Comment on Monetary Policy Analysis when Planning Horizons are Finite by Michael Woodford

Transcription

1 Comment on Monetary Policy Analysis when Planning Horizons are Finite by Michael Woodford Jennifer La O Columbia University and NBER August 21, Introduction In most macroeconomic models, time is infinite. Agents are endowed with rational expectations including the cognitive ability to solve complex infinite-horizon planning problems. This is a heroic assumption; but when does it matter? In Monetary Policy Analysis when Planning Horizons are Finite, Michael Woodford reconsiders this unrealistic feature, introduces a novel boundedrationality framework to address it, and explores under what circumstances this affects the policy conclusions of the standard New Keynesian paradigm. Woodford develops a new cognitive framework in which agents transform their infinite-horizon problem into a sequence of simpler, finite-horizon ones. The solution method used by the agent is to backwards induct over a finite set of periods given some perceived value function he has assigned to his perceived terminal nodes. This solution method seems quite natural; in fact, Woodford is motivated by a beautiful analogy to how state-of-the-art artificial intelligence (AI) programs play the games of chess or go. Take chess a game with a finite strategy space and thereby in theory solvable via backwards induction. In practice, however, the space of strategies is so large that solving the game in this fashion would require unfathomable processing power. Consider then the most effective AI programs. A typical decision-making process may be described as follows: at each turn, the machine looks forward at all possible moves for both itself and its opponent a finite number of turns, thereby creating a decision tree with finite nodes. It assigns a value to each of the different possible terminal nodes; these values may be based on past experience or data. Finally, given these terminal node I thank Guido Lorenzoni, Marios Angeletos, and Xavier Gabaix for their valuable comments and suggestions. jenlao@columbia.edu 1

2 values, the machine backward inducts along its decision tree to choose its optimal move for the current turn. 1 Inspired by this method, Woodford develops a similar cognitive behavioral approach for economic agents and applies it to the standard New Keynesian model. He shows that this feature helps resolve two well-known and controversial problems within the New Keynesian literature. The first is the problem of equilibria indeterminacy: Woodford s model with finite-planning horizons reduces the set to a unique equilibrium. 2 The second is the well-known problem of the unbounded and unrealistic effectiveness of monetary policy announcements at the zero lower bound what is known as the Forward Guidance Puzzle. In this discussion I begin in Section 2 with a simple example of a single-household finite-horizon planning problem in order to illustrate the basic cognitive framework. I then review how Woodford applies this approach to the standard New Keynesian model in section 3. In section 4 I compare Woodford s model to a few recent papers in the monetary literature that consider similar yet distinct departures from rational expectations. Finally, in section 5 I briefly discuss how these contributions help resolve certain puzzles in the New Keynesian literature, after which I conclude. 2 A Simple Example To illustrate the basic idea behind the bounded rationality framework proposed by Woodford in this paper, I begin with a simple example of a single household consumption-savings problem. Suppose time is discrete and infinite: t = 0, 1,...,. The objective of the household is to choose a consumption and asset holdings sequence {c t, a t+1 } t=0 to maximize lifetime utility given by subject to its per-period budget constraint, max c,a β t u (c t ) (1) t=0 c t + a t+1 = y t + (1 + r) a t, (2) where y t is income in period t, r is the real interest rate, and the initial asset level a 0 is given. Let the real interest rate be exogenous and equal to the discount rate: β (1 + r) = 1. We may assume the usual regularity conditions on utility: u > 0, u < 0, the Inada conditions, along with the 1 See e.g. the pioneering work at IBM research on chess machine Deep Blue, Campbell et al. (2002). 2 While the problem of the New Keynesian model admitting a large multiplicity of equilibria some with explosive paths is well-known, this issue has recently been revived by John Cochrane in a number of papers (see, e.g. Cochrane 2011, 2018). One contribution of Woodford s finite-horizon model may be to view it as a selection criterion over the set of infinite-horizon rational expectations equilibria: it selects the equilibrium that corresponds to the uniquelydetermined equilibrium of the finite-horizon planning economy as the horizon approaches infinity. 2

