Price Indices: Part 2

MEASUREMENT ECONOMICS

ECON 4700

COLI

P P1,P 0 ,uR( ) =C uR ,P 1( ) C uR ,P 0( )

UR: Is the label of the “reference” indifference curve.

COLI

• Note that to measure the cost-of-living we compare expenditures between two periods to maintain a constant standard of living (same indifference curve).

• Money is used to measure the change in welfare.

• Applicable only in situations where money and welfare are linked.

• This means that goods that are not obtained through the market system will be excluded from the index: health care, clean air, public parks.

• (NEW) In discussions about consumer welfare, few issues are more sensitive than the measurement of changes in the cost of living.

A simple graphical illustration of the COLI

• Measuring the cost of living is a politically loaded issue.

• Millions and billions of $ are involved.

• Not a problem if all prices rose by 5%

• But as we know this is not the case and the index number problem often comes up here.

A simple graphical illustration of the COLI (cont’d)

• Assume a base year (year 0)• A consumer consumes: • 3 m2/wk of shelter and 10 Kg/wk of food.• Unit prices in base year (t = 0)

– Shelter : $20/m2

– Food : $10/Kg

• Consumption of shelter and food generates a level of satisfaction I0.

• Unit prices in current year (t = 1)– Shelter : $24/m2

– Food : $60/Kg• By how much as the COL changed between year 0 and year 1?

A simple graphical illustration of the COLI (cont’d)

• A true measure of the COL would measure the change in the cost of achieving the same level of satisfaction (I.e., same indifference curve) as that of the base year.

• Total cost of attaining I0 in base year 0:

– C0 = (P1 x Q1) + (P2 x Q2)– C0 = ($20/m2 x 3 m2/wk) + ($10/Kg x 10 Kg/wk) =– $160/wk

• In the current year prices have changed:– C1 = ($24/m2 x 10 m2/wk) + ($60/Kg x 4 Kg/wk) =– $480/wk

I0

10

3

Shelter m2/wk

0

A graph

Food Kg/wk

C (S1,F1)

10

4

A (S0,F0)

16

B0

8

B1

20

8

With the increase in prices, the consumer, to achieve the same level of satisfaction as in year 0, will have to make some changes to his or her consumption pattern.

With the increase in prices, the consumer, to achieve the same level of satisfaction as in year 0, will have to make some changes to his or her consumption pattern.

Will “substitute” in favour of the product for which the price has gone up by the smaller proportion. Shelter has gone up 20%; Food has increased over 500%.

Will “substitute” in favour of the product for which the price has gone up by the smaller proportion. Shelter has gone up 20%; Food has increased over 500%.

A simple graphical illustration of the COLI (cont’d)

• A true cost-of-living: C1/ C0 = 480/160 =• 3.0 or 300• In other words, it cost three times as much to achieve

the original level of satisfaction with the current price regime compared to the old one.

• What is the problem with the COLI?• Must know the consumer’s indifference curve. • What can be done?• Thank you Mr. Laspeyres…

I0

10

3

Shelter m2/wk

0

Laspeyres to the rescue…

Food Kg/wk

C (S1,F1)

10

4

A (S0,F0)

16

B0

8

B1

20

8

Laspeyres index is the cost of buying the original bundle at the new prices divided by the cost of the same bundle at the original prices.

Laspeyres index is the cost of buying the original bundle at the new prices divided by the cost of the same bundle at the original prices.

CL = ($24 x 3) + ($60 x 10) = $672CL = ($24 x 3) + ($60 x 10) = $672

BL

CL/C0 = $672 / $160 = 4.2 or 420CL/C0 = $672 / $160 = 4.2 or 420

D

I1

A simple graphical illustration of the COLI (cont’d)

• Given the choice, a consumer would buy at point D.

• In other words, if the consumer has enough income to buy the original bundle at the new prices, he could afford to buy an even grater bundle.

• So the Laspeyres index measures the increase in expenditures necessary to achieve not the same but a higher level of satisfaction than the base year.

I1

Shelter m2/wk

0

Or Paasche to the rescue…

Food Kg/wk

18

9

B10

20

8

Paasche index is based on the comparison of buying the current year’s bundle at different prices.

Paasche index is based on the comparison of buying the current year’s bundle at different prices.

