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Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco H. G. Ferreira DECRG

Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

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Page 1: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

Lecture 3Understanding Inequality:

Structure and Dynamics

Course on Poverty and Inequality Analysis

Module 5: Inequality and Pro-Poor Growth

Francisco H. G. Ferreira

DECRG

Page 2: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

Roadmap

1. Distributions

2. The Determinants of Inequality: a conceptual overview

3. Inequality Decomposition Analysis

4. Income Distribution Dynamics: statistical analysis

5. Income Distribution Dynamics: towards economic decompositions

6. Summing up

Page 3: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.1. Distributions

• Social welfare, poverty and inequality summarize different features of a distribution.

• Distribution of welfare indicator per unit of analysis.

• Discrete: y = {y1, y2, y3, …., yN}

• Continuous: The distribution function F(y) of a variable y,defined over a population, gives the measure of that population for whom the variable has a value less than or equal to y.

y

dxxfyF0

)()(

hhr

i jij

hrKAI

x

y

Page 4: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

The density function: f(x)

Figure 2: Income Distributions for Brazil, Mexico and The United States

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 2 4 6 8 10 12Log income

Den

sity Brazil

Mexico

USA

Sources: PNAD/IBGE 1999, CPS/ADS 2000Note: Gaussian Kernel Estimates (with optimal window width) of the density functions for the distributions of the logarithms of household per capita incomes. The distribution were scaled so as to have the brazilian mean. Brazil and Mexico are urban areas only. Incomes were converted to US dollar at PPP exchange rates (see Appendix).

The distribution function y

dxxfyFp0

)(

Page 5: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

Figura 1. Brasil 1981-1995: Paradas de Pen

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50 60 70 80 90 100

População Acumulada %

Renda

1981

1990

1995

The quantile function: y=F-1(p)

Page 6: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

The Lorenz curve:

Figura 2. Brasil 1981-1995: Curvas de LorenzdedeLorenz lLLorenz deLorenz Curves

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

População Acumulada %

RendaAcumulada%

1981

1990

1995

y

y

dxxxfyL0

1)(

p

y

dFpL0

11

or

Page 7: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

The Generalized Lorenz curve:

Figura 3. Brasil 1981-1995: Curvas de Lorenz Generalizadas

0

50

100

150

200

0 10 20 30 40 50 60 70 80 90 100

População Acumulada %

Renda

1981

1990

1995

y

dxxxfyGL0

)(

Page 8: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.2. The Determinants of Inequality: a conceptual overview

• Inequality measures dispersion in a distribution. Its determinants are thus the determinants of that distribution. In a market economy, that’s nothing short of the full general equilibrium of that economy.

• One could think schematically in terms of: y = a.r

• This suggests a scheme based on assets and returns: – Asset accumulation– Asset allocation / Use– Determination of returns– Demographics– Redistribution

Page 9: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.2. The Determinants of Inequality: a conceptual overview

Box 1: Schematic Representation of Household Income Determination I (Z, w)

Investment in Human Capital P (X, Z, w) V(J) The Matching Function

D( p(X, Z, J), X, Z, J, w) Remuneration in the Labor Market G(, w) Household Formation

F(y) Redistribution H(y+t)

Page 10: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.2. The Determinants of Inequality: a conceptual overview

• Modeling these processes in an empirically testable way is quite challenging. – Though there are G.E. models of wealth and income distribution

dynamics

• Historically, empirical researchers have used ‘shortcuts’, such as:– decomposing inequality measures by population subgroups, and

attributing “explanatory power” to those variables which had large “between” components;

– Decomposing inequality by income sources, to understand which contributed most to inequality, and why;

– Decomposing changes in inequality into changes in group composition, group mean and group inequality.

Page 11: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.3 Inequality Decomposition Analysis:a. By Population Subgroups

n

i y

iy

nyE

12

111

);(

n

i iy

y

nE

1

log1

)0(

n

i

ii

y

y

y

y

nE

1

log1

)1(

n

ii yy

ynE

1

2

22

1)2(

Not all inequality measures are decomposable, in the sense that I = IW + IB. The Generalized Entropy class is.

Examples includeTheil – L

Theil – T

0.5 CV2

Page 12: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.3 Inequality Decomposition Analysis:a. By Population Subgroups

Let Π (k) be a partition of the population into k subgroups, indexed by j. Similarly index means, n, and subgroup inequality measures. Then if we define:

n

i y

jjB fyE

12

11

);(

k

jjjW yEwyE

1

;;

where 1

jjj fvw

n

nv jj

j n

nf j

j

Then, E = EB + EW.

