by chaunce8 » Thu Mar 28, 2013 2:07 pm
A saddle point of a function is a point in the domain of the function where the first derivative is zero, but the point is not a minimum or a maximum of the function. For example in the two dimensional space (x,y), the function f = f(x) = x^3 has a saddle point at x = 0.
In the three dimensional space the analogy to a saddle becomes clearer. See for example the graph of the function z = f(x,y) = x^2 – y^2 with a saddle point at (x,y) = (0,0):
http://www.wolframalpha.com/input/?i=saddle+points+x%5E2+-+y%5E2
In economic models that describe dynamic processes, for example in utility maximization in optimal growth models or when dealing with models that include the assumption of rational expectaions, saddle points may appear.
In such economic dynamic systems, a saddle point is a temporary equilibrium, but quite unstable. The economic system may come to a short term equilibrium at a saddle point, but will eventually move on from there to a more long term equilibrium at a minimum or maximum point of the function describing the dynamic system.
A well know model that involves saddle point stability is for example the Portfolio Balance Model of foreign exchange rate:
Total domestic wealth is made up of domestic residents' holdings of domestic money M, of bonds B, and of foreign currency assets F*. Domestic residents are assumed not to hold foreign currency M*. Therefore domestic non-bank private sector wealth measured in domestic currency is defined as:
W = M + B + EF*
where E is the exchange rate defined as domestic currency per unit of foreign currency.
Assume the equilibrium condition for each asset market as something like:
M = m(r,r* E[e]) ; B = b(r,r*, E[e]) and EF* = f(r,r*E[e])
where r is the domestic interest rate, r* the foreign interest rate and E[e] the expected change of the forex rate.
In very simplified terms, the total system is in a short term equilibrium (or on a saddle point stable state) if two of the three asset markets are in equilibrium.