A two-sector model of fluctuations

This model is a simplified two-sector version of a more general model of an economy composed of K -sectors (agents), which is similar to the one described more fully in Section 8.6.

A more completely analyzed model in Section 8.6 has patterns of output fluctuations and growth that are similar to those of models of more complex economic structures. This full version of the model has three novel features. First, it uses a stochastic adaptive scheme that is different from those in many adaptive models in general and in agent-based computational economic models in particular. Second, the notion of holding, or sojourn, times of continuous­time Markov chains is used to select the next sector (agent) who acts. Thirdly, it introduces new types or sectors in a way similar to that of changing the number of types in the Ewens sampling formula in the population-genetics literature. See Ewens (1990) or Section A.7 of the Appendix on the Ewens distribution. New sectors or type are created and sectors disappear randomly and in a way correlated with excess demands. Sectors with positive excess demands create new ones randomly as new branches. This feature is in line with the findings by Schmookler (1966). He found evidence in the U.S. patent data that invention and technical progress are strongly influenced by demand conditions. This last feature is absent in this simplified version.

Inthis simplified two-sector version, we examine the stationary (equilibrium) distribution of the sizes of the two sectors. Resources are stochastically allo­cated to the two sectors in response to excess demands or supplies of the sectors. The output of the model randomly fluctuates. More importantly, this model demonstrates what we believe to be a generic property that the level of aggre­gate activities is influenced stochastically by the patterns of demands.

Sector i has productivity coefficient c_i, which is exogenously given and fixed. In our simplified version, c₁ = c₂ is assumed. Sector i employs N_i units of a factor of production. It is a nonnegative integer-valued random variable. We call its value the size of the sector. When N_i (t) = n_i, i = 1, 2,..., K, the output of sector i is c_in_i, and the total output (GDP) of this economy is

i = 1, 2.

In a closed model n is fixed and Y remains constant, even though n1 and n2 = n - n 1 will fluctuate. In an open model n 1 and n are random numbers. We do not discuss the dynamics of the model here. However, it is convenient to introduce the notion of holding-time here, which is important in develop­ing the full version of the model in Section 8.6. We assume that the time it takes for sector i to adjust its size by one unit, up or down, T_i, is exponentially distributed,

4.2