ECON 504 2 LECTURE 1 TRANSVERSALITY AND STOCHASTIC LAGRANGE MULTIPLIERS CHRISTOPHER A SIMS PRINCETON UNIVERSITY SIMS PRINCETON EDU 1 E XAMPLE LQPY The ordinary LQ permanent income model has agents solving i h t 1 2 max E b Ct 2 Ct Cs Ws t 0 subject to Wt R Wt 1 Ct 1 Yt E b 5t Wt 0 t The solution for the simple case where Rb 1 and Yt is i i d with mean Y is well known to be Ct 1 b Wt b Y 2 A REASONABLE MODIFICATION OF LQPY Where does the limit on the growth rate of W in come from We believe that the agent should see constraints on making W large and negative i e borrowing a lot but why the constraint on positive accumulation at a high rate So replace by lim inf E R t Wt 0 a standard form for a no Ponzi condition Then the problem is no longer LQ and the standard solution is not optimal so long as Var Yt 0 and Yt 0 with probability one 3 W HY IS THE STANDARD SOLUTION NOT OPTIMAL It implies Wt Wt 1 Yt Y 1 So Et Wt 1 Wt i e Wt is a martingale Theorem A bounded martingale converges almost surely Since the changes in Wt always have the same nonzero variance W does not converge Therefore by the theorem it is unbounded both above and below In particular eventually it will get above 1 W R 1 Date March 27 2003 1 ECON 504 2 LECTURE 1 TRANSVERSALITY AND STOCHASTIC LAGRANGE MULTIPLIERS 2 Once Wt W we can set Ct 1 which delivers maximum possible satiation level utility forever and we can be sure that no matter how bad our luck in drawing Yt values we can avoid violating Wt 0 This has to be better than continuing with the standard solution which would at this point push C above 1 This deviation from the standard solution entails W increasing toward infinity at the rate b t which is why with imposed we do find the standard solution to be optimal 4 S TANDARD TVC AND OUR MODIFIED LQPY PROBLEM The Lagrange multiplier on the constraint in this problem is lt 1 Ct and the usual TVC is E0 b t lt Wt E0 b t 1 Ct Wt 0 t Since Wt is a random walk in this solution and has i i d increments its second moment is O t as is therefore E0 Ct Wt The conventional TVC is satisfied So this is a problem with concave objective function and convex constraints The standard solution satisfies all the Euler equations and the conventional TVC but it is not in fact an optimum In a standard finite dimensional problem a concave objective function and convex constraint sets imply that any solution to the FOC s is an optimum What s wrong here 5 N OTATION T HE M OST G ENERAL S ETUP Our practice things dated t are always known i e available for use as arguments of decision functions at t or later This convention differs from that in much of the growth literature and in the classic Blanchard Kahn treatment of linear RE models but it saves much confusion Also variables chosen at t are dated t A stochastic optimization problem in general form max E b t Ut Ct Z t 2 C0 t 0 subject to gt Ct Z t 0 t 0 where we are using the notation Cnm 3 Cs s m n 6 An implicit constraint Ct is adapted to Zt Each Ct is not a vector of real numbers but instead a function mapping the information available at t Z t into vectors of real numbers It is possible to eliminate the random variables and expectations from our discussion by considering the simplified special case where at each t there are only finitely many possible values of Z t Then the Ct decision function is just a long vector characterized by the list of values it takes at each possible value for Z t expectations are just weighted sums ECON 504 2 LECTURE 1 TRANSVERSALITY AND STOCHASTIC LAGRANGE MULTIPLIERS 3 7 L AGRANGIAN AND FOC S The Lagrangian for this problem E b tUt Ct Z t t 0 b t lt gt Ct Z t 4 t 0 The FOC s H C t U g b t Et b s t s b s t s lt s 0 C t s 0 C t s 0 t 0 5 8 N ECESSITY AND S UFFICIENCY Separating Hyperplane Theorem If V is a continuous concave function over a convex constraint set in some linear space and if there is an x with V x V x then x maximizes V over if and only if there is a continuous linear function f such that f x f x implies that x lies outside and f x f x implies V x V x 9 In a finite dimensional problem with x n 1 we can always write any such f as n f x fi xi 6 i 1 where the fi are all real numbers If the problem has differentiable V and differentiable constraints of the form gi x 0 then it will also be true that we can always pick fi V x xi 7 and nearly always write f x l j j g j x x x 8 with li 0 all i The nearly is necessary because of what is known as the constraint qualification ECON 504 2 LECTURE 1 TRANSVERSALITY AND STOCHASTIC LAGRANGE MULTIPLIERS 4 10 Kuhn Tucker Theorem sufficiency If V is a continuous concave function on a finite dimensional linear space V is differentiable at x gi i 1 k are convex functions each differentiable at x there is a set of non negative numbers li i 1 k such that g x V x li and x x i gi x 0 and li gi x 0 i 1 k then x maximizes V over the set of x s satisfying gi x 0 i 1 k 11 T HE FLY IN THE OINTMENT CONVERGENCE OF INFINITE SUMS Interpret V as given by the maximand in 2 x as being C the optimal C sequence and x as being a generic C sequence In our stochastic problem 6 8 become t t t U C Z t 0 0 Cs f C E bt 0 C s t 0 s 0 t t gt C 0 Z t0 E b t lt Cs 9 C s t 0 s 0 12 The version of 9 with orders of summation interchanged is Ut C t0 Zt0 Cs E bt C s t s s 0 E b t lt s 0 t s t t gt C 0 Z 0 Cs Cs 10 Using the law of iterated expectations together with the fact that Cs is a function of information known at s we can expand this expression to t t Ut C 0 Z 0 Cs E Es b t …