Barcelona Tutorial Ch V

Tina

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• Publié par: Phyon
• Nombre de lectures: 9
• Langue: English

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1
The Measurement of Business Capital, Income and Performance

Tutorial presented at the University Autonoma of Barcelona, Spain, September 21-22,
2005; revised December 2005.

Erwin Diewert,
Department of Economics,
University of British Columbia,
Vancouver, Canada, V6T 1Z1.
Email: [e-mail]

V. Constructing a Capital Stock for Inventories and the Measurement
of Inventory Change

1. Introduction
2. The SNA Treatment of Inventory Change
3. A Suggested Alternative Treatment of Inventory Change
4. Conclusion
Appendix: A Theoretical Treatment of Inventory Change

1. Introduction

The current System of National Accounts (SNA) treatment of inventory change in real
terms is very confusing to users. The problem is that it can happen that the value of
inventory change has a sign that is opposite to the sign of the corresponding constant
dollar inventory change. This means that the corresponding implicit price deflator is
meaningless. In this paper, the nature of the problem is explained and a solution to the
problem is suggested. In the Appendix, a theoretical framework that provides a unified
1treatment for measuring inventory change and the user cost of inventories is explained.
Appendix 2 gives some background information on the origins of the theoretical
framework used in Appendix 1.

In section 2, a simple 2 good, 4 period numerical example is introduced and it is
explained how a “typical” SNA treatment of inventory change works in the context of
this example. The example illustrates the problem described in the previous paragraph:
the current dollar aggregate inventory change has a sign opposite to the corresponding
constant dollar change.

In section 3, the same example is reworked using the methodological approach suggested
in Appendix 1. The suggested solution involves treating inventory change in a manner
that is symmetric to the current SNA treatment of exports and imports.

Section 4 concludes.

1 This methodology is based on Diewert and Smith (1994) and Diewert (2004; 36). The initial accounting
methodology can also be found in Diewert (2005; 21-23). Diewert and Lawrence (2005) used this
framework as well. The underlying model of production that is used in this chapter is the Hicks (1961) and
Edwards and Bell (1961) model explained in section 9.2 of chapter I. 2

2. The SNA Treatment of Inventory Change

Consider the following data on the end of period stocks of two inventory items for three
t tperiods, where p and q denote the price and quantity of stock n at the end of period t: n n

Table 1: Price and Quantity Data for Two Inventory Stocks

t t t t p p q q1 2 1 2
Period 0 1.0 1.0 200 200
Period 1 .9 2.0 260 150
Period 2 .8 3.0 310 110
Period 3 .7 4.0 330 100

tThus the price of the first stock p is slowly declining while the corresponding end of 1
tperiod stock q grows from 200 to 330 over the three periods. On the other hand, the 1
tprice of the second stock p quadruples over the three periods while the corresponding 2
tend of period stock q steadily falls from 200 to 100 over the three periods. These price 2
changes are more violent than what is usually observed over the course of a year but they
would not necessarily be unusual if we think of the first good as computer chip and the
second good as crude oil.

tThe end of period t SNA constant dollar stock of inventories, K , using the end of SNA
period 0 as the base period, can be defined as the following Laspeyres type quantity
aggregate:

t 0 t 0 t(1) K ≡ p q + p q ; t = 0,1,2,3. SNA 1 1 2 2

tNote that we use the inventory stocks q at the end of period t along with the prices of the n
0stocks at the end of period 0, p , in the above definition of the period t constant dollar n
stock of inventories. Thus for period 0 (the beginning of period 1), the constant dollar
stock coincides with the current dollar stock. The value of the current dollar stock of
tinventories at the end of period t, VK , is defined in the usual fashion as follows:

t t t t t(2) VK ≡ p q + p q ; t = 0,1,2,3. 1 1 2 2

t t tIf we divide VK by K , we obtain P , the end of period t SNA implicit price index SNA SNA
for the constant dollar stock of inventories:

t t t t t t t 0 t 0 t(3) P ≡ VK /K = [p q + p q ]/[p q + p q ] ; t = 0,1,2,3. SNA SNA 1 1 2 2 1 1 2 2

