# Would you predict that the value of a is negative or positive?

Suppose you work for a national pizza chain that is considering introducing a new product: hot dog s Show more Suppose you work for a national pizza chain that is considering introducing a new product: hot dog stuffed crust pizza with mustard drizzle. We will refer to this pizza as hd pizza. The chain is debating offering the product nationwide and wants you to look at the profitability of such a move. If it goes forward with the plan it would like to know the price it should charge. For the sake of simplicity suppose that the hd pizza is only available as a 16 inch pizza pie. Creating a Demand Model Before crunching any numbers we want to think of the factors that would affect how many hd pizzas people will want to buy. For each restaurant location you determine that the majority of customers live within 10 miles of the restaurant. We will call this 10 mile region the market area. What are some of the things that will affect the number of hd pizzas that buyers will want to purchase? Some possibilities: The price of hd pizza The number of people per pizza shop in the market area The average household income in the market area The presence of a major university in the market area Using these four factors we come up with the following demand equation: qd is the number of hd pizzas purchased at the restaurant (per month). P is the price of an hd pizza with no added toppings. N is the number of people in the market area divided by the number of pizza shops in the market area. M is the median household income (in thousands of dollars) in the market area. U equals 1 if a major university is located in the market area and 0 if there is not a major university in the area. The sign of each coefficient will tell you if the variable moves in the same direction or opposite direction of qd. If the coefficient is positive we assume that the variable moves in the same direction as qd. If the coefficient is negative we assume that the variable moves in the opposite direction of qd. For example if we assume that hd pizza is a normal good so that as income increases the quantity demanded of hd pizzas also increases then we are also assuming that c is positive. If we assume that hd pizza is an inferior good so that as income increases the quantity demanded of hd pizzas decreases then we are assuming that c is negative. 1. Would you predict that the value of a is negative or positive? Justify your answer. Do the same for the value of b and the value of d. Estimating the Demand Function In order to estimate the value of each coefficient you choose 50 randomly selected locations to offer the hd pizza for a month. You have the first 5 locations charge $10/pie the next 5 locations charge $11/pie the next 5 charge $12/pie and so on. Restaurant Location Quantity Demanded (hd pizzas / month) Price of HD Pizza Pie ($ / pie) Number of People in Market Area Number of Pizza Shops in Market Area Average Household Income in Market Area ($1000s) Major University? (1 for yes and 0 for no) 1 85 10 100000 7 49.1 0 2 79 10 12000 2 32.2 0 3 93 10 54000 6 41.2 1 4 95 10 275000 8 64.2 0 5 87 10 75000 5 60.1 0 6 77 11 123000 10 55.6 0 7 74 11 32000 4 38.7 0 8 90 11 47000 3 40.1 1 9 75 11 88000 7 39.2 0 10 78 11 120000 8 59.5 0 11 87 12 92000 2 63.4 0 12 94 12 500000 21 98.2 1 13 72 12 37000 2 53.2 0 14 73 12 43000 2 43.1 0 15 72 12 65000 3 47.2 0 16 55 13 4500 1 22.3 0 17 67 13 57000 2 36.5 0 18 57 13 21000 4 40.1 0 19 59 13 32000 4 48.2 0 20 63 13 67000 5 61.6 0 21 55 14 18000 1 37.2 0 22 52 14 31000 3 31.8 0 23 64 14 33000 1 43.2 0 24 50 14 6300 1 42.1 0 25 51 14 19000 2 37.8 0 26 47 15 15000 1 43.2 0 27 48 15 91000 6 52.1 0 28 45 15 22500 2 39.1 0 29 64 15 36000 2 45.8 1 30 42 15 2100 1 44.2 0 31 41 16 74000 4 41.4 0 32 39 16 35000 3 57.2 0 33 41 16 81000 5 52.3 0 34 42 16 17000 1 54.6 0 35 41 16 23000 2 71.2 0 36 28 17 9400 1 28.9 0 37 31 17 39000 4 32.6 0 38 54 17 430000 16 67.4 1 39 23 17 7400 3 18.5 0 40 45 17 27000 3 42.4 1 41 18 18 73900 14 23.6 0 42 36 18 18000 4 56.8 1 43 20 18 9000 1 23.4 0 44 20 18 14500 3 42.6 0 45 38 18 63000 5 33.2 1 46 24 19 52000 2 59.3 0 47 37 19 156000 7 58.2 1 48 34 19 81000 5 60.1 1 49 37 19 73000 4 69.9 1 50 17 19 14000 1 49.2 0 Open a new Excel spreadsheet and label your columns appropriately. Something like: Type in the data that is given in the table above. Now we dont have the values of N but we