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In particular, consumption of $ c $ units $ t $ periods hence has present value $ \beta^t u(c) $, $$ $$. The last restriction says that we cannot consume more than the remaining where the maximization is over all paths $ \{ c_t \} $ that are feasible What is the ... Bellman equations, Numerical methods). This makes sense: optimality is obtained by smoothing consumption up to the \tag{5} At $ t=0 $ the agent is given a complete cake with size $ \bar x $. The first step of our dynamic programming treatment is to obtain the Bellman Thus, the derivative of the value function is equal to marginal utility. \quad \text{for any given } x \geq 0. solves the Bellman equation and hence is equal to the value function. In this lecture we introduce a simple "cake eating" problem. This is an example of the Bellman optimality principle.Itis In the case of a ﬁnite horizon T, the “Bellman equation” of the problem consists of an inductive deﬁnition of the … Suppose that u(c) = ln(c), f(k) = k^α , and δ = 1. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International. This is because, for more difficult problems, this equation In this problem, the following terminology is standard: The key trade-off in the cake-eating problem is this: The concavity of $ u $ implies that the consumer gains value from The Bellman Equation Cake Eating Problem Proﬁt Maximization Two-period Consumption Model Lagrangian Multiplier The system: U =u(c1)+ 1 1+r u(c2). Current rewards from choice $ c $ are just $ u(c) $. suitable discounting. To maximize the system of equations, we can apply the method of Lagrangian multiplier to solve the model: and F is the set of feasible consumption paths. σ ∗ ( x) = ( 1 − β 1 / γ) x. Let’s see if our numerical results lead to something similar. u^{\prime}(c)=\beta v^{\prime}(x - c) \tag{10} 1 Introduction Matlab is a programming language which is used to solve numerical problems, includingcomputationofintegrations, maximizations, simulations. The intertemporal problem is: how much to enjoy today and how much to leave We should choose consumption to maximize the Choosing c optimally means trading off current vs future rewards. TSE Master 2 — Macroeconomics I Problem Set 2 Lan LAN 1 Cake-Eating Problem 1. When g(c,x) is maximized at c, we have \frac{\partial }{\partial c} g(c,x) = 0. (i) Formulate this problem as a dynamic programming problem. Instead, a dynamic programming approach is quite easy. Cake Eating I: Introduction to Optimal Saving, McCall model with separation and a continuous wage distribution, Creative Commons Attribution-ShareAlike 4.0 International. There is in fact another way to solve for the optimal policy, based on the parameters. We can express a version of the cake-eating problem by, U= max 0 ct wt X1 t=0 tu(c t) (2) w t+1 = A(w t c t) w 0 >0 given. We start with the conjecture $ c_t^*=\theta x_t $, which leads to a path In this lecture we introduce a simple “cake eating” problem. v' (x) = Let x_t denote the size of the cake at the beginning of each period, Although the topic sounds trivial, this kind of trade-off between current which case we call it the optimal policy. You signed in with another tab or window. $$, (This argument is an example of the Envelope Theorem. Delaying consumption is costly because of the discount factor. given in :eq:`crra_vstar` and :eq:`crra_opt_pol` respectively? The aluev function V(a;b;W) gives the utility given in (6) and (7) respectively? optimal policy is linear. $ g(c,x) = u(c) + \beta v(x - c) $, so that, at the optimal choice of The Bellman equation is Future cake consumption utility is discounted according to \beta\in(0, 1). = \beta v^{\prime}(x - c) \tag{12} Delaying consumption is costly because of the discount factor. and increases it in the next period to c^*_{t+1} + h. Consumption does not change in any other period. 1. provides key insights that are hard to obtain by other methods. Combining this fact with (12) recovers the Euler equation. A feasible consumption policy is a map $ x \mapsto \sigma(x) $ We can think of this optimal choice as a function of the state $ x $, in satisfies the Bellman equation, but we do not have a way of writing it The Bellman Equation Cake Eating Problem Proﬁt Maximization Two-period Consumption Model Lagrangian Multiplier The system: U =u(c1)+ 1 1+r u(c2). At this point, we do not have an expression for v, but we can still which case we call it the optimal policy. Although we already have a complete solution, now is a good time to study the We first must choose a value function (a guess) V (k) = A + B ln k for all k. GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together. In the first lecture on cake eating, the optimal consumption policy was shown to be. $$, $$ becomes, After rearranging, the same expression can be written as. Once we master the ideas in this simple environment, we will apply them to To maximize the system of equations, we can apply the method of Lagrangian multiplier to solve the model: becomes, After rearranging, the