C11 = ($24 x 7.5) + ($60 x 3) = $360C11 = ($24 x 7.5) + ($60 x 3) = $360

CP/C0 = $360 / $180 = 2.0 or 200CP/C0 = $360 / $180 = 2.0 or 200

3E (S11,F11)

7.5

C10 = ($20 x 7.5) + ($10 x 3) = $180C10 = ($20 x 7.5) + ($10 x 3) = $180

B11

15

6

I’

A simple graphical illustration of the COLI (cont’d)

• The problem with the Paasche index is that the consumer could have achieved this year’s indifference curve at the base year prices without having to have spent as much as C10.

• Alternatively, with the same budget he could have reached a higher indifference curve.

• Because C10 is greater than the cost of achieving I1 in year 0, the ratio C11/ C10 is smaller than the cost of maintaining I1.

A simple graphical illustration of the COLI: Conclusion

• The Laspeyres index overstates the increase in the cost of maintaining the constant level of satisfaction.

• The Paasche index understates the increase in the cost of maintaining the constant level of satisfaction.

• Remember the COLI = 3.0 or 300• 4.2 > 3.0 > 2.0• L > COLI > P

A more theoretical approach to illustrating the COLI.

• COL index numbers are devices for reducing the comparison between two complete price vectors such as p1 and p0 to a single scalar.

• If the two vectors are proportional to one another, so that p1 is 5 percent greater than p0 then there is no problem in saying that prices at 1 are 5 times greater than prices at 0.

• This is usually not the case however.

A more theoretical approach to illustrating the COLI

(Cont’d)• A COL index uses a measure of the standard of living as a reference.

• P(P1, P0, qR) = (P1 x qR) / (P0 x qR)

• This is the basket approach.

• But a single bundle is a restrictive interpretation of a constant standard of living.

• Best to use a specific indifference curve as the reference concept that should be held constant.

• The COLI is then the ratio of the minimum expenditures necessary to reach the reference indifference curve at two sets of prices.

• P(P1, P0, uR) = C(uR, P1) / C(uR, P0)

D0E0

A

C B

q1

q0

F0

q2

q1

• Original budget line: AB

• Original consumption basket: q0

• Price in p2 increases, p1 constant

• New budget line: AC

• The budget line necessary to buy q0 at

new prices cuts vertical axis at E0

• Identical standard of living can be

obtained at F0 with budget line D0

COLI: Illustration 1, base quantity weighted

Base quantity reference index (Laspeyres) = 0E0/0A

Base utility reference index = 0D0/0A

L > Coli

u0

u1

0

D0

E0

A

q1

q0

F0

q2

q1

u0

u1

0

D1

E1

A

C B

q1q0

F1

q2

q1

• Original budget line: AB

• Original consumption basket: q0

• Price in p2 increases, p1 constant

• New budget line: AC

• The budget line necessary to buy q1 at

new prices cuts vertical axis at E1

• Identical standard of living can be

obtained at F1 with budget line D1

COLI: Illustration 2, current quantity weighted

P = 0A/0E1

Coli = 0A/0D1

P < Coli

u0

u1

0

More on the diagrams

• The inequalities hold in general• Inequalities date back to Konus (1923).• L = 0E0/0A > COLI > P = 0E0/0A• The bundle q0 is one way of reaching u0 but not necessarily the

cheapest when prices are p1. • Hence p1 x q0, the cost of q0 at p1 is greater than or equal to c(u0,

p1), the minimum cost of u0 at p1. • By definition of q0, p0 x q0 is equal to c(u0, p0), hence…• P(p1, p0, q0) = p1 x q0 / p0 x q0 c(u0, p1) / c(u0, p0) = P(p1, p0, u0)

More on the diagrams

• Similarly…• … since q1 is one way of obtaining u1 at p0

• Hence p0 x q1 c(u1, p0). • So that since p1 x q1 = c(u1, p1),• P(p1, p0, q1) = p1 x q1 / p0 x q1 ≤ c(u1, p1) / c(u1, p0)

= P(p1, p0, u1)

The inequalities do not imply…

• The the “true” cost of living lies somewhere between the Paasche and Laspeyres indexes.

• There is no unique true index.• The base-weighted utility index which has the Laspeyres index

as an upper limit is a different number from the current-weighted utility index that is no less than the Paasche index.

• In fact, it is possible for the Paasche index to exceed the Laspeyres index.

• Only when preferences are homothetic (sufficient and necessary condition) will there exist a price index and that L > COLI > P

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