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Price Indexes
Four Major Price Indexes
GDP Deflator -- Bureau of Economic Analysis (BEA)
Consumer Price Indexes (CPI) – Bureau of Labor Statistics (BLS)
Personal Consumption Expenditure Price Index (PCEPI) -- BEA
Producer Price Indexes
GDP deflator=100*Nominal GDP/Real GDP
This is also called Implicit GDP deflator
It depends on how to compute real GDP
Fixed weight real GDP – changes when the base year changes
Chain weighted real GDP – does not depend on the choice of the base year
It measures the average price level of goods and services included in the GDP
To import the graph, click 'Export' and choose 'Export as pdf' or other choices
Then, in MS Word, insert-object-adobe-acrobat-pdfxmldocument
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Consumer Price Index (CPI)
The Consumer Price Index is a measure of the average price of a fixed market basket of goods and
services purchased by the average household in the U.S.
Published monthly by the Bureau of Labor Statistics (BLS) of U.S. Department of Labor
Increases in the CPI level serve as a measure of the consumer inflation rate.
Reference for escalation agreements - labor contracts and other payment agreements that are
indexed to inflation rely on the CPI.
Two types of Index
Fixed weight index (weights are fixed by the base year weights)
Chain weighted (or chained) index
Different kinds of CPI: there are more than one CPI
Target Population
CPIs by region, types of products, types of consumers, etc.
CPI-W:
It measures prices of consumer goods purchased by All Urban Wage Earners and Clerical Workers.
CPI-W is used to adjust Social Security benefit payments
Covers about 32% of U.S. population
Started in 1913
CPI-U:
It measures prices of consumer goods purchased by typical All Urban Consumers.
Most commonly reported CPI.
Inflation rate computed from CPI-U is called the "headline inflation" rate
CPI-U is used to adjust the personal income tax brackets to keep up with inflation.
Started from 1978
Covers about 80% of U.S. population
CPI-E:
It mesures prices of consumer goods purchased by elderly population (62 years and older)
Purchase pattern of elderly population is different from that of younger population, such as medical
care
Inflation rate of this index has been higher than the inflation rate from CPI-W or CPI-U
Long-run trend in price
To measure the long-run trend in prices, we need to exclude transitory price changes.
Two ways to do it:
Excludes items frequently subject to volatile prices
Exclude items which had excessive price changes
Core CPI:
Excludes food and energy from the basket whose prices are very volatile
Inflation rate computed from core CPI is called the core inflation rate.
Sticky Price CPI
Price index of a subset of goods and services included in the CPI that change price relatively
infrequently.
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This reflects expectations about future inflation better than flexible prices
Published monthly by the Federal Reserve Bank of Atlanta
16% Trimmed-Mean Consumer Price Index (CPI)
Computes monthly price changes of CPI components
Exclude components of 8% highest changes and 8% lowest changes
Computes price index of remaining 84% components
Computed by the Federal Reserve Bank of Cleveland.
Median Consumer Price Index
Similar to 16% trimmed Mean CPI
Excludes components of 49.5% highest and lowest changes in prices
Computes price index of remaining 1% components in the middle
Computed by the Federal Reserve Bank of Cleveland.
Personal Consumption Expenditure (PCE) Price Index
Constructed by the BEA from the components of personal consumer expenditure in GDP
current base year=2009
Chain weighted index
Uses the Fisher-Ideal index formula
-The inflation rate computed from PCEPI is used for the target inflation rate by the FRB
The PCE price index (PCEPI), also referred to as
the PCE deflator, PCE price deflator,
the Implicit Price Deflator for Personal Consumption Expenditures (IPD for PCE) by the BEA
the Chain-type Price Index for Personal Consumption Expenditures (CTPIPCE) by the FOMC
Difference between the chained CPI and chained PCEPI
- Different formula:
CPI: Laspeyres index
Chained CPI-U: Törnqvist index
Chained PCEPI: Chain weight Fisher (or also called the chained Fisher-Ideal) index
- Different scope (population): The PCEPI is broader in scope.
