The text below is an excerpt from my forthcoming book Finitary Probabilistic Methods in Econophysics, written with Ubaldo Garibaldi for CUP. It is a partial answer to the post of Jean-Phillippe Bouchaud (less poetic, however).
In Economics, distributional problems appear at least in the following frameworks
1. economic growth (and its dependence on firms’ sizes);
2. allocation of resources (in particular the distribution of wealth).
Our presentation emphasizes the positive (rather than the normative) aspect of distributional problems. However, there is a general policy problem worth mentioning which was highlighted by Federico Caffe’ in his lectures . When introducing the 1938 paper by A. Bergson , Caffe’ wrote “… when, in economic reasoning, the social wealth distribution is assumed “given”, this means that the existing distribution is accepted, without evaluating whether it is good or bad, acceptable or unacceptable … this must be explicitly done further clarifying that the conclusions are conditioned on the acceptabiliy of the distributional set-up”.
Two main probabilistic methods were used to derive/justify observed/empirical distributions:
1. the statistical equilibrium method (discussed in this book). According to this approach, the time evolution of an economic system is represented by an aperiodic, irreducible Markov chain and the distribution of relevant quantities is given by the invariant distribution of the Markov chain.
2. The diffusive (possibly non-equilibrium) method. According to this approach, the time evolution of an economic system is represented by a random walk (see also Sections 9.8 and 10.8).
The statistical equilibrium method was challenged from at least two points of view. Some scholars do not believe that economic systems can achieve any sort of equilibrium, including statistical equilibrium and, on the contrary, they claim that such systems are strongly out-of-equilibrium. Other scholars, and in particular, post-Keynesian economists, do not believe that the behaviour of macroeconomic systems can be explained in terms of the behaviour of interacting individuals. In other words, they challenge the microfoundation of macroeconomics, possibly including a probabilistic microfoundation in terms of statistical equilibrium. Our provisional reply to these objections is that the concept of statistical equilibrium may prove useful even in economics and may be helpful to describe/justify empirical distributional properties. In other words, before rejecting the usefulness of this concept, it is worth studying its implications and understand what happens even in simple models. As argued by John Angle (personal communication), it may well be the case that the economic dynamics is fast enough to let relevant variables reach statistical equilibrium even in the presence of shocks moving the economic system out of this equilibrium.
Another common objection is that in economics, at odds with physics, there is no conservation of wealth or of the number of workers or of any other relevant quantity. This is true, but the example discussed in Section 7.6 shows how to study a market where the number of participant is not fixed, but yet statistical equilibrium is reached. In other words, the absence of conserved quantities in itself is not a major obstacle.
The last objection we consider does not deny the usefulness of statistical equilibrium, but of a probabilistic dynamical description. When considering large macroeconomic aggregates or a long time evolution, fluctuations may become irrelevant and only the deterministic dynamics of expected values is important. In other words, stochastic processes may be replaced by difference equations or even by differential equations for empirical averages. To answer this objection, in many parts of this book we have pointed the attention of the reader to the phenomenon of lack of self averaging, which is often there in the presence of correlations (see Section 5.2.1). In other words, when correlations are there, it is not always possible to neglect fluctuations and the description of economic systems in terms of random variables and stochastic processes becomes necessary.
 F. Caffe’, Lezioni di politica economica, Bollati-Boringhieri, (1978).
 A. Bergson, A reformulation of certain aspects of welfare economics, Quarterly
Journal of Economics 52, 310-334, (1938).
For a more quantitative discussion on what I mean by “statistical equilibrium” and further references, you can consult the relevant page of the Reykjavik manifesto: