Abstract
The purpose of this chapter is to link inequality to the production sphere of the economy, in order to better understand the connection between technical efficiency and the problem of inequality.
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Notes
- 1.
- 2.
Let E = 0 in case nothing is produced, and E = 1 when we obtain maximum efficiency, i.e. when social production is maximized under technological and total income disposability constraints.
- 3.
The choice of considering time as a discrete variable is sometimes questioned by arguing that finite difference equations can generate spurious chaotic dynamics which are not present as solutions of differential equations obtained from finite difference equations when the time interval tends to zero. For a conspicuous example, let’s remember the logistic, or quadratic map. In continuous time, the solutions of the logistic equation are always a function of time, with the shape of a capital S, while in discrete time there are values of the parameter for which the solutions are completely chaotic.
- 4.
In the numerical applications in Part III, it is more practical to introduce a uniform income taxation scheme, managed by the P.A., applied to the incomes of economic agents who do not belong to the poorest category of society, and to redistribute the proceeds to the poor. Of course, the results obtained are the same.
- 5.
The term “ad personam” is here introduced to distinguish a negative v j from the income tax on y j , introduced in Chap. 5.
- 6.
For instance, given the positive parameter, κ j , function \({f}_{j}({x}_{j}) = 1/{\kappa }_{j}\log (1 + {\kappa }_{j}{x}_{j})\) verifies all the stated assumptions. Another function verifying the stated properties is \({f}_{j}({x}_{j}) = 2/{\kappa }_{j}(\sqrt{1 + {\kappa }_{j } {x}_{j}} - 1)\). These functions overburden the numerical calculations; so in Chaps. 8 and 9 we shall work with the standard function \({f}_{j}({x}_{j}) = {\kappa }_{j}{x}_{j}\), of course with \(0 < {\kappa }_{j} < 1\), frequently found in Keynesian macroeconomics.
- 7.
This condition implies that, in an optimal state, all after transfer incomes, s j s, are positive.
- 8.
Presently, it is not necessary to assign the initial total income value at the disposal of the economy. We shall see in the next section, and in Chap. 5, that an initial income value, \(\bar{Y }\), must be assigned when maximizing social welfare.
- 9.
All weights are equal to 1, because there are no real prices.
- 10.
- 11.
How individual property rights are obtained is outside the economic sphere here considered. Thus, let’s assume that whatever the starting point of initial income distribution (depending, among other things, on the hereditary laws ruling the economy) it is preserved throughout all the time periods.
- 12.
Let’s remember that the v j (t)s can be either negative or positive.
- 13.
On problems of equity, ability-to-pay taxes, equal sacrifice, …it is interesting to consider Musgrave and Musgrave (1984, Part 3).
- 14.
Of course, q is a tax rate. In Sect. A.9 of the Appendix two marginal tax rates are considered.
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Nicola, P. (2013). Inequality, Efficiency, and the Production Sphere. In: Efficiency and Equity in Welfare Economics. Lecture Notes in Economics and Mathematical Systems, vol 661. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30071-4_4
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