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EC201 Intermediate Macroeconomics 2011/2012

EC201 Intermediate Macroeconomics Lecture Outline:

- How to measure economic activity (GDP, Inflation rate, Unemployment rate, etc. etc.);

- Important identities in national accounting;

- Issues in aggregate income distribution;

Essential reading: Mankiw: Ch. 2 and 3 National Accounting When we talk about national accounting we talk about measuring the total quantity of

goods and services produced in an economy in a given period of time.

In the previous lecture we have seen the series of real GDP per capita in the US. To

create that series we need to create data and therefore we need to define what is the

GDP and how to calculate it.

GDP (Gross Domestic Product): the value of final output produced during a given

period of time within the borders of a given country.

Within the boarders means that if a firm is located in UK but the ownership is in

France, the production of that firm located in UK will be counted in the GDP of UK

and not in the GDP of France.

There are three approaches to measuring GDP, and all give exactly the same measure

of GDP:

1) Product approach (or value-added);

2) Expenditure approach;

3) Income approach;

To see how all these approaches work we consider a simple example.

Consider a very simple economy where there is a coconut producer, a restaurant,

some consumers and a government.

Lecture 2: National Accounting

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The coconut producer: owns all the coconut trees, harvests the coconuts and in

current year produces 10 million coconuts which are sold for $2 each, yielding total

revenue of $20 million.

He pays $5 million to his workers, $0.5 million in interest on a loan to some

consumers and $1.5 million in taxes to the government.

The following table summarises those data:

Table 1 Coconut Producer

Total revenue $20 million

Wages $5 million

Interest on Loan $0.5 million

Taxes $1.5 million

The restaurant: of the 10 million coconuts produced, 6 million go to the restaurant

that uses them to produce meals that are sold to the consumers. In this case the

coconut represents an Intermediate Good, a good that is produced and then used as

an input to another production process. All coconuts cost $2 therefore the total cost

for the restaurant from buying them is $12 million.

The remaining 4 million coconuts go to the consumers.

The restaurant sells $30 million in meals during current year. This represents its total

revenues.

The restaurant pays its workers $4 million and pays $3 million in taxes.

Table 2 Restaurant

Total revenue $30 million

Wages $4 million

Taxes $3 million

Now define the following:

After tax profits = Total Revenue – Wages – Interest – Cost of intermediate goods –

Taxes

This is what is left to the producer and the restaurant after costs and taxes have been

paid. Using the data from the previous tables we have:

After tax profits: 50 – 9 – 0.5 – 12 – 4.5 = $24 million

$24 million are the after tax profits of the two firms in the economy (the coconut

producer and the restaurant). It does matter for our calculations how much of the $24

million goes to the owner of the restaurant and how much to the coconut producer.

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Consumers: work for the producer of coconuts, the restaurant and the government

(notice that the owner of the restaurant and the producer of coconuts are consumers as

well). They earn $9 ($5+$4) million from the producer and the restaurant and $5.5

million from the government. They receive $0.5 million from interest on a loan to the

producer and $24 million of after tax profits. They consume directly 4 millions of

coconuts (10 millions minus the 6 millions sold to the restaurant).

Furthermore, they pay $1 million in taxes.

Table 3 Consumers

Wage Income $14.5 million

Profits distributed $24 million

Interest Income $0.5 million

Taxes $1 million

Government: it collects taxes to provide national security (military) to the economy.

It pays the army using the taxes collected. We assume no deficit (balanced budget),

and therefore total tax collected (=$5.5 million) is equal to total spending.

Table 4 Government

Total revenue $5.5 million

Wages $5.5 million

1) The product approach (or value added).

In this approach, to calculate the GDP: we add the value of all goods and services

produced and then subtract the value of all intermediate goods used in production.

We subtract the value of the intermediate goods to avoid double counting in the

calculation.

Using this approach the GDP is simply defined as the sum of value added to goods

and services across all productive units in the economy.

