Measurement of National Wealth Comparisons

1. Measurement 3 National wealth comparisons across time and space

2. 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.

3. 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

4. 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!

5. Theory: cost of living indexes (2) In general, true cost of living indexes: • Depend on utility levels, hence on expenditures or income level • 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

6. Theory: cost of living indexes (3) (2) continued: Taylor expansion of cost function: c(u0,p1) ≈ c(u0,p0) + Σiq0i(p1i-p0i) + ½ΣiΣks0ik(p1i-p0i)(p1k-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 index P(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/p0i)(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

7. Example: LES (Linear Expenditure System) u=Πi(qi-γi)βi with Σiβi =1 γ subsistence basket, (y-Σi piγi) ‘supernumerary’ income • piqi=piγi+ βi(y-Σi piγi) • ψ(y,p)= A.(y-Σi piγi)/Πipiβi where A=Πi γiβi c(u,p)=u.Πipiβi/A+Σi piγi  V1[p0]=(y1-Σip1iγi) Πi(p0i/p1i)βi+ Σip0iγi  V1[p1]= y1

8. Interpersonal comparisons:equivalence scales Individuals or households do not necessarily share the same preferences over products. Let a be a variable that indexes this heterogeneity Assume 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)

9. Aggregation over individuals A country is not an individual. Is the country average ‘like’ an individual consumer? In general, NO, except: • When preferences are homogenous (of course) • 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

10. Theoretical conclusions No unique way to define real expenditures indexes with knowledge of quantities and prices only, and not preferences Because: • 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 known With heterogeneity of consumers, welfare comparisons involve: • Equivalence scale issues • Aggregation (over individuals) issues

11. Burgernomics

12. Aggregate PPP indexes (1) Concept: “Volume of expenditures in a country” Axiomatics (“index numbers theory”): Commensurability: Independence to units of measurement Base 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

13. 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)

14. 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: • Fisher binary indexes for basic headings-country pairs over representative products in each country  non-transitive; • Made transitive through EKS agregation over all third countries, minimizing the distance from original binary comparisons (=characteristicity…) CPD: statistical inference; missing values extrapolated; confidence intervals EKS: index numbers problem for bilateral comparisons: no assumption of relative prices homogeneity within basic headings

15. 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 = Σhωh,a PPPh (arithmetic) - Paasche = 1/[Σhωh,a’ (1/PPPh)] (harmonic) - Fisher = √Laspeyre √ Paasche Other solution: Törnqvist= Πh PPPh(ωh,a+ωh,a’)/2 Non-transitive indexes again: made transitive by EKS method No 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 comparisons For structures, other method of aggregation: Geary-Khamis (GK) method that preserves additivity, but Gerschenkron effect

16. 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.

17. From PPP to growth Nominal GDP in t  Nominal GDP in t+1 GDP 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/

18. 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 • 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)

19. 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≠1 I1=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: Iεis 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

20. 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%) Bourguignon & Morrisson, AER,2002

21. 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)

22. Rome (2) Robert C. Allen, Oxford University, 2007: How Prosperous were the Romans? Evidence from Diocletian`s Price Edict (301 AD)

23. Rome (3) Robert C. Allen, Oxford University, 2007: How Prosperous were the Romans? Evidence from Diocletian`s Price Edict (301 AD)

24. 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 ]

25. Productivity and Growth Accounting Productivity of Labor: ! GDP per worker ≠ GDP per capita ! Producer prices ≠ Final demand prices Growth Accounting (Denison, Solow, Malinvaud…) Yt = AtKtαLt1-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)

26. 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

27. A Factor-only model?

28. Factor-only model in sub-samples

29. Growth and Capital (end) Less than ½ of variance for the factor-only model Result robust to: • Age composition of physical capital • Quality of education, health… • Sectoral composition of output However: • Heterogeneity of physical capital, • Complementarity phys./hum. capital • Government sector