Implicit Functions

1
Implicit Functions

• The equation y mx b is an explicit function
• y is considered the dependent variable
• x is the independent variable
• The equation f(x,y,m,b) 0 is an implicit
  function
• The relationships between the variables and the
  parameters are implicitly present in the equation
  but not explicitly stated

2
Derivatives from Implicit Functions

• In many circumstances, it will be helpful to
  compute derivatives directly from implicit
  functions
• If f(x,y)0, then its total differential of f1dx
  f2dy 0
• Thus,

3
Production Possibility Frontier

• Earlier example 2x2 y2 225
• Can be rewritten f(x,y) 2x2 y2 - 225 0
• Then, the opportunity cost trade-off between x
  and y is

Because fx 4x and fy 2y
4
Implicit Function Theorem

• It may not always be possible to solve implicit
  functions of the form g(x,y)0 for unique
  explicit functions of the form y f(x)
• Mathematicians have derived the necessary
  conditions
• In many economic applications, these conditions
  are the same as the second-order conditions for a
  maximum (or minimum)