National income = total planned expenditure
consumption fn. is behavioural and
a = autonomous consumption expenditure,
b = marginal propensity to consume
x, y and lambda are
endogenous
the only exogenous
var is c (constraint
parameter)
2 for 1b and d
With a constraint g(x,y)=c,
you are no longer free to
change x and
y independently, because
any small change in
x must be offset by a specific
change in
y to stay on the constraint,
making dx and
dy dependent. As a result,
the usual second-order test
that assumes independent
movements breaks down,
and the curvature must be
evaluated only along the
allowed (feasible) direction.
This is why the second-order
condition is expressed using
the Hessian of the
Lagrangian, which correctly
accounts for the dependence
between
dx and dy.
what does positive mean here?
More of good x → utility goes up
More of good y → utility goes up
what does Continuous mean?
Marginal utility changes smoothly
No sudden jumps or breaks