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In the first part (here), we introduced Bellman's idea of backward induction. But what if we consider now infinite time horizon ? Actually, the maths will be even more simple... and we will be able to use fixed pointed theorem to derive solutions.The mathematical frameworkHere, consider the following value functionand defineA sequence  is said to be an admissible solution for starting point x,if  and . If we reformulate the dynamic programming idea, we obtain that if  is a solution to problem , then for all , sequence  is a solution to problem . It comes that function v is a solution of Bellman's equationNote that is can be ssen as some fixed point resul, sincei.e.So far, it shouldn't be so hard....Frank Ramsey's model (discrete version)In 1928, Frank Ramsey wanted to understand the amount of savings in a dynamic perspective (in how much of its income should a nation save). Consider the following infinite horizon problem, where some planifier wants to maximizesubject to constraints ,   and .Before looking at dynamic programming answers, we might start with standard Lagrangian optimization techniques.Assuming concavity of utility function and production function, we should look only for interior solutions. Define the Lagrangian asThus, the first order conditions are then given byandAssume further some terminal condition, e.g.(also called transversality condition). If we combine those two conditions, and assume that the first constraint is saturated, we obtain the so-called Euler equation,It is also possible to use Bellman's equation: given the dynamic of the capitalThe first order condition statesBut since v is unknown, so is its derivative. But from the enveloppe theroem, we obtain something likewhereWe can then writei.e.and finallywhich is, Euler's equation.A specified model, with calculationsAs in the previous post (here), consider a log utility function, and a power production function, and . The dynamic is then and  Note that fixed points are hereandRecall that the value function is defined asA natural idea to derive the value function can be to iterate, i.e.starting with a simple function, e.g the null function, at step 0. At step n=1thusAt step n=2,i.e.The first order condition is thenand thus, we obtainthat can be plugged in the previous equation, i.e.At step 3, we start from that new expression, derive the first order condition, and we getandand so on...And finally, we can prove thati.e. . Assuming thatactually, we can prove that(and  has a form that can be explicited).