Quantity Theory of Money and Real Wage

In a classical model, increasing the money stock leads to an increase in the price level, however, real wages are not affected. Why is that? Well, the quantity theory of money holds in the long run due to the following reason. The long run aggregate supply curve (classical model) is; y = (k, N). We know that k (capital) is fixed and the labour market is in equilibrium. The long run aggregate supply curve solely depends on the labour market from the above mathematical equation.

When the central bank increases the nominal money stock as expansionary monetary policy, price levels go up in the short run. In the labour market wage will immediately clear the market. Explained:
Labour market:

MC=MB
W= P * MPn

As price increases, W has to increase by the same amount to maintain the equality of the equation and for the labour market to be in equilibrium. Wage will immediately clear the market and a new equation will form, with no output change sin the long run. If W & P increase equally, their quotient is constant. Thus, W/P = real wage – constant in the long run.

It’s not a complete and perfect answer, but it gives you a good understanding why real wages don’t rise.