Cost Minimizing Inputs: Perfect Substitutes

The Perfect Substitutes Cost Minimizing Inputs calculator computes the Perfect Substitutes cost minimizing factors to produce a quantity of goods. 

INSTRUCTION: Choose units and enter the following:

• (α)  Production Exponent of Labor
• (β)  Production Exponent of Capital
• (w)  Per unit cost of labor
• (r)  Per unit cost of capital
• (q)  Quantity to be produced.

Perfect Substitutes CMI: The calculator returns the factors (see below).  The total cost C*(q) is returned in U.S. dollars.  However, this can be automatically converted to other currency units via the pull-down menu.

• L*(q) 
• K*(q) 
• C*(q)  This is the total cost.

The Math / Science

The Perfect Substitutes Cost Minimizing Input formula is a function of labor (L), capital (K),  output elasticity (α),  output elasticity of capital (β).  The goal is to set factors such that as production increases, cost increase in the minimum way.   The formulas for Perfect Substitutes Cost Minimizing Input are:

if (a/w >= b/r ){
  L*(q)  = q/a
  K*(q)  = 0
  C*(q) = (w/a)*q
  }

if (a/w < b/r) {
   L*(q)  = 0
   K*(q) = q/b
   C*(q) = (r/b)*q
   }

where the cost minimizing inputs are:

• L*(q) = Labor factor
• K*(q)  = Capital factor
• C* =  Total Cost
• α =  Production Exponent of Labor
• β = Production Exponent of Capital
• w = per unit Cost of Labor
• r = per unit Cost of Capital
• q = Quantity to be Produced

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