (linear and power) to a particular marketing situation. These estimations are used to determine a profit-maximizing price, and to do so you need to know two things:

·        The variable cost to produce each unit of the product. (Call this UC.)

·        The product's demand curve. Simply put, a demand curve tells you the number of units of your product a customer will demand at each price. In short, if you charge a price of $p per unit, the demand curve gives you a number D(p), which equals the number of units of your product that will be demanded at price $p. Of course, a firm's demand curve is constantly changing and often depends on factors beyond the firm's control (such as the state of the economy and a competitor's price). Part III, “Forecasting,” addresses these factors.

When you know UC and the demand curve, the profit corresponding to a price of $p is simply (p – UC) * D(p). After you have an equation for D(p), which gives the quantity of the product demanded for each price, you can use the Microsoft Office Excel Solver feature to find the profit-maximizing price.

Price Elasticity

Given a demand curve, the price elasticity for demand is the percentage of decrease in demand resulting from a 1 percent increase in price. When elasticity is larger than 1, demand is price elastic. When demand is price elastic, a price cut will increase revenue. When elasticity is less than 1, demand is price inelastic. When demand is price inelastic, a price cut will decrease revenue. Studies by economists have obtained the following estimates of elasticity:

·        Salt: 0.1 (very inelastic)

·        Coffee: 0.25 (inelastic)

·        Legal fees: 0.4 (inelastic)

·        TV sets: 1.2 (slightly elastic)

·        Restaurant meals: 2.3 (elastic)

·        Foreign travel: 4.0 (very elastic)

A 1 percent decrease in the cost of foreign travel, for example, can result in a 4 percent increase in demand for foreign travel. Managers need to understand the price elasticity at each price point to make optimal pricing decisions. In the next section you will use price elasticity to estimate a product's demand curve.

Forms of Demand Curves

There are multiple forms of demand curves that you can use to analyze marketing data. Using q to represent the quantity demanded of a product, the two most commonly used forms for estimating demand curves are as follows:

·        Linear demand curve: In this case, demand follows a straight line relationship of the form q = a – bp. Here q = quantity demanded and p = unit price. For example, q = 10 – p is a linear demand curve. (Here a and b can be determined by using a method described in the “Estimating a Linear Demand Curve” section of this chapter.) When the demand curve is linear, the elasticity is changing.

·        Power demand curve: In this situation, the demand curve is described by a power curve of the form q = apb, a>0, b<0. Again, a and b can be determined by the method described later in the chapter. The equation q = 100p-2 is an example of a power demand curve. If demand follows a power curve, for any price, the elasticity equals –b. See Exercise 11 for an explanation of this important property of the power demand curve. Thus, for the demand curve q = 100p-2 the price elasticity of demand always equals 2.

Estimating a Linear Demand Curve

Suppose that a product's demand curve follows a linear demand curve. Given the current price and demand for a product and the product's price elasticity of demand, determining the product's demand curve is a simple matter. The following example illustrates how to fit a linear demand curve.

Suppose a product is currently selling for $100 and demand equals 500 units. The product's price elasticity for demand is 2. Assuming the demand curve is linear, you can determine the equation of the demand curve. The solution is in the Linearfit.xls file, which is shown in Figure 4.1.

Figure 4-1: Fitting linear demand curve

image

Given two points, you know that there is a unique straight line that passes through those two points. You actually know two points on your demand curve. One point is p=100 and q=500. Because elasticity of demand equals 2, a 1 percent increase in price results in a 2 percent decrease in demand. Thus, if p=101 (a 1 percent increase), demand drops by 2 percent of 500 (10 units) to 490. Thus p=101 and q=490 is a second point on the demand curve. You can now use the Excel Trendline to find the straight line that passes through the points (100,500) and (101,490).

Begin by entering these points in the worksheet in the cell range D5:E6 (refer to Figure 4.1). Then select the range D4:E6, and on the Ribbon, in the Charts group choose Scatter &cmdarr;Scatter with Only Markers.

1. Begin by entering the points in the Linearfit.xls worksheet in the cell range D5:E6 (refer to Figure 4.1).

2. Select the range D4:E6.

3. Go to the Insert tab and in the Charts group choose Scatter &cmdarr; Scatter with Only Markers. You should now see that the graph has a positive slope.

This would imply that higher prices lead to higher demand, which cannot be correct. The problem is that with only two data points, Excel assumes that the data points you want to graph are in separate columns, not separate rows.

