USAID - Poverty Target Tool
The purpose of this tool is to compute the change in daily, per capita expenditures that would be
necessary to reach a specific level of poverty incidence in the Feed the Future Zone of Influence. The
goal is to understand how much daily, per capita expenditures must rise to reach a specific desired level
of poverty incidence under a specific set of assumptions.
Assumptions
The poverty tool makes several very strong, likely unrealistic assumptions. The first two assumptions are
particularly important to understand as they limit the interpretation of the results from using the tool:
(1) The decrease in poverty incidence suggested is the same for all values of consumption expenditures;
that is, whatever programs affect the population represented in the baseline survey affect every
individual equally. The tool then estimates the fixed amount of additional daily expenditures that must
be achieved by every individual to reach the poverty target. If every individual is not equally affected by
projects working in the zone of influence (e.g. the poor are targeted) and other changes in economic
conditions over time, then the tool might give misleading suggestions.
(2) Given a mean and variance, the distribution of expenditure is completely defined and given by a lognormal distribution with the given parameters1. Empirical evidence on consumption expenditure
variables collected in many such surveys demonstrate that per capita expenditures are approximately
log-normal; however, the strict log-normality of per capita expenditures can usually be rejected in
statistical tests. In the tool, we provide an estimate of the overstatement/understatement of poverty
incidence based on that assumption relative to poverty incidence suggested by the actual data.
(3) The distribution of expenditures is ergodic and stationary. This is a regularity condition, essentially
allowing us to use a central limit theorem. The consequence is that the expenditure distribution does
not change over time. Given that research on poverty incidence over time demonstrates that the
distribution does change over time, this assumption is unrealistic.
Values to Input
The tool asks the user to input the following quantities:
1. Average daily per capita expenditures, in dollars per day (use PPP terms to convert from local
currency)—this number should be calculated from the FTF Baseline Survey
1
A shifted log normal to be exact, since the support is 𝑋 ∈ [ 𝑚𝑖𝑛(𝑋), +∞ )
2. Variance of daily per capita expenditures, in dollars per day (use PPP terms to convert from local
currency)—this number should be calculated from the FTF Baseline Survey
3. Poverty line, also in dollars per day (usually $1.25/person/day)
4. Poverty incidence at baseline (using poverty line listed above)
5. Desired poverty incidence at endline (using poverty line listed above)
Technical Notes
Given these assumptions, we set to find an amount C such that the proportion of poor changes from
𝑃𝑜𝑣𝑡 at baseline to 𝑃𝑜𝑣𝑡+1 at endline. Figure 1 shows a graphical representation of the procedure.
Let X be expenditures. The given parameters are:
The (estimated) expectation of X
:
𝐸[𝑋𝑡 ]
The (estimated) variance of X
:
𝑠𝑋2
The proportion of poor at baseline
:
𝑃𝑜𝑣𝑡
The (desired) proportion of poor at endline
:
𝑃𝑜𝑣𝑡
The poverty line that correspond to the above proportions
The assumption of log-normality :
If 𝑋~𝑙𝑜𝑔normal (μ, σ2 ) then 𝑙𝑛𝑋~normal (μ, σ2 ) and
1 2
(1) 𝐸[𝑋] = 𝑒 𝜇+2𝜎
2
(2) 𝑠𝑥2 = (𝑒 𝜎 − 1)𝑒 2𝜇+𝜎
2
Using these restrictions we can solve for 𝜇 𝑎𝑛𝑑 𝜎 2 .
Note that
1
(1) 𝜇 + 2 𝜎 2 = ln(𝐸[𝑋])
2
(2) ln(𝑒 𝜎 − 1) + 2𝜇 + 𝜎 2 = ln(𝑠𝑋2 )
Solving for 𝜇 in (1)
1
𝜇 = ln(𝐸[𝑋]) − 𝜎 2
2
and substituting in (2)
:
𝐿
2
ln(𝑒 𝜎 − 1) = ln(𝑠𝑋2 ) − 2 ln(𝐸[𝑋])
So that,
𝑠𝑋2
𝜎 = ln [1 +
]
(𝐸[𝑋])2
2
and
1
𝑠𝑋2
𝜇 = ln(𝐸[𝑋]) − ln [1 +
]
(𝐸[𝑋])2
2
Having characterized the distribution of expenditures, we need to find the amount C (see figure 1) so
that the change in poverty is achieved.
The proportion in poverty at baseline is given by the blue and red shaded area to the left of the poverty
line (orange vertical line), the poverty at endline is given by the red shaded area.
