My Opinion: I think an increase in unemployment leads to a decrease in savings. Similarly, an increase in savings may lead to unemployment mostly in the short-run. This is because saving is a withdrawal of income from the economy and so may reduce aggregate demand thereby lowering incentive to produce.
However, savings are likely to increase investments in the long-run thereby reducing unemployment. Refer to to the "Feldstein-Horioka Puzzle"
Consider also reading these papers
Hahn, L. A. (1946). The Effects of Saving on Employment and Consumption. Journal of Marketing, 11(1), 35–43. https://doi.org/10.2307/1246804
Berloffa, G., & Simmons, P. (2003). Unemployment Risk, Labour Force Participation and Savings. The Review of Economic Studies, 70(3), 521–539. http://www.jstor.org/stable/3648599
There are two major theories that seek to explain the phenomenon of unemployment. For the first of them, of neoclassical inspiration, unemployment arises due to wage rigidity, caused by the behavior of agents (unions, companies, government) in the labor market. The second strand, derived from Keynes' General Theory, demonstrates that the level of employment fundamentally depends on the aggregate demand of the economy, which, in turn, derives from two variables: the propensity of the community to consume, which determines the level of aggregate consumption; and the current volume of investments.
The neoclassical conception that supply creates its own demand implied the premise that the savings generated by individuals, corresponding to the unconsumed part of their income, will necessarily result, in the same period, in an equivalent volume of capital investments. However, Keynes warned that, in the real world, the individuals who save are not necessarily the same individuals who make the decisions to invest in new machinery, equipment, or to establish new businesses. The amount of current investment depends on what Keynes called the incentive to invest, which in turn is linked to the relationship between entrepreneurs' expectations of future market behavior and the level of interest rates.
Then, given the marginal propensity to consume, entrepreneurs will increase or decrease employment, through the level of investment. The investment incentive depends on the relationship between the marginal efficiency of capital and loan interest rates. Thus, there can only be one level of employment compatible with the equilibrium: Income = Consumption + Investment. Otherwise, global demand and global supply would be different. This level of employment may be below, but not above, full employment. This fact attests to the mutual relationship between the marginal propensity to consume and the incentives to invest. In view of this aspect, the level of employment is determined by the marginal propensity to consume plus investment, which also determines the level of real wages. If the marginal propensity to consume and the amount of investment result in insufficient effective demand, the volume of employment falls until it is below the supply of labor willing to work at the prevailing real wage. This leads to involuntary unemployment.
After a long search for a theory that accurately explains the relationship between unemployment and bank deposits, I found that no specific theoretical framework has been developed that can be used to predict the nature of the effects of employment and then unemployment for financial development. In addition, the empirical studies that focused on examining the impact of unemployment on bank deposits are almost non-existent. And because deposits are one of the indicators for measuring financial development, the relationship can be extracted through that. In this regard, the empirical studies concluded that there are three outcomes. The first and the most common trend is where financial development has been found to have a negative impact on unemployment, implying that as the financial sector gets more and more developed, unemployment trends downwards. The second, but less common, trend is where the development of the financial sector is found to worsen unemployment. Then, there is a third trend which confirms the neutrality effect of financial development on unemployment.
It is also possible to rely on the theory of the life cycle, the stage of inactivity.