MBA, P1 Sep Oct 2011 Prices & Markets Timothy Van Zandt Estimating the Elasticity of the Demand for Gasoline (Adapted from an exercise by Pushan Dutt, 2005) 1. Overview In this activity, you do a simple demand estimation. We will lead you through the steps without explaining the statistical methods of regression. Such methods are not part of this course; you will study them toward the end of UDJ. We have thus reduced the assignment to a mechanical spreadsheet exercise. So what what purpose does the assignment have? 1. To convince you that demand functions and elasticities are measurable, rather than purely abstract concepts. 2. To provide practice in manipulating and interpreting demand functions, particularly constant-elasticity demand functions. 3. To motivate your study of linear regression in UDJ by providing an example of an application. Even without getting into complicated statistical methods, you can probably think of improvements to the estimation below. Shouldn t we worry about a time trend? Aren t there other important variables that affect gasoline? How do we know we are estimating a demand rather than supply curve? The fun part of estimation is that it is always possible to refine the data and methods further, and requires judgments that make estimation part science, part art. 2. Getting started Retrieve the raw data: http://faculty.insead.edu/vanzandt/pm/session03/gasolinedata.xls Save this file to your hard disk, make a copy, and then open it up in Excel. Take a look at the sheet rawdata. It has data by year on US gasoline consumption (G), gasoline prices (PG), per-capita income (Y ), price of new cars (PNC), price of used cars (PUC), and population (POP). We want to estimate a function that relates demand for gasoline (the dependent variable) to the remaining variables (the independent variables). 3. Dealing with population size The demand measures total consumption in the United States. Of course, it rises when the U.S. population increases, and we can account for this by including population as an independent variable. However, it is natural to think that an increase in population just scales up the demand proportionately. That is, we should just look for determinants of
Prices & Markets Estimating theelasticity of thedemand forgasoline 2 per-capita consumption and ignore the population variable. Let s do this. You need to create a new variable, per-capita demand, to use as the independent variable. Do this as follows: 1. Insert a new column next to column B. (Click on the top of column C; select Insert Column. The old column C becomes column D (etc) and a new blank column C is created.) 2. In row 1 of this new column C, type in a name for the variable: GPC. 3. In row 2, enter a formula for calculating demand divided by population: =B2/H2. 4. Copy this formula to the rest of the column: (a) Highlight cell C2; (b) move the cursor to the lower-right corner until it turns into a +; (c) left-click and hold, while dragging the mouse to the bottom of the column; (d) release. 5. Reality check: Columns A H of your spreadsheet should have data. The value of cell C7 should be about 0.802367. 4. Log-linear specification In this exercise, you will estimate a log-linear (constant-elasticity) demand function. GPC = A PG B 1 Y B 2 PNC B 3 PUC B 4. The coefficients B 1,B 2,B 3,B 4 are the elasticities. For example, B 1 is the price elasticity of demand; 1 B 2 is the income elasticity of demand. Furthermore, the units used to measure each variable do not affect the values of these coefficients, which is why we can be quite casual about the units. So that we can use linear regression, we take logs of both sides of this equation: log GPC = log A + B 1 log PG + B 2 log Y + B 3 log PNC + B 4 log PUC. (1) We need to create a new version of the data set that contains the logs of the raw data (for the variables in equation (1)). We add 5 columns to the worksheet for this. 1. Click at the top of column I. Select Insert Column 5 times. 2. Add variable names to the columns: LGPC, LPG, LY, LPNC, LPUC. 3. Initiate the formulas for the columns, in row 2. For example, in cell I2 type =LOG(C2) to get the values of log GPC (i.e., LGPC) in column I (letter I, not number 1 ). (The Excel function LOG() calculates the logarithms in base 10. It does not matter which base we use, as long as we use the same base for all the data.) 4. Extend the formulas to the other rows: Highlight cells I2 M2; move the cursor to the lower-right corner of the highlighted area so that it becomes a +; left-click, hold, and drag the mouse to the last row of data; release. 1. Usually the negative value B 1 is called the elasticity; B 1 is then a positive number that equals the magnitude of the elasticity. In this course, we call B 1 the elasticity to make in-class discussion easier.