3 no-ponzi-game condition. Furthermore, let s make the simplifying assumption of no uncertainty: suppose income in each period is a known constant: y t = y, for all t. Infinite-Horizon Rational Expectations. Let us first consider the standard method to solving this problem: consider a rational agent who takes into account the entire infinite-horizon. Under this framework, the rational household s problem may be solved using dynamic programming as in Stokey et al. (1989). In particular, we can reformulate the sequence problem in (1) into the familiar Bellman equation seen here: V (a) = max u ( y + (1 + r) a a ) + βv ( a ). (3) a This is a stationary Bellman equation: the problem of the household is the same in any given period. The household enters a period with asset state a, then chooses its control a in order to maximize the functional equation in (3) where V (a ) is the household s continuation value of carrying a assets into the following period. The first thing to note about the recursive formulation of the infinite-horizon sequence problem is that the Bellman equation is a finite-horizon problem! That is, given some continuation value function V (a ), the problem of the household becomes a simple, finite, one-period-ahead backwards induction problem of choosing how much to consume and how much to save today. Thus, the beauty of the recursive formulation used in dynamic programming is that under certain conditions, one may transform an infinite-horizon problem into a finite-horizon one that may be solved using backwards induction. The only conceptual difference is that the stationary value function V (a) is the fixed point that satisfies the Bellman equation in all periods. It is straight-forward to show that the fixed-point solution to the Bellman equation in (3) is characterized by the following stationary consumption and savings policy functions, c = C (a) y + ra and a = A (a) a, t, (4) and a corresponding value function given by V (a) = 1 u (y + ra), t. (5) That is, the rational expectations infinite-horizon agent consumes her full income each period along with the annuity value of her wealth; as a result, her asset position remains constant. Her value function is simply the discounted value in utils of her constant consumption stream. Finite-Horizon Boundedly Rational Agent. Consider now the boundedly-rational agent proposed 3

4 by Woodford. Consider an agent who, despite facing the infinite-horizon sequence problem in (1), only has the ability to contemplate and process a finite horizon; suppose the agent s horizon is T <. The boundedly-rational agent thereby solves a sequence of finite-horizon problems. In any period t the agent s problem is to maximize the following objective T β t u (c t ) + β T +1 V T +1 (a T +1 ) (6) t=0 subject to the same per-period budget constraint given in (2). One may reformulate this finite horizon problem with a non-stationary Bellman Equation as follows: V t (a) = max u ( y + (1 + r) a a ) ( + βv t+1 a ). (7) a Given a terminal continuation value function V T +1 (a), the agent solves for his optimal path of consumption and assets via backwards induction according to equation (7). The question then becomes: where does this terminal continuation value come from? First, suppose the continuation value function is correct, i.e. it coincides with the stationary rational expectations infinite-horizon value function given in (5). Specifically, suppose the terminal value function were given by V T +1 (a) = V (a) = 1 u (y + ra) If this were the case, then it is clear that the backwards induction problem in (7) would coincide with the stationary Bellman Equation in (3) in all periods. As a result, the finite-horizon solution would be identical to the infinite horizon solution, and the boundedly-rational agent would behave exactly as if he were rational. Put more simply, this is a restatement of the fact that the standard Bellman equation is a backwards induction problem with a particular value function that which is the unique fixed point of the infinite-horizon s recursive formulation. But now suppose that the boundedly-rational agent s terminal continuation value does not coincide with the infinite-horizon one. For pedagogical purposes, suppose that the agent truly believes that period T is his terminal node and he dies the following period. Accordingly, let the perceived value of carrying assets into the following period after death be set equal to zero: V T +1 (a) = 0. With this terminal continuation value, the agent solves for his optimal path of consumption and assets over his finite lifespan via backwards induction. problem is given by consumption and savings policy functions: The solution to this C t (a) = y + T t+1 (1 + r) a and A t+1 (a) = T t a (8) T t+1 4

