Unequal inequalities. II

This paper analyzes properties of measures of inequality, applied to income inequalities but meaningful for practically any measure of dispersion in economics. We call n the number of persons, i the person's index, x_i person i's income, $\text{x}\text{= Σ(}\text{x}{}_{\mathrm{i}}\text{n}\text{)}$ the average income, x the vector of the x_i's or income distribution, I(x) a real-valued function of x which is the measure (or index) of inequality.Part I (Sects. I–V), which appeared in the last issue of this journal, analyzed several structures or properties, and specific forms, of I. We distinguished several I's: the measures of inequality per person (or “absolute”) I^a, per pound (or “relative”) $\text{I}{}^{\mathrm{r}}\text{=}\text{I}{}^{\mathrm{a}}\text{x}$, and total nI^a. We presented several possible properties of inequality measures, such as: I = 0 if all x_i's are equal (“zero at equality”), I > 0 otherwise (“positivity out of equality”), symmetry of I for x (“impartiality”), $\text{((}\text{?I}\text{?x}{}_{\mathrm{i}}\text{) ? (}\text{?I}\text{?x}{}_{\mathrm{i}}\text{))(x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{) > 0}$ for x_i ≠ x_j (“rectifiance” of the function I, or “transfers principle,” this being the strict form whereas the weak one is with sign ?), the fact that $\text{(}\text{?(}\text{x}\text{? I}{}^{\mathrm{a}}\text{)}\text{?}{}_{\mathrm{i}}\text{)}\text{(}\text{?(}\text{x}\text{? I}{}^{\mathrm{a}}\text{)}\text{?}{}_{\mathrm{j}}\text{)}$ does not depend upon x_k for k ≠ i,j (“welfare independence,” or, for short, “independence”). Rectifiance plus symmetry is Schur-convexity. Independence plus symmetry plus zero at equality implies that $\text{x}\text{≡}\text{x}\text{? I}{}^{\mathrm{a}}\text{= ?}{}^{\mathrm{?1}}\text{[(}\text{1}\text{n}\text{) Σ ?(x}{}_{\mathrm{i}}\text{)]}$ where $\text{x}$ is the “equal equivalent income”; and we will show that, these three properties being satisfied, the following ones are equivalent to each other: positivity out of equality, rectifiance, quasi-convexity, ?'s concavity.Part I largely focused on the study of six related specific measures of inequality, which in particular possess all the above properties: ?, α, and Ξ being positive parameters, they are $\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{=}\text{x}\text{+ξ ? [(}\text{1}\text{n}\text{∑ (x}{}_{\mathrm{i}}\text{+ ξ)}{}^{\mathrm{1?epsi;}}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>1</mtext><mtext>??</mtext>}}$, $\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{=}\text{x}\text{+ξ ? ∏ (x}{}_{\mathrm{i}}\text{+ ξ)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$, $\text{I}{}_{\mathrm{c}}{}^{\mathrm{r}}\text{=}\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{x}$, I_r=I_c^r for ξ=O, $\text{I}{}_{\mathrm{r}}{}^{\mathrm{a}}\text{=}\text{x}\text{I}{}_{\mathrm{r}}\text{=I}{}_{\mathrm{c}}{}^{\mathrm{a}}$ for ξ=, $\text{I}{}_{\mathrm{l}}\text{=(}\text{1}\text{α}\text{)}\text{log}\text{[(}\text{1}\text{n}\text{) ∑ eα·(x?x}{}_{\mathrm{i}}\text{)]}$ and $\text{I}{}_{\mathrm{l}}{}^{\mathrm{r}}\text{=}\text{I}{}_{\mathrm{l}}\text{x}$. Lower indices c, r, l respectively stand for “centrist,” “rightist,” and “leftist” measures of inequality. I_r and I_l are invariant under respectively equiproportional variation in, or equal addition to, all incomes; measures which have the first of these two properties are said to be “intensive.”We now consider different and more general measures, and other properties. We first reconcile the last two properties by dropping the “indepencence” one (Section VI.). Then, we analyze another mildly equalitarian property, the “principle of diminishing transfers” (Section VII). Section VIII turns to the relations between inequality measures and Lorenz and concentration curves. We then consider the effect on inequality of additions of incomes, and we analyze the properties of “diminishing equality” (Section IX). The effect of unions of populations is the topic of Section X. Finally, the last section (XI) presents the more general relations between the various structural properties of inequality measures.¹     

When is a tax neutral?

