Sensitivity analysis of poverty

# Sensitivity analysis of poverty
### R.Andres Castaneda (based on the slides of many others that teach this class before me)
### The World Bank
### 2019-07-16

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# Why checking for robustness?

- What do you think are the main reasons why poverty may not be robust?

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1. Arbitrariness of the choice of Poverty Index
2. Arbitrariness of the choice of Poverty Line
3. Equivalence of scale used
4. Sampling error

- What else?

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# Elasticity of FGT poverty measure with respect to per capita consumption

`$$\eta = \frac{\Delta\%FGT_\alpha(x,z)}{\Delta\%PCE}$$`
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- Shift the distribution without changing its relative shape

- Poverty line remains constant

- What would happen if there is a concentration of marginal poor around the poverty line?

- .red[What are the main reasons why the consumption aggreagte may change?]

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## Graphical representation of change in PCE
<img src="./img/pce_shift.png" width="1500" style="display: block; margin: auto auto auto 0;" />

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## The same is true if we change the poverty lines
- Each component of the absolute poverty line (i.e. CBN) is a function of several variables that affects results
  
  - Food component
    
    - “Reference” Group could lead to different poverty lines even using the same method;
    
    - Nutritional requirement
    
    - Average or median of the “reference” population not only for quantities but also for prices
  
  - Nonfood component
    
    - How do we define the interval (“close”) to the poverty line?
    
    - Lower poverty line & upper poverty line
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## Graphical representation of change in poverty lines
<img src="./img/pl_shift.png" width="1500" style="display: block; margin: auto auto auto 0;" />

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# Poverty Dominance Analysis

- Dominance analysis is a comparison of multiple distributions.

- It allows us to compare results for all poverty lines when comparing across time/subpopulations

- Three relevant poverty curves:

- Poverty incidence curves

- Poverty deficit curves

- Poverty severity curves
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## Poverty incidence curves
<img src="./img/pov_inc_curve.png" width="1629" style="display: block; margin: auto auto auto 0;" />

???
The height is the headcount ratio when the poverty line is set at a particular level of the welfare aggregate (i.e. income or consumption). In other words, it is the CDF

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- If the poverty incidence curve for distribution A is above that for B for all poverty lines (up to `$z_{max}$`)...
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- ... there is more poverty in A than in B for all poverty measures and all poverty lines (up to `$z_{max}$`)

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### Graphically it looks like this
<img src="./img/fst_ord_dom.png" width="1500" style="display: block; margin: auto auto auto 0;" />

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### but, 
- What if the poverty incidence curves intersect?

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### but, 
- What if the poverty incidence curves intersect?

- Ambiguous poverty ranking

- What could you do?

- Restrict range of poverty lines

- Restrict class of poverty measures
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### Poverty Deficit curves

- Area under poverty incidence curve

- The height is proportional to the poverty gap measure, the larger the height the larger the poverty gap measure for a given poverty line or level of welfare aggregate

- Each point gives average poverty gap = poverty gap index times the poverty line `$z$`

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### Graphically, it look like this
<img src="./img/pov_deficit_curve.png" width="1828" style="display: block; margin: auto;" />

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- If the poverty deficit curve for distribution A is above that for B for all poverty lines (up to `$z_{max}$`)...

- ...there is more poverty in A than in B for all poverty measures which are strictly decreasing and weakly---convex in welfare aggregate (i.e. consumption or income) of the poor such as the Poverty Gap and Severity but not Headcount Ratio

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<img src="./img/def_curve1.png" width="70%" style="display: block; margin: auto;" />
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### But, 
- What if the poverty deficit curves intersect?

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<img src="./img/def_curve2.png" width="70%" style="display: block; margin: auto;" />

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### But, 
- What if the poverty deficit curves intersect?

- Ambiguous poverty ranking
--
- What could you do?

- Restrict range of poverty lines

- Restrict class of poverty measures
--
- .blue[... And so on and so forth with poverty severity]

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# Poverty dominance analysis - Recommendations

- First Order: construct the poverty incidence curves up to highest admissible poverty line for each - distribution
  - Do not cross `$\rightarrow$` Unambiguous comparison (2nd and 3rd Order holds too)
  - Do cross `$\rightarrow$` perform Second Order Dominance test
- Second Order: build poverty deficit curves and restrict range of proper measures
  - Do not cross `$\rightarrow$` Unambiguous comparison for higher order poverty indexes 
  - Do cross `$\rightarrow$` perform Third Order Dominance Test
- Third Order:  create poverty severity curves 
  - Do not cross `$\rightarrow$` Unambiguous comparison
  - Do cross `$\rightarrow$` nothing left to do

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layout: true

# Equivalence of Scales
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- Two options for the arbitrary approack
  - OECD - Modified scale

`$$AE_{OECD} = 1 + 0.5 \times (N_A - 1) + 0.3 \times N_C$$`
Where, 
  - `$N_A =$`  Total number of adults
  - `$N_C =$`  Totol number of children
    
  - LSMS (National Research Council '95)

`$$AE_{LSMS} = (N_A - \alpha N_C)^\theta$$`

Where, 
  - `$\alpha =$` Adult equivalent `$\in (0,1)$`
  - `$\theta =$` Economies of scale `$\in (0,1)$`
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## To check the sensitivity of the paremeters

- Select a "pivot" household which is unaffected by changes in parameters

- It could be any _household composition_ that would be chosen as modal type: `$(N_{A0};N_{A0})$`
  - Ex: Two adults and two children

`$$x^* = 
\frac{x}
     {(N_A + \alpha N_C)^\theta} 
\frac{(N_{A0} + \alpha N_{C0})^\theta}
     {(N_{A0} + N_{C0})}$$`

- `$x^*$` is always equal to per capita expenditure for the reference household because `$N_A = N_{A0} \text{ and } N_C = N_{C0}$`

- For all other household, 
  - `$1^{st}$` term if expenditure per adult equivalent
  - `$2^{nd}$` term is a constant ($N_{A0} \text{ and } N_{C0}$ and fixed)

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