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 N2, 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.