Page 13: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

An Example from Brazil

The Rise and Fall of Brazilian Inequality: 1981-2004

Page 14: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

A cross-country example: Race and ethnicity decompositions.

Source: WDR 2006

Page 15: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.3 Inequality Decomposition Analysis:a. By Population Subgroups

The methodology was developed by:Bourguignon, F. (1979): "Decomposable Income Inequality Measures",

Econometrica, 47, pp.901-20.Cowell, F.A. (1980): "On the Structure of Additive Inequality Measures",

Review of Economic Studies, 47, pp.521-31.Shorrocks, A.F. (1980): "The Class of Additively Decomposable

Inequality Measures", Econometrica, 48, pp.613-25.

Reviewed in:Cowell, F.A.and S.P. Jenkins (1995): "How much inequality can we

explain? A methodology and an application to the USA", Economic Journal, 105, pp.421-430.

Example from:Ferreira, F.H.G., Phillippe Leite and J.A. Litchfield (2001): “The Rise and

Fall of Brazilian Inequality: 1981-2004”, World Bank Policy Research Working Paper #3867.

Page 16: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.3 Inequality Decomposition Analysis:b. By Income Sources

• Shorrocks A.F. (1982): “Inequality Decomposition by Factor Components, Econometrica, 50, pp.193-211.

• Noted that could be written as:

n

ii yy

ynE

1

2

22

1)2(

2122)2( ff

ff EEE

Correlation of income source with total income

Share of income source

Internal inequality of the source

Page 17: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.3 Inequality Decomposition Analysis:b. By Income Sources

Source: Ferreira, Leite and Litchfield, 2006.

ff

f

f

ff Total Household Income per

Capita

Total Earnings from

Employment*

Total Income from Self-

Employment**

Total Employer Income***

Total Social Insurance

Transfers #

All Other Incomes ##

Mean 393.88 196.06 60.76 44.12 76.82 16.11E(2) 1.618 2.101 6.801 43.301 6.925 23.090Correlation with household income ( f)

1 0.569 0.310 0.598 0.443 0.299

Relative mean ( f) 1 0.498 0.154 0.112 0.195 0.041Absolute factor contribution (S f)

1.618 0.522 0.158 0.561 0.289 0.088

Proportionate factor contribution (sf)

1 0.323 0.098 0.347 0.179 0.054

E(2), yf>0 1.618 1.365 1.991 2.115 1.923 6.567Pop share with yf>0 1 0.717 0.341 0.060 0.326 0.300

Table 4: The Contribution of Income Sources to Total Household Income Inequality in 1981, 1993 and 2004.

2004

f

f

Page 18: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsa. Scalar decompositions

Mookherjee, D. and A. Shorrocks (1982): "A Decomposition Analysis of the Trend in UK Income Inequality", Economic Journal, 92, pp.886-902.

))y( ( )f - v( +

f )( - + f G(0) +

)G(0f

= G(0)

jjj

k

j=1

jjj

k

j=1jj

k

j=1

jj

k

j=1

log

log

Pure inequality

Group Size

Relative means

Page 19: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

The (obligatory) example from Brazil…

The Rise and Fall of Brazilian Inequality: 1981-2004

Observed Proportional change in E(0)

a b c d a b c d a b c d

Age 0.112 -0.003 0.000 0.002 -0.139 -0.002 0.000 0.017 -0.044 -0.003 0.000 0.019

Education 0.110 0.000 0.043 -0.035 -0.089 0.001 0.019 -0.053 0.011 0.001 0.088 -0.136

Family Type 0.120 -0.005 0.015 -0.004 -0.138 -0.005 0.022 0.005 -0.039 -0.004 0.040 -0.032

Gender 0.116 -0.005 0.000 0.000 -0.120 -0.004 0.000 0.000 -0.018 -0.009 0.000 -0.001

Race n.a. n.a. n.a. n.a. -0.101 -0.003 0.001 -0.021 n.a. n.a. n.a. n.a.

Region 0.141 -0.003 -0.003 -0.024 -0.118 -0.001 -0.001 -0.005 0.012 -0.005 -0.004 -0.028

Urban/rural 0.178 0.005 -0.032 -0.040 -0.104 0.002 -0.014 -0.009 0.054 0.017 -0.048 -0.049

Table 5. A Decomposition of Changes in Inequality by Population Subgroups.

1981-1993 1993-2004 1981-2004

-0.035

Note: Term a is the pure inequality effect; terms b and c are the allocation effect; term d is the income effect.Source: Authors’ calculations from PNAD 1981, 1993 and 2004.

