Measurement 3

National wealth comparisons

across time and space

Real expenditures

I want to compare 2 situations:- One with total expenditures y0 and price vector p0

- The other with y1 and p1

- y can be GDP or GNI of a country, or individual expenditures

- p is the vector of relative prices of products(market prices not producer prices)

Problem: we do not know the preferences of the countries or individuals

Reference book: Deaton & Muellbauer, Economics and Consumer Behavior, 1980, Cambridge U. Press, Part Three.

Theory: cost of living indexes (1)The same consumer (i.e. same preferences):

Max u=u(q1,...qi,...,qm) subject to Σi piqi = y

Dual: Min y = Σi piqi subject to u=u(q1,...qi,...,qm)

Cost function y=c(u;p1,...pi,...,pm ) = minimum income needed to reach utility level u at prices p

Fix a reference utility u0. True cost-of-living index: P(p, p0;u0) = c(u0,p)/c(u0,p0)

Measures how income should move to maintain reference utility level u0 when prices go from p0 to p

Theory: Cost of living and real expenditures

Indirect utility ψ=ψ(y,p), utility level reached under budget constraint (y,p)

Money metric of utility at reference prices p0: V=c[ψ(y,p);p0]

= minimum income needed to reach utility attained with y at p, when prices are p0 instead of p.

If p= p0, V=y; then, rewrite: V = y. c[ψ(y,p);p0]/c[ψ(y,p);p] = y / P(p, p0;ψ(y,p))

Real expenditures at reference prices p0:V1[p0] = y1/P(p1, p0;ψ(y1,p1)) = y1/P(p1,p0;u1) V0 [p0] = y0

But may also be defined at reference prices p1: V1 [p1] = y1

V0 [p1] = y0/P(p1, p0;ψ(y0,p0)) = y0/P(p1,p0;u0)

So, having measured expenditures y, let’s look for true cost of living indexes!

Theory: cost of living indexes (2)

In general, true cost of living indexes: (1) Depend on utility levels, hence on

expenditures or income level(2) Depend on all relative prices through

substitution effects

(1): Except if preferences are homothetic ie u(θq)=F[θu(q)] (with F monotone increasing function) c(u,p)=u.b(p) P(p1, p0,u) = b(p1)/b(p0)= P(p1, p0) (2): Except if no (or little) substitution effects

Theory: cost of living indexes (3)

(2) continued: Taylor expansion of cost function: c(u0, p1) ≈ c(u0, p0) + Σiq0

i(p1i- p0

i) + ½ΣiΣks0ik(p1

i- p0i)(p1

k- p0k)

[For the 2nd RHS term: recall that p0.q0=c(u0, p0) hence ∂c(u0, p0)/∂pi=q0i ]

So that if s0ik ≈0:

P(p1, p0,u0) ≈ p1.q0/p0.q0 = Laspeyres indexP(p1, p0,u1) ≈ p1.q1/p0.q1 = Paasche index

Even with substitution effects, it can be shown that we always have: Paasche ≤ P(p1,p0;u1) and Laspeyres ≥ P(p1,p0;u0)

So that: V0[p1] ≥ y0/Laspeyres and V1[p0] ≤ y1/Paasche

Fisher index = √Laspeyre.√ Paasche

Törnqvist index = Πi (p1i/p0

i) (ω1i+ ω0i)/2 with ω budget shares in the 2 situations

(exact formula for log of cost function quadratic in log prices and log utility at reference utility u*= √u0.√u1; good approximation by Taylor expansions)

This index makes clear that cost-of-living indexes differ between rich and poor if preferences are not homothetic

Example: LES (Linear Expenditure System)

u=Πi(qi-γi)βi with Σiβi =1γ subsistence basket, (y-Σi piγi) ‘supernumerary’ income

piqi=piγi+ βi(y-Σi piγi)ψ(y,p)= A.(y-Σi piγi)/Πipi

βi where A=Πi γiβi

c(u,p)=u.Πipiβi/A+Σi piγi

V1[p0]=(y1-Σip1iγi) Πi(p0

i/p1i)βi + Σip0

iγi

V1[p1]= y1

Interpersonal comparisons:equivalence scales

Individuals or households do not necessarily share the same preferences over products.

Let a be a variable that indexes this heterogeneityAssume we can write: u = u(q;a)=ψ(y,p;a)Then cost function is c = c(u,p;a)At reference prices and characteristics p0 and a0 :

V[p0;a0]= y / m[p0;a0]P[p0]With P=c[ψ(y,p;a),p;a]/c[ψ(y,p;a),p0;a]And: m=c[ψ(y,p;a),p0;a]/c[ψ(y,p;a),p0;a0]m is an equivalence scale

In the LES example: Let γ vary with household composition a: γ(a)

Aggregation over individuals

A country is not an individual. Is the country average ‘like’ an individual consumer?

