An Aggregate Supply Function:
A Mathematical Formulation
 

To derive an aggregate supply function, we need a production function, a factor market, and an assumption regarding price expectation.

Production Function

Assume a short-run production function with fixed capital and constant technology such that

where     y is output Assume that F is twice differentiable with The slope of the production function is Dy / DN   =    Fn  >  0
                =    Marginal product of labor To know how output will be produced, we need to know how much labor will be employed.  Thus, we have to examine the labor market or the demand and supply of labor.

 

Labor Demand

Assume firms facing competitive market so that price is given. Assume also that firms maximize profit.

where    p is profit Change in profit results from change in the sale of output and cost of input, or change in the amount of labor, such that Thus, firms choose to hire labor such that profit is maximized (ie no change in profit) or That is demand price of labor = value of marginal productivity of labor, PFN.

Under the assumption of diminishing marginal product of labor, Fnn < 0, the demand for labor curve is negatively sloped.

That is, DW     =     PFnnDN     <     0     as     Fnn     <     0

 

Labor Supply

Assume perfect competition on the labor side of the market for labor, and that workers supply labor for money for what it will buy in the eyes of the workers.

That is

where    yL = real labor income Assume further that workers maximize their utility from the mix of real labor income and leisure. where    U = Utility H = maximum amount of time available and UyL, US > 0

Workers will choose to supply labor such that utility is at maximum or

as under perfect competition  Dw = 0

and by definition  DH = 0

To maximize utilityworkers will continue to supply labor till there is no change in their utility, such that

or which is the marginal rate of substitution of leisure for income.

We can solve  w  =  US/UyL such that

or or We shall assume that the substitution effect of income for leisure is stronger than the income effect of an increase in wage rate, at least at low values of w so that there is a positive relationship between w and N. Thus, the supply for labor is upwardly sloped. That is, Equilibrium in the labor market Or That is or Rewriting Substituting this into the production function Dy  =  [FN / {PegN - PfN}] [f(N)DP - g(N)DPe]
 
  Price Expectation

Assume that price expectation takes the following form:

where         DPe = PePDP

and

Substituting this price expectation into the labor market equilibrium equation

        DN  =  [1 / {PegN - PfN}] [f(N)DP - g(N) PePDP]

as   Peg(N)  =  Pf(N)  in equilibrium

Thus,

            DN  =  [1 / {PegN - PfN}] [1 - (P / Pe)PeP] f(N)DP
 

Aggregate Supply

Substituting this into the production function to obtain

        Dy  = [FN / {PegN - PfN}] [1 - (P / Pe)PeP] f(N)DP

Which gives an aggregate supply curve

        DP  =  [{PegN - PfN} / [FNf(N){1 - (P / Pe)PeP}]]Dy

whose slope is

        DP / Dy  =  [PegN - PfN] / FNf(N)[1 - (P / Pe)PeP]

Note: the slope of the aggregate supply curve depends, among the value of many parameters, the value of PeP.

If PeP is 1, which can be interpreted as workers having perfect foresight or having no money illusion, or information is costless, the aggregate supply curve will be a vertical line as

        DP / Dy  =  [PegN - PfN] / [FNf(N)(1 - 1)]

 

 
 

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