VPC in Farm Inventories 1 

Value of Physical Change in Farm Inventories: 

How Does One Reconcile Annual and Subannual Estimates? 

Andrew Baldwin, 

Industry Measures and Analysis Division, 

Statistics Canada 

Andy.Baldwin@statcan.ca

(613)951-3675 

R.H. Coats Building, 22-B 

Ottawa ON K1A 0T6 

Canada 

Background of the Issue in Canada 

For Statistics Canada, there has been a substantial difference in the calculation of the 

value of physical change (VPC) of farm inventories by different divisions within the 

agency at current prices. Historically, three divisions have used a single annual price to 

calculate the annual VPC in inventories, while one division has used different prices to 

measure the VPC for each quarter, with the annual VPC calculated simply as the sum of 

the quarterly estimates. More specifically, Agriculture Division has used the annual 

approach to estimate VPC as a component of agricultural value added, Input-Output 

Division has used the same approach in its own calculations of gross output and value 

added for the Input-Output Tables, while the Industry Measures and Analysis Division 

has necessarily used the same approach, since the Input-Output estimates of gross output 

and value added at current prices are its benchmarks, from which it calculates gross 

output and value added in volume terms. 

Until this year, only Income and Expenditure Accounts Division, which is responsible for 

quarterly measures of gross domestic expenditure, favoured the sum-of-quarters approach 

in its estimates of farm inventories, which is one of two components of the aggregate 

business investment in inventories, the other being (obviously) non-farm inventories. In 

this approach, quarterly VPC is valued at current quarter prices and the annual VPC is 

simply the sum of the quarterly estimates. 

This difference in approach is primarily an issue relating to current price measures, since 

even IEAD agrees that the annual base year price used to calculate volume measures of a 

Laspeyres type must be calculated as some average of quarterly prices themselves, and 

cannot be calculated implictly from the annual VPC at current prices and the change in 

inventories measured in tonnes or some other physical unit of measure. However, the 

choice of approach does have implications for System of National Accounts price and 

volume estimates, especially for industry value added, as will be discussed later. 
 VPC in Farm Inventories 2 

In all probability, of all the official national accounting agencies in the world, only the 

Canadian System of National Accounts (hereafter CSNA) has ever published two quite 

different measures of farm VPC at current prices. The differences between the two 

methods are not trivial, even at the national level. Moreover, GDP is calculated annually 

on a provincial basis using both the expenditure and value added approaches. For a small 

province with a large agriculture sector like Saskatchewan or Prince Edward Island, the 

discrepancy between the two is often more important in value than many other individual 

components of GDP. 

To some extent, this has been a problem, creating an important discrepancy between the 

value added and sum-of-expenditures measures of gross domestic product, which, as 

simple textbook examples show, should be equal. At the same time, it is one of those 

problems that is nice to have, since it only exists because of the good quality and fine 

detail of the underlying Canadian industrial statistics. A 1996 OECD document found 

that half of the 18 countries surveyed had no independent estimates of VPC, lacking the 

information to do so. Other countries that do have independent estimates of inventory 

change may adopt the annual approach as much out of necessity as conviction, since they 

simply lack the detailed quarterly information to use the sum-of-quarters approach. Even 

the United States Bureau of Economic Analysis (hereafter BEA) has essentially adopted 

the annual approach by default, since for the most part it lacks the detailed quarterly 

information to implement the sum-of-quarters approach. 

At a meeting on December 24 that included the three directors of SNA divisions that 

calculate farm VPC, it was decided that IOD would move to IEAD’s sum-of-quarters 

approach starting with the update of the 2002 I-O Tables in 2005, a decision that has now 

been implemented. This inevitably meant that IMAD was also locked into the sum-ofquarters appproach for its industry value added estimates. 

Oddly enough, a main reason for making this change seems to have been a desire for 

greater consistency in the farm VPC estimates, a goal which was easily achievable simply 

by having IEAD switch to the annual approach for its annual estimates, which would not 

only have ensured that all four divisions used the same approach within SNA, but would 

have improved comparability of our estimates with the BEA’s. 