5 Note that C t (a) > C (a) and A t+1 (a) < A (a) for all t, and that A T +1 (a) = 0. That is, if the boundedly-rational agent believes that he dies in exactly T periods, then it is optimal for him to consume a constant amount every period but an amount that is greater than that of the rational infinite-horizon agent. As a result, the boundedly-rational agent plans to eat into his life savings until he has none left following the last period of his finite life. From the perspective of the agent in period t, his optimal asset path should appear as in the first panel of Figure 1: monotonically decreasing over time so that it is exactly equal to zero at terminal date T. Figure 1: Finite Horizon Asset Plans in the Simple Example But now consider what happens in the following period, at date t + 1. When the boundedlyrational agent enters this period, he now realizes that in fact the world doesn t end for him at time T, it instead ends at date T + 1! With his newfound extra period of life, he must plan for his bright future: he performs the same backward induction argument as before, but now with a lower incoming value of assets. He again chooses to consume a constant amount each period and his revised plan for asset holdings is represented in the second panel of Figure 1 as a new, monotonically decreasing curve that it is exactly equal to zero at terminal date of T + 1. This means that in every consecutive period, the agent wakes up and realizes that he has one more period of life. Every day he thus chooses a new declining asset path that is slightly at odds with the one he chose the day before. As a result, the boundedly-rational agent s actual asset path becomes the upper envelope of his sequence of perceived asset paths; see the last panel of Figure 1. This simple example thereby illustrates that there are in fact two key deviations from rationality for the boundedly-rational agent. The first is that in order for his actions to deviate from those of the rational agent, it must be that his continuation values at his perceived terminal nodes differ from those of the rational infinite-horizon agent. The second deviation is that the boundedly-rational agent acts as if he is truly solving a finite horizon problem. But this is another fiction: the agent doesn t realize that tomorrow he will wake up to face a new finite-horizon problem and will devise 5

6 a new optimal course of action that may not correspond to his last. 3 3 Application to the New Keynesian Framework Woodford s novel bounded-rationality framework of finite-horizon planning can be applied to any model of economic actors. In this paper he applies this approach to the standard New Keynesian DSGE model. Consider the standard infinite-horizon New Keynesian model with infinitely-lived, utility-maximizing households and monopolistically-competitive price-setting firms. 4 Rationality is typically assumed on both the side of households and firms. By log-linearization around the steady state, the standard model reduces to the following familiar set of three equations: 1. The Euler Equation (or, the modern IS curve) ỹ t = E t ỹ t+1 σ (i t E t π t+1 ) (9) where ỹ t is the output gap at time t, i t is the nominal interest rate, and E t π t+1 is expected next-period inflation. 2. The New Keynesian Philips Curve π t = βe t π t+1 + κỹ t (10) 3. And a Central Bank Reaction Function, i.e. monetary policy. The first equation is the household s standard log-linearized Euler equation. The second is the result of staggered nominal price-setting by the forward-looking firms. The third equation closes the model with some specification for monetary policy. This is typically stated in the form of a Taylor rule, but for the purposes of this discussion I abstract from the details of this equation. Woodford applies his bounded-rationality framework of finite planning horizons to the agents in a typical New Keynesian model: both households and firms. As in the standard model, households earn income and choose their optimal plans for consumption and savings to maximize lifetime expected utility. But unlike the standard model, here households have finite planning horizons: they choose a plan for consumption and savings for only k periods in the future. In Woodford s general formulation, households are endowed with a value function defined over their control variables and all possible exogenous states at their perceived terminal planning node, t + k. In order to solve for their optimal plan, they backward induct using these k-period ahead continuation values. 3 This feature is also true for the chess-playing AI programs: at each turn they may choose a new course of action that may be at odds with their previous plan. 4 See, for example, the standard model developed in Woodford (2003b). 6