From our example: the coconut producer does not use any intermediate good,

therefore the value added of his production is $20 million (his total revenues).

For the restaurant, the value added is its total revenues minus its cost of intermediate

goods: $30 million – $12 million = $18 million.

Also the government is producing something (national security) and therefore we

need to include that in our calculations. However, we have a problem here. We don’t

have a price for the good “National Security” (classical example of a public good).

The standard practice in this case is to evaluate the national security services at the

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cost of the inputs to production. In our case, the only input was labour, and the total

cost was $5.5 million. Therefore the value added for the government is $5.5 million.

Using the production approach (or value added), the value of the GDP in our

economy for the current year is:

GDP = 20 + 18 + 5.5 = $43.5 million

2) The expenditure approach

Here the GDP is defined as: the total spending on all final goods and services

produced in the economy in a given period of time.

Notice: the word final in the definition implies that WE DO NOT COUNT spending

on intermediate goods.

What is total expenditure?

Total Expenditure = C + I + G + NX

C is total expenditure in consumption

I is investment expenditure

G is government expenditure

NX is net exports (= Exports in goods and services – Import in goods and services)

We include exports because they are produced within the country we are considering

and we subtract imports because goods produced abroad are included in C, I and G

and we don’t want to include them in the calculation of GDP (remember “within

borders”). In some models we will see we shall consider the case of a closed

economy. By definition, a closed economy is an economy that does not have trade

with other countries, therefore, NX=0 in this case. We will do this because it is

simpler to analyse a closed economy than an open economy.

From our example, using the expenditure approach we have that I = 0 and NX = 0.

There is no investment in our example and no international trade.

The GDP is then given by: C + G

Total consumption is $38 million, $8 million on coconuts and $30 million at the

restaurant. Government expenditure is $5.5 million.

Therefore: GDP=C+I+G+NX=$43.5 million

3) The Income approach

In this case the GDP is defined as the sum of all income received by economic agents

contributing to production.

Income includes the profits of firms.

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The income of the consumers is $14.5 million in wages. Then we need to add $0.5

million of income on the interest on a loan. Then we need to include the income given

by profits after tax of $24 million. Finally we need to include the taxes paid by the

producers since they are income for the government. We do not include the taxes paid

by consumers since they are receiving back those taxes as wages (this is just a transfer

and not a contribution to production).

Using this method: GDP = 14.5 + 0.5 + 24 + 4.5 = $43.5 million

Therefore, the GDP is equivalent to the total income in the economy. Let Y denoting

the total income of the economy, using the definition of the total expenditure we have:

Y C I G NX= + + +

This IDENTITY is called Income – Expenditure identity.

It is the basic identity in national accounting.

The idea is: the total quantity produced (Y) in an economy must be ultimately sold.

The different components of aggregate expenditure:

a) Consumption: the value of all goods and services bought by households

It includes:

durable goods: last a long time, ex: cars, home appliances

nondurable goods: last a short time, ex: food, clothing

services: work done for consumers, ex: dry cleaning, air travel.

b) Investment: spending on new capital (the factor of production)

It includes:

business fixed investment: Spending on plant and equipment that firms will use to

produce other goods & services.

residential fixed investment: Spending on housing units by consumers and landlords.

inventory investment: The change in the value of all firms’ inventories.

c) Government spending: includes all government spending on goods and services.

It excludes transfer payments (e.g., unemployment insurance payments, pensions),

because they do not represent spending on goods and services.

d) Net Exports: The value of total exports (X) minus the value of total imports (M).

Suppose now that a firm:

produces $10 million worth of final goods

but only sells $9 million worth.

Does this violate the expenditure = output identity?

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The answer is NO, because unsold output adds to inventory, and thus counts as

inventory investment – whether intentional or unplanned. Thus, it’s as if a firm

“purchased” its own inventory accumulation. We have just seen the inventory enters

in the definition of Investment and therefore they counted as expenditure.