However, the move did not eliminate differences in the volume measure of farm VPC by 

IEAD and IMAD, since IEAD, unlike any other official entity in the world, calculates 

chain Fisher volume estimates with quarterly links, while IMAD calculates chain Fisher 

volume estimates with annual links, as prescribed by the System of National Accounts 

1993 manual. While it does make IOD’s farm VPC at current prices identical with that of 

IOD, IOD’s estimate will no longer be the same as Agriculture Division’s. Now Statistics 

Canada will have two different estimates of farm VPC appearing as components of value 

added by the agriculture sector, where previously there was only one. Also, the Canadian 

estimates for value added by the agriculture sector will be less comparable than they once 
 VPC in Farm Inventories 3 

were with those of the United States, as well as with other countries that use the annual 

approach.1

Data Are Much Better for the Annual Approach than for the Sum-of-Quarters 

Approach 

As good as they are, the Statistics Canada data are not equally good for calculating both 

annual and quarterly VPCs. While they do permit for reasonable estimates of quarterly 

inventory change, unlike those of many other countries, the annual data are much 

stronger than the quarterly, which in itself would seem to dictate treating the annual 

estimates as benchmarks to which quarterly estimates would be adjusted. 

Briefly put, the main source of information on crop inventories comes from a Farm 

Stocks Survey conducted three times a year for the dates December 31, March 31 and 

July 31 by Crops Section of Agriculture Division. The survey covers cereals, oilseeds and 

specialty crops. 

So for crops, while there is reliable information on annual change in physical stocks of 

the most important crops (i.e. the change from December 31 of one year to the next), only 

the first quarter has similarly reliable information. Since one can difference stocks 

between March 31 and December 31, the first quarter estimates are on the same footing 

with the annual estimates. The same cannot be said for the other quarters. 

A farm supply-disposition (also called supply-demand) balance sheet reflecting activity 

on farms before grain enters commercial channels is used to generate stock estimates for 

the missing dates: June 30 and September 30. In addition to the opening inventory 

estimates themselves, supply consists of production, for which monthly estimates are 

available. In addition to the closing inventory estimates, disposition consists of deliveries, 

seed use, feed, waste and dockage.2

 The largest part of the deliveries are licenced grain 

deliveries for which weekly Canadian Grain Commission estimates are available, albeit 

with substantial quality problems and data lags that require adjustment. Seed use 

information is available from another survey conducted by Crops Section of Agriculture 

Division. 

Feed, waste and dockage, however, is a residual category for which there are no monthly 

survey estimates. One of the main factors driving monthly consumption of feed is the 

 

1

 The present author was co-author of a submission to Canada’s SNA Steering Committee with a 

recommendation that CSNA terminology regarding chain volume measures, which largely copy those of 

the BEA, be replaced by others terms, which were not only more appealing, but more consistent with SNA 

1993 or IMF standards. At that time, the decision was made to stay with the existing terminology because 

of the very high importance attached to CSNA being consistent with its American counterpart, even at the 

expense of being inconsistent with SNA 1993. Obviously the deference extended to the BEA in these 

matters of terminology does not extend to questions of methodology; the minutes of the December 24 

meeting make no mention of BEA practice. 

2

Dockage is any material that must be removed from a commercial grain crop so that it may be assigned 

the highest grade for which it qualifies. See Cat.No.22-001-XIB, Vol.82, No.1, p.6. 
 VPC in Farm Inventories 4 

monthly numbers of livestock, so that the feed estimates, and hence the crop stock 

estimates, are only as good as the monthly head counts for livestock. Residual estimation 

of the closing inventories for June 30 and September 30 is thus much less reliable 

because one of the categories required for this residual estimation is also a residual 

category. 

If for crops, only first quarter estimates of physical changes in stocks are as reliable as 

annual estimates, for livestock categories, there is no quarter of the calendar year for 

which changes in head counts are available from surveys, as they are on an annual basis, 

except for hogs. The Livestock Survey makes a count of all livestock for January 1 and 

July 1, and of hogs for April 1 and October 1. Thus, unlike the Crop Survey all survey 

dates are for the first day of a calendar quarter, however only hogs are surveyed four 

times a year. And since the April and October hog surveys only have a quarter of the 

sample of the January livestock survey, even for hogs the quartery estimates of physical 

change in stocks are not of the same quality as the annual estimates. 

