Stochastic macro-equilibrium: a microfoundation for the Keynesian economics

Okun (1963) found that a decline of the unemployment rate by one percent raises the growth rate of real GDP by three percent. The Okun coefficient three is much larger than the elasticity of output with respect to labor which is supposed to be equal to the labor share, and roughly one third. This finding demonstrates that there always exists significant under-employment of labor other than unemployment in the macroeconomy: see Okun (1973).

We must note that this result does not ensue only from the assumption that the firm’s individual demand curve is downward-sloping. If there is no kink in the perceived demand curve, an increase in sales at the unchanged price may be followed by an increase in price leaving output and labor employment little changed. With the kinked individual demand curve, the marginal revenue becomes discontinuous. Under this assumption, a shift in demand is all absorbed in changes in output and labor employment leaving the price unchanged.

To be precise, it is to be realized in the sense of expected value. In physics, variance is normally so small relative to expected value that we practically always observe the expected value.

In physics, \(\beta \) is normally positive. This difference arises because workers strive for job sites with higher productivity, not the other way round (Iyetomi 2012). In physics, \(\beta \) is equal to the inverse of temperature, or more precisely, temperature is defined as the inverse of \({\partial S}/{\partial D}\) when \(S\) is the entropy and \(D\) energy. Thus, negative \(\beta \) means the negative temperature. It may sound odd, but the notion of negative temperature is perfectly legitimate in such systems as the one in the present analysis; see Appendix E of Kittel and Kroemer (1980).

This method has been time and again successful in natural sciences when we analyze object comprising many micro elements. Economists might be still skeptical of the validity of the method in economics saying that inorganic atoms and molecules comprising gas are essentially different from optimizing economic agents. Every student of economics knows that behavior of dynamically optimizing economic agent such as the Ramsey consumer is described by the Euler equation for a problem of calculus of variation. On the surface, such a sophisticated economic behavior must look remote from “mechanical” movements of an inorganic particle which only satisfy the law of motion. However, every student of physics knows that the Newtonian law of motion is actually nothing but the Euler equation for a certain variational problem; particles minimize the energy or the Hamiltonian!. It is called the principle of least action: see Chapter 19 of Feynman (1964)’s Lectures on Physics, Vol. II. Therefore, behavior of dynamically optimizing economic agent and motions of inorganic particle are on a par to the extent that they both satisfy the Euler equation for respective variational problem. The method of statistical physics can be usefully applied not because motions of micro units are “mechanical,” but because object or system under investigation comprises many micro units individual movements of which we are unable to know.

When the number of potential jobs with high productivity is limited, behavior of economic agents necessarily becomes correlated; If good jobs are taken by some workers, it becomes more difficult for others to find such jobs. Galibaldi and Scalas (2010) suggest that we study the problem by Markov model with such constraints. The present analysis precisely does it by introducing ceilings on \(n_j \).

Like many others, Postel-Vinay and Robin (2002) assume that there is no ceiling for job-sites with high productivity. On this assumption as well as the assumption that the product market is perfectly competitive, they regard a decreasing number of workers with high productivity jobs as a consequence of less recruitment efforts made by high productivity firms than by low productivity firms. This is a strange interpretation. There is no reasonable reason why high productivity firms make less recruitment efforts than low productivity firms. A more plausible assumption is that firms are monopolistically competitive facing the downward-sloping individual demand curve rather than the price takers in the product market, and that jobs with high productivity are limited in number by such demand constraints. Suppose, for example, that the automobile industry is a high productivity industry. It would be unreasonable to argue that the level of employment in the industry is only the outcome of job matching, and that a limited size of employment is due to a lack of the firms’ recruitment efforts. Plainly, the size of car producers is given by their capacity which is, in turn, determined by the level of demand for cars.

This result is obtained by the method of Gibbs’ canonical ensemble. Gibbs established the statistical mechanics by introducing the concept of “canonical ensemble” which is a collection of macro states, \(Y\) in our present case. Suppose that there are \(K\) possible levels of \(Y\) denoted by \(Y_1,\ldots , Y_K\). For the moment, we reinterpret \(n_k \) as the number of cases where \(Y\) takes the value \(Y_k \) (\(k=1,\ldots ,K)\). The sum of \(n_k,\, N\) is given. Then, \(n_k \) satisfies Eq. (2). We assume that the average of \(Y\) is equal to constant \(D\): \(\sum _{k=1}^K {Y_k \left( {{n_k }/N} \right) } =D\). Replacing \(c_k \) by \(Y_k /N\), we observe that this equation is equivalent to Eq. (5). Thus, we can apply the exactly same entropy maximization as we did in the basic model in Sect. 3. It leads us to Eq. (15).

It is called the grand canonical partition function in physics.

\(\mu \) is called the chemical potential in physics, and measures the marginal contribution in terms of energy of an additional particle to the system under investigation.

To be precise, more workers are on the “job sites” with higher productivity because productivity may differ across job sites even within a single firm. The method I used in this paper should apply to “job sites” rather than firms, because in general, a firm is a cluster of job sites with different productivity levels.