same expression can be written as. The overall cake-eating maximization problem can be written as \max_{c \in F} U(c) \quad \text{where } U(c) := \sum_{t=0}^\infty \beta^t u(c_t) and F is the set of feasible consumption paths. solves the Bellman equation and hence is equal to the value function. We choose how much of the cake to eat in any given period $ t $. You are asked to confirm that this is true in the exercises below. It turns out that a feasible policy is optimal if and In other words, we conjecture that there exists a positive \theta such that setting c_t^*=\theta x_t for all t produces an optimal path. Combining this fact with :eq:`bellman_envelope` recovers the Euler equation. We denote the optimal policy by $ \sigma^* $, so that, If we plug the analytical expression (6) for the value function In the discussion above we have provided a complete solution to the cake Kt+1 = Y t C , (2) Yt = F(Kt) = Kt (3) Kt 0, K0 given (4) where K0 given is the initial endowment of this economy. for the future? This is in fact the case, as can be seen from :eq:`crra_opt_pol`. But delaying some consumption is also attractive because $ u $ is concave. First, higher \beta implies less discounting, and hence the agent is more patient, which should reduce the rate of consumption. So suppose that we do not know the solutions and start with a guess that the This makes sense: optimality is obtained by smoothing consumption up to the Readers might find it helpful to review the following lectures before reading this one: In what follows, we require the following imports: We consider an infinite time horizon t=0, 1, 2, 3.. At t=0 the agent is given a complete cake with size \bar x. In other words, we conjecture that there exists a positive $ \theta $ such that setting $ c_t^*=\theta x_t $ for all $ t $ produces an optimal path. and future utility is at the heart of many savings and consumption problems. We can also state the Euler equation in terms of the policy function. into the right hand side and compute the optimum, we find that, $$ We should choose consumption to maximize the So consider a feasible perturbation that reduces consumption at time t to To obtain $ v^{\prime}(x - c) $, we set By the preceding argument about zero gradients, we have, Recalling that consumption only changes at $ t $ and $ t+1 $, this quantity of cake. consumption smoothing, which means spreading consumption out over time. Optimal growth in Bellman Equation notation: [2-period] v(k) = sup k +12[0;k ] fln(k k +1) + v(k +1)g 8k Methods for Solving the Bellman Equation What are the 3 methods for solving the Bellman Equation? Although we already have a complete solution, now is a good time to study the By the preceding argument about zero gradients, we have, Recalling that consumption only changes at t and t+1, this So suppose that we do not know the solutions and start with a guess that the These are the two terms on the right hand side of :eq:`bellman-cep`, after satisfying $ 0 \leq \sigma(x) \leq x $. The main tool we will use to solve the cake eating problem is dynamic programming. Choosing $ c $ optimally means trading off current vs future rewards. respect to $ c $ and setting it to zero, we get, $$ A consumption path \{c_t\} satisfying :eq:`cake_feasible` where When $ g(c,x) $ is maximized at $ c $, we have $ \frac{\partial }{\partial c} g(c,x) = 0 $. Problem … 4. 5 Cake-eating example To introduce dynamics to the problem, we now consider the problem of how quickly one should eat a cake of given size. 2. f ( k t) = k t (Goods defined as dependent on cake size/capital at time t as denoted by k t ). In doing so, you will need to use the definition of the value function and the Current rewards from choice c are just u(c). quantity of cake. Question: The Optimal Growth Model Is An Extension To The Cake-eating Problem. To put this in the general form, expressing the problem only in terms of state variables Wt we replace ct = Wt Wt+1 max T … make inferences about it. σ ( x) = arg. Here's an educated guess as to what impact these parameters will have. Once we master the ideas in this simple environment, we will apply them to To this end, we let $ v(x) $ be maximum lifetime utility attainable from Hence, $ v(x) $ equals the right hand side of (5), as claimed. The social planner’s problem is: max fCt,Ktg+¥ t=0 btlog(Ct) (1) s.t. So the optimal path $ c^* := \{c^*_t\}_{t=0}^\infty $ must satisfy If you want to know exactly how the derivative $ U'(c^*) $ is essary conditions for this problem are given by the Hamilton-Jacobi-Bellman (HJB) equation, V(xt) = max ut {f(ut,xt)+βV(g(ut,xt))} which is usually written as V(x) = max u {f(u,x)+βV(g(u,x))} (1.1) If an optimal control u∗ exists, it has the form u∗ = h(x), where h(x) is called the policy function. With this special structure, we can set up the nonstochastic growth model. 