Mathematically2,
𝐿
𝑃𝑜𝑣𝑡 = ∫
𝑑𝐹(𝑋)
min(𝑋)
And
𝐿
𝑃𝑜𝑣𝑡+1 = ∫
𝑑𝐹(𝑋)
C+min(𝑋)
Note that since the distribution is not changing, and the endline distribution is just a shifted log-normal
distribution of the previous one, we can write
𝐿−𝐶
𝑃𝑜𝑣𝑡+1 = ∫
𝑑𝐹(𝑋)
min(𝑋)
Performing the integration,
2
Note that 𝑃𝑜𝑣𝑡 does not involve C. However, this measure serves to gauge how well does the log-normal
distribution approximates the expenditure distribution. In the do file attached this is called the error, since is really
𝐿
𝑃𝑜𝑣𝑡 = ∫
𝑑𝐹̃ (𝑋) where 𝐹̃ is the empirical CDF.
min(𝑋)
𝑃𝑜𝑣𝑡+1 = 𝐹(𝐿 − 𝐶) − 𝐹(min(𝑋))
Where F is the cumulative distribution function of the log-normal, by assumption. Note that
𝐹(min(𝑋) = 0 by definition.
𝑃𝑜𝑣𝑡+1 = 𝐹(𝐿 − 𝐶) = Φ (
ln(𝐿 − 𝐶) − 𝜇
)
𝜎
Solving for C,
Φ−1 (𝑃𝑜𝑣𝑡+1 ) =
ln(𝐿 − 𝐶) − 𝜇
𝜎
𝐶 = 𝐿 − 𝑒 𝜇+𝜎Φ
−1 (𝑃𝑜𝑣
𝑡+1 )
Summary
The expected value of expenditures at follow up with a poverty line of $1.25 PPP is
𝐸[𝑋𝑡+1 ] = 𝐸[𝑋𝑡 + 𝐶] = 𝐸[𝑋𝑡 ] + 𝐶
Where
𝐶 = 1.25 − 𝑒 𝜇+𝜎Φ
𝜎 2 = ln [1 +
−1 (𝑃𝑜𝑣
𝑡+1 )
𝑠𝑋2
]
(𝐸[𝑋])2
1
𝑠𝑋2
𝜇 = ln(𝐸[𝑋]) − ln [1 +
]
(𝐸[𝑋])2
2
And
Φ−1 is the inverse of the CDF of a normal(0,1)
Final Notes on Excel Sheet
Given this solution, is straight forward to calculate these amounts in Excel. The attached excel file
(USAID-Tool-Excel-Calculation) computes these values for the given target at endline and for 8 values
below this in decreases of 2.5 percentage points.
In the example given3, the baseline poverty at the $1.25 PPP line is 40.07% and we want to decrease it
to 20%. The average per-capita per day is $1.72 (2005 PPP) with a variance of $1.24 (2005 PPP).
The mean per-capita per day expenditure we would observe at endline if poverty decreased to 20%
would be $2.09 (2005 PPP) (in cell C27). If instead we want poverty at endline to be 10%, the average
per-capita per day income observed would be $2.29 (2005 PPP).
The sheet also plots these values.
$ 2005 PPP
Avg. Expenditure and Change at Endline
$3.50
$3.00
$2.50
$2.00
$1.50
$1.00
$0.50
$-
Increment
Mean
Poverty at Endline
Notes on the Syntax
The dofile computes the above by simulating the expenditure distribution under the assumptions
described above.
It uses 100,000 observations and verifies that the assumptions are not violated. The results are place in
a ‘results’ folder. The file gives a more detailed calculation than the excel and computes the deviation
between the targeted decrease in poverty and the achieved under the simulation (Error).
The results are the following:
1. A summary text file named: Summary-Results.txt
/* *********************************************************************
Summary of Results
********************************************************************* */
3
Uses the values calculated from the Feed the Future Impact Evaluation Baseline Survey in Honduras (2012)
Baseline :
Mean PC per day expenditure is $1.7203297 PPP with variance 1.2445709
Poverty at baseline is .40068412 and target endline level is .2
The poverty line is S1.25 PPP
Simulation:
Mean PC per day expenditure is $1.717 PPP with variance 1.245
Poverty at baseline is .405 and target endline level is .202
The poverty line is S1.25 PPP
Results:
Increasing expenditures by $.37 PPP PC/Day
changes poverty by -.203
The target mean expenditure per capita per day is $2.09 PPP
The poverty under the 1.25 line goes from .405 to .202
Error : -.0023 (Negative -> Achieved reduction is greater under the
simulation)
The desired change is -.20068412 vs. the simulated change -.203
/* *********************************************************************
END
********************************************************************* */
2. Graph of the expenditure distributions before and after.
3. Graph of the log- expenditure distributions before and after.
4. Cumulative distribution before and after the change
5. A graph with the average mean expenditure for lower levels of poverty at baseline
6. A graph with the increment (C) for lower levels of poverty at baseline
Figure 1 Graphical Description of the Procedure
.4
.6
Distribution of Expenditures
.2
Density
𝑃𝑜𝑣𝑡
𝑃𝑜𝑣𝑡+1
0
C
0
2
4
X
Expenditures-Before
Poor at Baseline
Poverty Line $1.25 PPP
kernel = epanechnikov, bandwidth = 0.0772
Expenditures-After
Poor at Endline
Increment amount C
6