4. To ensure Excel understands that the individual points are in separate rows, click inside the graph.

5. On the Ribbon click the Design tab in the Chart Tools section.

6. Click Switch Row/Column in the Data section of the Design tab.

NOTE

Note that by clicking the Select Data button, you can change the source data that generates your chart.

7. Next, right-click one of the points and then click Add Trendline.

8. Click the Linear button and click Display Equation on Chart option.

9. Click Close in the Format Trendline box.

You see the straight line plot, complete with the equation referred to in Figure 4.1. Because x is price and y is demand, the equation for your demand curve is q = 1500 – 10p. This equation means that each $1 increase in price costs 10 units of demand. Of course, demand cannot be linear for all values of p because for large values of p, a linear demand curve yields negative demand. For prices near the current price, however, the linear demand curve is usually a good approximation to the product's true demand curve.

Estimating a Power Demand Curve

Recall that for a linear demand curve the price elasticity is different for each price point. If the marketing analyst believes that elasticity remains relatively constant as price changes, then she can use a power demand curve (which has constant price elasticity) to model demand for a product.

Again assume that a product is currently selling for $100 and demand equals 500 units. Assume also that the product's price elasticity for demand is known to equal 2. (In Chapters Chapter 31 and Chapter 32 you will learn some more advanced methods to estimate price elasticity). You can fit a power demand curve to this information by performing the following steps in the Powerfit.xls file, as shown in Figure 4.2.

Figure 4-2: Fitting Power Demand Curve

image

1. After naming cell E3 as a, enter a trial value for a.

2. In cell D5, enter the current price of $100. Because elasticity of demand equals 2, you know that the demand curve has the form q=ap-2 where a is unknown.

3. In cell E5, enter a formula that computes the demand when the unit price equals $100. The demand is linked to your choice of the value of a in cell E3 with the formula a*D5^-2.

4. Now use the Global Seek command to determine the value of a. This formula makes the demand for price $100 equal to 500 units. Goal Seek enables you to find a value of a cell in a spreadsheet (called the By Changing Cell) that makes a formula (called the Set Cell) hit a wanted Value. In the example you want to change cell E3 so that the formula in cell E5 equals 500. Set cell E5 to the value of 500 by changing cell E3.

5. To invoke Goal Seek, switch to the Data tab and select What-If analysis from the Data Tools Group, then choose Goal Seek from the drop down arrow. Fill the dialog box as shown in Figure 4.3. With these settings, Excel changes the changing cell (E3) until the value of the set cell (E5) matches your desired value of 500.

Figure 4-3: Goal Seek dialog box

image

A value for a = 5 million yields a demand of 500 at a price of $100. Thus, the demand curve (refer to Figure 4.2) is given by q = 5,000,000p-2. For any price, the price elasticity of demand on this demand curve equals 2.

Using the Excel Solver to Optimize Price

Often in marketing you want to maximize profit, minimize, or optimize some other objective. The Excel Solver is a powerful tool that you can use to solve many marketing (and other!) optimization problems. The Solver is an add-in. To activate the Solver proceed as follows:

For Excel 2010 or Excel 2013:

1. Select File and then Options.

2. Select Add-Ins, click Go, check the Solver add-in, and select OK.

3. Now click the Data tab and you should see the Solver add-in on the right side of the Ribbon.

For Excel 2007:

1. Click the Office Button (the oval in the left side of the ribbon) and choose Excel Options.

2. Selecting Add-Ins, click Go, check the Solver add-in, and select OK.

3. Now click the Data tab and you should see the Solver add-in on the right side of the Ribbon.

In this chapter the examples work with the Excel 2010 Solver, which is more powerful than the Solver included with previous versions of Excel. If you select Solver from the Data tab, the Solver window appears, as shown in Figure 4.4.

Figure 4-4: Solver window

image

To define a Solver model, you must specify in the Solver dialog box the following three parts of the model:

·        Set Objective or Target Cell: The objective cell contains the goal you want to maximize (like profit) or minimize (production cost).

·        Changing Variable Cells: These cells are the cells that you can change or adjust to optimize the target cell. In this chapter the changing cells will be each product's price.

·        Constraints: These are restrictions on the changing cells. For example, you might want to constrain the price for each of your products to be within 10 percent of the competitor's price.

