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Elasticity
Problem
Economists speak of "price elasticity" when changes in price of a commodity are coupled with changes in demand for that item. Price elasticity is defined as follows:
(% change in demand)/(% change in price)
When price elasticity is -1, a change in price has no effect on total revenue, because it is wholly counterbalanced by a change in demand. To what extent is the demand for cigarettes elastic? Do smokers alter their purchasing patterns when cigarette prices change? In one study changes in cigarette consumption in different states were calculated before and after tax changes (see Lyon & Simon, 1968). The positive observations (implying an increase in demand when the price rises) run against all theory, but they can be considered to be the result simply of measurement errors and can be treated as they stand. The median price elasticity being -.511, an economist would say that demand is relatively inelastic. This implies that an increase in price would lead to an increase in total revenue, because demand would not drop by as much as prices would increase. The median is used to reduce dependence on observations at either extreme, some of which, as noted, seem questionable.
73 Observations of Price Elasticity of Demand (Computed From Cigarette Sales Data Preceding And Following State Tax Changes)
| 1.725 |
-0.142 |
-0.377 |
-0.713 |
-1.066 |
| 1.139 |
-0.174 |
-0.383 |
-0.724 |
-1.118 |
| 0.957 |
-0.234 |
-0.385 |
-0.734 |
-1.145 |
| 0.863 |
-0.24 |
-0.393 |
-0.749 |
-1.146 |
| 0.802 |
-0.251 |
-0.444 |
-0.752 |
-1.157 |
| 0.517 |
-0.277 |
-0.482 |
-0.753 |
-1.282 |
| 0.407 |
-0.301 |
-0.511 |
-0.766 |
-1.339 |
| 0.304 |
-0.302 |
-0.538 |
-0.805 |
-1.42 |
| 0.204 |
-0.302 |
-0.541 |
-0.886 |
-1.443 |
| 0.125 |
-0.307 |
-0.549 |
-0.926 |
-1.478 |
| 0.122 |
-0.328 |
-0.554 |
-0.971 |
-2.041 |
| 0.106 |
-0.329 |
-0.6 |
-0.972 |
-2.092 |
| 0.031 |
-0.346 |
-0.613 |
-0.975 |
-7.1 |
| -0.032 |
-0.357 |
-0.644 |
-1.018 |
|
| -0.1 |
-0.376 |
-0.692 |
-1.024 |
|
Note: Table adapted from Lyon & Simon, 1968, p. 891.
Our task is to estimate a confidence interval for the measure of price elasticity. This will enable us to answer the questions, "How reliable is our estimate of price elasticity? How much might it differ from one sample to the next?"
Resampling Procedure
If we had time and money, we could take lots of additional samples and learn how much the median changes from sample to sample. Since we lack the power to generate new samples by experimentally altering state tax laws, we need a hypothetical universe from which to draw samples. What is our best guess about what such a universe might look like? It is the observed sample. So, we can create our hypothetical universe by simply copying our original sample over and over until we have, say, millions of copies of observation #1, millions of copies of observation #2, etcetera.
Now we can draw samples to see how they behave and to learn how variable the mean is from one to the next. Actually, even on a computer, replicating millions of observations is tedious, so we use a shortcut - in drawing each sample, we simply replace each observation after selecting it. This achieves the same effect as replicating an infinitely large universe from our sample. This procedure - drawing a sample with replacement from the original sample - is called the bootstrap.
- Take 73 pieces of paper and write on them the observed values.
- Mix the pieces up, draw out one piece, write down its number, and replace. Do this 73 times.
- Calculate the median of this resample, and record it on a scoreboard.
- Repeat (2) and (3) many times.
- Examine the recorded data to determine the 2.5% to 97.5% interval.
Computer Implementation In Resampling Stats
READ file "cigaret.dat" elast
MEDIAN elast medelas
let Resampling Stats calculate the median value of
these data
REPEAT 1000
SAMPLE 73 elast elast$
a bootstrap sample (note that the "$" indicates
a bootstrap counterpart to the observed sample)
MEDIAN elast$ medelas$
the median of the simulated sample
SCORE medelas$ scrboard
save the answer to a vector called "scrboard"
END
HISTOGRAM scrboard
PERCENTILE scrboard (2.5 97.5) interval
PRINT medelas interval
Results
Frequency histogram of median resampled elasticity
medelas = -0.511 [observed value]
interval = -0.692 -0.357 [estimated 95 % confidence interval]
Conclusion
From the simulation results, we are 95% confident that the correct value for elasticity lies between -0.69 and -0.36.
References
Lyon, H.L., & Simon, J.L. (1968, November). Price elasticity of the demand for cigarettes in the United States. American Journal of Agricultural Economics, 50, 888-895.