A proportional income tax is said to be neutral if variations in the tax rate do not affect the optimal composition of the consumption commodities of a utility maximizing consumer. It is shown that neutrality obtains if and only if the indirect utility function of the consumer is groupwise homothetic. If the income tax rates on labor and property incomes are allowed to differ, then neutrality obtains if and only if the indirect utility function of the consumer is homothetically separable. If, in addition, it is required that the proportional income tax be neutral with respect to not only consumption commodities but also to leisure, then 'the indirect utility function must have the form: $\text{V = V(f(w)+H}{}^1\text{(p))}$, where w and p are the normalized prices of leisure and consumption respectively, and H¹(p) is homogeneous of degree one in p.     

Sufficient conditions in free final time optimal control problems. A comment

By an application of sufficient conditions, assume that an optimal pair (χ_τ(·), u_τ(·)) and an adjoint function p_τ(·) were found in the control problem in question with the final time τ fixed but arbitrary. Then a sufficient condition for one of these pairs, say , to be optimal in the corresponding free final time problem is that the Hamiltonian, with (χ_τ(t), u_τ(τ?), p_τ(τ), τ) inserted, is nonnegative (nonpositive) to the left (right) of $\text{τ}{}^1$.     

A new test for normality

Burr (1942) type XII distribution $\text{?(u)=kc u}{}^{\mathrm{c?1}}\text{(1+u}{}^{\mathrm{c}}\text{)}{}^{\mathrm{-(k+1)}}\text{u?0, k > 0, c > 0}$ is considered. Particular values of k and c give β₁ ? 0 and β₂ ? 3. Using this fact tests for normality of observations and regression disturbances are constructed.u.1. Introduction     

A note on the generalised measures of risk aversion

This paper is concerned with generalised scalar measures of risk aversion. A measure R which may meaningfully be applied to both unidimensional risks (risk in income or wealth) and multidimensional risks has been constructed. In case of identical preferences, we have also constructed an alternative measure of risk aversion $\text{R}{}^1$ which is shown to be related to the Khilstrom-Mirman measure. This relationship explains the nature of the Khilstrom-Mirman measure.     

Optimal control problems from second-order difference equations

We examine solutions x?l_∞(Rⁿ) of the equations $\text{F}{}_{\mathrm{t}}\text{(}\text{x}{}_{\mathrm{t?1}}\text{,}\text{x}{}_{\mathrm{t}}\text{,}\text{x}{}_{\mathrm{t+1}}\text{) = 0}$ and derive conditions for the existence of objective functions G_t so that $\text{x}$ solves max_x?l∞ΣG_t(x_t, x_t+1). We specialize the conditions to time autonomous equations and apply them to a competitive industry in temporary equilibrium. The objective function in the example is of the cost-benefit form: consumer surplus minus opportunity cost of labor minus industry-wide cost of investment.     

Disequilibrium theory,waiting costs,and saving behavior in centrally planned economies: A queueing-theoretic approach

An expected-utility maximizer, obliged to make a single purchase from two alternatives, finds that a queue forms for the good that would have been selected had both goods been available. An M/M/1 queue is posited and for each period of delay the maximum additional payment $\text{b}{}^1$ that the consumer would incur to join the queue is determined. This has implications for saving. Not only the availability of the good, but also the delay in obtaining it, the alteration in its arrival rate, and the “stampede effect” determine saving behavior.     