The Excel Solver has been greatly revamped and improved in Excel 2010. The primary change is the presence of the Select a Solving Method drop-down list. From this list you must select the appropriate solution engine for your optimization problem. You can choose from the following options:

·        The Simplex LP engine is used to solve linear optimization problems. A linear optimization problem is one in which the target cell and constraints are all created by adding together terms of the form (changing cell)*(constant). Most marketing models are not linear. An exception is the classic advertising media selection model discussed in Chapter 35, “Media Selection Models.”

·        The GRG Nonlinear engine is used to solve optimization problems in which the target cell and/or some of the constraints are not linear and are computed by using typical mathematical operations such as multiplying or dividing changing cells, raising changing cells to a power, using exponential or trig functions involving changing cells, and so on. The GRG engine includes a powerful Multistart option that enables users to solve many problems that were solved incorrectly with previous versions of Excel. The Multistart option will be used extensively throughout this book.

·        The Evolutionary engine is used when your target cell and/or constraints contain nonsmooth functions that reference changing cells. For example, if your target cell and/or constraints contain IF, SUMIF, COUNTIF, SUMIFS, COUNTIFS, AVERAGEIF, AVERAGEIFS, ABS, MAX, or MIN functions that reference the changing cells, then the Evolutionary engine probably has the best shot at finding a good solution to your optimization problem. The Evolutionary engine is extensively used throughout this book.

After you have input the target cell, changing cells, and constraints, what does Solver do? A set of values for the changing cells is a feasible solution if it meets all constraints, and the Solver essentially searches through all feasible solutions and finds the set of feasible solution changing cell values (called the optimal solution) that has the best value for the target cell (largest in a maximization and smallest in a minimization). If there is more than one optimal solution, the Solver stops at the first one it finds.

Pricing Razors (No Blades!)

Using the techniques described in the “Estimating a Linear Demand Curve” section, it's easy to determine a demand curve for the product that's originally purchased. You can then use the Microsoft Office Excel Solver to determine the original product price that maximizes the sum of the profit earned from razors. Then you can show how the fact that purchasers of razors also buy blades reduces the profit maximizing price for razors.

Suppose that you currently charge $5.00 for a razor and you sell 6 million razors. Assume that the variable cost to produce a razor is $2.00. Finally, suppose that the price elasticity of demand for razors is 2, and the demand curve is linear. What price should you charge for razors?

You can determine a demand curve (assuming a linear demand curve), as shown in Figure 4.5. (You can find this data and the chart on the no blades worksheet in the file razorsandblades.xls.) Two points on the demand curve are price = $5.00, demand = 6 million razors and price = $5.05 (an increase of 1 percent), demand = 5.88 million (2 percent less than 6 million).

Figure 4-5: Optimizing razor price: no blades

image

1. Begin by drawing a chart and inserting a linear trend line, as shown in the section “Estimating a Linear Demand Curve.” You'll find the demand curve equation is y = 18 –2.4x. Because x equals price and y equals demand, you can write the demand curve for razors as follows:

equation

2. Associate the names in cell C6 and the range C9:C11 with cells D6 and D9:D11.

3. Next, enter a trial price in D9 to determine demand for that price in cell D10 with the formula =18−2.4*price.

4. Determine in cell D11 the profit for razors by using the formula =demand*(price–unit_cost).

5. You can now use Solver to determine the profit-maximizing price. The Solver Parameters dialog box is shown in Figure 4.6. Choose to maximize the profit cell (cell D11) by changing the price (cell D9).

Figure 4-6: Solver window for razor price: no blades

image

6. The model is not linear because the target cell multiplies together two quantities—demand and (price–cost)—each depending on the changing cell. Therefore choose the GRG Nonlinear option. Solver finds that by charging $4.75 for a razor, you can maximize the profit. (The maximum profit is $18.15 million.)

Incorporating Complementary Products

Certain consumer product purchases frequently result in the purchase of related products, or complementary productsTable 4.1 provides some examples:

Table 4.1 Examples of Complementary Product

Original Purchase

Tie-in Complementary Product

Men's suit

Tie or shirt

Inkjet printer

Printer cartridge

Xbox console

Video game

Cell phone

Case

If the profit from complementary products is included in the target cell, the profit maximizing price for the original product will decrease. Suppose that the average purchaser of a razor buys 50 blades and you earn $0.15 of profit per blade purchased. You can use the Excel Solver to determine how this changes the price you should charge for a razor. Assume that the price of a blade is fixed. (In Exercise 3 at the end of the chapter, the blade price changes.) The analysis is in the blades worksheet of the razorsandblades.xls file, as shown in Figure 4.7.

