CobbCobb--DouglasDouglas--Eventually!Eventually!

Alan A. PowellAlan A. Powell Keith R. Keith R. McLarenMcLarenKen R. Pearson Maureen T. Ken R. Pearson Maureen T. RimmerRimmer

MonashMonash UniversityUniversity

TOPICS COVERED IN PAPERTOPICS COVERED IN PAPER•• historical review of directlyhistorical review of directly

additive preferences (LES)additive preferences (LES)•• Engel flexibility, [Engel] Engel flexibility, [Engel]

regularity, convergence to Cobbregularity, convergence to Cobb--Douglas Douglas ——differing opinionsdiffering opinionsabout desirability of the latterabout desirability of the latter•• implicit indirect additivity implicit indirect additivity ——

AIDADSAIDADSAAn IImplicitly DDirectly AAdditive DDemand SSystem

TOPICS [2]TOPICS [2]•• problems in CGE problems in CGE simssims wherewhere

there are v. large changes inthere are v. large changes inincomeincome•• insertion of AIDADS into ainsertion of AIDADS into a

CGE model & its calibration CGE model & its calibration ——ORANIORANI--G G →→ ORANAIDADORANAIDAD•• demonstration of demonstration of ORANAIDADORANAIDAD’’ssability to maintain regularityability to maintain regularitythroughout a large (esp. negative)throughout a large (esp. negative)change in per capita incomeschange in per capita incomes

TOPICS [3]TOPICS [3]

•• effectively global regularityeffectively global regularityof AIDADS [in the Appendix]of AIDADS [in the Appendix]

Main ideas to be covered in thispresentation:

Engel Flexibility, [Engel] Regularity,Engel Flexibility, [Engel] Regularity,Convergence to what as income Convergence to what as income →→ ∞∞??

Relevance for CGE?Relevance for CGE?If large changes in per capita If large changes in per capita income are involved, Engel income are involved, Engel flexibility and regularity are flexibility and regularity are needed to keep the consumer needed to keep the consumer demand system valid over the demand system valid over the course of a simulationcourse of a simulation

Constant budget shares (Cobb-Douglas)

AIDS - a luxury good

AIDS - a necessity

Regular but inflexible

Irregular and inflexible

…… but AIDS is but AIDS is muchmuch more more flexible than CDflexible than CD

Linear Expenditure System

a luxury

LES - a necessity

Regular but not very flexible

Why not?Why not?

In the LES,budget shares and expenditure elasticitiesare monotonic in total expenditure

0

0.5

1

1.5

2

2.5

3

1 3 5 7 9 11 13 15 17 19 21 23 25

Log of total expenditure

Expenditure Elasticity

Typical monotonic convergence of expenditure elasticities to unity in Engel

rank 2 demand systems

Engel FlexibilityEngel Flexibility——DefinitionsDefinitions• Extremely inflexible

All budget shares are invariant to changes in total expenditure and to changes in relative prices; all total expenditure elasticities are equal to one, and all own price elasticities are equal to minus one; all cross price elasticities are equal to zero — COBB-DOUGLAS utility function

Engel Flexibility [2]Engel Flexibility [2]• Very inflexible

All expenditure elasticities are globally constant and equal to unity, but with budget shares varyingas functions of relative prices —all homothetic utility functions,all homothetic utility functions,including CESincluding CES

Engel Flexibility [3]Engel Flexibility [3]• Somewhat inflexible

Budget shares vary with changes in total expenditure at any given setting of relative prices, while expenditure elasticities vary among commodities and change with changes in total expenditure and changes in relative prices. Total expenditure elasticities are MONOTONIC in real total expenditure ——all demand systems of Engel rank 2all demand systems of Engel rank 2 (whichmeans virtually all demand systemsvirtually all demand systems commonly in use, including the LESLES).

Engel Flexibility [4]Engel Flexibility [4]• Flexible

Engel elasticities & budget shares are not necessarily monotonic in total expenditure at any given setting of relative prices —Engel rank 3 demand systems, Engel rank 3 demand systems, including AIDADSincluding AIDADSAAn IImplicitly DDirectly AAdditive DDemand SSystem

MonotonicMonotonic

NonNon--monotonicmonotonic

NonNon--parametric empirical parametric empirical work with household survey work with household survey data establishes that such data establishes that such

nonnon--monotonic behavior is a monotonic behavior is a factual attribute of the data:factual attribute of the data:Lewbel (1991), Lewbel (1991), RimmerRimmer & &

Powell (1994)Powell (1994)

Convergence to Cobb-Douglas with increasing per capita income — 1st opinion

(a) As the real income of a consumer becomes indefinitely large, re-mixing the consumption bundle becomes irrelevant: having chosen the ultimately satisfying budget shares at any given set of relative prices, the superlatively wealthy continue to allocate additional income in the same proportions. With very large and increasing per capita income, ultimately the utility function becomes indistinguishable from Cobb-Douglas.

2nd opinion(b) Consumer demand systems in which the income elasticities monotonically approach one (from above, in the case of luxuries; from below, in the case of neces-sities) are unsatisfactory both theor-etically and empirically. For instance, a necessity with a low (< 1) income elasticity may very well become less elastic with further increases in income.

Budget share

Per capita real expenditure

India 1975 Engel elas = 0.485

USA 1975 Engel elas = 0.124

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

3 8 13 18 23 28 33 38 43 48 53

natural logarithm of real per capita total expenditure

Engel elas1

Engel elas2

Engel elas3

Figure 1, page 7D-11, Vol. 3

AIDADS

AIDADS can accommodate inferiority

Budget shares

AIDADS does converge to CD

EVENTUALLY!

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45 50

share 3

share 2

share 1

natural logarithm of per capita real expenditure

Figure 2, page 7D-11, Vol. 3

AIDADS again

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45 50

share 3

share 2

share 1

natural logarithm of per capita real expenditure-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

3 8 13 18 23 28 33 38 43 48 53

natural logarithm of real per capita total expenditure

Engel elas1

Engel elas2

Engel elas3

The end