1. Sub-Regional Workshop on International Merchandise Trade Statistics Compilation and Export and Import Unit Value Indices 21 – 25 November Guam Reef & Olive Spa Resort, Tamuning, Guam Price Indices and UVIs Presentation by Rens Hendriks Economic Statistician, Pacific Community

2. 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 0 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0
The Laspeyres index
𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 0 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0
L = Laspeyres
t = current period
0 = base period
𝑃 𝐿 𝑡/0 = Laspeyres price index between current and base period
p = price
q = quantity
i = item
𝑖=1 𝑛 = sum over item 1 to n

3. The Laspeyres index
The Laspeyres index compares the costs of buying the same goods today as we bought yesterday:
$5,00
$5,50

4. An example 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 0 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0
𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 0 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0
𝑃 𝐿 𝑡/0 = €6.00 €5.00 = 1.2
Base period (t = 0)
Item
Quantity
Price
Expenditure
Apples (i = 1) 2
€ 1.00
€ 2.00
Pears (i = 2)
€ 1.50
€ 3.00
Total
€ 5.00
Comparison period (t = 1)
Price
Expenditure
Apples (i = 1)
€ 1.50
€ 3.00
Pears (i = 2)
Total
€ 6.00

5. Exercise 1
Calculate the Index in exercise 1

6. A more practical way
In practice, the index as it is written is very hard to calculate
Detailed quantities are usually not available.
Luckily, the index can also alternatively be written as:
𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝑖 0 𝑝 𝑖 𝑡 𝑝 𝑖 0
That is: The index can be calculated as the weighted average of the price changes of the individual products

7. Laspeyres index - example
Prices
Jan
Feb
Mar
Apples
$3.29
Pears
$1.99
$2.19
Oranges
$2.79
$2.49
Price relatives
Weights
Contribu- tions
1.00
20%
0.20
1.10
70%
0.70
0.77
0.89
10%
0.10
0.09
Index
1.07
1.06

8. Exercise 2
Calculate the Index in exercise 2

9. Sampling of products
"Simple random sampling" by Dan Kernler - Own work. Licensed under CC BY-SA 4.0 via Commons -

10. Elementary indices 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝑖 0 𝑝 𝑖 𝑡 𝑝 𝑖 0 Weights
𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝑖 0 𝑝 𝑖 𝑡 𝑝 𝑖 0
Weights
Apples
Pears
Price relatives
Pink ladies
Granny smiths
Bartlet
D’anjous
Mismatch between price relatives and weights

11. Elementary indices Apples 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝐸𝐴 0 𝑝 𝑖 𝑡 𝑝 𝑖 0 𝑃 𝐸𝐴 𝑡/0
Pears
𝑃 𝐸𝐴 𝑡/0 But how to calculate?
𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝐸𝐴 0 𝑝 𝑖 𝑡 𝑝 𝑖 0 𝑃 𝐸𝐴 𝑡/0
EA: Elementary aggregate
Pink ladies
Granny smiths
Bartlet
D’anjous

12. The Jevons index 𝑃 𝐽 𝑡/0 = 1 𝑛 𝑝 𝑖 𝑡 1 𝑛 1 𝑛 𝑝 𝑖 0 1 𝑛 Toothpaste Jan
𝑃 𝐽 𝑡/0 = 1 𝑛 𝑝 𝑖 𝑡 1 𝑛 𝑛 𝑝 𝑖 𝑛
Toothpaste
Jan
April
Colgate
€ 1.99
€ 1.49
Aquafresh
€ 2.49
€ 2.59
Signal
€ 2.19
€ 2.99
Geometric mean prices
€ 2.21
€ 2.26
Jevons index =
1.021
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑚𝑒𝑎𝑛 𝐽𝑎𝑛= 1.99∙2.49∙ =2.21
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑚𝑒𝑎𝑛 𝐴𝑝𝑟𝑖𝑙= 1.49∙2.59∙ =2.26
𝑃 𝐽 𝑡/0 = =1.021

13. Exercise 3 Calculate the Elementary Indices for pears and apples
Using the elementary indices, calculate the index for ‘fruit’

14. Summary The process just treated results in ‘pure’ price indices
The prices of a fixed basket of goods is followed through time.
A variety of methods is available to deal with situations where the representativity of the sample changes
Downsides:
The process is labour intensive and expensive

15. Alternative
Replace the elementary indices based on samples with unit value indices.
For trade: unit value indices are a by product of IMTS compilation.
Benefits:
Cheap and easy to produce
In theory full coverage: no sampling error
However, they must be used with caution because:
Often issues with data quality
Unit value indices can have bias.
Requires extensive outlier detection routines

16. The Unit Value Index
𝑃 𝑈𝑉𝐼 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 𝑡 𝑖=1 𝑛 𝑞 𝑖 𝑡 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0 𝑖=1 𝑛 𝑞 𝑖 0
Formula looks complicated
However, really it is simply the average price in the current period divided by the average price in the previous period:
𝑃 𝑈𝑉𝐼 𝑡/0 = 𝑝 𝑡 𝑝 0

17. Example - Apples Jan Feb Apples Price Quantity Value Red delicious
$2.00 4
$8.00
$2.20 5
$11.00
Anjou
$1.50
$6.00
$1.65 3
$4.95
Golden Delicious
$1.80
$7.20
$1.89
$5.67
Total 12
$21.20 11
$21.62
Average Price
$21.2/12 =
$1.77
$29.81/15 =
$1.97
Unit value Index
$1.99/$1.77 =
1.11

18. Unit Value Bias 𝑃 𝐿 𝑡/0 = $8∙1.1+$6∙1.1+$7.2∙1.05 $21.20 =1.08 Jan Feb
Price relatives
Apples
Price
Quantity
Value
Red delicious
$2.00 4
$8.00
$2.20 5
$11.00
1.1
Anjou
$1.50
$6.00
$1.65 3
$4.95
Golden Delicious
$1.80
$7.20
$1.89
$5.67
1.05
Total 12
$21.20 11
$21.62
Average Price
$21.2/12 =
$1.77
$29.81/15 =
$1.97
Unit value Index
$1.99/$1.77 =
1.11
𝑃 𝐿 𝑡/0 = $8∙1.1+$6∙1.1+$7.2∙1.05 $21.20 =1.08

19. Exercise 4
Exercise 4 has example data for imports of vehicles for quarters
For each quarter calculate
The total value of imports
The total quantity of vehicles imported
The unit value of the imports (value/quantity)
Calculate the index (Q2 UVI / Q1 UVI)
Tip: Use a pivot table to do the calculations quickly
Do you think the index could be biased?