To summarise: we have now seen that GDP measures

total income

total output

total expenditure

the sum of value-added at all stages in the production of final goods

Another measure of total income:

The GDP is the most used measure for total income of an economy. However, there

are other measures of total income that can be used:

Gross National Product (GNP) = GDP + Net Factor Payments (NFP)

The GNP measures the value of output produced by domestic factors of production,

regardless of whether the production takes place. Remember that the GDP instead

measures total income earned by domestically-located factors of production,

regardless of nationality

Net factor payments = factor payments from abroad – factor payments to abroad.

For example, if a UK firm is managed by Italian residents, then the profits created by

this firm will be counted in the Italian GNP but not in the Italian GDP.

In general the difference between GDP and GNP is not particularly large.

Another way to write the income-expenditure identity

The income-expenditure identity implies

Y C I G NX= + + +

Define with YD the disposable income of the private sector:

Y Y NFP TD = + −

where Y is the aggregate income, NFP are the net factor payments from abroad and T

are the taxes. (There may be other factors that affect the disposable income, for

example, the interest rate households obtain from government bonds, the transfer of

money from government, etc. etc.. Adding those elements will not affect the final

result)

Private Saving is then defined as:

S Y CP D= −

Government saving is defined as:

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S T GG = −

If SG is positive, the government is running a surplus, while if it’s negative it’s

running a deficit.

The National Saving is then defined as:

S S S Y C T GP G D= + = − + − 1)

By the definition of Y D , we have that Y T Y NFPD + = +

Therefore, we can rewrite equation 1) as

S Y NFP C G= + − − 2)

Substitute into 2) the definition of Y given by the Income-Expenditure identity:

S C I G NX NFP C G I NX NFP= + + + + − − = + +

Thus, national saving must be equal investment plus net exports plus net factor

payments from abroad.

The quantity NX + NFP is called Current Account (CA), it is a measure of the

balance of trade in goods and service with other countries.

Therefore, our income-expenditure identity can be written also as:

S I CA= +

In a closed economy, where NX = 0, and NFP = 0, the income-expenditure identity

implies: S I=

National saving must be equal to aggregate investment.

Real vs. nominal GDP

GDP is the value of all final goods and services produced.

nominal GDP measures these values using current prices.

real GDP measure these values using the prices of a base year.

Changes in nominal GDP can be due to:

changes in prices.

changes in quantities of output produced.

Changes in real GDP can only be due to changes in quantities, because real GDP is

constructed using

constant base-year prices.

Real GDP is a better measure of the well-being of an economy. Suppose that all the

prices of goods double. Then the GDP will double as well, however, quantities are the

same of before, therefore, the economy is not better-off even if the GDP has double.

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How to compute nominal and real GDP:

2006 2007 2008

P Q P Q P Q

good A $30 900 $31 1,000 $36 1,050

good B $100 192 $102 200 $100 205

nominal GDP: multiply Ps & Qs from same year

2006: $46,200 = $30 × 900 + $100 × 192

2007: $51,400

2008: $58,300

real GDP multiply each year’s Qs by 2006 Ps

2006: $46,200

2007: $50,000

2008: $52,000 = $30 × 1050 + $100 × 205

Notice that Real GDP in 2007 and 2008 is lower than GDP in those years. This is

because the price effect (inflation) has been taken out by using a base year for the

prices (2006).

Notice that the choice of the base year matters for calculations.

For example, if we choose 2007 as the base year, the real GDP in 2007 is $51,400,

while in 2008 is $53,460.

Inflation Rate

The inflation rate is the percentage increase in the overall level of prices.

One measure of the price level is the GDP deflator, defined as:

×Nominal GDPGDP deflator = 100

Real GDP

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Using the same example as before:

Nominal GDP Real GDP GDP

deflator

Inflation

rate

2006 $46,200 $46,200 100.0 n.a.

