My text Economics by Hall, Lieberman, gives a model of $\text{AE}$ (aggregate expenditure) in the short term as a function of $\text{GDP}$, $\text{MPC}$ (marginal propensity to consume), and autonomous expenditure $\text{AE}_0$.

$$\text{AE} = \text{GDP} \cdot \text{MPC} + \text{AE}_0$$

Equilibrium is assumed to occur where aggregate expenditure equals GDP. The solution is $\text{GDP}^* = \frac {\text{AE}_0} {1-\text{MPC}}$. So far there has been no reference to the employment level. The text assumes GDP is an increasing function of employment level (this function is called the aggregate production function), so this short term equilibrium $\text{GDP}^*$ is actually associated with some short term employment level equilibrium $E^*$.

However, there is an alternate concept of employment level equilibrium $E^{**}$ (full employment) which the level at which the labor market clears (which is obtained by the intersection of supply and demand curves of the labor market.) My text has some things to say about this matter.

... it would be quite a coincidence if our equilbrium GDP happened to be the output level at which the entire labor force were employed.

In other words, $E^* \ne E^{**}$ in general. However, the text goes on to say something I find mysterious.

In the short-run macro model, the economy can overheat because spending is too high. As long as spending remains high, production will exceed potential output and employment will be unusually low.

So we can have an "overheating" situation where $E^* \gt E^{**}$. How have these extra employees been enticed to enter the market? According to the labor supply curve, it must be that wages have risen. But there are no employers willing to hire these additional employees according to the labor demand curve. This seems to be a contradiction.

How can we have a situation where short term equilibrium employment exceeds full employment?


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