can use Excels fill operation to quickly get these values. Note that each value of N is just the number of people divided by the number of shops. To get the value of N for the first restaurant wed divide the number in cell D2 (column D row 2) by the number in E2 (column E row 2). To get the value of N for the third restaurant wed divide the entry in D3 by the entry in E3. Hit the Enter or Return key. This will put the result of dividing the entry in D2 by the entry in E2 in H2. We could continue to type =D3/E3 in H3 =D4/E4 in H4 and so on but we can save time by using Excels fill operation. Select cells H2 to H51 select Fill and then select Down. Next you need to activate Excels multiple regression feature. Open the File menu (or press Alt+F) and select Options. Click Add-Ins on the left side of the window. Click Go next to the Manage: Add-ins option at the bottom of the window. In the new window check the box next to Analysis ToolPak then click OK. Now that the feature is activated lets use it. Click the Data tab and then select Data Analysis. In columns J K L M and N put the data for qd P N M and U (in that order). For the N data click paste values. This is the order that appear in the demand equation. In the Data Analysis window select Regression and click on OK. In the next window select elements J1 through J51 for the Y values. Select the elements K1 through N51 for the X values. Make sure the Labels box is checked. On the next couple of pages I am going to show many of the tasks in the guided project with example numbers. Your numbers will be different you will use the numbers that you get after running the regression as instructed above. *Example* Suppose that after you run the regression Excel returns the following: Then your demand equation would be: Suppose further that the national median household income is $47000 (M = 47) and that on average N = 13000. We can use this to get two demand curves for hd pizza: one for markets with a major university (U = 1) and one for markets without a major university (U = 0). Demand in market with major university (U = 1): If we solve this for P we can graph the demand curve. Go to www.wolframalpha.com and type in the string shown below: Which produces the demand curve for hd pizza in a market with a major university: Demand in a market without a major university (U = 0): If we solve this for P we can graph the demand curve. Which produces the demand curve for hd pizza in a market without a major university: Note that this curve is slightly to the left of the demand curve of the market with a major university. In other words removing a major university from a market causes the demand for hd pizza to decrease. To find the profit-maximizing quantity we set MR = MC and solve for q. For a linear demand equation we get the equation for marginal revenue by multiplying the qd term by 2. Using the demand for a market with a university: Assume that MC = 7. That is each hd pizza adds $7 to total cost regardless of how many hd pizzas you produce. So the profit-maximizing quantity is found by setting that expression for MR equal to 7 and solving for q. To find the profit-maximizing price plug that quantity into the demand curve. So your recommendation to the firm would be to set a price of $17.31 and expect to sell around 87 pizzas if the market contains a university. How much monthly profit should each restaurant in such a market expect to make from the sale of hd pizzas? Using the demand for a market without a university: Assume that MC = 7. That is each hd pizza adds $7 to total cost regardless of how many hd pizzas you produce. So the profit-maximizing quantity is found by setting that expression for MR equal to 7 and solving for q. To find the profit-maximizing price plug that quantity into the demand curve. So your recommendation to the firm would be to set a price of $16.45 and expect to sell around 79 pizzas if the market doesnt contain a university. How much monthly profit should each restaurant in such a market expect to make from the sale of hd pizzas? 2. For the example on this page explain what is wrong with each argument. a. Producing 79 hd pizzas per month doesnt maximize profit. If the restaurant produces 100 pizzas it would make $16.45(100) $7(100) = $945 of monthly profit. b. A price of $16.45 doesnt maximize profit. If the restaurant charged $50 per pizza itd make $50(79) $7(79) = $3397 of monthly profit. Show less