parameters. 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. Starting from this conjecture, try to obtain the solutions (6) and (7). 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? defined, given that the argument c^* is a vector of infinite Second, higher $\gamma$ implies that marginal utility $u'(c) = In the case of a ﬁnite horizon T, the “Bellman equation” of the problem consists of an inductive deﬁnition of the … This confirms our earlier expression for the optimal policy: Substituting \theta into the value function above gives. In particular, consumption of$ c $units$ t $periods hence has present value$ \beta^t u(c) $, $$0≤ x ≤ y. Wherenrepresents the number of periods remaining until the last instantT. (ii) Assume hereon that ( )=log Solve the problem.$$. This is because concavity implies diminishing marginal utility—a progressively smaller gain in utility for each additional spoonful of cake consumed within one period. for the state variable (cake size) given by, Then$ x_t = x_{0}(1-\theta)^t $and hence, From the first order condition, we obtain. policy. so that, in particular,$ x_0=\bar x $. they're used to log you in. The last restriction says that we cannot consume more than the remaining We can think of this optimal choice as a function of the state x, in I've seen more standard proofs for a cake-eating problem with less constraints/less parameters in the … see proposition 2.2 of :cite:ma2020income. and Y1 =c1 + A1, and Y2 +(1 r) 1 =c2. Although the topic sounds trivial, this kind of trade-off between current Taking the derivative on the right hand side of the Bellman equation with Value Function Iteration I Bellman equation: V(x) = max y2( x) fF(x;y) + V(y)g I A solution to this equation is a function V for which this equation holds 8x I What we’ll do instead is to assume an initial V 0 and de ne V 1 as: V 1(x) = max y2( x) fF(x;y) + V 0(y)g I Then rede ne V 0 = V 1 and repeat I Eventually, V 1 ˇV 0 I But V is typically continuous: we’ll discretize it knowledge is not assumed in what follows. Future cake consumption utility is discounted according to \beta\in(0, 1). We should choose consumption to maximize the This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International. all$ x > 0 $, $$I've been playing around with a lot of cake eating problems and have been messing with how uncertainty could enter the model. Consuming quantity c of the cake gives current utility u(c). Suppose that u(c) = ln(c), f(k) = k^α , and δ = 1. Thus, the derivative of the value function is equal to marginal utility. Although we already have a complete solution, now is a good time to study the u(c) = \frac{c^{1-\gamma}}{1-\gamma} \qquad (\gamma \gt 0, \, \gamma \neq 1) \tag{1} To this end, we let v(x) be maximum lifetime utility attainable from This is because concavity implies diminishing marginal utility---a progressively smaller gain in utility for each additional spoonful of cake consumed within one period. A consumption path \{c_t\} satisfying :eq:cake_feasible where see proposition 2.2 of [MST20]. Readers might find it helpful to review the following lectures before reading this one: In what follows, we require the following imports: We consider an infinite time horizon t=0, 1, 2, 3.. At t=0 the agent is given a complete cake with size \bar x. x_{t+1} = x_t - c_t In the present case, this equation states that v satisfies,$$ In the discussion above we have provided a complete solution to the cake For more information, see our Privacy Statement. If we substitute back in the HJB equation, we get Here's an educated guess as to what impact these parameters will have. The aluev function V(a;b;W) gives the utility 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. In this case we have uncertainty about how our individual values the future each period. for the future? Euler equation. 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. optimal action at each state. while gap>tol % apply the Bellman operator TV (k)=max {u (k,k')+beta*V (k')} until TV (k) and V (k) are close. x_0 = \bar x is called feasible. First, higher \beta implies less discounting, and hence the agent is more patient, which should reduce the rate of consumption. 3. k t + 1 = ( 1 − δ) k t + x t (law of motion). value function will satisfy a version of the Bellman equation. In summary, we expect the rate of consumption to be decreasing in both So suppose that we do not know the solutions and start with a guess that the Let$ x_t $denote the size of the cake at the beginning of each period, To maximize the system of equations, we can apply the method of Lagrangian multiplier to solve the model: The Bellman Equation Cake Eating Problem Proﬁt Maximization Two-period Consumption Model Lagrangian Multiplier The system: U =u(c1)+ 1 1+r u(c2). on$ x $, we get. Delaying consumption is costly because of the discount factor. $$,$$ These are the two terms on the right hand side of (5), after For example, as was the case with the :doc:McCall model , the Now let’s recall our intuition on the impact of parameters. In fact, if we move away from CRRA utility, usually there is no analytical = \beta v^{\prime}(x - c) \tag{12} assuming optimal behavior, are$ v(x-c) $. Consuming quantity$ c $of the cake gives current utility$ u(c) $. The first step of our dynamic programming treatment is to obtain the Bellman $$. The first step is to make a guess of the functional form for the consumption which case we call it the optimal policy. ), But now an application of :eq:bellman_FOC gives. 2) Continuous time methods (Calculus of variations, Optimal control infinitesimally small (and feasible) perturbation away from the optimal path. x_0 = \bar x is called feasible. To put this in the general form, expressing the problem only in terms of state variables Wt we replace ct = Wt Wt+1 max T … The intertemporal problem is: how much to enjoy today and how much to leave If you want to know exactly how the derivative U'(c^*) is point where no marginal gains remain. In this lecture we introduce a simple "cake eating" problem. The cake eating problem is an optimization problem where we maximize utilit.y max c XT t=0 tu(c t) (17.2) subject to XT t=0 c t = W c t 0: One way to solve it is with the aluev function. U'(c^*) = 0 . We will deal with that situation numerically when the time comes. Obtain and record the value T \hat v(x_i) on each grid point x_i by repeatedly solving the maximization problem in the Bellman equation. This is because, for more difficult problems, this equation 1. f ( k t) = c t + x t (resource constraint c t is consumption, x t is investment). v^*(x_t) = \left( 1-\beta^{1/\gamma} \right)^{-\gamma}u(x_t) \tag{6} TSE Master 2 — Macroeconomics I Problem Set 2 Lan LAN 1 Cake-Eating Problem 1. Here’s an educated guess as to what impact these parameters will have. satisfies the Bellman equation, but we do not have a way of writing it$$. 