7 A similar procedure is applied to the firms. As in the standard model, monopolisticallycompetitive firms choose nominal prices in a staggered fashion à la Calvo. Applying his framework, Woodford assumes that firms plan ahead for only k periods in the future and backwards induct using a value function defined over their controls and all possible exogenous states at date t + k. Households and firms each period are assumed to make optimal plans conditional on their k- period ahead continuation values. Aside from the two deviations from rationality alluded to above in my simple example, households and firms are rational or knowledgeable in all other senses: they know all current and past states as well as past realizations of all endogenous variables, they have the correct perception of the conditional probabilities of future exogenous states and the law of motion of endogenous outcomes conditional on these states, and they can perform their backwards induction operations given perceived value functions without error. What then matters is the determination of the terminal node continuation value. Again, if the continuation value function were the same as that of the fully rational infinite-horizon agent, then despite their constrained planning abilities, the boundedly-rational agents would still behave rationally; in this case the model would simply reduce to the standard one. Woodford begins by endowing agents with a specific value function for their terminal node, one that is non-state dependent. In particular he assumes the value function that would arise in the perfect-foresight steady state equilibrium. This seems like a fairly natural assumption: it is as if the economy had been resting for a prolonged period of time at its steady state with a constant inflation rate and zero real disturbances. Households and firms have had enough time to learn their proper values as functions over their controls in steady state, but not how these values should behave in response to shocks. As a result, when choosing their optimal plan they use a terminal node continuation value which is perfectly flat across exogenous states. 5 Woodford furthermore considers a beautiful extension in which he allows for heterogeneity in the length of planning horizons. Rather than imposing that all households and firms have the same finite planning horizon, he supposes that a fraction (1 ρ) ρ k of the population have planning horizon of length k = 0, 1,..., for some parameter ρ (0, 1). This extension with heterogeneity and unbounded support of finite planning horizons is quite useful as it reduces the state space of the model back to that of the original rational expectations framework. In particular, the equilibrium characterization elegantly results in a modified Euler Equation (or modern IS curve) given by ỹ t = ρe t ỹ t+1 σ (i t ρe t π t+1 ), (11) 5 Woodford also considers a version of his model where agents learn over time about their continuation values. This is a very interesting extension which could in itself be the basis of an entirely separate paper. 7

8 and a modified New Keynesian Phillips curve given by π t = ρβe t π t+1 + κỹ t. (12) Comparing equations (11)-(12) to their counterparts in the standard rational expectations framework (9)-(10), one observes that the only real difference between the two models is that the boundedly-rational one features aggregate attenuation of expectations of future variables: future expectations of inflation and the output gap are dampened by the parameter ρ (0, 1). The economy on the whole acts as if it were myopic. The forces behind this attenuation are fairly intuitive. Consider first the agents with the longest planning horizons, those with k. These are the rational types: conditional on the equilibrium law of motion for endogenous variables, these agents behave with near rationality. Thus, if the entire population were to consist of these agents, as it does in the limit as ρ approaches one, then the model indeed converges to the standard rational expectations equilibrium. Consider now the most boundedly-rational types, the agents with the shortest planning horizons: those with k = 0 or k = 1. These agents are effectively making static decisions. Because their continuation values are completely independent of shocks, these values do not fully and accurately reflect the future. As a result, these agents need not form expectations of future variables and they behave as if the economy were perpetually in steady state. If one were to increase the population size of these most boundedly-rational types, i.e. decrease ρ towards zero, then the economy would behave on the whole with greater myopia. This would be not only due to the greater presence of the behavioral types, but also due to the general equilibrium expectations formed by the more rational types. In summary, equations (11) and (12) replace equations (9) and (10) of the standard rational expectations model, respectively. These two equations along with a suitable central bank reaction function provide a complete system of three equations per period in three unknowns, (ỹ t, π t, i t ), and thereby fully characterize the equilibrium of this model. 4 Bounded Rationality in Monetary Models: A Brief User s Guide Consider again Woodford s original question: do boundedly-rational agents finite-planning capacities matter within the context of the standard New Keynesian model? The answer he provides is undoubtedly yes, it does matter. To the extent to which the population of households and firms have short, finite planning horizons and continuation values do not fully reflect the future, the current impact of movements in future expectations within the standard Euler equation and New Keynesian Phillips curve are attenuated. Rather than focus on the implications of this feature, 8

