# How Economists Measure Price Elasticity

October 19, 2008 5 Comments

(See also this post about converting quarterly data into monthly using cubic splines interpolation.)

(Note: In this post, we’re ignoring the famous problem of endogeneity bias when trying to estimate demand elasticities. It turns out simultaneity isn’t much or an issue in this case — the paper discussed below addresses the issue and ends up with very similar results — so we’ll delay a discussion of instrumental variables for another post.)

Like most concepts in economics, price elasticity is easy to talk about but hard to measure.

In this post, I’ll show you how Chamberlain Economics measures demand elasticities in the real world. We’ll develop a simple theory, write it down mathematically, find some data and crunch the numbers in Excel. At the end, I’ll hand over a spreadsheet with our own elasticity estimates for retail gasoline that replicate the numbers from a well-known recent econometric study.

**Start with Theory**

Our goal is to estimate the price elasticity of demand for retail gasoline. The first step is to start with a theory about the demand for gas.

The simplest theory is that we know gasoline — like everything else — should have a demand curve. What should it look like? In the simplest case, it should be driven by two things: the price of gas, and how much income people have. If gas prices rise consumption should fall; conversely, if income goes up gas consumption should rise also.

So there’s our theory. Now, let’s write it down mathematically. If gas demand is a function of prices and income, one way we can write it is like this:

**G = a*P + b*Y**

Where:

G = Gallons of gas demanded per year

P = The price of gas

Y = Average income in the economy

a, b = Coefficients for the magnitude of the impact of prices and income on gas demand. (Note: According ot our theory, the “a” coefficient on prices should be a negative number and the “b” coefficient on income should be positive.)

Now that we’ve got a theory, the next step is to translate it into a form we can estimate in the real world. Think of the theory as an architect’s drawing — it’s a guide, but our goal is to actually to build it with hammer and nails.

To do this, think about real-world factors that might complicate our simple theory. For one, we should probably control for population by using per capita figures. Next, we should control for inflation by inflation-adjusting everything. Finally, we should control for seasonal variation somehow, since gas demand always peaks in summer and slows in winter.

Taking these messy details into account, here’s how we translate our theory into a relationship we can actually estimate. Economists call this “specifying the model”:

**Gij = A + a*Pij + b*Yij + ei + eij**

Where:

G = Per capita gas demand in month i and year j

A = The y-intercept term in our linear demand curve

Pij = The inflation-adjusted gas price in month i and year j

Yij = Real per capita disposable personal income in month i and year j

a, b = Coefficients on price and income

ei = A dummy variable for the month of the year to control for seasonal variation (there are actually eleven of these, one for each month January through November; they’re one if it’s the month in question and zero otherwise); this is called “seasonal fixed effects”

eij = A mean-zero random error term for month i and year j.

This way of specifying our model is called “linear”. This isn’t the only way to do it — at the end I’ll mention another way called “double log” that has some advantages. But for now, we’re ready to collect some data and run a regression.

**Go Find the Data**

The best source for data is always official government sources. Here’s the data we’ll use for this:

1. Gallons of gas demanded: We’ll use data from the U.S. Energy Information Administration. It’s called “product supplied”. The numbers are in barrels, so you’ll have to multiply them by 42 to convert them to gallons:

http://tonto.eia.doe.gov/dnav/pet/hist/mgfupus1m.htm

2. Gas prices: We’ll use numbers from the U.S. Bureau of Labor Statistics here. It’s from their “average price data” series, and it’s the monthly retail price of gas:

http://www.bls.gov/cpi/home.htm

3. Income: We’ll use data from the U.S. Bureau of Economic Analysis for this one. It’s called “disposable personal income”, and it comes from line 26 on Table 2.1 from the National Income and Product Account (NIPA) tables:

http://www.bea.gov/national/nipaweb/SelectTable.asp?Selected=N#S2

4. Something to inflation-adjust prices and income: For this one, we’ll use the implicit price deflator for GDP from the Bureau of Economic Analysis. It’s on line 1 of NIPA Table 1.1.9:

http://www.bea.gov/national/nipaweb/SelectTable.asp?Selected=N#S1

5. Population figures to turn gallons and income into per capita figures: This is a hard one, because we need monthly figures and the U.S. Census Bureau only produces annual figures. Also, it’s hard to piece together a consistent series from before and after each decennial census. Thankfully I’ve done the hard work for you — the Excel files below include a monthly population series I put together myself.