2007 51,400 50,000 102.8 2.8%

2008 58,300 52,000 112.1 9.1%

Notice that the GDP deflator identifies an index that measures the overall price

LEVEL in a given year.

Inflation rate is the rate of change of that index from one year to the following.

Example with 3 goods

For good i = 1, 2, 3

Pit = the market price of good i in month t

Qit = the quantity of good i produced in month t

NGDPt = Nominal GDP in month t

RGDPt = Real GDP in month t

The GDP deflator is a weighted average of prices.

The weight ⎟⎟⎠

⎞⎜⎜⎝

⎛

t

it

RGDPQ

on each price reflects that good’s relative importance in GDP.

Note that the weights change over time

Another measure for the inflation rate: CPI

CPI = Consumer Price Index

It is based on a fixed basket of goods that are normally an important part of

household’s consumption.

CPI in any month equals:

= tt

t

NGDPGDP deflatorRGDP

+ += 1t 1t 2t 2t 3t 3t

t

P Q P Q P QRGDP

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

1t 2t 3t1t 2t 3t

t t t

Q Q QP P PRGDP RGDP RGDP

Cost of basket in that monthCost of basket in base period

100 ×

10

Example with 3 goods

For good i = 1, 2, 3

Ci = the amount of good i in the CPI’s basket

Pit = the price of good i in month t

Et = the cost of the CPI basket in month t

Eb = the cost of the basket in the base period

The CPI is a weighted average of prices.

The weight on each price reflects that good’s relative importance in the CPI’s basket.

Note that the weights remain fixed over time.

CPI vs. GDP Deflator

prices of capital goods

included in GDP deflator (if produced domestically)

excluded from CPI

prices of imported consumer goods

included in CPI

excluded from GDP deflator

the basket of goods

CPI: fixed

GDP deflator: changes every year

In the following figure we plot the CPI and the GDP deflator for the US economy

from 1950 to 2005.

t

b

ECPI in month E

=t 1t 1 2t 2 3t 3

b

P C + P C + P CE

=

31 21t 2t 3t

b b b

CC CP P PE E E⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

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Although the two series display a similar pattern, there are some relevant differences

in some years. Those differences are due to the elements we listed in the previous

page. In general the CPI index is the most common way to measure inflation. This is

because it is based on the consumption of households and therefore is a better

measure of the cost of living in an economy.

By its construction the CPI index may overstate the rate of inflation:

Introduction of new goods: The introduction of new goods makes consumers better

off but it does not reduce the CPI, because the CPI uses fixed weights.

Unmeasured changes in quality: Quality improvements increase the value of the

dollar, but are often not fully measured.

Unemployment Rate

Categories of the population (POP)

Employed (E): working at a paid job

Unemployed (U): not employed but looking for a job

Labour Force (L): the amount of labour available for producing goods and

services; all employed plus unemployed persons

not in the labour force (NILF): not employed, not looking for work (for

example, full-time students)

unemployment rate: percentage of the labour force that is unemployed.

U.S. adult population by group, June 2007

-3%

0%

3%

6%

9%

12%

15%

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

GDP deflator CPI

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Number employed = 146.1 million

Number unemployed = 6.9 million

Adult population = 231.7 million

E = 146.1, U = 6.9, POP = 231.7

L = E +U = 146.1 + 6.9 = 153.0

NILF = POP – L = 231.7 – 153 = 78.7

unemployment rate

U/L x 100% = (6.9/153) x 100% = 4.5%

Stock Variables vs Flow variables

It is sometimes important to distinguish between those two kinds of variables.

A stock is a quantity measured at a point in time.

E.g.,

“The U.S. capital stock was $26 trillion on January 1, 2006.”

A flow is a quantity measured per unit of time.

E.g., “U.S. investment was $2.5 trillion during 2006.”

