A puzzle about the monetary expression of labor time : an equilibrating mechanism or just a coincidence?

In this paper, we report a puzzling result about the monetary expressions of labor time (MELTs) of the productive and unproductive sectors. Since part of the aggregate value produced in productive sectors is transferred to unproductive sectors, the productive sector's MELT is a measure of value realized in productive sectors while the unproductive sector's MELT is a measure of value transferred to unproductive sectors. Using the national income data for the U.S. economy during 1987-2016 and for the Korean economy during 1993-2016, it is found that the MELT of the aggregate productive sector and the MELT of the aggregate unproductive sector have been moving in a very close lockstep in both countries during the entire sample periods. We build a model which explicitly formalizes the unproductive sector as not producing any value but making the value production process efficient, and find that the co-movement of the two MELTs is not an optimal condition. We also suggest some ex post implications of it, including what the puzzling result implies on the relation between unproductive sector and capital accumulation.