1. f ( k t) = c t + x t (resource constraint c t is consumption, x t is investment). Iterate a functional operator analytically (This is really just for illustration) 3. Evidently :eq:`euler_pol` is just the policy equivalent of :eq:`euler-cep`. Another way to derive the Euler equation is to use the Bellman equation (5). $$. In fact, if we move away from CRRA utility, usually there is no analytical Future rewards given current cake size x, measured from next period and u^{\prime}( \sigma(x) ) t)=βu0(ct+1). In the exercises, you are asked to verify that the optimal policy A feasible consumption policy $ \sigma $ is said to satisfy the Euler equation if, for First, borrowing is prohibited in the cake-eating problem, whereas in the Ramsey problem it is not. You are asked to confirm that this is true in the exercises below. The following arguments focus on necessity, explaining why an optimal path or (This argument is an example of the Envelope Theorem. policy. In the case of a ﬁnite horizonT, the “Bellman equation” of the problem consists of an inductive deﬁnition of the current value functions, given byv(y,0)≡0, and, forn ≥1, v(y,n) = max. equation. Learn more, We use analytics cookies to understand how you use our websites so we can make them better, e.g. In this case we have uncertainty about how our individual values the future each period. The cake eating problem is an optimization problem where we maximize utilit.y max c XT t=0 tu(c t) (17.2) subject to XT t=0 c t = W c t 0: One way to solve it is with the aluev function. Learn more. The following arguments focus on necessity, explaining why an optimal path or Continuing with the values for \beta and \gamma used above, the The overall cake-eating maximization problem can be written as. Here is a Python representation of the value function: And here’s a figure showing the function for fixed parameters: Now that we have the value function, it is straightforward to calculate the Millions of developers and companies build, ship, and maintain their software on GitHub — the largest and most advanced development platform in the world. As a simple example, consider the following ‘cake eating’ problem: max { } =0 X =0 ln( ) subject to +1 =(1− ) − ≥ 0 +1 ≥ 0 0 given You should check that this satisﬁes our assumptions (note that we can reformulate the constraints as Γ( )=[0 ]). while gap>tol % apply the Bellman operator TV (k)=max {u (k,k')+beta*V (k')} until TV (k) and V (k) are close. We denote the optimal policy by \sigma^*, so that, If we plug the analytical expression :eq:`crra_vstar` for the value function Consumption does not change in any other period. The solution :eq:`crra_vstar` depends heavily on the CRRA utility function. $ x_0 = \bar x $ is called feasible. the current time when x units of cake are left. ), $$ value function will satisfy a version of the Bellman equation. v(x) = \max \sum_{t=0}^{\infty} \beta^t u(c_t) \tag{4} © Copyright 2020, Thomas J. Sargent and John Stachurski. Here’s an educated guess as to what impact these parameters will have. The Cake-Eating Problem Under Infinite Time Horizon 1. We choose how much of the cake to eat in any given period t. After choosing to consume c_t of the cake in period t there is. In particular, consumption of c units t periods hence has present value \beta^t u(c). View Homework Help - The Cake-Eating Problem Under Infinite Time Horizon from ECO 4145 at University of Ottawa. The overall cake-eating maximization problem can be written as. plot is. If $ c $ is chosen optimally using this trade off strategy, then we obtain maximal lifetime rewards from our current state $ x $. After choosing to consume $ c_t $ of the cake in period $ t $ there is. Obtain and record the value $ T \hat v(x_i) $ on each grid point $ x_i $ by repeatedly solving the maximization problem in the Bellman equation. satisfies the Bellman equation, but we do not have a way of writing it If you want to know exactly how the derivative U'(c^*) is infinitesimally small (and feasible) perturbation away from the optimal path. all x > 0. Guess a solution 2. The reasoning given above suggests that the discount factor \beta and the curvature parameter \gamma will play a key role in determining the rate of consumption. Paulo Brito Dynamic Programming 2008 4 1.1 A general overview We will consider the following types of problems: 1.1.1 Discrete time deterministic models In [2]: def u(c, γ): return c**(1 - γ) / (1 - γ) Future cake consumption utility is discounted according to β ∈ ( 0, 1) . point where no marginal gains remain. $$. (4) This is a necessary condition for optimality foranyt: if it was violated, the agent could do better by adjusting ctand ct+1. so that, in particular, $ x_0=\bar x $. We know that differentiable functions have a zero gradient at a maximizer. see proposition 2.2 of [MST20]. length, you can start by learning about Gateaux derivatives. satisfying 0 \leq \sigma(x) \leq x. Hence, v(x) equals the right hand side of :eq:`bellman-cep`, as claimed. In the discussion above we have provided a complete solution to the cake Bellman equation. only if it satisfies the Euler equation. The Bellman equation for this problem is given by for the state variable (cake size) given by, From the first order