The CPI covers out-of-pocket expenditures by urban consumers.
The PCEPI also covers spending by rural consumers; nonprofit institutions that serve consumers;
medical care and insurance funded by government and employers; and imputed financial
services. There are items that are included in PCEPI and excluded from the CPI
- Different weights
Weight differences reflect differences in: scope, item definitions, and sources of expenditure
data
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Comptation Methods
CPI – Laspeyres index formula
Fixed Weight methods (base weighted method)
The method that has been used for a long time in the past
Tends to overestimate the effect of price change on cost of living (welfare)
Chain Weighted CPI (or Chained CPI or Chain-Linked CPI)
C-CPI-U:
Newly indtroduced CPI.
Corrects the intrinsic bias in fixed weight price indexes.
Weights are updated every month
Started in August, 2002
How to compute CPI
Laspeyres Index - CPI
Paasche Index
Fisher (Ideal) Index – PCEPI (chained)
Tornqvist Index – chained CPI-U
Cost of Living Index
Laspeyres Index
∑𝑛𝑖=1 𝑃𝑖𝑡 𝑄𝑖0
𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑎𝑠𝑒 𝑏𝑎𝑠𝑘𝑒𝑡
𝐿𝐼𝑡 =
=
𝑏𝑎𝑠𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑎𝑠𝑒 𝑏𝑎𝑠𝑘𝑒𝑡 ∑𝑛𝑖=1 𝑃𝑖0 𝑄𝑖0
The Laspeyres index compares the cost of the base year basket at old and new prices.
The weights multiplied to each price are fixed at the quantities in base period.
This type of CPI is called the fixed weight index.
It overestimates of the impact of price changes on the cost of living
The CPI is usually reported as a number that is multiplied by 100. Hence, the base period CPI is equal
to 100.
The numerator of the CPI can be written as
𝑃
𝑃𝑖𝑡 𝑄𝑖0 = 𝑃𝑖0 𝑄𝑖0 𝑃 𝑖𝑡
𝑖0
Hence,
∑𝑛 𝑃 𝑄
𝑃𝑖0 𝑄𝑖0
𝑖=1 𝑃𝑖0 𝑄𝑖0
𝐿𝐼𝑡 = ∑𝑛𝑖=1 𝑃𝑖𝑡 𝑄𝑖0 = ∑𝑛𝑖=1 ∑𝑛
𝑖=1 𝑖0 𝑖0
𝑃
𝑃
× 𝑃 𝑖𝑡 = ∑𝑛𝑖=1 𝑆𝑖0 × 𝑃𝑖𝑡
𝑖0
𝑖0
𝑃𝑖0 𝑄𝑖0
𝑖=1 𝑃𝑖0 𝑄𝑖0
𝑆𝑖0 = ∑𝑛
where Si0 is the share of expenditure on item i in the base period.
The Laspeyres index is thus a weighted arithmetic mean of the ratio of current prices to base prices
with weights equal to the share of expenditure on each item in base period.
The price ratio is called the price relative, or growth of price.
This formula is used to use the data from the Consumer Expenditure Survey (called the Household
Budget Surveys in some other countries). This survey does not ask how many packs of beer a
household bought. but , it asks how much they spent.
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The Consumer Expenditure Survey (CE) program consists of two surveys, the Quarterly Interview
Survey and the Diary Survey, that provide information on the buying habits of American consumers,
including data on their expenditures, income, and consumer unit (families and single consumers)
characteristics. The survey data are collected for the Bureau of Labor Statistics by the U.S. Census
Bureau.
Paasche Index
𝑃𝐼𝑡 =
∑𝑛𝑖=1 𝑃𝑖𝑡 𝑄𝑖𝑡
𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑏𝑎𝑠𝑘𝑒𝑡
= 𝑛
𝑏𝑎𝑠𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑏𝑎𝑠𝑘𝑒𝑡 ∑𝑖=1 𝑃𝑖0 𝑄𝑖𝑡
The Paasche index compares the cost of the new basket at old and new prices.