Figure 4-7: Optimizing razor price: blade profit included

image

To maximize profit perform the following steps:

1. Use the Create from Selection command in the Defined Names group on the Formulas tab to associate the names in cells C6:C11 with cells D6:D11. (For example, cell D10 is named Demand.)

NOTE

Recall that cell D10 of the no blades worksheet is also named Demand. When you use the range name Demand in a formula Excel simply refers to the cell named Demand in the current worksheet. In other words, when you use the range name Demand in the blades worksheet, Excel refers to cell D10 of that worksheet, and not to cell D10 in the no blades worksheet.

2. In cells D7 and D8, enter the relevant information about blades.

3. In D9, enter a trial price for razors, and in D10, compute the demand with the formula 18-2.4*price.

4. Next, in cell D11, compute the total profit from razors and blades with the following formula:

demand*(price–unit_cost)+demand*blades_per_razor*profit_per_blade

Here the demand*blades_per_razor*profit_per_blade is the profit from blades.

5. The Solver setup is exactly as shown earlier in Figure 4.6: Change the price to maximize the profit. Of course, now the profit formula includes the profit earned from blades.

Profit is maximized by charging only $1.00 (half the variable cost!) for a razor. This price results from making so much money from the blades. You are much better off ensuring that many people have razors even though you lose $1.00 on each razor sold. Many companies do not understand the importance of the profit from tie-in products. This leads them to overprice their primary product and not maximize their total profit.

Pricing Using Subjectively Estimated Demand Curves

In situations when you don't know the price elasticity for a product or don't think you can rely on a linear or power demand curve, a good way to determine a product's demand curve is to identify the lowest price and highest price that seem reasonable. You can then try to estimate the product's demand with the high price, the low price, and a price midway between the high and low prices. This approach is based on a discussion in the book Power Pricing, by Robert Dolan. Given these three points on the product's demand curve, you can use the Microsoft Office Excel Trendline feature to fit a quadratic demand curve with the following equation:

1 c04-math-001

Fitting a quadratic demand curve in this manner enables the slope of the demand curve to either become steeper or flatter, which is much more realistic than the linear demand curve that requires the slope to remain constant.

For any three specified points on the demand curve, values of ab, and c exist that makes Equation 1 exactly fit the three specified points. Because Equation 1 fits three points on the demand curve, it seems reasonable to believe that the equation can give an accurate representation of demand for other prices. You can then use Equation 1 and Solver to maximize profit, which is given by the formula (price–unit cost)*demand. The following example shows how this process works.

Suppose that a drugstore pays $0.90 for each unit of ChapStick it orders. The store is considering charging from $1.50 through $2.50 for a unit of ChapStick. It thinks that at a price of $1.50, it can sell 60 units per week. (See the ChapStickprice.xls file.) At a price of $2.00, it thinks it can sell 51 units per week and at a price of $2.50, 20 units per week. To determine what price the store should charge for ChapStick, perform the following steps.

1. Begin by entering the three points with which to chart your demand curve in the cell range E3:F6.

2. Select E3:F6, click the Scatter option on the Charts group on the Ribbon, and then select the first option for a Scatter chart.

3. Right-click a data point and select Add Trendline.

4. In the Format Trendline dialog box, choose Polynomial, and select 2 in the Order box to obtain a quadratic curve of the form of Equation 1. Then select the option Display Equation on Chart. Figure 4.8 shows the required Trend Curve Settings. The chart containing the demand curve is shown in Figure 4.9.The estimated demand curve Equation 2 is as follows:

2 c04-math-002

Figure 4-8: Pricing with a quadratic demand curve

image

Figure 4-9: ChapStick demand curve

image

5. Next, insert a trial price in cell I2. Compute the product demand by using Equation 2 in cell I3 with the formula =–44*price^2+136*price–45. (Cell I2 is named Price.)

6. Compute the weekly profit from ChapStick sales in cell I4 with the formula =demand*(price–unit_cost). (Cell E2 is named Unit_Cost and cell I3 is named Demand.)

7. Use Solver to determine the price that maximizes profit. The Solver Parameters dialog box is shown in Figure 4.10. The price is constrained to be from the lowest through the highest specified prices ($1.50 through $2.50). If you allow Solver to consider prices outside this range, the quadratic demand curve might slope upward, which implies that a higher price would result in larger demand. This result is unreasonable, which is why you constrain the price.