0.107 -0.128

Page 20: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsb. A More Disaggregated Look

• In practice, decompositions of changes in scalar measures suffer from serious shortcomings:– Informationally inefficient, as information on entire distribution is

“collapsed” into single number.– Decompositions do not ‘control’ for one another.– Can not separate asset redistribution from changes in returns.

• With increasing data availability and computational power, studies that decompose entire distributions have become more common.– Juhn, Murphy and Pierce, JPE 1993– DiNardo, Fortin and Lemieux, Econometrica, 1996

Page 21: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsb. The Oaxaca-Blinder Decomposition

• These approaches draw on the standard Oaxaca-Blinder Decompositions (Oaxaca, 1973; Blinder, 1973)

• Let there be two groups denoted by r = w, b.

• Then and

• So that

• Or:

• Caveats: (i) means only; (ii) path-dependence; (iii) statistical decomposition; not suitable for GE interpretation.

irririr Xy

wiwyw X bibyb X

bibiwbwiwybyw XXX

wibiwbwibybyw XXX

“returns component” “characteristics component”

Page 22: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsb. Juhn, Murphy and Pierce (1993)

irririr Xy irirrir XF 1

001

010' iiii XFXy

Juhn, Murphy and Pierce (1993):

where

Define:

001

110" iiii XFXy

Then: 0' ii yFIyFI

ii yFIyFI '"

ii yFIyFI "1 Observed charac. Component.

Returns component

Unobserved charac. component

Page 23: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsb. DFL and BFL

How and why does fA(y) differ from fB(y)?

dTTyyf CC ,)(

dTTTygyf CCC

One could decompose fB(y) - fA(y) into:

yfyfyfyfyfyf BAg

BABAg

AB

dTTTygyf ABBAg where

A similar (but distinct) decomposition would be obtained with

dTTTygyf BABA

Page 24: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsb. DiNardo, Fortin and Lemieux (1996)

dTTTTygyf AABA

dTTTygyf BABA

Essentially, DFL propose estimating a counterfactualincome distribution such as:

By appropriately reweighing the sample, as follows.

where AtT

BtTT

Page 25: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsb. Bourguignon, Ferreira and Lustig (2005).

Partition the set of covariates T into V and W, where V can logically depend on W.

dVdWWVWVygyf CCC ,,

dVdWWWvhWVvhWVvhWVygyf CCCCCC ...,,, 2,122111

Define a counterfactual distribution fsAB(y; ks, A). I.e.

dVdWWWvhWVvhWVvhWVygyf AAABAsBA ...,,, 2,122111

Replace the joint distribution of covariates by the appropriate product of conditional distributions, and the joint distribution of W.*

*: Note that the order of conditioning will affect interpretation.

Page 26: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsb. Bourguignon, Ferreira and Lustig (2005).

For each counterfactual distribution fs, the difference between fA and fB can be decomposed as follows:  

And it follows that:

yfyfyfyfyfyf sBAsAB

yyyyyy sq

Bq

Aq

sq

Aq

Bq

Page 27: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsb. Bourguignon, Ferreira and Lustig (2005).

• Estimate - for each country or time period - simple econometric models of earnings, occupational structure, education and fertility choices.

• Simulate the effects of importing the parameter estimates of each model from county A into country B (individually or jointly).

• Decompose distributional differences into:– Price effects– Occupational structure effects– Endowment (or population characteristic) effects

Page 28: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.4. Income Distribution Dynamicsb. Brazil, 1976-1996. (Ferreira and Paes de Barros, 1999)

Level 1: y = G (V, W, ; )  Aggregation rule: 

Earnings: Occupational Choice:  Level 2: V = H (W, ; ) Education: MLE (EA, R, r, g, nah; )

 Fertility: MLC ( nch E, A, R, r, g, nah; )

  

hn

ih

J

j

jhi

jhi

hh yyI

ny

10

1

1

ijhijj

hi xy )(log

sj

ZZ

Z

s

i jisi

si

ee

eP

Page 29: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

Comparing g(p) and gs(p) (i): The price effect.

Figure 15a: A Complete Decomposition

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

Percentiles

Dif

fere

nces

of

log

inco

mes

alphas and betas 1996-1976

Source: "Pesquisa Nacional por Amostra de Domicilios" (PNAD), 1976 and 1996.

Page 30: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

Comparing g(p) and gs(p) (ii): The price effect and the occupational structure effect combined.

Figure 15b: A Complete Decomposition

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

Percentiles

Dif

fere

nces

of

log

inco

mes

alphas and betas alphas, betas, gammas 1996-1976

Source: "Pesquisa Nacional por Amostra de Domicilios" (PNAD), 1976 and 1996.