It is often the case that a flow variable measures the rate of change in a corresponding

stock variable. Here are some examples:

Stock Flow a person’s wealth ↔ a person’s annual saving n. of people with college degrees ↔ n of new college graduates this year the govt debt ↔ the govt budget deficit The Neoclassical theory of income distribution

We consider a closed economy. When we consider income distribution in a given

economy we may look at the personal distribution of income. In this case we talk how

income is distributed to the population of the economy. For example, 50% of UK

population holds 93% of the total income of UK. We can also talk about the

functional distribution of income. In this case we want to know how the aggregate

income is divided among the factors of production, meaning, workers, owners of

capital (capitalists) and owners of land etc. etc. Here we consider the issue of the

functional distribution of income. Our main question is: what determines the

economy’s total output/income and how total income is distributed among the factors

of production?

In defining the GDP from an accounting point of view we have seen what an

economy produces. It produces consumption goods, investment goods ect. etc.

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Now we ask: how we produce those goods. The economy’s output of goods and

services depends on the quantity of inputs (or factors of production) and the ability to

turn inputs into output. In economics the way inputs are transformed into outputs is

defined by a production function. Here we will consider only two inputs, labour and

capital. So our problem is to find how aggregate income is divided between workers

and capitalists.

Define with Y the total output produced (real output) and with K and L the amount of

inputs available in the economy. K is for the level of Capital and L for Labour.

A production function is defined as: Y F K L= ( , )

The inputs K and L are transformed into Y through the function F. The production

function reflects the available technology for turning capital and labour into output.

On of the main properties of a production function is related to the concept of Returns

to Scale. When we talk about changing the scale we talk about the effects on output of

changing ALL the inputs of production (therefore, we may assume we are in the long-

run). Mathematically, the idea of returns to scale is related to the idea of homogeneity

of a function. A function f(X) is homogeneous of degree k if it’s true that:

f rX r f Xk( ) ( )=

where r is a constant.

Now we can provide a definition for the returns to scale of a production function:

a) Constant returns to scale: if we increase all the inputs of production by an

amount r, the total output will increase by the same amount r. In terms of

homogeneity, constant returns to scale are equivalent to say that the

production function is homogeneous of degree 1;

b) Decreasing returns to scale: if we increase all the inputs by r, the total output

increases by an amount less than r;

c) Increasing returns to scale: if we increase all the inputs by r, the total output

increases by an amount larger than r;

Example: the Cobb-Douglas production function

Y F K L K L= =( , ) α β

where α and β are two positive constants.

To check the degree of homogeneity:

( )),(

)()(),()( LKFr

LKrLrKrrLrKrLrKFβα

βαβαββααβα

+

+

=

===

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The Cobb-Douglas production function is homogenous of degree α + β (it is exactly

the k in our previous definition).

Therefore, we have constant returns to scale when α + β = 1, decreasing returns when

α + β < 1 and increasing returns when α + β > 1.

An important property of homogenous functions is given by the Euler’s theorem:

Consider a function of n variables f x x xn( , , ... , )1 2 homogenous of degree k, then it is

possible to show that:

kf x x x x fx

x fx

x fxn n

n

( , , ... , ) ...1 2 11

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=∂∂

+∂∂

+ +∂∂

The Euler’s theorem tells us that we can rewrite a homogenous function in terms of its

partial-derivatives.

Why is this important for us?

The partial derivatives of the production function we are considering have a nice

economic interpretation:

∂∂

=F K L

K( , ) Marginal productivity of Capital (MPK)

∂∂

=F K L

L( , ) Marginal productivity of Labour (MPL)

In competitive markets, from profit maximisation conditions, it is true that:

MPK rP

= and MPL WP

=

where r is the cost of renting capital, W is the nominal wage and P is the aggregate

price level. Therefore, r/P is the REAL cost of capital and W/P is the REAL wage.