Once you’ve compiled these data in columns in an Excel sheet, you’re ready to run your regression. When you do, you should find something like this:

For 2000-2007, the coefficient on gas prices should be about -1.07, and the coefficient on income should be about 0.0007. Using the formula for price elasticity of E = (Average price over the period/Average quantity over the period)*(price coefficient), that implies a price elasticity of demand of about **-0.048** and an income elasticity of about **0.51**.

And that’s about what we’d expect. We know short-run demand for gas is inelastic, and has a negative relationship with prices. And we know that income should be positively related to gas demand, which it is.

The above method is based on a well-known 2006 study from Hughes, Knittel and Sperling which you can read at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=930730.

For those who’d like to see the final product, here are Chamberlain Economics’ own elasticity estimates for gasoline. The first file uses the linear specification above. The second one uses a “double log” specification, which basically takes the log of the data. The big advantage of the latter is that the regression coefficients are also the price and income elasticities, which is handy:

Price elasticity of demand for gasoline: Linear model.

Price elasticity of demand for gasoline: Double log model.

If you want the full data as STATA files instead, here they are:

Price elasticity of demand for gasoline: Linear model.

Price elasticity of demand for gasoline: Double log model.

Questions? Give us a call here.

This is a great intro. Thanks for posting the data files!

I’m writing a term paper on the incidence of a gas tax moratorium (let me know if you’re interested in a copy once I’ve finished) and this post has been really useful. Thank you for making it publicly available.

I am attempting to estimate the short-run price elasticity of demand for gasoline and I’m going to follow your methodology with updated data (through August 2008).

I noticed an error with your treatment of GDP deflators (it may also apply to your treatment of monthly income). GDP deflators are quarterly, so obviously some extrapolation is required to arrive at monthly values for the purposes of our regression. The way you did this is (as far as I can tell) to assume the GDP deflator increases linearly (fair assumption) and then arrange your monthly numbers so that the GDP deflator for the first month of any given quarter is equal to the reported GDP deflator for that quarter.

This is unacceptable because quarterly GDP deflators are a report of the average GDP deflation over that entire quarter, not the average GDP deflation over the first month of that quarter (as you have it in your data). For example, consider Q12002. The reported quarterly GDP deflator is 103.568. You have in your data: January 2002: 103.658, February 2002: 103.691, March 2002: 103.815. These three values average to 103.721, which is well above the GDP deflator for Q12002, 103.568. Essentially, your method overestimates monthly GDP deflators.

A better method is to calculate monthly GDP deflators such that such that the GDP deflator increases linearly but the three monthly GDP deflators for any given quarter average to that quarter’s reported GDP deflator. So, I compute GDP deflator for January, February, and March 2002 as follows (bracketed periods represent the GDP deflator over that period, for example [March2002] = the GDP deflator for March 2002):

[January2002] = [Q12002] – ([Q12002] – [Q42001])/3

[February2002] = [Q12002]

[March2002 = [Q12002] + ([Q22002] – [Q12002])/3

This method produces monthly GDP deflator data which is consistent with the reported quarterly GDP deflator data. I applied the same method to calculate monthly disposable income from quarterly values (with slight variation for quarters in which monthly disposable income declined instead of rose).

Good luck with Chamberlain Economics, and I hope it’s alright if I list this post as a reference for my paper (along with the rather famous paper it mentions, obviously).

Another note: the US Census Bureau has monthly population estimates here:

http://www.census.gov/popest/national/NA-EST2007-01.html

Joseph –

Good luck with the paper and thanks for your notes. You’re right that different methods of data interpolation will lead to different estimates of monthly figures from quarterly actuals. Here’s a good overview of two common interpolation methods — quadradic and cubic spline method — from the UK’s official statistical office:

http://www.statistics.gov.uk/STATBASE/Product.asp?vlnk=2184

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