Agriculture Division provides the CSNA with monthly estimates of inventory change on 

a timely basis for all livestock categories, and these are in fact used by the present author 

in calculating volume estimates of livestock production. Although there are no published 

or experimental estimates of VPC at current prices for livestock based on a sum-ofmonths approach all the data exist to calculate it. 

This is not true for crops, for which only end-of-quarter stock estimates are presently 

available, but since marketings of grain are available on a weekly basis, there is no reason 

one could not calculate end-of-month stock estimates for crops and apply a sum-ofmonths approach to VPC estimation. 

Official Guidelines on the Measurement of VPC 

This measurement issue is one on which the SNA93 manual fails to be prescriptive, 

providing some ammunition for both sides in the dispute, although on balance it would 

seem its dictums provide more support for the annual approach. Generally speaking, the 

SNA93 manual seems to operate in an annual estimation framework and national 

accountants who must construct monthly or quarterly estimates are often left with no 

clear guidance. On the other hand, the IMF Quarterly National Accounts Manual seems 

to opt clearly and unambiguously for the sum-of-quarters approach. 

According to SNA93, the basic principle of VPC measurement is “that output should be 

recorded at the time it is produced and valued at the same price whether it is immediately 

sold or otherwise used or entered into inventories for sale or use later. No output is 

recorded when goods produced previously are withdrawn from inventories and sold or 

otherwise used. It follows that entries into inventories must be valued at the basic prices 

prevailing at the time of entry, while withdrawals must be valued at the prices at which 

they are then sold.” (paragraph 6.58). SNA93 also says, and this is the crucial point: 

“When prices are stable, the measurement of changes in inventories is relatively simple. 

However, when there is inflation, significant price increases may occur while goods are 
 VPC in Farm Inventories 5 

held in inventory. Holding gains accruing on goods held in inventory after they have been 

produced must not be included in the value of output”.

The IMF Quarterly National Accounts Manual provides an example of how the quarterly 

VPC for coal inventories would be calculated using end-of-quarter information on book 

values at current prices and a monthly price index for coal. The change in inventories for 

2001 Q1 is evaluated at first quarter prices and the possibility that there might indeed be 

some advantage to be gained in evaluating the VPC at average 2001 prices is not even 

considered. (For Canada at any rate, the calculation of farm VPC is much simpler than in 

the IMF manual’s example and similar ones relating to non-farm inventories. The big 

bugbear of such estimates is usually to obtain good estimates of the inventory stocks in 

volume terms, using monthly price indexes and information on the turnover period of 

inventories. But for Canadian farm inventories, physical estimates of livestock and crops 

by category are available from survey information. It is only the revaluation of the 

changes in these physical stocks that is at issue.) 

The IMF manual says that: “...higher frequency data reduce the possibility of uneven 

price and volume movements within the period. As a consequence, the annual sum of the 

quarterly valuation adjustments may be superior to annually calculated ones, unless there 

is some other compelling difference, such as differences in coverage or detail [emphasis 

added]. Similarly, if monthly data are available, the calculation should generally be done 

on a monthly basis for use in quarterly estimates.” (See section 3.139). Thus, although the 

IMF manual favours the sum-of-quarters approach over the annual approach for 

measuring annual change, they judge it inferior to a sum-of-months approach, and 

presumably, as long as the coverage and detail were the same, they would judge a sumof-months approach inferior to a sum-of-days approach. 

They seem to be unperturbed by the problems that high-frequency valuation can create, 

which include the incorporation of holding gains in the valuation estimates, contrary to 

SNA93 principles, or the creation of negative implicit prices for VPC for commodities, 

even at the most detailed level. They are not explicitly concerned that price detail might 

dry up as they go from the annual to the quarterly frequency or from the quarterly to the 

monthly, although this may be a consideration when they write “as long as the coverage 

and detail are the same”. 

