Linear Regression – Estimating Demand Elasticities

U.S. Sugar Consumption 1896-1914

H. Schultz (1933). “A Comparison of Elasticities of Demand Obtained by Different Methods,” Econometrica, Vol.1, #3,pp. 274-308.H. Schultz (1925). “Appendix 2,” Journal of Political Economy, Vol.33, #6, pp.634-637.

Problem Description

• Dependent Variable: Consumption per Capita (Q)• Independent Variables:

o Real Price (P), BLS adjusted, 1913=100, all commoditieso Year (t), centered around 1905

• Models:o Additive:

o Multiplicative:

o Linearized Multiplicative Model:

0 * * 0P tQ P t E

0 ** ** 1tP tQ P e E

*0 0ln ln ln ln ** * * ***P t P tQ P t Q P t

Elasticities of Demand (Ignoring error terms)

0 0

10 1

0

Controlling for Time:

% Change in

% Change in

Model 1:

1Model 2: tP

tP

QP t

PP QP t

P t P t

tP QPt

Q Q Q P Q

P P P Q P

PQ P P

P Q P t P t

Q PP e

P Q P e

t P

Effects of Time Shift (Ignoring Error terms)

0

Controlling for Price:

Model 1:

1Model 2:

Pt

t

tt t t

Q

t

Q QP e Q

t Q t

Datayear Sugar UPrice Pop PrIndx Q P t ln(Q) ln(P)1896 2195 4.532 70.240 66 62.50 6.87 -9 4.1352 1.92671897 2319 4.503 71.574 66 64.80 6.82 -8 4.1713 1.92031898 2243 4.965 72.943 67 61.50 7.41 -7 4.1190 2.00291899 2327 4.919 74.345 75 62.60 6.56 -6 4.1368 1.88081900 2486 5.320 76.258 82 65.20 6.49 -5 4.1775 1.86991901 2657 5.050 77.351 80 68.70 6.31 -4 4.2297 1.84251902 2874 4.455 78.956 84 72.80 5.30 -3 4.2877 1.66841903 2856 4.638 80.564 84 70.90 5.52 -2 4.2613 1.70861904 3099 4.772 82.311 84 75.30 5.68 -1 4.3215 1.73711905 2948 5.256 83.631 86 70.50 6.11 0 4.2556 1.81021906 3208 4.515 84.310 91 76.10 4.96 1 4.3320 1.60171907 3353 4.649 86.529 96 77.50 4.84 2 4.3503 1.57751908 3568 4.957 87.882 91 81.20 5.45 3 4.3969 1.69511909 3649 4.765 89.218 94 81.80 5.07 4 4.4043 1.62321910 3752 4.972 91.961 97 81.60 5.13 5 4.4018 1.63431911 3754 5.345 94.798 95 79.20 5.63 6 4.3720 1.72751912 3925 5.041 96.556 99 81.30 5.09 7 4.3981 1.62771913 4192 4.278 98.173 100 85.40 4.28 8 4.4473 1.45351914 4212 4.683 99.929 98 84.30 4.78 9 4.4344 1.5641

Note: On Figures 2A-2C, Schultz is using Q/10. Plots in this series are based on Q

Regression – Model 1SUMMARY OUTPUT

Regression StatisticsMultiple R 0.9765R Square 0.9536Adjusted R Square 0.9478Standard Error 1.8159Observations 19

ANOVAdf SS MS F P-value

Regression 2 1083.92 541.96 164.36 0.0000Residual 16 52.76 3.30Total 18 1136.68DV=Q

Coeff Std Err t Stat P-value Lower 95%Upper 95%Intercept 92.88 5.76 16.12 0.0000 80.67 105.09P -3.34 1.01 -3.31 0.0044 -5.48 -1.20t 0.92 0.15 6.05 0.0000 0.60 1.24

t Year elastcity dQ/dt-9 1896 -0.37169 0.92-8 1897 -0.36303 0.92-7 1898 -0.40097 0.92-6 1899 -0.33448 0.92-5 1900 -0.32512 0.92-4 1901 -0.30934 0.92-3 1902 -0.24451 0.92-2 1903 -0.25388 0.92-1 1904 -0.25983 0.920 1905 -0.28152 0.921 1906 -0.21446 0.922 1907 -0.20582 0.923 1908 -0.23479 0.924 1909 -0.21251 0.925 1910 -0.21293 0.926 1911 -0.23592 0.927 1912 -0.2065 0.928 1913 -0.16615 0.929 1914 -0.18723 0.92

^

92.88 3.34 0.92Q P t

Regression – Model 2

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.9738R Square 0.9482Adjusted R Square 0.9418Standard Error 0.0264Observations 19

ANOVAdf SS MS F P-value

Regression 2 0.2045 0.1023 146.5135 0.0000Residual 16 0.0112 0.0007Total 18 0.2157

Coeff Std Err t Stat P-value Lower 95%Upper 95%Intercept 4.7683 0.1459 32.6822 0.0000 4.4590 5.0776ln(P) -0.2727 0.0843 -3.2369 0.0052 -0.4513 -0.0941t 0.0124 0.0022 5.6281 0.0000 0.0077 0.0171

t Year elasticity dQ/dt-9 1896 -0.2727 0.7731-8 1897 -0.2727 0.7841-7 1898 -0.2727 0.7762-6 1899 -0.2727 0.8126-5 1900 -0.2727 0.8251-4 1901 -0.2727 0.8417-3 1902 -0.2727 0.8937-2 1903 -0.2727 0.8950-1 1904 -0.2727 0.89920 1905 -0.2727 0.89241 1906 -0.2727 0.95652 1907 -0.2727 0.97483 1908 -0.2727 0.95594 1909 -0.2727 0.98705 1910 -0.2727 0.99636 1911 -0.2727 0.98347 1912 -0.2727 1.02328 1913 -0.2727 1.08649 1914 -0.2727 1.0673

^

^4.7683 0.2727 0.0124

ln 4.7683 0.2727 ln 0.0124

t

Q P t

Q e P e