Figure 4-10: Solver window for quadratic demand curve example

image

You find that the drugstore should charge $2.04 for a ChapStick unit. This yields sales of 49.4 units per week and a weekly profit of $56.24.

NOTE

The approach to pricing outlined in this section requires no knowledge of the concept of price elasticity. Inherently, the Solver considers the elasticity for each price when it determines the profit-maximizing price.

NOTE

For the quadratic demand model to be useful, the minimum and maximum prices must be consistent with consumer preferences. A knowledgeable sales force should be able to come up with realistic minimum and maximum prices.

Using SolverTable to Price Multiple Products

The approach developed to price a product in the preceding section can be extended to enable a company to easily price hundreds or thousands of products. The only information required for each product is the unit cost, estimated demand for lowest possible price, estimated demand for an intermediate price, and estimated demand. Using the SolverTable add-in written by Chris Albright (available for download at www.kelley.iu.edu/albright/Free_downloads.htm) you can easily fit the quadratic demand curve to each product. SolverTable is an Excel add-in that enables you to easily vary the inputs to a Solver model and track a wanted set of outputs.

After fitting the demand curves, you can use Solver to set a price for each product to maximize the total product generated from all products by completing the following steps (the Data worksheet in the Fittingmultipledemandcurves.xls file shows the estimated demand for three products at a low ($1.10), medium ($1.30), and high ($1.50) price):

1. Use an HLOOKUP function that keys off the entry in cell A11 to place the demands for each product in E14:E16. The Solver model chooses constant a, a coefficient b for price, and a coefficient c for price2 that exactly passes through the demand points for each product.

2. Use SolverTable to loop through each product by changing the value in cell A11 in the range 1, 2, …, where n = number of products.

3. Name the range F6:H10 Lookup.

4. Copy the formula =HLOOKUP($A$11,Lookup,C14) from E14 to E15:E16 to pull the demand for each product corresponding to the product index in A11.

5. Copy the formula =E$3+$E$4*D14+$E$5*(D14^2) from F14 to F15:F16 to compute the forecasted demand based on the quadratic demand curve based on the values in E3:E5.

6. Copy the formula =(E14-F14)^2 from G14 to G15:G16 to compute the squared error in the demand forecast for each price. For each price the estimation error is simply actual demand minus demand estimated from quadratic demand curve.

7. In cell G12 use the formula =SUM(G14:G16) to compute the sum of the squared estimation errors.

8. If you minimize G12 by changing E3:E5, Solver finds the values of ab, and c that make the sum of squared errors equal to 0. You minimize the sum of squared errors instead of minimizing the sum of errors because if you do not square errors, then the sum of positive and sum of negative errors cancels each other out. Minimizing the sum of squared errors mimics the action of the Trend Curve Polynomial option described in the preceding section. The appropriate Solver Window is shown in Figure 4.11.

Figure 4-11: Solver window for SolverTable example

image

9. Choose the GRG Nonlinear option because the target cell is not constructed by adding up terms of the form (changing cell)*constant.

10. For Solver to obtain a correct solution to this problem, go to Options and check Use Automatic Scaling; then go to the GRG tab and select Central Derivatives.

11. Select Solve and you find the answer (see Figure 4.12). The Sum of Squared errors is 0 to 26 decimal places and the quadratic demand curve is 73.625+195price+87.5price^2. Because ab, and c can be negative, do not check the Make Unconstrained Variables Non-Negative box.

Figure 4-12: Use of SolverTable for price optimization

image

Using SolverTable to Find the Demand Curve for All Products

After you have a Solver model in a worksheet, you can use SolverTable to “loop through” the values of one (in a One-way SolverTable) or two (in a Two-way SolverTable) inputs and track any wanted outputs. To get started, assuming you have installed SolverTable, select SolverTable from the Ribbon, choose a One-way Table, and fill in the SolverTable dialog box, as shown in Figure 4.13.

Figure 4-13: SolverTable settings

image

To begin Solver place a 1 in the input cell (A11) and run the Solver to track the output cells (ab, and c and Sum of Squared Errors [SSE]). Then Solver places a 2 in A11 and finally a 3 in A11. The results are placed in a new worksheet (STS_1), as shown in Figure 4.14.

Figure 4-14: SolverTable results

image

You can now set up a Solver model that determines the profit maximizing price for each product. Proceed as follows:

1. Enter trial prices for each product in the range F5:F7.

2. Based on the prices in F5:F7, compute the demand for each product by copying the formula =B5+C5*F5+F5^2*D5 from G5 to G6:G7.