Those two conditions simply say that in competitive markets the inputs of production

are paid at their marginal productivity. Now we can see how total income is

distributed among the inputs of production (workers and capitalists). Assume that the

production function defining the total output is homogenous of degree 1 (= constant

returns to scale). By the Euler’s theorem we can write the production function as

(remember that here k = 1):

F K L K FK

L FL

K MPK L MPL( , ) = ∂∂

+∂∂

= × + ×

Multiply each side of the above equation by the aggregate price level P.

Then ),( LKFP× is the VALUE of total production (the GDP of our economy),

while we must notice that MPK P r× = and P MPL W× = .

Using those facts we can rewrite the above equation as:

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PY K r L W= × + × 3)

Equation 3) says that the total income in the economy is equal to the sum of factors

payments. This result is called the “Exhaustion Product Theorem”.

Therefore, if there are:

a) Constant returns to scale;

b) Competitive markets;

Total output is divided between the payments to capital and the payments to labour,

depending on their marginal productivities.

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Mathematical Appendix

1) Discrete percentage changes

Consider a variable Y for which you have values for different period of times (t).

Denote with Yt the value of Y in period t (so Yt+1 denotes the value of Y in period t+1

and so on).

a) The discrete percentage change (or the growth rate) in the value of Y between

period t and period t+1 is given by:

1001 ×⎟⎟⎠

⎞⎜⎜⎝

⎛ −= +

t

tt

YYYg or 10011 ×⎟⎟

⎠

⎞⎜⎜⎝

⎛−= +

t

t

YYg

The difference tt YY −+1 is called first-difference, and it is denoted with tt YYY −=Δ +1 .

Therefore, the percentage change of Y between period t and period t+1 can be written

as: 100×Δ

tYY

If we let the difference between tt YY −+1 to be really small, that is if we take the

following limit:

YdY

YY=

Δ→Δ 0

lim

where dY is an infinitesimal change in Y, we obtain the expression YdY that is called

the instantaneous growth rate of Y.

b) For any variables X and Y:

the percentage change in (X × Y ) ≈ percentage change in X + percentage change

in Y

Proof:

Consider a function XYYXf =),(

The total differential of that function is defined as:

dYY

YXfdXX

YXfYXdf∂

∂+

∂∂

=),(),(),(

The total differential tells me: what is the change in the function ( ),( YXdf ) given

small changes in the variables X and Y ( dYdX , ).

The term X

YXf∂

∂ ),( indicates the partial derivative of the function with respect the

variable X. Given that our function is XYYXf =),( , by basic calculus we know that:

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YX

YXf=

∂∂ ),( and X

YYXf

=∂

∂ ),(

Therefore, our total differential can be written as:

XdYYdXYXdf +=),(

Using the fact that XYYXf =),( , we can rewrite the above expression as:

XdYYdXXYd +=)(

Dividing each side of the above expression by (XY) we obtain:

YdY

XdX

XYXYd

+=)( A1)

The expression XYXYd )( is the instantaneous growth rate of (XY), while

XdX and

YdY

are the instantaneous growth rates of X and Y respectively.

Expression A1) says that the growth rate of XY is equal to the growth rate of X plus

the growth rate of Y. This is true if the change in variables is small (remember: d

denotes a very small change). However, when we consider discrete changes (when we

use Δ instead of d) expression A1) still hold approximately.

End of the proof.

Example: suppose that the real GDP has increased by 2% from 2005 to 2006. Suppose

that inflation measured by the GDP deflator has increased by 3% in the same period.

What is the growth rate of nominal GDP between 2005 and 2006?

We know that:

Nominal GDP = Real GDP × GDP Deflator

Therefore:

Percentage change in Nominal GDP ≈ 2 + 3 ≈ 5%.

c) For any variables X and Y:

the percentage change in (X /Y ) ≈ percentage change in X – percentage change in

Y

Example. Suppose that population has increased by 1% between 2005 and 2006,

while real GDP has increased by 2% during the same period. What is the percentage

change of the real GDP per capita?

