Abstract
This chapter treats the various computation methods offered by the literature to estimate the alternative measures of well-being outlined in Chap. 2, including the Vartia’s, Taylor’s and Breslaw’s approximations and the ordinary differential equations method. It also proposes alternative estimation methods and studies the relation between these methods and the underlying demand systems.
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Notes
- 1.
See also Dumagan and Mount (1991) for the general form of preferences and a third order of approximation. Note that, for the rest of the book, higher orders refer to the case of homothetic functions (i.e. \(\frac {\partial x_i}{\partial m} = \alpha _i\)).
- 2.
Note that McKenzie and Pearce (1976) derive the formulae with changes in nominal income, while Hicks considers only the change in prices. For the second term, we have that \(\frac {\partial ^2 e(p,u)}{\partial p_k \partial p_j} = \frac {\partial h_k (p,u)}{\partial p_j}\). Also, note that the second term is based on the Slutsky equation, so that \(\frac {\partial h_k (p,u)}{\partial p_j}=\frac {\partial x_k(p,u)}{\partial p_j} + x_k\frac {\partial x_j}{\partial m}\).
- 3.
- 4.
When initial prices are equal to 1, a marginal increase in income by one unit will change the bought quantities. However, the sum of changes in quantities is equal to 1.
- 5.
We have that:
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\(S_{CV} =\Delta x_1^S+\Delta x_2^S\) and C = m(1 + Δp 1)α
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\(\Delta x_1^S = (\alpha C/(1+\Delta p_1))-x_1^a\) and \(\Delta x_2^S = (1-\alpha )C/(1)-x_2^a\)
Also, we have that:
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\(S_{EV} =\Delta x_1^S+\Delta x_2^S\) and E = m∕((1 + Δp 1)α)
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\(\Delta x_1^S = (\alpha E/(1))-x_1^a\) and \(\Delta x_2^S = ((1-\alpha ) E/(1))-x_2^a\)
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- 6.
See Bacon (1995) for a simple spreadsheet approach to the use of the Vartia’s method.
- 7.
Breslaw and Smith (1995) claim that their algorithm converges faster than the Vartia’s algorithm, while the work done by Sun and Xie (2013) shows the superiority of the latter. We tested the performance of the two algorithms both programmed in Stata, and we found the Breslaw and Smith algorithm to be faster.
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Araar, A., Verme, P. (2019). Theory and Computation. In: Prices and Welfare. Palgrave Pivot, Cham. https://doi.org/10.1007/978-3-030-17423-1_3
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DOI: https://doi.org/10.1007/978-3-030-17423-1_3
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