In general, NO, except:(1) When preferences are homogenous (of course)(2) For very particular preferences: “quasi-homotheticity” of

utility function = Gorman polar forms like LES with heterogenous γ but homogenous β

In other cases, summation often introduces some quantities linked to the inequality of the distribution of income - like the “Almost Ideal Demand System” (AIDS) for instance

Theoretical conclusions

No unique way to define real expenditures indexes with knowledge of quantities and prices only, and not preferences

Because:1) The cost-of-living can be measured at utility or income

levels 0 or 1 (non-homotheticity)2) If prices differ, with substitution effects the true index is

unknown, only bounds are knownWith heterogeneity of consumers, welfare comparisons

involve:1) Equivalence scale issues2) Aggregation (over individuals) issues

Burgernomics

Year 2007 USA Euro zone Brazil China Indonesia South Africa

Big Mac price in local currency 3.57* 3.37 € 7.50 R$ 12.5Y 18,700R 16.9R

Actual exchange rate (..../USD) 1 0.63 1.58 6.83 9,152 7.54

Big Mac price in current dollars 3.57 5.34 4.73 1.83 2.04 2.24

PPP adjustment 1 1.50 1.32 0.51 0.57 0.63

Source: http://www.economist.com/finance/displaystory.cfm?story_id=11793125 *: average of New-York, Chicago, Atlanta and San Francisco

Imagine the dollar value comes back to 1 euro: Everybody carries on eating one Big Mac Big Mac in Euro Zone is still 3.37€ but now 3.37$ PPP adjustment with US is now 0.94 only

Aggregate PPP indexes (1)

Concept: “Volume of expenditures in a country”Axiomatics (“index numbers theory”):Commensurability: Independence to units of

measurementBase country invariance: Independence to the

numeraire (USD or Euros or Naira)Transitivity, Characteristicity, Additivity,

Gerschenkron effect…

2 programs: World Bank ICP program + Eurostat-OECD program

PPP: 1st level of aggregation (1)

Basic headings h: finest detail for which expenditures shares are available; includes i=1,…m products

Ex.: Restaurants (includes Big Mac)

CPD method (ICP program World Bank):Within each basic heading h:

Product i in country j (j=1,…n countries in comparison)

ln pij = α2 D1j + α2 D2j+…+ αm Dmj + β1Di1+…+ βnDin + uij

Dij: dummy =1 if product i is priced in country j

PPP between j and k = exp(βj- βk)

PPP: 1st level of aggregation (2)

CPD: Commensurable, independent from base country, transitive:- unit changes in the coefficients α- exp(βj- βk) = exp(βj- βl).exp(βl- βk)

But assumes constant relative prices across countries within basic headings = exp(ai-ai’)

CPRD extension: include dummies for “representativeness” of the product i in the country j

More complex “EKS” methodology used by Eurostat-OECD: 1) Fisher binary indexes for basic headings-country pairs over

representative products in each country non-transitive; 2) Made transitive through EKS agregation over all third countries,

minimizing the distance from original binary comparisons (=characteristicity…)

CPD: statistical inference; missing values extrapolated; confidence intervalsEKS: index numbers problem for bilateral comparisons: no assumption of

relative prices homogeneity within basic headings

PPP: 2nd level of aggregation (within regions)

Pair of countries a & a’ in region A. PPPh = price level a’/a for basic heading h. Let’s compute aggregate PPP(a’,a):

- Laspeyres = Σh ωh,a PPPh (arithmetic)- Paasche = 1/[Σh ωh,a’ (1/PPPh)] (harmonic)- Fisher = √Laspeyre √ PaascheOther solution: Törnqvist= Πh PPPh

(ωh,a+ωh,a’)/2

Non-transitive indexes again: made transitive by EKS methodNo reference price or volume structure: no Gerschenkron effect i.e.

sensitivity of the share of country in regional GDP to reference price or reference volumes

But not additive: GDP ≠ C + I + G + X-M … when aggregates are PPP measures

Better suited for level comparisons than for structures comparisonsFor structures, other method of aggregation: Geary-Khamis (GK)

method that preserves additivity, but Gerschenkron effect

PPP: 3rd level of aggregation (between regions)

Chain indexes:PPP(a’,b’) = PPP(a’,a).PPP(a,b).PPP(b,b’)

a of region A, b of B: “ring countries” in ICP PPP(a,b) links regions A and B

specific investment in price data collection in reference countries in order to extend the comparability of baskets

Choice of chaining through reference countries results from cost considerations

Transitivity preserved, Additivity is definitely lost

Ring countries in ICP 2003-2006: Brazil; Cameroon; Chile; Egypt; Estonia; Hong Kong, China; Japan; Jordan; Kenya; Malaysia; Oman; Philippines; Senegal; Slovenia; South Africa; Sri Lanka; United Kingdom; and Zambia.