The Paasche index underestimate the impact of price changes on the cost of living
The Paasche index can be written as
∑𝑛 𝑃𝑖𝑡 𝑄𝑖𝑡
𝑃𝐼𝑡 = ∑𝑛𝑖=1
𝑖=1 𝑃𝑖0 𝑄𝑖𝑡
𝑃𝑖0 𝑄𝑖𝑡
𝑖=1 𝑃𝑖0 𝑄𝑖𝑡
= ∑𝑛𝑖=1 ∑𝑛
×
𝑃𝑖𝑡
𝑃𝑖0
≡ ∑𝑛𝑖=1 𝑆𝑖0𝑡 ×
𝑃𝑖𝑡
𝑃𝑖0
𝑃𝑖0 𝑄𝑖𝑡
𝑖=1 𝑃𝑖0 𝑄𝑖𝑡
𝑆𝑖0𝑡 = ∑𝑛
Si0t is the share of expenditure on item i when the current basket is purchased at the base period prices.
This share is cometimes called a hybrid weight as it mixes values in two periods.
Fisher (Ideal) Index
It is the geometric mean of the Laspeyres and Paasche indices
𝐹𝐼 = √𝐿𝐼 × 𝑃𝐼
The Fisher index is symmetric in treating the base year and current year equally.
It corrects somewhat the upward bias of the Laspeyres index and the downward bias of the Paasche
index
Walsh Index
This index uses the geometric mean of quantities as the weight to prices
∑𝑛 𝑃 √𝑄𝑖0 𝑄𝑖𝑡
𝑊𝐼 = ∑𝑛𝑖=1 𝑃 𝑖𝑡
𝑖=1 𝑖0 √𝑄𝑖0 𝑄𝑖𝑡
Geometric Laspeyres price index
𝑃𝑖𝑡 𝑆𝑖0
)
𝑃𝑖0
𝐺𝐿𝐼𝑡 = ∏𝑛𝑖=1 (
𝑃𝑖0 𝑄𝑖0
𝑖=1 𝑃𝑖0 𝑄𝑖0
𝑆𝑖0 = ∑𝑛
𝑙𝑛(𝐺𝐿𝐼𝑡 ) = ∑𝑛𝑖=1 𝑆𝑖0 × [𝑙𝑛(𝑃𝑖𝑡 ) − 𝑙𝑛(𝑃𝑖0 )]
𝑃
𝑙𝑛 (𝑃 𝑖𝑡 ) = 𝑙𝑛(𝑃𝑖𝑡 ) − 𝑙𝑛(𝑃𝑖0 ) ≈
𝑖0
𝑃𝑖𝑡 −𝑃𝑖0
𝑃𝑖0
is the rate of price change, i.e., inflation rate.
Geometric index is the exponential transformation of weighted average of inflation rate between
the base period and the current period.
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Geometric Paasche Index
𝑃
𝐺𝑃𝐼𝑡 = ∏𝑛𝑖=1 (𝑃𝑖𝑡 )
𝑆𝑖0𝑡
𝑖0
𝑃𝑖0 𝑄𝑖𝑡
𝑖=1 𝑃𝑖0 𝑄𝑖𝑡
𝑆𝑖0𝑡 = ∑𝑛
𝑙𝑛(𝐺𝑃𝐼𝑡 ) = ∑𝑛𝑖=1 𝑆𝑖0𝑡 × [𝑙𝑛(𝑃𝑖𝑡 ) − 𝑙𝑛(𝑃𝑖0 )]
Chained Price Indexes
Price indexes discussed above are defined for a fixed base period. We can use them for adjacent
periods, essentially moving the base period each time. This is achieved by using period t-1 and t instead
of period 0 and t. They are called the chain-linked or chained price indexes.