A Hypothetical Example 

Now let us consider an example formulated by Kishori Lal, the former director general of 

the CSNA, in an earlier paper on the inventory change problem. In the first quarter a 

farmer produces 100 tonnes of a crop, and the market price per tonne is $10, the same 

price as in the year previous. Thus the value of output is $1,000. The farmer has no 

opening inventory as of the beginning of the first quarter. There is no opening inventory 

on January 1st, at the beginning of the year and of the first quarter. No output is sold and 

the entire output is added to inventory. If the VPC is calculated based on the same quarter 

prices, this addition to inventory is valued at $1,000. 
 VPC in Farm Inventories 6 

In the second quarter, there is no production, but the entire first quarter crop is sold at the 

much higher second quarter price of $15. The second quarter VPC reflects a withdrawal 

from inventory which is valued at $1,500. Because the second quarter price is so much 

higher this VPC estimate includes a holding gain of $500 from the goods held in 

inventory. 

In the third quarter, the farmer produces 200 tonnes but the price per tonne falls to $8. He 

sells 80 tonnes in the same quarter, leaving an inventory of 120 tonnes at the end of the 

third quarter, for a VPC of $960 ($8 x 120). This stock is entirely sold off at the much 

higher quarterly price of $12, creating a holding gain of $480 (($12-$8) x 120). 

Calculating the sum of the quarterly VPCs, one has an annual VPC of -$980, although 

there is no change in the farmer’s physical stocks of his crop at all between the beginning 

and the end of the year. He started and ended the year with no inventories. 

The problem, of course, lies in the holding gains, which for the year are equal in value 

but opposite in sign to the VPC. This example, is of course, somewhat unrealistic as 

appies to Canadian agriculture where the winter is harsh and very few farmers harvest a 

winter and a fall crop, but it still illustrates in a very nice way the strengths and 

weaknesses of the different approaches. 

All that is required to remove any holding gains from the quarterly VPC estimates is to 

evaluate the stocks at the same price in all quarters, i.e. at the annual price. The average 

sales price for the year is $11.93 per tonne (based on a sales value $3,580 with 300 

tonnes sold). It can be seen that using this price for all quarters, the inventory 

accumulation of the first quarter is valued at $1,193 and is matched with a decumulation 

of the same value in the second quarter. The same thing happens in the third and fourth 

quarters where a $1,432 VPC in the third quarter is matched by a -$1432 VPC in the 

fourth quarter, so that the annual VPC takes a zero value. 

In his own paper Lal showed the annual VPC generated by the annual approach, which he 

admitted was intuitively more attractive than the negative VPC generated by the sum-ofquarters approach. However, he neglected to show the quarterly estimates that would be 

generated using an annual price, which are also intuitively much more satisying than 

those generated using the quarterly approach. The inventory pattern in the first half of the 

year and the second half of the year is like the year as a whole. There are big swings in 

stocks within each semesters, but zero stocks when they begin and end.. For the first half 

of the year, the quarterly approach shows a VPC of -$500, while the annual approach 

shows a zero VPC. Again for the second half of the year, the quarterly approaach shows a 

VPC of -$480, while the annual approach shows a VPC of $0. 

IEAD spokesmen have rightly pointed out that they do not know the annual price at the 

beginning of the year. Using their quarterly approach until an annual estimate became 

available would mean considerable revisions in their quarterly VPC estimates. However, 

it is neither necessary nor desirable to calculate estimates for the first three quarters of the 

year valuing each quarter at its own prices. This can introduce massive holding gains into 
 VPC in Farm Inventories 7 

the estimates that will have to be removed when annual prices are later substituted. It 

would make much better sense to calculate the VPC for all quarters at the first quarter 

prices. Then the first quarter estimates would be consistent with the quarterly approach, 

while the remaining quarters would be calculated so that there are no holding gains in the 

estimates. 

Of course, if minimization of revisions is the goal, some other set of prices than first 

quarter prices may more closely approximate the final annual prices. The table shows 

estimates based on replacing the average first quarter price by the average of first and 

second quarter prices in the second quarter and the average price for the first three 

quarters in the third quarter. (Unfortunately since there are no sales in the first quarter, a 

sales-weighted average of first and second quarter prices is equivalent to using first 

quarter prices.) The VPC’s generated by these cumulative prices for the current year are 

shown in Table 1. It can be seen that after three quarters, there is little change between 

the VPC estimates based on partial year information and the final estimates based on four 

quarters of data. 