9 allow me to make a slight detour and compare Woodford s work to certain recent models that also depart from the standard model. Consider first the sparsity model of Gabaix (2014). In his bounded rationality framework, Gabaix introduces agents with limited capacity to pay attention to all variables in the world. Agents choose optimally which variables to pay attention to, subject to a linear cost of attention. The linearity results in agents optimally choosing to pay attention to some variables but to pay zero attention to others. Hence agents simplified versions of the world are sparse. In Gabaix (2016) he applies his sparsity framework to the typical New Keynesian paradigm. His application results in the following modified Euler equation and New Keynesian Phillips curve: ỹ t = m h E t ỹ t+1 σ (ĩ t E t π t+1 ) π t = βm f E t π t+1 + κỹ t where m h, m f [0, 1]. Similar to Woodford, the Gabaix (2016) sparsity version of the standard New Keynesian model also features a dampening of future expectations in the Euler equation and the Phillips curve. This macro attenuation is the result of cognitive discounting in the agents perceived law of motion of exogenous states. One need not even stray away from rational expectations in order to generate such features consider the recent work of Angeletos and Lian (2016). Standard macroeconomic models not only endow agents with rational expectations but they also impose common knowledge. This means that agents not only share the same information about present and future shocks, but that they also face zero uncertainty about the general equilibrium reaction to these shocks equivalently, they face zero uncertainty about the actions of others. Beginning with the seminal work of Morris and Shin (2002) and Woodford (2003a), a large literature has explored the aggregate implications of relaxing common knowledge in macro environments with strategic interactions. 6 One of the main lessons from this literature is that strategic complementarity in actions leads agents to put greater weight on higher-order beliefs; higher-order beliefs in turn are more anchored to agents priors, and thereby dampen the equilibrium impact of aggregate shocks. Angeletos and Lian (2016) show that a certain type of dynamic strategic interaction emerges quite naturally in the standard New Keynesian model. In both the consumption-savings decisions of individual households as well as in the forward-looking behavior of price-setting firms, optimal decisions today depend positively on expectations of future decisions of others. The further an event 6 For example, Angeletos and La O (2010) and Angeletos and La O (2013) demonstrate how in a real business cycle model devoid of nominal frictions, demand externalities lead to strategic complementarity among firms. If one then relaxes common knowledge and allows firms to have heterogeneous information, strategic complementarity leads them to place more weight on common sources of information, thereby opening the door for sentiment-driven business cycles while dampening the aggregate effects of TFP shocks. 9