Chain-Linked Laspeyres Index
n
CLIt 
P Q
P
i 1
it 1
it
i 1
n
Qit 1
it -1
Chain-Linked Paasche Index
n
CPIt 
P Q
i 1
n
it
P
i 1
it
Qit
it -1
Chain-Linked Fisher Index
CFIt  CLIt  CPIt
Chain-Linked Walsh Index
n
CWI t 
P
Qit 1Qit
P
Qit 1Qit
i 1
n
i 1
it
it -1
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Chain-Linked Törnqvist Index
The BLS uses the following formula to compute the chained CPI-U
Monthly index
 P 
   it 
i 1  Pit -1 
n
CTI t 1, t
Si
Si 
1
Sit 1  Sit 
2
Sis 
PisQis
n
P Q
i 1
is
s  t  1 or t
is
Long term index
 P 
 CTI 0,t 1    it 
i 1  Pit -1 
n
CTI 0,t
Si
Taking natural log of this equation, we can write
n
 P 
ln( CTI 0,t )  ln( TI 0,t 1 )   Si ln  it 
i 1
 Pit 1 
This shows that the inflation rate computed from the TI is equal to the weighted average of the
inflation rate of individual prices where the weights are the average of the expenditure shares in two
periods.
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Problems of Fixed Weight Laspeyres and Paasche Indexes
Fixed weight indexes – does not consider substitution effects
Substitution Biases:
Substitute a relatively more expensive item with a relatively less expensive item
When the price of a product in the consumer basket increases substantially, consumers tend to
substitute lower-priced alternatives. For example, if a freeze in Florida causes the price of oranges to
skyrocket, consumers may substitute Texas grapefruits for Florida oranges. Since the CPI is a fixedweight price index, it would not accurately predict the impact of the price increase on the consumer's
budget.
The Laspeyres index overstates while the Paasche index understates the change in the cost of living
Some economists estimate that the bias of the Laspeyres index overstates the CPI by about 1% per
year.
Biases of Laspeyres and Paasche Indexes
Consider the case of two goods. A consumer buys a utility maximizing bundle Q0 subject to the budget
constraint (budget line) B0 when prices are P0=(P10,P20). The utility level is represented by the indifference
curve U0. The expenditure E0=P0Q0 is the denominator term of the Laspeyres index.
Prices changed to Pt=(P1t,P2t). If income stays the same at E0, the new budget line is Bt and the new
equilibrium bundle is Qt. The new utility level Ut is lower than U0.
How do we measure the effect of the price change on consumer welfare?
Suppose we offer an additional income to compensate for the loss of utility. What is the minimum amount
of additional income we have to offer? An additional income will shift the budget line Bt outward. If a
sufficient amount of additional income is offered to shift the budget line to 𝐵𝑡∗ , the consumer purchases
the optimum bundle 𝑄𝑡∗ and utility level is back to the initial level U0.
Therefore, the dollar value of the effect of rising prices can be measured by the difference between the
initial budget E0 before the price increase and the income level Et* corresponding to the budget line 𝐵𝑡∗ .
Note that Et* is the minimum income level to reach the initial utility level U0 at new prices:
𝐸𝑡∗ = min(𝑃1𝑡 𝑄1𝑡 + 𝑃2𝑡 𝑄2𝑡 | 𝑈 = 𝑈0 )
The difference 𝐸𝑡∗ − 𝐸0 is called the Compensating Variation (CV).
The ideal cost of living index is the ratio
𝐸∗
𝐼𝑑𝑒𝑎𝑙 𝐶𝑂𝐿𝐼 = 𝐸𝑡
0
The numerator 𝐸𝑡0 = ∑𝑛𝑖=1 𝑃𝑖𝑡 𝑄𝑖0 of the Laspeyres index is the expenditure to buy the initial bundle at
the new prices.
Is 𝐸𝑡∗ greater than Et0? The answer is no. This is seen easily by noting that the initial bundle Q0 is outside
of the budget line 𝐵𝑡∗ . Any bundle outside of 𝐵𝑡∗ requires a higher expenditure.
Another way to look at this is as follows: Suppose we offer an additional income so that the consumer can
buy the initial bundle if he/she wants to. The budget line with this income will pass through Q0 and has
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the same slope as the slope of budget line with new prices. (see the budget line 𝐵𝑡0 in the figure below).