Note that the SNA implicit price index for the inventory stock is a Paasche price index
between period t and 0.

tThe SNA constant dollar value of inventory change for period t, ΔK , can be defined SNA
in a straightforward manner as the difference between the end of period t and beginning
of period t constant dollar stocks defined above by (1):
3
t t t 1−(4) ΔK ≡ K − K t = 1,2,3 SNA SNA SNA
0 t 0 t 0 t 1 0 t 1− − = p q + p q − [p q + p q ] using (1) 1 1 2 2 1 1 2 2
0 t t−1 0 t t−1 = p [q − q ] + p [q − q ] 1 1 1 2 2 2
0 t 0 t = p Δq + p Δq 1 1 2 2

t t t 1−where Δq ≡ q − q is the difference in the closing and opening stock of inventory n n n
item n over period t. Note that the last equation in (4) shows that the aggregate change in
the constant dollar change in inventories is equal to the sum of the individual item
changes, using the end of period 0 prices as weights.

tThe (approximate) SNA current dollar value of inventory change for period t, ΔVK , SNA
tcan be defined as the sum of the individual item changes, Δq , weighted by the average n
t−1 t 2of the beginning and end of period prices, (1/2)p + (1/2)p : n n

t t−1 t t t−1 t t(5) ΔVK ≡ [(1/2)p + (1/2)p ]Δq + [(1/2)p + (1/2)p ]Δq ; t = 1,2,3. SNA 1 1 1 2 2 2

tThe corresponding implicit price index for the SNA inventory change, ΔP , is obtained SNA
t tby dividing the value series ΔVK defined by (5) by the constant dollar series ΔK SNA SNA
defined by (4):

t t t(6) ΔP ≡ ΔVK /ΔK t = 1,2,3 SNA SNA SNA
t 1 t t t 1 t t 0 t 0 t− − = (1/2){[p + p ]Δq + [p + p ]Δq }/{p Δq + p Δq }. 1 1 1 2 2 2 1 1 2 2

tThe above definition for the change in stocks price index, ΔP , looks a bit strange at SNA
t tfirst sight but if the weights Δq and Δq are positive, it can be seen that it is a perfectly 1 2
reasonable price index that compares an average of the beginning and end of period t
prices with the base prices (which are the end of period 0 prices for the inventory
3components).

The above definitions are used to construct the value, price and quantity of end of period
t t tinventory stocks (VK , P and K respectively) for periods 0,1,2 and 3 and the SNA SNA
t t tvalue, price and quantity of the change in inventory stocks (ΔVK , ΔP and ΔK SNA SNA SNA
respectively) for periods 1-3 using the data in Table 1. The results are listed in Table 2
below.

Table 2: Values, Prices and Quantities for Aggregate Inventories at Period 0 Prices

2 This is not quite the theoretically correct measure of inventory change that is suggested in the System of
National Accounts 1993 on pages 130-131 but is regarded as an approximation that is frequently used as
the following quotation indicates: “This suggests that even when prices are changing a good approximation
to the PIM may be obtained by taking the difference between the quantity of goods held in inventory at the
beginning and at the end of the accounting period and valuing the difference at the average prices
prevailing within the period. This measure, which may be described as the “quantity” measure, is widely
used in practice and is sometimes mistakenly considered to be the theoretically appropriate measure under
all circumstances.” SNA 1993, page 131. For a more complete discussion of the SNA theoretically correct
measure of inventory change, see Hill (2005). However, Hill (2005) notes that the theoretically correct
method suffers from the same problems that arise using the approximate method.
3 0 However, note that when we set t equal to zero, in general, ΔP will not equal unity; i.e., the index does SNA
not satisfy the identity test. 4

t t t t t t VK P K ΔVK ΔP ΔK SNA SNA SNA SNA SNA
Period 0 400 1.000 400 _____ _____ _____
Period 1 534 1.302 410 −18.0 −1.800 10
Period 2 578 1.376 420 −57.5 −5.750 10
Period 3 631 1.467 430 −20.0 −2.000 10

At first glance, the values, prices and quantities for the aggregate inventory stock look
reasonable, with the values growing fairly quickly due to rapid increases in the price of