Multiplicative Demand Functions
ROW/COL B C D E F G H I
5 For demand equations we can also use the following:
6
7 Q = aPcAdYe
8 This is called a multiplicative demand function.
9 The exponents are the elasticities of the variables.
10
11
12 Ln(Q) = a + c*ln(P) + d*ln(A) + e*ln(Y) 
13
14
15 Some data:
16
17 Q P A Y
18 500 105.00 100.00 500.00
19 1800 90.00 195.00 576.54
20 900 75.00 95.00 600.00
21 1400 130.00 145.00 706.23
22 1800 59.00 120.00 750.00
23 1200 114.00 169.00 810.00
24 1300 105.00 95.00 795.00
25 975 165.00 152.00 815.00
26 2000 134.00 202.00 900.00
27
28 A linear fit  using the above data
29
30 Q= f + g*P + h*A + i*Y 
31
32
33 SUMMARY OUTPUT
34
35 Regression Statistics
36 Multiple R 0.888864171
37 R Square 0.790079515
38 Adjusted R Square 0.664127224
39 Standard Error 283.3424136
40 Observations 9
41
42 ANOVA
43   df SS MS F Significance F
44 Regression 3 1510807.61 503602.5 6.2728475 0.0379091
45 Residual 5 401414.617 80282.92
46 Total 8 1912222.22      
47
48   Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
49 Intercept -142.008306 572.243073 -0.24816 0.8138798 -1613.00355 1328.986939
50 X Variable 1 -9.98905184 3.65624537 -2.73205 0.0411796 -19.3877144 -0.590389268
51 X Variable 2 8.198651835 2.7095159 3.025873 0.0292191 1.23363086 15.16367281
52 X Variable 3 1.933355427 0.88602668 2.182051 0.0809034 -0.34424493 4.210955787
53
54 Estimating the results
55
=+$C$49+$C$50*C57+$C$51*D57+$C$52*E57
56 Q P A Y Estimated Q Actual Error
57 500 105.00 100.00 500.00 596 500 96 <=+F57-G57
58 1800 90.00 195.00 576.54 1672 1800 -128
59 900 75.00 95.00 600.00 1048 900 148
60 1400 130.00 145.00 706.23 1114 1400 -286
61 1800 59.00 120.00 750.00 1702 1800 -98
62 1200 114.00 169.00 810.00 1671 1200 471
63 1300 105.00 95.00 795.00 1125 1300 -175
64 975 165.00 152.00 815.00 1032 975 57
65 2000 134.00 202.00 900.00 1916 2000 -84
66
67
68 Now try the non-linear fit
69
70 Take the logarithms of the data
71
72 ln(Q) ln(P) ln(A) ln(Y)
73 6.2146 4.6540 4.6052 6.2146
74 7.4955 4.4998 5.2730 6.3570
75 6.8024 4.3175 4.5539 6.3969
76 7.2442 4.8675 4.9767 6.5599
77 7.4955 4.0775 4.7875 6.6201
78 7.0901 4.7362 5.1299 6.6970
79 7.1701 4.6540 4.5539 6.6783
80 6.8824 5.1059 5.0239 6.7032
81 7.6009 4.8978 5.3083 6.8024
82
83 The multiplicative fit
84
85 SUMMARY OUTPUT
86
87 Regression Statistics
88 Multiple R 0.889270086
89 R Square 0.790801287
90 Adjusted R Square 0.665282059
91 Standard Error 0.251221433
92 Observations 9
93
94 ANOVA
95   df SS MS F Significance F
96 Regression 3 1.19286622 0.397622 6.3002402 0.03759558
97 Residual 5 0.31556104 0.063112
98 Total 8 1.50842726      
99
100   Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
101 Intercept -1.81142129 3.01281415 -0.60124 0.573897 -9.55609396 5.933251388
102 X Variable 1 -0.84307889 0.32376373 -2.60399 0.048017 -1.67533869 -0.010819089
103 X Variable 2 0.922450092 0.34014869 2.711902 0.0421803 0.04807146 1.796828719
104 X Variable 3 1.266559782 0.51433052 2.462541 0.0570503 -0.05556676 2.588686326
105
106
107 Remember that the above results are in logs, so the answer will be the log of sales and you would need to take the anti-log (exp()) of the number ot get the resulting quantity sold
108
109
110
111
=+$C$101+$C$102*C113+$C$103*D113+$C$104*E113
112 ln(Q) ln(P) ln(A) ln(Y) Estimated ln(Q) Estimated exp(Q) Actual error
113 6.2146 4.6540 4.6052 6.2146 6.3841 592 500 92 <=+G113-H113
114 7.4955 4.4998 5.2730 6.3570 7.3105 1496 1800 -304
115 6.8024 4.3175 4.5539 6.3969 6.8514 945 900 45
116 7.2442 4.8675 4.9767 6.5599 6.9842 1079 1400 -321
117 7.4955 4.0775 4.7875 6.6201 7.5518 1904 1800 104
118 7.0901 4.7362 5.1299 6.6970 7.4099 1652 1200 452
119 7.1701 4.6540 4.5539 6.6783 6.9242 1017 1300 -283
120 6.8824 5.1059 5.0239 6.7032 7.0081 1106 975 131
121 7.6009 4.8978 5.3083 6.8024 7.5716 1942 2000 -58
122
123 =EXP(F113)
124
125