3. Enter the unit cost for each product in H5:H7, and compute the profit for each product as (price-unit cost)*demand by copying the formula =(F5-H5)*G5 from I5 to I6:I7.

4. Compute the total profit for all products in cell I2 with the formula =SUM(I5:I7).

5. Now use Solver (see Figure 4.15) to choose the prices in F5:F7 to maximize total profit (I2). This, of course, chooses the profit maximizing price for each product.

Figure 4-15: Finding profit maximizing price for each product

image

You find that you should charge $1.39 for Product 1 and $1.37 for Products 2 and 3. The SolverTable provides a powerful tool that can enable the marketing analyst to quickly estimate a demand curve for several products and determine a profit maximizing price for each product.

Summary

In this chapter you learned the following:

·        Price elasticity of demand is the percentage decrease in demand for a 1 percent increase in price.

·        Given the current price and demand and price elasticity, you can use the Excel Trend Line feature to fit a linear demand curve (Demand = a -b*price). For a linear demand curve price, elasticity is different at every point on the demand curve.

·        Given the current price and demand and price elasticity, you can use the Excel Goal Seek feature to fit a power or constant elasticity demand curve (Demand =a(price)-b)

·        Given a demand curve you can use Solver to find a profit maximizing price by maximizing (price – unit cost) * demand.

·        If you do not know the price elasticity, you can use the Polynomial portion of the Trend Line feature to fit a quadratic to three points on the demand curve.

·        Using the SolverTable add-in you can quickly fit the demand curve for a huge number of products and then use Solver to find the profit maximizing price for each product.

Exercises

1. Your company charges $60 for a board game it invented and has sold 3,000 copies during the last year. Elasticity for board games is known to equal 3. Use this information to determine a linear and power demand curve.

2. You need to determine the profit-maximizing price for a video game console. Currently you charge $180 and sell 2 million consoles per year. It costs $150 to produce a console, and the price elasticity of demand for consoles is 3. What price should you charge for a console?

3. Now assume that, on average, a purchaser of your video game console buys 10 video games, and you earn $10 profit on each video game. What is the correct price for consoles?

4. In the razorsandblades.xls file example, suppose the cost to produce a blade is $0.20. If you charge $0.35 for a blade, a customer buys an average of 50 blades from you. Assume the price elasticity of demand for blades is 3. What price should you charge for a razor and for a blade?

5. You manage a movie theater that can handle up to 8,000 patrons per week. The current demand, price, and elasticity for ticket sales, popcorn, soda, and candy are given in Figure 4.16. The theater keeps 45 percent of ticket revenues. Unit cost per ticket, popcorn sales, candy sales, and soda sales are also given. Assuming linear demand curves, how can the theater maximize profits? Demand for foods is the fraction of patrons who purchase the given food.

Figure 4-16: Data for Exercise 5

image

6. A prescription drug is produced in the United States and sold internationally. Each unit of the drug costs $60 to produce. In the German market, you sell the drug for 150 euros per unit. The current exchange rate is 0.667 U.S. dollars per euro. Current demand for the drug is 100 units, and the estimated elasticity is 2.5. Assuming a linear demand curve, determine the appropriate sales price (in euros) for the drug.

7. Suppose it costs $250 to produce a video game console. A price between $200 and $400 is under consideration. Estimated demand for the game console is shown in the following table. What price should you charge for game console?

Price

Demand

$200

50,000

$300

25,000

$400

12,000

8. Use the demand information given in Exercise 7 for this exercise. Each game owner buys an average of 10 video games. You earn $10 profit per video game. What price should you charge for the game console?

9. You want to determine the correct price for a new weekly magazine. The variable cost of printing and distributing a copy of the magazine is $0.50. You are thinking of charging from $0.50 through $1.30 per copy. The estimated weekly sales of the magazine are shown in the following table. What price should you charge for the magazine?

Price

Demand (in Millions)

$0.50

2

$0.90

1.2

$1.30

.3

10. Given the following information in the table for four products, find the profit-maximizing price for each product.

image

11. (Requires calculus) Show that if the demand for a product is given by the power curve q = ap-b, then for any price a, a 1 percent increase in price will decrease demand by 1 percent.

12. For the demand curve q =100p-2 show that for p = 1, 2, 4, 8, and 16 that a 1 percent increase in price will result in approximately a 2 percent decrease in demand.