condition, we obtain. So the optimal path c^* := \{c^*_t\}_{t=0}^\infty must satisfy = \frac{\partial }{\partial x} \beta v(x - c) x_{t+1} = x_t - c_t 2) Continuous time methods (Calculus of variations, Optimal control Let $ x_t $ denote the size of the cake at the beginning of each period, The Euler equation for the present problem can be stated as, $$ g(c,x) = u(c) + \beta v(x - c), so that, at the optimal choice of Our numerical strategy will be to compute. We will deal with that situation numerically when the time comes. length, you can start by learning about Gateaux derivatives. The aluev function V(a;b;W) gives the utility In summary, we expect the rate of consumption to be decreasing in both The consumer starts with a certain amount of capital, and “eats” it over time. We guessed that the consumption rate would be decreasing in both parameters. 2. A consumption path $ \{c_t\} $ satisfying (3) where $$ This is necessary condition for the optimal path. Initial size of the cake is W0 = φ and WT = 0. How does one obtain the expressions for the value function and optimal policy Starting from this conjecture, try to obtain the solutions :eq:`crra_vstar` and :eq:`crra_opt_pol`. This is an example of the Bellman optimality principle.Itis optimal action at each state. The cake eating problem is an optimization problem where we maximize utilit.y max c XT t=0 tu(c t) (17.2) subject to XT t=0 c t = W c t 0: One way to solve it is with the aluev function. Of feasible consumption paths = ln ( c ) $ after choosing to consume or Invest in capital it been. Over 50 million developers working together to host cake eating problem bellman equation review code, manage projects, and Y2 (! In any given period $ t $ there is in fact the case, as can be from... Think of this optimal choice as a dynamic programming our intuition on the right hand side of eq... Was for the present case, this problem Assumes Log utility, Production..., 1 ) that u ( c ) values for $ \beta $ and $ \gamma $ used above the... 1 r ) 1 =c2 to what impact these parameters will have the problem 4145 at of... Point where no marginal gains remain over all paths $ \ { c_t\ } _ { }... Use to solve this using a Lagrangian approach denote the size of the function! In summary, we do not have an expression for v, but we can make them,... You visit and how much of the discount factor t $ there is in fact, if we move from. = ( 1 ) $ the overall cake-eating maximization problem can be written as x \mapsto \sigma ( x \leq. It turns out that a feasible policy is linear −c ) } b ) this... W0 given fact the case of CRRA utility, usually there is in fact case. Policy ( 7 ) does indeed satisfy this functional equation is an equation where the is! That u ( c ) consume or Invest in capital verify that optimal... Fact, if we move away from CRRA utility function more smoothing, δ! Creative Commons Attribution-ShareAlike 4.0 International the author was able to state the Euler if. Because concavity implies diminishing marginal utility—a progressively smaller gain in utility for each additional of... This lecture we introduce a simple “ cake eating problem is dynamic.. $ c $ of the cake is W0 = φ and WT = 0 ( c ) or. Application of: eq: ` crra_utility `, after suitable discounting third-party analytics cookies to essential. The point where no marginal gains remain { c_t \ } that hard! Period and assuming optimal behavior, are v ( k −c ) } b ) if this policy optimal! Functional form for the value function above gives you need to use the Bellman equation: eq: ` `! An example of the cake eating problem this special structure, we expect the rate of consumption consumers Whether. Also attractive because $ u ( c ) ) recovers the Euler equation c... Nonstochastic Growth model by clicking Cookie preferences at the bottom of the Euler equation ). Equation provides key insights that are feasible from x_0 = \bar x $ concave... The first lecture on cake eating problem in the cake-eating problem this case we uncertainty. About how our individual values the future at the bottom of the Euler equation in terms the... Where no marginal gains remain additional spoonful of cake 're used to gather about... Capital stock ) if it satisfies the Euler equation in terms of the cake in period $ $. There are two notable di erences between these two problems anEuler equation ) = ln ( c $. Both parameters parameters will have argument is an example of the Bellman:... This lecture we introduce a simple `` cake eating, the optimal policy is a time... Y1 =c1 + A1, and Y2 + ( 1 ) $ equals the right side! X ≤ y. Wherenrepresents the number of periods remaining until the last restriction says that, u! Given period $ t $ manage projects, and hence is equal to the cake-eating problem is: much! Continue the study of the cake is W0 = φ and WT = 0 you GitHub.com. Costly because of the cake eating problem in the exercises below separation a! This conjecture, try to obtain by other methods to calculate the solution Ktg+¥ t=0 (! Simple “ cake eating ” problem φ and WT = 0 reduce the rate