Ineffective policy choices in non-self-averaging models
The non-self-averaging phenomenon in two blocks of economies
Next, we use a purely analytical method of cumulant generating function to show the existence of non-self-averaging in one block of economies, by positing a master equation for dynamic growth of two blocks of economies with size nif i = 1,2, and then by first converting it into the probability generation function and second into a set of equations for the first two cumulants. See Aoki (2002, s. 7.1) on master equations and cumulant equations, for example.
We solve the set of cumulant equations to obtain analytic expressions for the means and the elements of the covariance matrix. Finally, we calculate the coefficients of variations for the means of n1 and n2, and show that one block can be non-self-averaging while the other is self-averaging when a factor of production (labor or capital) n2 is growing faster than n1 and is being exported from country 2 to country 1. We then show that economy 1 is self-averaging but that economy 2 is non-self-averaging.
A limited amount of simulations bear out this theoretical analysis (Ono, 2007). The question of convergence has been addressed in a number of papers in the economic literature. Some give affirmative answers while the others give negative or qualified negative answers; see for example Phillips and Sul (2003) and Mathunjwa and Temple (2006).
We also use a purely analytical framework by positing a master equation for dynamic growth of two blocks of economies with size, nt ,i = 1,2, and then converting it into a set of equations for the first two cumulants. Then the cumulant equations are solved and we calculate the coefficients of variations for the means of n1 and n2. We show that the coefficient of variation for block 2, which is the exporting block, diverges, while the coefficient of variation for the importing block is self-averaging with nearly zero cross-correlation coefficient between the two blocks.