Prices & Markets Estimating theelasticity of thedemand forgasoline 3 5. Reality check: You should have data in columns A M. The first rows of columns I M should look roughly like this: 5. The regression (for other than Office Mac 2008 users) 1. If you cannot see Data Analysis in the Tools menu, select Tools Add-Ins ; then check the Analysis ToolPak button. 2. Select Tools Data Analysis and choose Regression. 3. Click inside the Input Y-Range field in the pop-up window; then highlight the data for the dependent variable LPGC, including the first row that contains the variable name: cells I1:I37. 4. Click inside the Input X-Range field in the pop-up window; then highlight the data for the independent variables LPG, LY, LPNC, andlpuc, including the first row that contains the variable names: cells J1:M37. 5. Tick the Labels box. (This tells Excel that the first row of data contains the names of the variables.) 6. Select New Worksheet Ply. 7. Select OK. 6. The regression: for Office Mac 2008 users Microsoft removed Visual Basic and support for the Data Analysis add-in from Office Mac 2008. You therefore have four options: 1. Figure out how to mimic the above using Excel s built-in functions (it is possible, but tedious). 2. Use some other linear regression software. 3. Do the exercise with a classmate who uses a different version of Excel, print out two copies, and put your name on one of them. 4. Print out just the sheet showing that you created the logged data, and write Microsoft is Evil on the top. If you find a good workaround, please let us know.
Prices & Markets Estimating theelasticity of thedemand forgasoline 4 7. Interpreting the results Look at the first two columns of the last table on the new worksheet. It should look something like this. These are our estimates of the coefficients, i.e., of the elasticities. Let s first check that they have the signs we expect: 1. B 1 is negative: price of gasoline goes up demand goes down. 2. B 2 is positive: per-capita income goes up per-capita demand goes up. 3. B 3 and B 4 are negative: price of cars goes up demand for gasoline goes down. This is all as expected. The own-price elasticity of demand, B 1, is about 0.06 pretty inelastic!
Prices & Markets Estimating theelasticity of thedemand forgasoline 5 Appendix: Optional advanced exercise on short-run vs. long-run elasticities It s likely that when price changes, demand will not change immediately because it takes time for you to adjust behavior (organize a car pool; buy a more fuel efficient car). So we want short-run and long-run estimates of elasticities. How do we do that? It s quite simple actually. Inertia in consumption behavior means that consumption today depends not only on current prices but also on the previous level of consumption. Thus, we include as an independent/explanatory variable a lagged consumption term: log GPC = log A+B 1 log PG+B 2 log Y +B 3 log PNC+B 4 log PUC+B 5 log PrevGPC, (2) where PrevGPC is the gasoline consumption in the previous year. Create the data for the lagged value as follows: 1. Insert a new column (N) and label it LPrevGPC. 2. In cell N3, enter the formula =I2. That is, it is the previous year s value of LPGC. Extend this formula down to cell N37 in the usual way. Cell N2 is empty. Run the regression as before, but include the new variable. You have to start with the second year of data (row 3) because, for the first year (row 2), you do not have the previous year s gasoline consumption (N2 is empty). This makes it more complicated to include the labels (row 1), so we skip them and untick the Labels box. The coefficient on LPG is the short-run price elasticity of demand and the coefficient on LY is the short-run income elasticity of demand. Let s think about how to calculate the long-run elasticity from our coefficients. First, what do we mean by long-run elasticity? It is the change in long-run demand in response to price, and long-run demand means the value that demand would converge to if the independent variables stopped changing and people had forever to adjust their consumption decisions. Let s focus on how demand responds to own price and write the demand function as GPC = A PG B 1 PrevGPC B 5. Given PG, the long-run demand LR_GPC is the value at which GPC = PrevGPC and hence demand stops adjusting. Thus, it solves LR_GPC = A PG B 1 LR_GPC B 5 LR_GPC 1 B 5 = A PG B 1 LR_GPC = A 1 1 B 5 PG B 1 1 B 5. The exponent on PG is B 1 /(1 B 5 ); this is the long-run elasticity of demand. The value of B 5 should be between 0 and 1 (or there is a problem with either the data or your estimation). Hence, the long-run elasticity is greater than the short-run elasticity (as we would expect).