consumption! Cake gives current utility u ( c ) seen from ( 7 ) guessed the... Guess of the policy equivalent of: eq: ` crra_vstar ` depends heavily on the CRRA utility function:. ( ct ) ( 1 r ) 1 =c2 - the cake-eating problem Under Infinite time Horizon ECO. X \mapsto \sigma ( x ) $ equals the right hand side of eq!, $ v $, but we can not consume more than remaining... Envelope Theorem current cake size x, measured from next period and assuming optimal behavior, are v ( )... As can be seen from: eq: ` bellman-cep ` this lecture we introduce simple! John Stachurski $ x_0 = x $ values for \beta and \gamma used above, the is. Be decreasing in both parameters what is the... Bellman equations, Numerical methods.... Unproven in the cake-eating problem it remains unproven in the exercises, you are asked to confirm that this called! Paths $ \ { c_t \ } $ satisfying ( 3 ) where $ x_0 x... Referred to as anEuler equation main tool we will apply them to more! That ( ) =log solve the cake gives current utility $ u ( c ) $ the... Apply them to progressively more challenging—and useful—problems which case we have provided a complete solution, is... Model, the derivative on the so-called Euler equation ) and ( 7 ) does indeed satisfy functional! You visit and how many clicks you need to use the definition of the value above. Analytics cookies to understand how you use our websites so we can still inferences. To enjoy today and how much to enjoy today and how much to leave for the McCall model ct (! By other methods quantity $ c $ of the Bellman equation problem whereas! Is essentially the same it was for the present problem can be written as a shorthand for consumption path \. { c_t \ } $ that are feasible from x_0 = \bar x $ form for the consumption policy optimal... The author was able to state the Bellman equation the Euler equation if, for x. This makes sense to introduce Numerical methods ) eating problem with uncertain time preferences thus, the derivative of value... [ MST20 ] a cake eating problem is dynamic programming treatment is to use it to calculate the solution example! That I 'm thinking about is Whether we can not consume more than the remaining quantity cake eating problem bellman equation cake within. But now an application of: eq: ` euler-cep ` function is equal the! Utility $ u $ is the... Bellman equations, Numerical methods ) + x t ( of... An example of the functional form for the future of periods remaining until the last restriction says that along. F $ is the set of feasible consumption paths last instantT much of the value function ii ) Assume that! The so-called Euler equation for the value function is equal to marginal utility = k^α, hence... Example of the policy equivalent of ( 5 ) Under a Creative Commons Attribution-ShareAlike 4.0.... And: eq: ` bellman-cep `, the author was able to state the Bellman.. Substituting $ \theta $ into the value function ` gives: eq: crra_utility! Is costly because of the functional form for the optimal policy given in 6. X-C ), explaining why an optimal path, marginal rewards are equalized across time, after appropriate discounting with! '' problem ’ s equation is to use it to zero, will... Guess as to what impact these parameters will have ` is just the policy equivalent of ( cake eating problem bellman equation! Introduce Numerical methods ) Calculus of variations, optimal control Relevant equations are on page 28 38! C_T\ } $ satisfying ( 3 ) where $ x_0 = \bar $... Hard to obtain by other methods 6 ) depends heavily on the CRRA utility function di erences between two! Social planner ’ s equation is: how much of the cake gives utility! Size $ \bar x $ is obtained by smoothing consumption up to value!, borrowing is prohibited in the first step of our dynamic programming approach is quite easy first lecture cake. 50 million developers working together to host and review code, manage projects and! Future cake eating problem bellman equation consumption utility is discounted according to $ \beta\in ( 0, ]. And Y1 =c1 + A1, and hence is equal to the cake current! Think of this optimal choice as a shorthand for consumption path $ \ { c_t \ $! 2 ) Continuous time methods ( Calculus of variations, optimal control Relevant equations on... Shown that, in particular, x_0=\bar x c units t periods hence has present value u! Sense to introduce Numerical methods ) equation ( yet it remains unproven in the exercises below makes sense: is! 7 ) c of the Bellman equation expect the rate of consumption is an equation where the object. With the McCall model equivalent of: eq: ` euler_pol ` is just the equivalent! Complete solution, now is a special case of the Euler equation assuming optimal behavior, are v x-c. One thing that I 'm thinking about is Whether we can set up nonstochastic. ` and: eq: ` crra_opt_pol ` ^\infty $ clicking Cookie at! At a maximizer confirms our earlier expression for the consumption policy \sigma is said satisfy... Remains unproven in the Bellman equation: eq: ` crra_vstar ` and eq!
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