10 is in the future, the more iterations of forward-looking general equilibrium behavior are needed, the more agents must form beliefs over what other agents will do, and hence the greater the anchoring of their actions to the prior. As a result, the Angeletos and Lian (2016) model aggregates to the following modified Euler equation and New Keynesian Phillips curve: ỹ t = ΛE t ỹ t+1 σ (ĩ t λe t π t+1 ) π t = βγe t π t+1 + κγỹ t where Λ, λ, Γ, γ [0, 1]. Thus, while the Angeletos and Lian (2016) model features no departure from rationality, it still generates a similar aggregate weakening of future expectations. Third, consider the recent paper by Farhi and Werning (2017). These authors depart from the standard New Keynesian framework in two ways. First, they allow for incomplete markets: households face idiosyncratic income risk and occasionally-binding borrowing constraints as in a Bewley-Aiyagari-Huggett economy. Second, agents are boundedly-rational in the form of k-level thinking. 7 Similar to Angeletos and Lian (2016), this latter feature implies a lack of common knowledge: agents must not only form beliefs about future shocks, but must also forecast their general equilibrium effects. This amounts to forming beliefs of the beliefs of other agents. As with informational frictions, k-level thinking attenuates these higher-order beliefs. Thus, a similar aggregate dampening of future expectations must also arise in Farhi and Werning (2017). 5 Implications for Forward Guidance and Conclusion Woodford s model of boundedly-rational agents with finite-planning horizons is a novel departure from the standard rational expectations New Keynesian paradigm. While this departure may differ from those featured in Gabaix (2016), Angeletos and Lian (2016), and Farhi and Werning (2017), it is clear that all four of these papers generate similar (albeit non-identical) aggregate implications. In particular they all work towards mitigating the current impact of future expectations. Attenuation of future beliefs is useful for a number of reasons. First, Woodford illustrates how this feature helps resolve the indeterminacy of equilibria in the standard model. He furthermore demonstrates how this feature may also resolve the well-known Forward Guidance Puzzle, namely, the excessive and unreasonable power of monetary policy announcements regarding interest rate changes in the far future. In fact, all four of the models considered tackle this issue essentially by killing the extreme forward-looking nature of the standard rational expectations paradigm. Instead, 7 Similar deviations from rational expectations in macroeconomic settings can be found in Evans and Ramey (1992; 1995; 1998) and Garcia-Schmidt and Woodford (2015). 10

11 with aggregate dampening of future expectations, announcements of interest rate changes in the far future have negligible effects today. There appears to be an urgency in the monetary literature to liberate models from the inordinate amount of forward-looking behavior embedded in the Euler Equation and the New Keynesian Phillips Curve. Woodford s work as well as the three other papers mentioned above compose a movement towards replacing the old New Keynesian model with a new New Keynesian model: the same, familiar set of three equations but with mitigated effects of future beliefs. Moving forward, it would appear to me that empirically distinguishing between these models may be nearly impossible given that they offer such similar time-series predictions. Rather, it would perhaps be more fruitful to tease out reduced-form estimates of these aggregate attenuation parameters from the macro data, e.g. along the lines of previous empirical work by Gali and Gertler (1999) and Gali et al. (2005). One hopes that these exciting recent developments in theory will spur new empirics. References Angeletos, G.-M. and J. La O (2010): Noisy Business Cycles, in NBER Macroeconomics Annual 2009, Volume 24, University of Chicago Press, (2013): Sentiments, Econometrica, 81, Angeletos, G.-M. and C. Lian (2016): Forward Guidance without Common Knowledge, NBER Working Paper No , 5 Campbell, M., A. J. Hoane, and F. hsiung Hsu (2002): Deep Blue, Artificial Intelligence, 134, Cochrane, J. H. (2011): Determinacy and Identification with Taylor Rules, Journal of Political Economy, 119, (2018): Michelson-Morley, Fisher, and Occam: The Radical Implications of Stable Quiet Inflation at the Zero Bound, NBER Macroeconomics Annual, 32, Evans, G. W. and G. Ramey (1992): Expectation Calculation and Macroeconomic Dynamics, The American Economic Review, 82, (1995): Expectation calculation, hyperinflation and currency collapse, in The New Macroeconomics: Imperfect Markets and Policy Effectiveness, ed. by H. D. Dixon and N. Rankin, Cambridge University Press. 7 (1998): Calculation, Adaptation, and Rational Expectations, Macroeconomic Dynamics,