The other point is that it is possible to calculate VPC estimates according to the quarterly 

approach and benchmark them to annual estimates based on the annual approach. For the 

current year, this should only be done once the annual estimate is available, although then 

the quarterly indicator series should be consistently based on the quarterly approach over 

the benchmarking range. The estimates in Table 1 were calculated just by assigning each 

quarter a share of the annual discrepancy between the annually-measured VPC and its 

sum-of-quarters counterpart based on the ratio of the absolute value of its VPC to the sum 

of absolute values of the VPCs for the year. Of course, more sophisticated benchmarking 

procedures would probably be used in a production context. 

If one recalls the discussion of the data sources in a previous section, all the data 

currently exists to calculate quarterly livestock VPC using a sum-of-months approach 

rather than a quarterly approach, and could surely be created to do so for crops. Given 

that the weaknesses of the monthly stock estimates are essentially the same as the 

quarterly ones in relation to the much superior annual estimates, it can be safely assumed 

that data considerations are not responsible for IEAD opting for a quarterly approach to 

measuring quarterly VPC rather than a sum-of-months approach. That division does not 

seem to share the view of the IMF Quarterly Manual that “a quarterly estimate from the 

sum of monthly data ... will be better than one calculated from quarterly data” although 

they obviously endorse the view that “the annual estimate would be better if made as a 

sum of the quarters than if made from annual data”. (See the IMF Manual, section 3.140). 

Therefore, IEAD appears to be concerned about the perverse effects on measured 

quarterly VPC one would encounter using the sum-of-months approach due to inventory 

changes in different months within the same quarter being evaluated at different prices. 

However, it is quite unconcerned about the perverse effects on measured annual VPC that 

currently prevail using the sum-of-quarters approach due to inventory changes in 

different quarters within the same year being evaluated at different prices. 
 VPC in Farm Inventories 8 

In the same way, it has implemented chain volume measures of GDP with quarterly 

linking in spite of SNA93 guidelines that call for annual linking. The rationale for this is 

that one must ensure that the first quarter estimates of quarterly growth are true measures 

of volume change, which would not happen with once-a-year chaining with annual links. 

(It would be possible with once-a-year chaining with fourth-quarter links.) It doesn’t 

matter that in doing this, the annual measures are not true measures of volume change, as 

they are in all countries that have followed the SNA93 guidelines. It is a different issue, 

but one that reveals a general prejudice in favour of quarterly estimates as opposed to 

annual estimates. 

An Actual Example 

We will now look at an actual example of VPC measurement which shows another 

perverse result possible using the sum-of-quarters approach: a negative annual implicit 

price for a commodity.3

Table 2 shows a summary of the Canada Supply/Disposition table from Agriculture 

Division for barley 1988. The upper half of the table lists physical change by quarter in 

tonnes, the bottom half multiplies the quantities by quarterly average prices to estimate 

VPC. The quarterly prices are weighted averages of surveyed monthly prices for 

marketings (deliveries) of barley, with weighting based on marketings. Similarly, the 

annual unit price of $86.72 is a weighted average of the quarterly prices. 

The two columns on the right list annual estimates. SUM OF QTS represents the IEAD 

method, while ANNUAL CALC is the Agriculture Division method, which until this year 

was also used by IOD. 

In quantity terms, the two figures are, of course, identical, both showing a reduction of 

1.9 million tonnes. Valued in thousands of dollars, multiplying the physical change by the 

annual price of barley gives a VPC of $169.5 million. (The calculation is off slightly 

because the Can. figure is a sum of 10 provinces; this table shows only the Canada 

totals.) 

By contrast, summing the VPCs over quarters yields +$87,541. The implicit price for the 

quarterly method is $-46.02 per tonne, while the range of prices is $56.50 to $116.58. 

