ECON4309/ECON6309 Economic Measurement Tutorial Assignment #3 2023
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ECON4309/ECON6309 Economic Measurement
Tutorial Assignment #3
2023
Problem 1
Suppose the consumer’s utility function is defined as f(q) ≡ [q · Aq]1/2 where A = A\ and A−1 exists. For the vector of commodity prices p ≫ 0, use calculus to solve the following constrained minimization problem:
min{p · q : [q · Aq]1/2 = 1} ≡ c(p)
q
Show that c(p) = [p · A−1p]1/2 .
Problem 2
The arithmetic conditional democratic cost of living indexes can be written as follows:
PD(∗) (p1(0) , . . . ,pH(0),p 1(1) , . . . ,pH(1),u,e1 , . . . ,eH ) ≡ 
[ 
]
H
= 工 
P (ph h(0),ph(1),uh ,eh )
h=1
(1)
(2)
where P (phh(0),ph(1),uh ,eh ) is the conditional cost of living index for household h. Instead of taking an arithmetic average of the individual household indexes, we could take an equally weighted harmonic mean. That is, define the harmonic conditional democratic cost of living index as follows:
PD(∗)H (p1(0) , . . . ,pH(0),p 1(1) , . . . ,pH(1),u,e1 , . . . ,eH ) ≡ { 
[Ph (p ,p ,u
,eh )]−1 }−1
(3)
1. Work out an observable bound for the Paasche-type harmonic conditional democratic cost of living index PD(∗)H (p1(0) , . . . ,pH(0),p1(1) , . . . ,pH(1),u1 ,e1(1) , . . . ,eH(1)).
2. Express this bound in expenditure share form and compare your answer to the similar form for the conditional plutocratic cost of living index.
Problem 3
The theory of the cost of living suffers from several problems. Which problem do you regard as being most of concern, and why?
Problem 4
Use the price data in Table 9.1 of the CPI Manual to calculate harmonic mean and CSWD elementary price indexes. Use the three different methods as in the table (month-to-month, chained month-to-month, direct index on January). Compare and discuss the results.
Table 9.1 Calculation of price indices for an elementary aggregate1
|
January February March April May June |
July |
||||||
|
Prices |
|
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|
Item A |
6 00 6 00 7 00 6 00 6 00 6 00 |
6 60 |
|||||
|
Item B |
7.00 7.00 6.00 7.00 7.00 7.20 |
7.70 |
|||||
|
Item C |
2.00 3.00 4.00 5.00 2.00 3.00 |
2.20 |
|||||
|
Item D |
5.00 5.00 5.00 4.00 5.00 5.00 |
5.50 |
|||||
|
Arithmetic mean prices |
5 00 5 25 5 50 5 50 5 00 5 30 |
5 50 |
|||||
|
Geometric mean prices |
4.53 5.01 5.38 5.38 4.53 5.05 |
4.98 |
|||||
|
Month-to-month price ratios |
1.10 |
||||||
|
Item A |
1.00 |
1.00 |
1.17 |
0.86 |
1.00 |
1.00 |
|
|
Item B |
1 00 1 00 0 86 1 17 1 00 1 03 |
1 07 |
|||||
|
Item C |
1.00 1.50 1.33 1.25 0.40 1.50 |
0.73 |
|||||
|
Item D |
1.00 1.00 1.00 0.80 1.25 1.00 |
1.10 |
|||||
|
Item A |
1 00 |
Current-to-reference-month (January) price ratios 1 00 1 17 1 00 1 00 1 00 |
1 10 |
||||
|
Item B |
1.00 1.00 0.86 1.00 1.00 1.03 |
1.10 |
|||||
|
Item C |
1.00 1.50 2.00 2.50 1.00 1.50 |
1.10 |
|||||
|
Item D |
1.00 1.00 1.00 0.80 1.00 1.00 |
1.10 |
|||||
|
Carli index – the arithmetic mean of price ratios Month-to-month index 100.00 112.50 108.93 101.85 91.25 113.21 |
100.07 |
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|
Chained month-to-month index 100.00 112.50 122.54 124.81 113.89 128.93 |
129.02 |
||||||
|
Direct index on January |
100.00 112.50 125.60 132.50 100.00 113.21 |
110.00 |
|||||
|
Dutot index – the ratio of arithmetic mean prices Month-to-month index 100.00 105.00 104.76 100.00 90.91 106.00 |
103.77 |
||||||
|
Chained month-to-month index 100.00 105.00 110.00 110.00 100.00 106.00 |
110.00 |
||||||
|
Direct index on January |
100.00 105.00 110.00 110.00 100.00 106.00 |
110.00 |
|||||
|
Jevons index – the ratio of geometric mean prices = geometric mean of price ratios Month-to-month index 100.00 110.67 107.46 100.00 84.09 111.45 |
98.70 |
||||||
|
Chained month-to-month index 100.00 110.67 118.92 118.92 100.00 111.45 |
110.00 |
||||||
|
Direct index on January 100.00 110.67 118.92 118.92 100.00 111.45 1 All price indices have been calculated using unrounded figures. |
110.00 |
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2023-04-26