Elasticity of Y with respect to X is the proportional change in Y per proportional change in X: ELASTICITIES AND DOUBLE-LOGARITHMIC MODELS Y X A O 1 This

Elasticity of Y with respect toX is the proportional change inY per proportional change in X:

ELASTICITIES AND DOUBLE-LOGARITHMIC MODELS

0 52

XYdXdY

XdXYdY

elasticity Y

X

A

Ox

y

1

This sequence defines elasticities and shows how one may fit nonlinear models with constant elasticities. First, the general definition of an elasticity.


ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS

0 52

XYdXdY

XdXYdY

elasticity

OAA

of slopeat tangent the of slope

Y

X

A

Ox

y

2

Re-arranging the expression for the elasticity, we can obtain a graphical interpretation.



0 52

XYdXdY

XdXYdY

elasticity

OAA


Y

X

A

Ox

y

3

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.


4


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.

0 52

XYdXdY

XdXYdY

elasticity

OAA


Y

X

A

O

1elasticity

XY

0 52

5


In this case the tangent at A is steeper than OA and the elasticity is greater than 1.

A

O

1elasticity


XYdXdY

XdXYdY

elasticity

OAA


Y

X

6


In general the elasticity will be different at different points on the function relating Y to X.

XYdXdY elasticity

OAA


Y

xO X

XY 21

21

2

21

2

)/(

/)(

X

XX

A

7


In the example above, Y is a linear function of X.

xO

A

XY 21

XYdXdY elasticity

OAA


21

2

21

2

)/(

/)(

X

XX

Y

X

8


The tangent at any point is coincidental with the line itself, so in this case its slope is

always 2. The elasticity depends on the slope of the line joining the point to the origin.

xO

A

XY 21

XYdXdY elasticity

OAA


21

2

21

2

)/(

/)(

X

XX

Y

X

9


OB is flatter than OA, so the elasticity is greater at B than at A. (This ties in with the

mathematical expression: (1/X) + 2 is smaller at B than at A, assuming that 1 is positive.)

x

A

O

B

XY 21

XYdXdY elasticity

OAA


21

2

21

2

)/(

/)(

X

XX

Y

X


21

XY

10

However, a function of the type shown above has the same elasticity for all values of X.


21

XY

121

2 XdXdY

11

For the numerator of the elasticity expression, we need the derivative of Y with respect to X.


21

XY

121

2 XdXdY

11

1 2

2

X

XX

XY

12

For the denominator, we need Y/X.

13


Hence we obtain the expression for the elasticity. This simplifies to 2 and is therefore constant.

21

XY

121

2 XdXdY

11

1 2

2

X

XX

XY

211

121

2

2

elasticity

X

XXYdXdY

14


By way of illustration, the function will be plotted for a range of values of 2. We will start with a very low value, 0.25.

Y

X

21

XY 25.02

15


We will increase 2 in steps of 0.25 and see how the shape of the function changes.

21

XY 50.02

Y

X

16


21

XY 75.02

Y

X

17


When 2 is equal to 1, the curve becomes a straight line through the origin.

21

XY 00.12

Y

X

18


21

XY 25.12

Y

X

19


21

XY 50.12

Y

X

20


Note that the curvature can be quite gentle over wide ranges of X.

21

XY 75.12

Y

X

21


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

XY 00.22

Y

X

22


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.

21

XY

X

X

XY

loglog

loglog

loglog

22

1

1

2

2

23


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.

21

XY

X

X

XY

loglog

loglog

loglog

22

1

1

2

2

'' 2'1 XY

1'1 log

log'

,log' where

XX

YY

24


The coefficient of X' will be a direct estimate of the elasticity, 2.

21

XY

X

X

XY

loglog

loglog

loglog

22

1

1

2

2

'' 2'1 XY

1'1 log

log'

,log' where

XX

YY

25


The constant term will be an estimate of log 1. To obtain an estimate of 1, you calculate

exp(b1'), where b1' is the estimate of 1'. (This assumes that you have used natural logarithms, that is, logarithms to base e, to transform the model.)

21

XY

X

X

XY

loglog

loglog

loglog

22

1

1

2

2

'' 2'1 XY

1'1 log

log'

,log' where

XX

YY

26


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).

0

2000

4000

6000

8000

10000

12000

14000

16000

0 20000 40000 60000 80000 100000 120000 140000 160000

FDHO

EXP

. reg FDHO EXP

Source | SS df MS Number of obs = 869---------+------------------------------ F( 1, 867) = 381.47 Model | 915843574 1 915843574 Prob > F = 0.0000Residual | 2.0815e+09 867 2400831.16 R-squared = 0.3055---------+------------------------------ Adj R-squared = 0.3047 Total | 2.9974e+09 868 3453184.55 Root MSE = 1549.5

------------------------------------------------------------------------------ FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- EXP | .0528427 .0027055 19.531 0.000 .0475325 .0581529 _cons | 1916.143 96.54591 19.847 0.000 1726.652 2105.634------------------------------------------------------------------------------

27


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.

. reg FDHO EXP



28


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.

. reg FDHO EXP



29


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.

30


Here is the regression line plotted on the scatter diagram

0

2000

4000

6000

8000

10000

12000

14000

16000

0 20000 40000 60000 80000 100000 120000 140000 160000EXP

FDHO

31


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.

5.00

6.00

7.00

8.00

9.00

10.00

7.00 8.00 9.00 10.00 11.00 12.00 13.00

LGFDHO

LGEXP

. g LGFDHO = ln(FDHO)

. g LGEXP = ln(EXP)

. reg LGFDHO LGEXP

Source | SS df MS Number of obs = 868---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000Residual | 184.579612 866 .213140429 R-squared = 0.3138---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167

------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754------------------------------------------------------------------------------

32


Here is the result of regressing LGFDHO on LGEXP. The first two commands generate the logarithmic variables.


. g LGEXP = ln(EXP)

. reg LGFDHO LGEXP



33


The estimate of the elasticity is 0.48. Does this seem plausible?


. g LGEXP = ln(EXP)

. reg LGFDHO LGEXP



34


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.


. g LGEXP = ln(EXP)

. reg LGFDHO LGEXP



35


The intercept has no substantive meaning. To obtain an estimate of 1, we calculate e3.16, which is 23.8.

48.08.23ˆ48.017.3ˆ EXPOHFDLGEXPHODLGF

36


Here is the scatter diagram with the regression line plotted.

5.00

6.00

7.00

8.00

9.00

10.00

7.00 8.00 9.00 10.00 11.00 12.00 13.00

LGFDHO

LGEXP

37


Here is the regression line from the logarithmic regression plotted in the original scatter diagram, together with the linear regression line for comparison.

0

2000

4000

6000

8000

10000

12000

14000

16000

0 20000 40000 60000 80000 100000 120000 140000 160000EXP

FDHO

38


You can see that the logarithmic regression line gives a somewhat better fit, especially at low levels of expenditure.

0

2000

4000

6000

8000

10000

12000

14000

16000

0 20000 40000 60000 80000 100000 120000 140000 160000EXP

FDHO

39


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).

0

2000

4000

6000

8000

10000

12000

14000

16000

0 20000 40000 60000 80000 100000 120000 140000 160000EXP

FDHO

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

Documents

Elasticity of Y with respect to X is the proportional change in Y per proportional change in X: ELASTICITIES AND DOUBLE-LOGARITHMIC MODELS Y X A O 1 This