Classical Model - Algebraic Example

A Quadratic Production Function

We assume that output is
determined by a production function where labor **N** is the only variable
factor of production.

**Y = a + b N - c N ^{2}, b > 0, c > 0.**

The marginal product of labor is **MPN = b - 2 c N**.
The labor demand curve will then be **W/P = b - 2 c N**. The maximum
output occurs at **N = b/(2c)**.

A Fixed Labor Supply (The Simple Classical Model)

Suppose the supply of labor is fixed at **N***. For the quadratic production function
above, substituting **
N*** for **N** shows that **W/P = b - 2 c N***. Output is then
given by **Y = a +b N* - c N* ^{2}**.

Business cycles could be caused by either a change in **
N*** or by a change in the parameters of the production function. For example, changes in **b** cause
changes in **Y** and in **W/P**.

Linear Labor Supply (The Classical Model)

The labor supply function need not be linear for the Classical Model, but that case does provide an algebraically convenient mate to the quadratic production function.

Suppose the labor supply is given by **W/P = d + e N**, with **d > 0** and **e
< 0**. (This might look more intuitive as **N = (W/P - d)/e**.)
In equilibrium, it must be the case that **d + e N = b - 2 c N**. We
can solve this equilibrium condition for **N = (b-d) / (2c+e)**.

Business cycles could be caused by changes in any of the
supply and demand parameters. For example, an increase in **b** caused by a positive technology shock
causes an increase in **N**. Output then goes up for two reasons, **N** is bigger and
the production function is shifted upward.

An increase in **d**, which represents an upward shift in
the labor supply curve, causes a decrease in **N** and, hence, **Y**.