Definition
K-T theorem in standard form
In the standard form of nonlinear program:
- maximize F(x)
- inequality constraints
, for example, 
- write Lagrange as
Suppose 
, then there exists a Lagrange multiplier 
such that for all l 
N
with equality if 
Note
(1) N is total dimension / total number of 
/ total number of goods
(2) l, k belongs to N (can be regarded as 1, 2 for 
)
(3) Some utility functions will not always be solvable using KT conditions, for example quasi-linear utility.
Implication
(1) the Lagrange multiplier 
is:
- the shadow value of wealth
- the marginal utility of wealth at the optimum
- how much does utility change with a small increase in wealth (spent automatically).
(2) For strictly interior solutions (all , equality holds), we have:
Let 
, the marginal utility of dimension l, then the above equation can be written as:
Interpretation
At an interior solution, the MRS (marginal substitution rate, slope of the indifference curve) at the optimum is equal to the ratio of prices.
The indifference curve is tangent to the budget constraint.
Proposition
Suppose u is continuously differentiable at strictly quasi-concave, then there is a unique solution to the UMP characterized by the Kuhn-Tucker (KT) First order conditions (FOCs):
(1) 
(2) 
Moreover, suppose that the preference are locally non-satiated, and that any optimum must be strictly interior; then the optima are uniquely characterized by
and 