From PPP to growthNominal GDP in t Nominal GDP in t+1GDP in t GDP in t+1 at t prices (What is called growth; Laspeyres

index)GDP PPP in t GDP PPP in t+1 (at current international prices)

Either I compare GDP with current price and structure of expenditure: better suited for comparison across space

Or I fix price and expenditure structure in t and let GDP move with national prices: better suited to comparisons across time at least in the short run

I may also chain indexes across time: but again, I loose additivity (growth of GDP does not fit with growth of components)

See Penn World Tables 6.2, http://pwt.econ.upenn.edu/

Introducing income inequality (1)

From the Lorenz curve of income, ie income shares of each household, or of each decile of the population

compute the GNI earned by each household or decile Each country is represented by N households or 10

deciles instead of GNI

(1)Directly compare countries income distributions (Session 12 on inequality - Generalized Lorenz curves; cf. literature on world income inequality)

(2) Or else compute a Bergson-Samuelson (utilitarist) social welfare index instead of GNI per capita:

W(u1,…, uN) = W(y1/m1P1,…, yN/mNPN)

Introducing income inequality (2)

Consider for instance the Atkinson-Kolm inequality index:Iε=1-[(1/N).Σn(yn/ŷ)1- ε/(1-ε)] 1/(1- ε) for ε≠1I1=1-exp[(1/N).Σnlog(yn/ŷ)] for ε=1

where ŷ is mean income (or GNI PPP per capita)

Corresponding social welfare function:Wε = ŷ [1- Iε]

= [(1/N).Σn(yn)1- ε/(1-ε)] 1/(1- ε) ε=0: average income (utilitarist)ε=+∞: minimum income (Rawlsian)ε= society aversion for inequality, or individual risk aversion under the veil of

ignorance

Wε measures the « equivalent-income » of an equal distribution: Iε is the share of total income I am ready to loose to reach an equal distribution with the same welfare as with ŷ and prevailing inequalities

Bourguignon & Morrisson, AER,2002

Introduce life expectancy through “lifetime income”: LY = Σt=0,…L y/(1+ δ)t ≈(1-e-δL)/δ with δ discount factor (for instance 2%)

Rome (1)

Measure welfare in Rome in comparison with…:- GDP? Rather impossible- Height stature of skeletons?

(See Health & nutrition sessions)

- Unskilled laborer’s household purchasing power: Wages and prices in denarii ( silver grams)

from the Diocletian edict (maximum prices for inflation control)

Bare bone basket

Robert C. Allen, Oxford University, 2007: How Prosperous were the Romans? Evidence from Diocletian`s Price Edict (301 AD)

Rome (2)

Robert C. Allen, Oxford University, 2007: How Prosperous were the Romans? Evidence from Diocletian`s Price Edict (301 AD)

Rome (3)

Robert C. Allen, Oxford University, 2007: How Prosperous were the Romans? Evidence from Diocletian`s Price Edict (301 AD)

See also…

Angus Deaton’s papers on growth and poverty measurement, and on PPP indexes at the micro level: http://www.princeton.edu/~deaton/papers.html

In particular:A. Deaton, 2003.

"Measuring Poverty in a Growing World" (or "Measuring Growth in a Poor World")

[ “Current statistical procedures in poor countries understate the rate of poverty reduction, and overstate growth in the world”]

A. Deaton, J. Friedman, and V. Alatas, 2004. "Purchasing Power Parity Exchange Rates from Household Survey Data: India and Indonesia“

[ PPP computed from household surveys reveal very different from World Bank or Penn World Tables estimates and reevaluate the poverty differential between India and Indonesia, in favor of India ]

Productivity and Growth Accounting

Productivity of Labor: ! GDP per worker ≠ GDP per capita ! Producer prices ≠ Final demand prices

Growth Accounting (Denison, Solow, Malinvaud…)

Yt = AtKtαLt

1-α

ΔlnYt = ΔlnAt+αΔlnKt+(1-α)ΔlnLt

Demographic growth and “Demographic dividend” (Dependence ratio evolutions)

Capital accumulation ΔlnAt = “Solow residual” (regression analysis)

or “Total factor productivity” (TFP) (accounting)

GDP level accounting(Long-term growth)

Caselli (Handbook of Econ. Growth)

Y = A Kα (Lh)1-α y = A kα h1-α = A yKH

y : GDP per worker PPP

k: Kt = (1-δ) Kt-1 + It and K0=I0/(g0+δ)(I: Investment PPP; K0 steady-state Solow capital stock)

h: h=exp[φ(s)] i.e. Mincerian equation(s: average n. of years of education; φ piecewise linear with decreasing returns)

α =0.3 ; δ = 0.06

A Factor-only model?

Factor-only model in sub-samples

Growth and Capital (end)

Less than ½ of variance for the factor-only modelResult robust to:- Age composition of physical capital- Quality of education, health…- Sectoral composition of outputHowever: Heterogeneity of physical capital, Complementarity phys./hum. capital Government sector