Price Changes in Factor and Product Markets I think something is wrong in my reasoning, and I can't really find out what it is

Lets say the supply of labor decreases in a market. That would mean that the wage rate would increase in that market. Firms hiring out of that market will experience an increase in their marginal cost, and the supply in the market for the product which uses that labor will shift left, increasing the price of that product. But that would mean that the marginal revenue product of labor would increase because of that increased price and shift the demand for labor right. At that point the wage rate would increase and everything would cycle through again.

Where am I going wrong? How does the wage rate stabilize?
Let's


distinguish exogeneous from endogenous factors, 
distinguish partial versus general equilibrium analysis, 
consider here the labor market of one product, i.e., a partial analysis,
consider that this labor market is perfectly competitive.


In your reasoning, the decrease in labor supply, which typically leads to an increase in the wage rate, would come from an exogenous shock. This shock causes an endogenous process that stabilizes the wage rate at a new partial equilibrium. This is a situation where the product of marginal labor income equals the wage rate. 

In a competitive labor market, the wage has two properties:


It adjusts to make supply and demand for labor equal.
It equals the value of the marginal product of labor.


Suppose the labor supply decreases in our market. The supply of labor shifts to the left. At the initial wage, say $w_1$, the quantity of labor supplied falls behind the quantity of labor demanded. The wage increases from $w_1$ to $w_2$, and the firm reduces its demand for labor. As the firm lays off workers, the marginal product of a worker increases, so the value of marginal product increases. In the new (partial) equilibrium, the wage is equal to the value of the marginal product of labor and both are higher than before. 

Regarding your comment and the stability of this new equilibrium


  I think where I am confused is how does the exogenous shock to the labor market cause the wage rate to stabilize? Because the firm will now get higher prices in the product market, shouldn't the demand for labor increase and wouldn't the wage rate go up again and the process would repeat again and keep on cycling upwards?


The above equilibrium is a stable partial equilibrium, but general equilibrium forces exist. As a result of the shock, the firm will initially get higher prices in the product market, but we can assume that if it is a competitive market, demand for its product will decrease. This will lower the prices of its products. Demand for labour will shift to the left ... wages will go down (not up). However, we will need a general equilibrium model to better understand these changes.