Page 31: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

Comparing g(p) and gs(p) (i): Price, Occupation, Education and Fertility effects.

Figure 15: A Complete Decomposition

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

Percentiles

Dif

fere

nces

of

log

inco

mes

alphas and betas alphas, betas, gammas

mu(d), alphas, betas, gammas mu(d), mu(e), alphas, betas, gammas

1996-1976

Source: "Pesquisa Nacional por Amostra de Domicilios" (PNAD), 1976 and 1996.

Page 32: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.5. Income Distribution Dynamics: a. towards economic decompositions?

• Generalized Oaxaca-Blinder decompositions such as those discussed above, whether parametric or semi-parametric, suffer from two shortcomings:

– Path-dependence

– The counterfactuals do not correspond to an economic equilibrium. There is no guarantee that those counterfactual incomes would be sustained after agents were allowed to respond and the economy reached a new equilibrium.

Page 33: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

(a) Partial Equilibrium Approaches

• The first step towards economic decompositions, in which the counterfactual distributions may be interpreted as corresponding to a counterfactual economic equilibrium, is partial in nature.

• One example comes from attempts to simulate distributions after some transfer, in which household responses to the transfer (in terms of child schooling and labor supply) are incorporated.

– Bourguignon, Ferreira and Leite (2003)

– Todd and Wolpin (2005)

– (These two papers differ considerably in how they model behavior. Todd and Wolpin are much more structural.)

Page 34: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

(b) General Equilibrium Approaches

• However, a number of changes which are isolated in statistical counterfactuals – such as changes in returns to education, or in the distribution of years of schooling – are likely to have general equilibrium effects.

• Similarly, certain policies one might like to simulate may require a general equilibrium setting.

• There are two basic approaches to generate GE-compatible counterfactual income distributions (and thus counterfactual GICs):

– Fully disaggregated CGE models, where each household is individually linked to the production and consumption modules. E.g. Chen and Ravallion, 2003, for China.

– “Leaner” macroeconomic models linked to microsimulation modules on a household survey dataset. E.g. Bourguignon, Robilliard and Robinson, 2005, for Indonesia.

Page 35: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

Distributional Impact of China’s accession to the WTO. (Chen & Ravallion, 2003)

GE-compatible counterfactual GICs corresponding to a specific policy.

Page 36: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

(b) General Equilibrium Approaches (continued).

Macro model

Linkage AggregatedVariables

(prices, wages, employment levels)

Household income micro-simulation model

• In the Macro-Micro approach, some key counterfactual linkage variables are generated in a “leaner” macro model, whose parameters may have been calibrated or estimated from a time-series, and then fed into sector-specific equations estimated in the household survey, to generate a counterfactual GIC.

Page 37: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

The distribution of the impacts of the 1999 Brazilian devaluation (Ferreira, Leite, Pereira and Pichetti, 2004)

Figure 7 - Comparison betweenActual Observed Changes &

Experiment 1 - using Representative Households Groups (RHG)Experiment 2- using Pure Micro Simulation model

Experiment 3 - using Full Macro-Micro Linkage model

-6%

-4%

-2%

0%

2%

4%

6%

8%

0 10 20 30 40 50 60 70 80 90 100

Percentiles

Lo

g d

iffe

ren

ce

Actual Experiment 1 - RHG

Experiment 2 - Pure Micro Simulation Experiment 3 - Full Macro-Micro Linkage

Percent changes between 1999 and 1998 in Nominal Income (in Reais, R$) / Month for each percentile of the distribution in Brazil

Page 38: Lecture 3 Understanding Inequality: Structure and Dynamics Course on Poverty and Inequality Analysis Module 5: Inequality and Pro-Poor Growth Francisco

3.6. Summing up

• There has long been an interest in understanding changes in (or differences across) income distributions.

• Static and dynamic decompositions of certain measures of inequality (by population subgroup or income source) can shed some light on the “structure” of inequality, and on the importance of covariates.

• But decompositions of scalar indices are inherently informationally constrained. Disaggregated statistical decompositions based on entire counterfactual distributions (parametrically or semi-parametrically) help shed more light on changes (and to separate the effects of returns, participation and composition effects).

• A step beyond this sort of statistical analysis is to build counterfactual distributions that correspond to economic equilibria. If sensibly estimated, these would allow inference of causality – and hence policy simulations.

– Some progress on simple partial equilibrium models.

– Harder with general equilibrium approaches, where CGEs or macro models are subject to many criticisms.