Download Mesurement 3

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) + 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
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
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)
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 known
With heterogeneity of consumers, welfare comparisons
involve:
1) Equivalence scale issues
2) Aggregation (over individuals) issues
Burgernomics
Year 2007
USA
Euro zone
Brazil
China
Indonesia
Big Mac price in local currency
3.57*
3.37 €
7.50 R$
12.5Y
18,700R
South
Africa
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
1
1.50
1.32
0.51
0.57
0.63
PPP adjustment
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
measurement
Base country invariance: Independence to the
numeraire (USD or Euros or Naira)
Transitivity, Characteristicity, Additivity,
Gerschenkron effect…
2 programs: World Bank ICP program + EurostatOECD 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)
2)
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
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 √ Paasche
Other solution: Törnqvist= Πh PPPh(ω +ω )/2
h,a
h,a’
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
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 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/
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≠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: 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
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
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 = 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)
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 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