A new approach to the incorporation of attributes into consumer theory

A framework is developed which can be used to analyze a large class of consumption situations which cannot be handled by Lancaster's model. To achieve this end, a certain “expansion” of attribute space is performed. Subsequently, we formally define the notions of mixability and effective number of goods$\text{n}$. This new model degenerates into Lancaster's model for the special case when $\text{n}\text{= 1}$.     

Extreme value statistics and recurrence intervals of NYMEX energy futures volatility

Energy markets and the associated energy futures markets play a crucial role in global economies. It is of great theoretical and practical significance to gain a deeper understanding of extreme value statistics of the volatility of energy futures traded on the New York Mercantile Exchange (NYMEX). We investigate the statistical properties of the recurrence intervals of daily volatility time series of four NYMEX energy futures, which are defined as the waiting times τ between consecutive volatilities exceeding a given threshold q. We find that the recurrence intervals are distributed as a stretched exponential $P_q\left(\tau \right)\sim e^{{\left(\mathit{a\tau }\right)}^{-\gamma }}$, where the exponent γ decreases with increasing q, and there is no scaling behavior in the distributions for different thresholds q after the recurrence intervals are scaled with the mean recurrence interval $\overline{\tau }$. These findings are significant under the Kolmogorov–Smirnov test and the Cramér–von Mises test. We show that the empirical estimations are in nice agreement with the numerical integration results for the occurrence probability W_q(Δt|t) of a next event above the threshold q within a (short) time interval after an elapsed time t from the last event above q. We also investigate the memory effects of the recurrence intervals. It is found that the conditional distributions of large and small recurrence intervals differ from each other and the conditional mean of the recurrence intervals scale as a power law of the preceding interval $\overline{\tau }\left({\tau }_0\right)/\overline{\tau }\sim {\left({\tau }_0/\overline{\tau }\right)}^{\beta }$, indicating that the recurrence intervals have short-term correlations. Detrended fluctuation analysis and detrending moving average analysis further uncover that the recurrence intervals possess long-term correlations. We confirm that the “clustering” of the volatility recurrence intervals is caused by the long-term correlations well known to be present in the volatility. Our findings shed new lights on the behavior of large volatilities and have potential implications in risk management of energy futures.     

Partial identification in binary response models with nonignorable nonresponses

This study investigates the identification of parameters in semiparametric binary response models of the form y=1(x^′β+v+ε>0)$y=1(x^{\prime }\beta +v+\epsilon >0)$ when there are nonignorable nonresponses. We propose an estimation procedure for the identified set, the set of parameters that are observationally indistinguishable from the true value β$\beta$, based on the special regressor approach of Lewbel (2000). We show that the estimator for the identified set is consistent in the Hausdorff metric.     

Dual interpersonal comparisons of utility and the welfare economics of income distribution

Interpersonal comparisons can be of utility levels and/or of utility differences. Comparisons of levels can be used to define equity in distributing income. Comparisons of differences can be used to construct an additive Bergson social welfare function over income distributions. When both utility levels and utility differences are compared, one can require the constructed additive Bergson social welfare function to indicate a preference for more equitable income distributions. This restricts the form of both the individual utility functions and the optimal distribution of income. The form of these restrictions depends on whether the levels and differences of the same utility functions are being compared.     

The Debreu private ownership economy with complementary commodities and prices

Let E_p be a Debreu private ownership economy in which there are some complementary commodities. It means that all commodity bundles are contained in the proper subspace V   of commodity–space [R]l(l∈{1,2,…})${\mathfrak{[R]}}^l\left(l\in \left\{1,2,\dots \right\}\right)$. The production plans maximizing the producers' profits do not have to satisfy the dependency in the qualities of commodities seen in the consumers' plans. It may cause no-existence an equilibrium in economy E_p. The competitive leads, however, to adjustment the production plans to improve the consumers' satisfaction. The procedure of change the production system covering the consumers' requirements is presented. As a result, the model of the private ownership economy with complementary commodities and prices, being the simplification of the initial model, is elaborated. Consequently, the necessary condition for the existence of an equilibrium in the economy E_p is proved.