1. ELASTICITIES AND DOUBLE-LOGARITHMIC MODELS
Elasticity of Y with respect to
X is the proportional change in
Y per proportional change in X: A Y O X
This sequence defines elasticities and shows how one may fit nonlinear models with constant elasticities. First, the general definition of an elasticity. 1

2. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
Elasticity of Y with respect to
X is the proportional change in
Y per proportional change in X: A Y O X
Re-arranging the expression for the elasticity, we can obtain a graphical interpretation. 2

3. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
Elasticity of Y with respect to
X is the proportional change in
Y per proportional change in X: A Y O X
The elasticity at any point on the curve is the ratio of the slope of the tangent at that point to the slope of the line joining the point to the origin. 3

4. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
Elasticity of Y with respect to
X is the proportional change in
Y per proportional change in X: A Y O X
In this case it is clear that the tangent at A is flatter than the line OA and so the elasticity must be less than 1. 4

5. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
Elasticity of Y with respect to
X is the proportional change in
Y per proportional change in X: A Y O X
In this case the tangent at A is steeper than OA and the elasticity is greater than 1. 5

6. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY A O X x
In general the elasticity will be different at different points on the function relating Y to X. 6

7. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY A O X x
In the example above, Y is a linear function of X. 7

8. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY A O X x
The tangent at any point is coincidental with the line itself, so in this case its slope is always b2. The elasticity depends on the slope of the line joining the point to the origin. 8

9. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY B A O X x
OB is flatter than OA, so the elasticity is greater at B than at A. (This ties in with the mathematical expression: (b1 /X) + b2 is smaller at B than at A, assuming that b1 is positive.) 9

10. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
However, a function of the type shown above has the same elasticity for all values of X. 10

11. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
For the numerator of the elasticity expression, we need the derivative of Y with respect to X. 11

12. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
For the denominator, we need Y/X. 12

13. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
Hence we obtain the expression for the elasticity. This simplifies to b2 and is therefore constant. 13

14. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY X
By way of illustration, the function will be plotted for a range of values of b2. We will start with a very low value, 0.25. 14

15. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY X
We will increase b2 in steps of 0.25 and see how the shape of the function changes. 15

16. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY X 16

17. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY X
When b2 is equal to 1, the curve becomes a straight line through the origin. 17

18. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY X 18

19. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY X 19

20. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY X
Note that the curvature can be quite gentle over wide ranges of X. 20

21. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELSY X
This means that even if the true model is of the constant elasticity form, a linear model may be a good approximation over a limited range. 21

22. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
It is easy to fit a constant elasticity function using a sample of observations. You can linearize the model by taking the logarithms of both sides. 22

23. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
You thus obtain a linear relationship between Y' and X', as defined. All serious regression applications allow you to generate logarithmic variables from existing ones. 23

24. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
The coefficient of X' will be a direct estimate of the elasticity, b2. 24

25. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
The constant term will be an estimate of log b1. To obtain an estimate of b1, you calculate exp(b1'), where b1' is the estimate of b1'. (This assumes that you have used natural logarithms, that is, logarithms to base e, to transform the model.) 25

26. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
FDHO
EXP
Here is a scatter diagram showing annual household expenditure on FDHO, food eaten at home, and EXP, total annual household expenditure, both measured in dollars, for 1995 for a sample of 869 households in the United States (Consumer Expenditure Survey data). 26

27. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
. reg FDHO EXP
Source | SS df MS Number of obs =
F( 1, 867) =
Model | Prob > F =
Residual | e R-squared =
Adj R-squared =
Total | e Root MSE =
FDHO | Coef. Std. Err t P>|t| [95% Conf. Interval]
EXP |
_cons |
Here is a regression of FDHO on EXP. It is usual to relate types of consumer expenditure to total expenditure, rather than income, when using household data. Household income data tend to be relatively erratic. 27

28. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
. reg FDHO EXP
Source | SS df MS Number of obs =
F( 1, 867) =
Model | Prob > F =
Residual | e R-squared =
Adj R-squared =
Total | e Root MSE =
FDHO | Coef. Std. Err t P>|t| [95% Conf. Interval]
EXP |
_cons |
The regression implies that, at the margin, 5 cents out of each dollar of expenditure is spent on food at home. Does this seem plausible? Probably, though possibly a little low. 28

29. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
. reg FDHO EXP
Source | SS df MS Number of obs =
F( 1, 867) =
Model | Prob > F =
Residual | e R-squared =
Adj R-squared =
Total | e Root MSE =
FDHO | Coef. Std. Err t P>|t| [95% Conf. Interval]
EXP |
_cons |
It also suggests that $1,916 would be spent on food at home if total expenditure were zero. Obviously this is impossible. It might be possible to interpret it somehow as baseline expenditure, but we would need to take into account family size and composition. 29

30. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
FDHO
EXP
Here is the regression line plotted on the scatter diagram 30

31. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
LGFDHO
LGEXP
We will now fit a constant elasticity function using the same data. The scatter diagram shows the logarithm of FDHO plotted against the logarithm of EXP. 31

32. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source | SS df MS Number of obs =
F( 1, 866) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
LGFDHO | Coef. Std. Err t P>|t| [95% Conf. Interval]
LGEXP |
_cons |
Here is the result of regressing LGFDHO on LGEXP. The first two commands generate the logarithmic variables. 32

33. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source | SS df MS Number of obs =
F( 1, 866) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
LGFDHO | Coef. Std. Err t P>|t| [95% Conf. Interval]
LGEXP |
_cons |
The estimate of the elasticity is Does this seem plausible? 33

34. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source | SS df MS Number of obs =
F( 1, 866) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
LGFDHO | Coef. Std. Err t P>|t| [95% Conf. Interval]
LGEXP |
_cons |
Yes, definitely. Food is a normal good, so its elasticity should be positive, but it is a basic necessity. Expenditure on it should grow less rapidly than expenditure generally, so its elasticity should be less than 1. 34

35. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source | SS df MS Number of obs =
F( 1, 866) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
LGFDHO | Coef. Std. Err t P>|t| [95% Conf. Interval]
LGEXP |
_cons |
The intercept has no substantive meaning. To obtain an estimate of b1, we calculate e3.16, which is 23.8. 35

36. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
LGFDHO
LGEXP
Here is the scatter diagram with the regression line plotted. 36

37. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
FDHO
EXP
Here is the regression line from the logarithmic regression plotted in the original scatter diagram, together with the linear regression line for comparison. 37

38. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
FDHO
EXP
You can see that the logarithmic regression line gives a somewhat better fit, especially at low levels of expenditure. 38

39. ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
FDHO
EXP
However, the difference in the fit is not dramatic. The main reason for preferring the constant elasticity model is that it makes more sense theoretically. It also has a technical advantage that we will come to later on (when discussing heteroscedasticity). 39

40. Copyright Christopher Dougherty 1999-2001
Copyright Christopher Dougherty This slideshow may be freely copied for personal use.