We know that:

Real GDP per capita = (real GDP)/Population

Therefore:

Percentage change in real GDP per capita ≈ 2 – 1 ≈ 1%.

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2) Index Numbers

In statistics index numbers are used to describe the behaviour over time of some

interesting variables. In particular, they are useful when we are interested in the

growth rates of those variables over a certain period of time. They are also useful to

compare series of numbers of different size. We normally use index numbers with a

fixed base. For example, consider the following series for GDP in UK from 1997 to

2002 (values are in million of pounds).

Year GDP

1997 864710

1998 891684

1999 916639

2000 951256

2001 971565

2002 988338

The Index Number is defined as:

Index Number current period = Current Period Value Base Period Value

×100

First, we need to decide the base period we want to use. For example, use the first

year of the GDP series as the base year. The index number for the GDP is then given

by:

Year GDP Index

1997 100

1998 103.11

1999 106

2000 110

2001 112.35

2002 114.29

Some of the calculations are:

GDP index 1997 = GDP value 1997 GDP value 1997

× = =100 864710864710

100 100

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GDP index 1999 = GDP value 1999 GDP value 1997

× = =100 916639864710

100 106

Notice that the index number is a pure number (it does not depend on any unit of

measure). Furthermore, the index number does not have any meaning by itself. The

fact that the GDP index in 1998 is 103.11 does not mean anything.

What it is meaningful is that now we can easily compare the behaviour of the series

with respect the base year.

Using the GDP index we can say that in 1998 the GDP was 3.11% higher than in

1997. Or we can say that the GDP in 2002 is 14.3% higher than in 1997, and so on.

Notice that the index number series in the previous table does not tell you the year-to-

year growth rate of the variable. We can calculate the year-to-year growth in the GDP

using the original series or using the index number series.

For example, from 1999 to 2000, the GDP has increased by:

GDP - GDP GDP2000 1999

1999

× =−

=100 951265 916639916639

100 3 77%.

or using the GDP index series

110 106106

100 3 77%−= .

The most important index numbers used in economics are probably the price indexes.

There are two main ways to calculate a price index:

1) Laspeyers Index:

Consider a set I of good and services, that is I={1,2,…i,…n}

Denote with:

tip , the price of good i in period t.

tiq , the quantity of good i sold in period t

0,ip the price of good i in period 0 (the base year)

0,iq the quantity of good i sold in period 0

The Laspeyres index in period t is defined as:

Pp qp qL t

i t i

i i,

, ,

, ,

( )( )

= ×∑∑

0

0 0

100

The numerator is the cost of the bundle of I goods and services in period 0 evaluated

at prices at t. The denominator is the cost of the bundle in period 0 (the base year).

20

Example: suppose you have a basket of goods containing 20 pizzas and 10 compact

discs in year 2002. The prices of a pizza and a compact disc are given by the

following table:

pizza CDs

2002 $10 $15

2003 $11 $15

2004 $12 $16

2005 $13 $15

We can calculate the Laspeyres index with base year 2002.

In 2002 the index is given by:

PL ,200210 20 15 1010 20 15 10

100 100=× + ×× + ×

× =

In 2003:

PL , .200311 20 15 1010 20 15 10

100 105 7=× + ×× + ×

× =

and so on.

The Consumer price index is an example of a Laspeyres index.

The Laspeyers index of our basket of goods is summarised in the following table,

where we reported also the year-to-year inflation.

Year Index Inflation rate

2002 100.0 n.a.

2003 105.7 5.7%

2004 114.3 8.1%

2005 117.1 2.5%

2) Paasche Index

Differently from the Laspeyers index, here the basket of goods is changing over time.

The formula is:

Pp qp qP t

i t i t

i i t,

, ,

, ,

( )( )

= ×∑∑ 0

100

The GDP deflator is an example of a Paasche Index.

Notice that in order to calculate the Paasche index using the previous example we

should know the quantities sold in each year.