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Mathematical Formulation of Poverty Index 294 4. Analysis of data Using the values tabulated in Table 1, the poverty index can be calculated from the following equation, derived from Eq.6. This is given as, PI = {k1 k2 (f/e)} (10) HHS (2007) gives poverty guidelines. These are given in Table 2 below. Table 2: 2007 HHS Poverty Guidelines Persons 48 Contiguous In family or household States and D.C. Alaska Hawaii 1 $10,210 $12,770 $11,750 2 $13,690 $17,120 $15,750 3 $17,170 $21,470 $15,750 4 $20,650 $25,820 $23,750 5 $24,130 $30,170 $27,750 6 $27,610 $34,520 $31,750 7 $31,090 $38,870 $35,750 8 $34,570 $43,220 $39,750 The e values are obtained from Table 1 for each of the states in USA. That includes 48 contiguous States and D.C. and the states of Alaska and Hawaii. The k1 values can be obtained from Table 2 for the corresponding state. There exists no specific data for the median family size like the median family income tabulated in Table 1. The indirectly derived median family size as per the existing data existing in literature is found to be 4.5. This implies that 50% population has family size more than 4.5 and 50% less. As this number seems to be high, a median family size is taken as 2 (k2 f=2) for calculation of poverty index. K f represents the modified value of actual f using the constant of proportionality. 5. Application of Optimization principles to poverty index To get an optimal value of poverty index, one can apply the principles of NLPP (Nonlinear Programming problem) to Eq. 8 which is a non-linear equation. A Non-linear Programming problem (NLPP) can be solved by successive linearization of the linear Programming problem (LPP). There are several techniques available for solving LPP problems for standard and non-standard LPP (Hillier and Liebermann, 2000; Putcha, 2007). Alternatively, one can use the principles of Lagrange Multiplier (Revelle et al., 2004) to get an initial optimal value of Poverty index. Then, the value of the poverty index can be refined by standard techniques of NLPP. Lagrange formulation: For this the optimization problem can be formulated as follows: Minimize p = k1 k2 f/e (10) s.t. f = 13690 (12) The numbers 2 and 8 have been picked based on HSS(2007). The corresponding Lagrange function L can be written as, L = p + λ1 (f-8) + λ2 (e-13690) (13) The solution of the above optimization problem can be found by the solution of simultaneous equations obtained by taking partial derivatives of EQ. 13 with respect to p, e, λ1 and λ2. This will give optimal values of poverty index as well as optimal values of the family income (e) and the family size for the minimum poverty index.
295 Chandra Putcha 6. Results The poverty indices are tabulated in Table 3 below for various states. Table 3: Poverty indices for various states in USA State Poverty index Alabama 0.55 Alaska 0.49 Arizona 0.49 Arkansas 0.60 California 0.42 Colorado 0.42 Connecticut 0.35 Delaware 0.43 District of Columbia 0.44 Florida 0.50 Georgia 0.48 Hawaii 0.44 Idaho 0.53 Illinois 0.43 Indiana 0.49 Iowa 0.49 Kansas 0.48 Kentucky 0.56 Louisiana 0.56 Maine 0.51 Maryland 0.35 Massachusetts 0.36 Michigan 0.47 Minnesota 0.40 Mississippi 0.62 Missouri 0.52 Montana 0.53 Nebraska 0.48 Nevada 0.44 New Hampshire 0.38 New Mexico 0.56 New York 0.44 North Carolina 0.52 North Dakota 0.49 Ohio 0.48 Oklahoma 0.57 Oregon 0.48 Pennsylvania 0.47 Rhode Island 0.42 South Carolina 0.50 South Dakota 0.54 Tennessee 0.54 Texas 0.52 Utah 0.47 Vermont 0.47 Virginia 0.40 Washington 0.42 West Virginia 0.62 Wisconsin 0.45 Wyoming 0.47 Puerto Rico 1.34
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