12 Farhi, E. and I. Werning (2017): Monetary Policy, Bounded Rationality, and Incomplete Markets, NBER Working Paper No , 5 Gabaix, X. (2014): A Sparsity-Based Model of Bounded Rationality, The Quarterly Journal of Economics, 129, (2016): A Behavioral New Keynesian Model, NBER Working Paper No , 5 Gali, J. and M. Gertler (1999): Inflation dynamics: A structural econometric analysis, Journal of Monetary Economics, 44, Gali, J., M. Gertler, and J. David Lopez-Salido (2005): Robustness of the estimates of the hybrid New Keynesian Phillips curve, Journal of Monetary Economics, 52, Garcia-Schmidt, M. and M. Woodford (2015): Are Low Interest Rates Deflationary? A Paradox of Perfect-Foresight Analysis, NBER Working Paper Morris, S. and H. S. Shin (2002): Social Value of Public Information, The American Economic Review, 92, Stokey, N. L., R. E. L. Jr., and E. C. Prescott (1989): Recursive Methods in Economic Dynamics, Harvard University Press, Cambridge. 2 Woodford, M. (2003a): Imperfect Common Knowledge and the Effects of Monetary Policy, Knowledge, Information, and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps. 4 (2003b): Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. 4 12

13 Appendix: Proofs for the Simple Example The Infinite-Horizon Problem. Consider the infinite horizon version of the Bellman equation in (3). We may guess and verify that the value function takes the form in (5). Given this guess, the Bellman equation may thereby be written as V (a) = max a u ( y + (1 + r) a a ) + β 1 u ( y + ra ) Taking the first order condition of this expression with respect to a we get, u ( y + (1 + r) a a ) + β 1 ru ( y + ra ) = 0. Using the assumption that β (1 + r) = 1, this reduces to u ( y + (1 + r) a a ) = u ( y + ra ) Solving this expression for a we obtain the following policy function for assets: a = A (a) = a Similarly, for consumption we have c = y + (1 + r) a A (a) = y + ra, thereby verifying the policy functions found in (4) as well as the value function in (5). QED. The Finite-Horizon Problem. Consider the finite horizon version of the Bellman equation in (7). We may guess and verify that the value function takes the following form. T t+1 V t (a) = ( u y + ) (1 + r) a T t+1 We next prove by induction that the value function and the policy functions above are correct for all t T ; that is, we assume this is correct for t + 1 and show that it is correct for t. First consider terminal period t = T and suppose the household enters the period with assets a T. Given that the value for this household in period T is zero: V T +1 (a T +1 ) = 0, the Bellman equation in (7) implies that V T (a T ) = u (y + (1 + r) a T ). This verifies the value function in (13) holds for period t = T. Next consider any date t < T. Assuming (13) is true for time t + 1, the Bellman equation in (13) 13

14 (7) written for time t is given by V t (a) = max u ( y + (1 + r) a a ) + β 1 ( βt t a u y + ) (1 + r) a. T t Taking the first order condition of this expression with respect to a we get u ( y + (1 + r) a a ) ( T t + β u y + ) (1 + r) a (1 + r) = 0. T t T t Using the assumption that β (1 + r) = 1, this reduces to u ( y + (1 + r) a a ) ( = u y + ) (1 + r) a T t Solving this expression for a we obtain the following policy function for assets: a = A t+1 (a) = T t T t+1 a, Similarly, for consumption we have ( ) c = C t (a) = y + (1 + r) a A t+1 (a) = y + T t+1 (1 + r) a Finally, plugging these policy functions into the Bellman equation in (7), we verify that the value function at time t indeed satisfies (13). QED. Alternative way to obtain the Infinite-Horizon Solution. Finally, consider the limit of the finite horizon solution as T. In this limit, the finite horizon value function in (13) converges to V t (a) V (a) and the finite-horizon consumption and asset policy functions in (8) converge to C t (a) C (a) = y + ra, A t (a) A (a) = a. Therefore, the consumption and asset policy functions converge to the stationary infinite-horizon functions found in (4). QED. 14