Bergson-Samuelson welfare function

The Bergson-Samuelson welfare function (also Bergson welfare function ) describes a welfare function in which the social evaluation of a condition depends solely on the evaluations of the affected individuals and not on other factors. One speaks synonymously of an individualistic social welfare function .

presentation

The Bergson-Samuelson welfare function is basically based on individual rankings. All individuals in a society have individual preference orders in the model framework - they can consistently say for any two alternatives (for example goods or options for action) whether they prefer one to the other or whether they consider both to be equal. (See in detail the article Order of preference. ) The orders of preference, in turn, can be represented by utility functions with appropriate construction . This is a real-valued function that assigns a numerical value to each alternative, following the principle that an alternative that is evaluated better than another also receives a higher numerical value and two equally evaluated alternatives receive the same numerical value - utility functions give one Order of preference, therefore, just a simple mathematical representation.

In addition to these individual orders of preference, however, there should also be a social ranking that expresses the assessment of conditions on a social level. This overall social ranking is also represented by a function, namely by a social welfare function (SWF) - it is the social counterpart to the individual utility function.

The fundamental idea of ​​the Bergson-Samuelson welfare function is to construct this social hierarchy from the individual preference orders of the members of society - and only from this. According to this it is inadmissible to base a social evaluation on arguments other than individual evaluations. Formally:

Definition (Bergson 1938): Let the utility function of an individual i from a society with n members . Here an allocation with the equipment of person i. Then a welfare function is called a Bergson-Samuelson welfare function . ${\displaystyle u_{i}(\mathbf {x} )}$ ${\displaystyle 1\leq i\leq n}$ ${\displaystyle \mathbf {x} =(\mathbf {x} _{1},\ldots ,\mathbf {x} _{n})}$ ${\displaystyle \mathbf {x} _{i}}$ ${\displaystyle W(\mathbf {x} ):=F\left[u_{1}(\mathbf {x} ),\ldots ,u_{n}(\mathbf {x} )\right]}$

The utility value for society as a whole according to a Bergson-Samuelson welfare function always results as a function of the individual utility functions of all members of society. A non-individualistic social welfare function, on the other hand, could contain further arguments in addition to the individual benefit, for example external value judgments by the government or by scientists. Note that this does not make a statement as to whether the individual benefits of all members of society actually flow into social welfare: A welfare function that is identical to the individual benefit of a single “dictator” also belongs to the class of Bergson-Samuelson welfare functions.

General isoelastic SWF

The isoelastic welfare function represents an important class of Bergson-Samuelson welfare functions. It follows the form

${\displaystyle W(\mathbf {x} )={\frac {1}{1-\rho }}\sum _{i=1}^{n}\alpha _{i}\left[u_{i}(\mathbf {x} )\right]^{1-\rho }}$with , and ${\displaystyle \alpha _{i}>0\;\forall i}$ ${\displaystyle 1\leq i\leq n}$ ${\displaystyle \rho \geq 0,\rho \neq 1}$

and has important special cases:

• So follows for the so-called general utilitarian SWF ${\displaystyle \rho =0}$

${\displaystyle W^{AU}(\mathbf {x} )=\sum _{i=1}^{n}\alpha _{i}\cdot u_{i}(\mathbf {x} )}$,

those for the special utilitarian SWF (often just for short: utilitarian SWF ) ${\displaystyle \alpha _{i}=1\;\forall i,1\leq i\leq n}$

${\displaystyle W^{SU}(\mathbf {x} )=\sum _{i=1}^{n}u_{i}(\mathbf {x} )}$

transforms. While the special utilitarian SWF directly sums up the individual benefits across all members of society, the general utilitarian SWF weights them beforehand with a - person-specific - factor.

• For and follows the Maximin SWF in the limit value (also Rawls' welfare function according to Rawls 1971) ${\displaystyle \rho \rightarrow \infty }$ ${\displaystyle \alpha _{i}=1\;\forall i,1\leq i\leq n}$

${\displaystyle W^{M}(\mathbf {x} )=\min \left\{u_{1}(\mathbf {x} ),\ldots ,u_{n}(\mathbf {x} )\right\}}$.

According to this, social welfare is only measured according to the level of the individual welfare of the worst off.

• For the limit value follows the Nash SWF ${\displaystyle \rho \rightarrow 1}$

${\displaystyle W^{N}(\mathbf {x} )=\prod _{i=1}^{n}\left[u_{i}(\mathbf {x} )\right]^{\alpha _{i}}}$.

literature

• Abram Bergson: A reformulation of certain aspects of welfare economics. In: The Quarterly Journal of Economics. 52, No. 2, 1938, pp. 310-334, doi : 10.2307 / 1881737
• Friedrich Breyer and Martin Kolmar : Basics of economic policy. 3rd edition Mohr Siebeck, Tübingen 2010, ISBN 978-3-16-150193-7 .
• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 .

Web links

• “Social welfare functions” on mikrooekonomie.de

Individual evidence

1. ↑ See Mas-Colell / Whinston / Green 1995, p. 117.
2. ↑ See Breyer / Kolmar 2010, p. 66 f .; Mas-Colell / Whinston / Green 1995, p. 829.
3. ↑ See for example Jehle / Reny 2011, p. 284.
4. ↑ See Beyer / Kolmar 2010, p. 68; here contrary to Mas-Colell / Whinston / Green 1995, p. 827 f., which understand as general-utilitarian SWF (generalized utilitarian) and as pure utilitarian SWF (purely utilitarian). ${\displaystyle W(\mathbf {x} )=\sum _{i=1}^{n}g_{i}\left[u_{i}(\mathbf {x} )\right]}$ ${\displaystyle W(\mathbf {x} )=\sum _{i=1}^{n}\alpha _{i}u_{i}(\mathbf {x} )}$
5. ↑ See Breyer / Kolmar 2010, p. 68; Jehle / Reny 2011, p. 287.
6. ↑ John Rawls : A Theory of Justice . Belknap Press, Cambridge 1971.
7. ↑ See Breyer / Kolmar 2010, p. 72; Mas-Colell / Whinson / Green 1995, p. 827 and 829
8. ↑ See Breyer / Kolmar 2010, p. 73.