This result is, of course, again due to the incorporation of holding gains in the annual 

estimate of the VPC at current prices, given the strong increase in barley prices in the 

fourth and especially the second quarter when there was a depletion of stocks. It is 

another argument for benchmarking quarterly VPC estimates based on the quarterly 

approach to annual estimates based on the annual approach. 

 

3

 This example is taken from a memo by Richards and Miller, 1997. 
 VPC in Farm Inventories 9 

Table 1: Hypothetical Example of Quarterly Inventory Change (See Lal (2001)) 

 

Y0 Q1 Q2 Q3 Q4 Y1 

Output quantity 100 0 200 0 300

Sale quantity 0 100 80 120 300

Unit sale price 10 10 15 8 12 11.93

Cumulative unit sale price 10 15 11.89 11.93

Output value 1,000 0 1,600 0 2,600

Sale value 0 1,500 640 1,440 3,580

Beginning-of-period quantity 0 100 0 120 0

End-of-period quantity 100 0 120 0 0

Start-of-period book value 0 1,000 0 960 0

End-of-period book value 1,000 0 960 0 0

Change in book value 1,000 -1,000 960 -960 0

VPC (valued at quarterly prices) 1,000 -1,500 960 -1,440 -980

Inventory holding gain 0 500 0 480 980

VPC (valued at Q1 prices for all quarters) 1,000 -1,000 1,200 -1,200 0

VPC (valued at annual prices) (Q2 update) 1,500 -1,500 0

VPC (valued at annual prices) (Q3 update) 1,189 -1,189 1,427 1,427

VPC (valued at annual prices) (final; Q4 

update) 1,193 -1,193 1,432 -1,432 0

VPC (quarterly approach; benchmarked est.) 1,200 -1,200 1,152 -1,152 0
 VPC in Farm Inventories 10 

Table 2: Actual Example of Inventories for Barley 

Value of Physical Change of Farm Stocks, 1988 

Barley for Canada 

 Sum of Annual 

 Q1 Q2 Q3 Q4 Quarters Estimate 

 

Beginning Inventory 10,511,354 7,102,375 3,834,276 8,979,115 

(tonnes) 

Ending Inventory 7,102,375 3,834,276 8,979,115 8,609,303 

(tonnes) 

Physical Change -3,408,979 -3,268,099 5,144,839 -369,812 -1,902,051 -1,902,051

(tonnes) 

Avg. FCR Price per 

Tonne $56.50 $76.21 $104.21 $116.58 

 

$86.72

Value of Physical Change -$192,766 -$246,795 $564,428 -$37,326 $87,541 -$169,511

in Inventories (,000) 

 

Implicit Price 

(VPC/Tonnes) -$46.02 $86.72
 VPC in Farm Inventories 11 

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Dollars”, mimeo, Vancouver: Univeristy of British Columbia Department of Economics, 

April 30, 2006 

Diewert, W. Erwin and Anne-Marie Smith, Productivity Measurement for a Distribution 

Firm, Cambridge, MA: National Bureau of Economic Research Working Paper No.4812 

Ehemann, Christian. 2005. “An Alternative Estimate of Real Inventory Change for 

National Economic Accounts.” International Journal of Production Economics 93-94, 

101-110.

Hill, Peter, Inflation Accounting: A Manual on National Accounting under Conditions of 

High Inflation, Paris: OECD, 1996 

Lal, Kishori, “Aggregating Sub-Annual Current Price Value of Changes in Inventories to 

Annual Totals under Conditions of Inflation: An Inherent Dilemma”, mimeo, Systems of 

National Accounts, Statistics Canada, March 2001 

Mayerhauser, Nicole T. “Description of Change in Farm Inventories in US GDP”, 

Washington: National Income and Wealth Division, Bureau of Economic Analysis, 

memo to André Picard, Stastics Canda, January 26, 2005 

Picard, André, “Real Net Exports and Real Change in Inventories”, Ottawa: Income and 

Expenditure Accounts Division, Statistics Canada, memo to A. Baldwin, October 1, 2004 

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———, “The Measurement of Chained-Dollar Change in Private Inventories”, mimeo, 

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Canada, June 10, 1997