bordered Hessian Method of Optimization for 2nd order condition ofa utility function with constraint

Bordered Hessian is a matrix method to optimize an objective function f(x,y) where there are two factors. the word optimization is used here because in real life there are always limitations ( constraints ) which we have to consider and we have to maximize ( if it is output or utility function ) or minimize ( if it is a cost function ) with respect to the limitation.

So in this video we wii be building one mathematical derivation example of utility maximization from consumption of two goods x and y with the constraint of total income available (M) and prices of these goods (p1) for good x and (p2) for good y with parallel solution.

So as a rational human we want to get as much utility as possible from consumption of x and y but our resources are limited so we will try to maximize utility from our scarce resources.

Max f(x,y) subject to g(x,y) = M

Utility (objective function) = U = u(x,y) = 2xy

2xy means that both goods are complement for us, it can be seen as assume we do not buy x ( x= 0 ) then to total utility function also becomes zero. And the coefficient 2 represent that for each pair of x and y ( means x = 1 and y = 1 to x = 2 and y = 2 ) we get twice the utility from each higher pair. This specification of utility function is imaginary as utility cannot be measured in reality but it is in consensus that for each individual this function can be different.
Budget (constraint) = I = g(x,y) , I = p1x + p2y , 90 = 3x + 4y

Also we have budget constraint, here nothing is imaginary where income can be measured and prices of good x and good y are also measured from the market. Following table will show parallel solving of bordered hessian with left side is mathematical derivation and right side is solving the example with values.
Solving these gives the values of x y that optimizes the Lagrange function. By optimization means within the limit of total income and the given prices we can at most get x units of good x and y units of good y.

And the value of λ will represent shadow cost; it tells how much objective (utility) will increase if we increase one unit of budget (resources).
But optimized does not mean that the utility is now at its maximum or minimum it only tells that we can afford x and y units of goods with this income and prices.

Now Bordered Hessian Determinant

Consider 2 variable x, y model with one constraint so hessian will be of 3×3 order will all of its components are the second derivative of the functions defined above




whichissimplytheplainHessian
Fxx Fxy
Fyx Fyy
bordered by the first derivatives of the constraint with zero on the principal diagonal.The order of a bordered principal minor is determined by the order of the principal minor being bordered.Hence H above represents a second bordered principal minor H2 ,because the principal minor being bordered is 2x2.

If all the principal minors are negative, the bordered Hessian is
Negative definite, and a negative definite Hessian always satisfies the sufficient condition for a relative maximum
#BorderedHessian #optimisation