The consumer can buy Q0 if she wants to since it is on the budget line. But she will not buy it because of
the change in relative price. The cost of buying the initial bundle at new prices, i.e., corresponding to the
budget line 𝐵𝑡0 is greater than 𝐸𝑡∗ . Since this is the numerator term of the Laspeyres index, it overstates
the effect of price increase.
Paasche Index understates the change in living cost
To see this, consider increases in prices in the figure below. Price increases reduce the utility level from
U0 to Ut. This decrease in utility caused by price changes is equivalent to the decrease in utility caused by
a reduction of income from E0 to 𝐸𝑡∗ , where the latter is the income level corresponding to the budget line
𝐵𝑡∗ . The difference in income levels, 𝐸𝑡∗ − 𝐸0 , is called the Equivalent Variation (EV).
The Paasche index budget line lies above the equivalence variation budget line, and hence it indicates a
smaller reduction in income and the Paasche index understates the effect of price increases.
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Laspeyres Index
𝐵𝑡∗
𝐵𝑡0 =Laspeyres index budget line
𝑄𝑡∗
Q0
Qt
U0
Ut
Bt
B0
Paasche Index
Q0
𝐵𝑡∗
Qt
U0
𝐵𝑡0 =Paasche Index budget line
Bt
𝑄𝑡∗
U1
B0
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Producer Price Index
http://www.bls.gov/news.release/pdf/ppi.pdf
http://www.bls.gov/ppi/ppifaq.htm
The Producer Price Index (PPI) of the Bureau of Labor Statistics (BLS) is a family of indexes that measures
the average change over time in prices received (price changes) by producers for domestically
produced goods, services, and construction.
The PPI used to be called the Wholesale Price index before 1978. The PPI often preceeds the CPI.
PPIs measure price change from the perspective of the seller. This contrasts with other measures, such
as the Consumer Price Index (CPI). CPIs measure price change from the purchaser’s perspective.
More than 10,000 PPIs for individual products and groups of products are released each month. PPIs are
available for the products of virtually every industry in the mining and manufacturing sectors.
The PPI classifies goods into two categories: Final Demand good and Intermediate Demand good. This is
called the FD-ID system. Before 2014, the primary classification was by stage of processing (SOP)
Final demand includes goods, services, and construction which are sold for personal consumption,
capital investment, government purchases, and export.
Intermediate demand includes goods, services, and maintenance and repair construction sold to
businesses, excluding capital investment. BLS publishes two parallel treatments of intermediate
demand, each constructed from the identical set of commodity price indexes. The first treatment
organizes commodities according to commodity type, and the second organizes commodities using a
stage-based, production flow model.
PPIs are constructed using selling prices reported by establishments of all sizes, selected by probability
sampling, with the probability of selection proportionate to size. Individual items and transaction
terms also are chosen by probability proportionate to size. BLS strongly encourages cooperating
companies to supply actual transaction prices at the time of shipment to minimize the use of list
prices. Prices submitted by survey respondents are effective on the Tuesday of the week containing
the 13th day of the month. The survey is conducted via Internet, mail, and fax.
The target set of items differs between CPI and PPI in a number of areas: (1) the CPI includes imports,
while imports are excluded from PPI; (2) owners' equivalent rent is included in CPI, but not in PPI;
(3) the CPI only includes components of personal consumption that are directly paid for by the
consumer, whereas the PPI includes components of personal consumption that are not paid for by
the consumer (for example, medical services paid for by government or insurance companies); (4)
the PPI includes exports, while the CPI does not; (5) the PPI includes government purchases, while
the CPI does not; and (6) the PPI includes sales to businesses as inputs to production, including
capital investment, whereas the CPI does not.
The price collected for an item included in the PPI is the revenue received by its producer. Sales and
excise taxes are not included in the price because they do not represent revenue to the producer.
The price collected for an item included in the CPI is the out-of-pocket expenditure by a consumer for
the item. Sales and excise taxes are included in the price because they are necessary expenditures by
the consumer for the item.
The formula used to calculate PPIs is a modified Laspeyres index. The base period quantity in the formula
is the quantity of the commodity shipped during the base period.