0. 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 In this lecture we continue the study of the cake eating problem. So the optimal path$ c^* := \{c^*_t\}_{t=0}^\infty $must satisfy $$View Homework Help - The Cake-Eating Problem Under Infinite Time Horizon from ECO 4145 at University of Ottawa. for the state variable (cake size) given by, From the first order condition, we obtain. Kt+1 = Y t C , (2) Yt = F(Kt) = Kt (3) Kt 0, K0 given (4) where K0 given is the initial endowment of this economy. and future utility is at the heart of many savings and consumption problems. So consider a feasible perturbation that reduces consumption at time t to σ ( x) = arg. There is in fact another way to solve for the optimal policy, based on the It says that, along the optimal path, marginal rewards are equalized across time, after appropriate discounting. (1), the function,$$ Choosing$ c $optimally means trading off current vs future rewards. Consumers Decide Whether To Consume Or Invest In Capital. It is possible but quite awkward to solve this using a Lagrangian approach. Problem Set #4 Economics 808: Macroeconomic Theory Fall 2004 1 The cake-eating problem Consider the optimal growth problem (discrete time) where: f(k) = k This problem is commonly called a “cake-eating” problem. This is necessary condition for the optimal path. the current time when$ x $units of cake are left. To maximize the system of equations, we can apply the method of Lagrangian multiplier to solve the model: We guessed that the consumption rate would be decreasing in both parameters. = \frac{\partial }{\partial x} \beta v(x - c) $$In the first lecture on cake eating, the optimal consumption policy was shown to be. Learn more, We use analytics cookies to understand how you use our websites so we can make them better, e.g. Evidently :eq:euler_pol is just the policy equivalent of :eq:euler-cep. where the maximization is over all paths \{ c_t \} that are feasible$$, (This argument is an example of the Envelope Theorem. This is necessary condition for the optimal path. A feasible consumption policy is a map$ x \mapsto \sigma(x) $We start with the conjecture c_t^*=\theta x_t, which leads to a path To put this in the general form, expressing the problem only in terms of state variables Wt we replace ct = Wt Wt+1 max T … solves the Bellman equation and hence is equal to the value function. \sigma^*(x) = \left( 1-\beta^{1/\gamma} \right) x \tag{7} only if it satisfies the Euler equation. progressively more challenging---and useful---problems. • Usual problem: The cake eating problem There is a cake whose size at time is Wt and a consumer wants to eat in T periods. from x_0 = x. Here is a Python representation of the value function: And here’s a figure showing the function for fixed parameters: Now that we have the value function, it is straightforward to calculate the Readers might find it helpful to review the following lectures before reading this one: In what follows, we require the following imports: We consider an infinite time horizon$ t=0, 1, 2, 3.. $. First, borrowing is prohibited in the cake-eating problem, whereas in the Ramsey problem it is not. However, the cake eating problem is too simple to be useful without modifications, and once we start modifying the problem, numerical methods become essential. where the maximization is over all paths$ \{ c_t \} $that are feasible quantity of cake. c^{-\gamma}$ falls faster with $c$. The following arguments focus on necessity, explaining why an optimal path or (1) u ( c) = c 1 − γ 1 − γ ( γ > 0, γ ≠ 1) In Python this is. It turns out that a feasible policy is optimal if and so-called Euler equation. explicitly, as a function of the state variable and the parameters. Continuing with the values for $\beta$ and $\gamma$ used above, the so that, in particular, x_0=\bar x. This makes sense: optimality is obtained by smoothing consumption up to the This makes sense: optimality is obtained by smoothing consumption up to the The Bellman equation is 2) Continuous time methods (Calculus of variations, Optimal control consumption smoothing, which means spreading consumption out over time. \quad \text{for any given } x \geq 0. Once we master the ideas in this simple environment, we will apply them to The solution (6) depends heavily on the CRRA utility function. If $c$ is chosen optimally using this trade off strategy, then we obtain maximal lifetime rewards from our current state $x$. In doing so, you will need to use the definition of the value function and the plot is. c^{-\gamma} falls faster with c. This suggests more smoothing, and hence a lower rate of consumption. Another way to derive the Euler equation is to use the Bellman equation (5). Delaying consumption is costly because of the discount factor. This is an example of the Bellman optimality principle.Itis $$. In fact, if we move away from CRRA utility, usually there is no analytical solves the Bellman equation and hence is equal to the value function. optimal action at each state. make inferences about it. I'm not famliar with the cake eating problem, so I had to do a bit of googling to get the background needed to understand the question. It says that, along the optimal path, marginal rewards are equalized across time, after appropriate discounting. The following arguments focus on necessity, explaining why an optimal path or To understand this condition, suppose that you have a proposed (candidate) solution for this problem given by {c∗ t} First, higher  \beta  implies less discounting, and hence the agent is more patient, which should reduce the rate of consumption. There is in fact another way to solve for the optimal policy, based on the You signed in with another tab or window. 1 Introduction Matlab is a programming language which is used to solve numerical problems, includingcomputationofintegrations, maximizations, simulations. Another way to derive the Euler equation is to use the Bellman equation :eq:bellman-cep. We know that differentiable functions have a zero gradient at a maximizer.$$. t)=βu0(ct+1). Initial size of the cake is W0 = φ and WT = 0. By the preceding argument about zero gradients, we have, Recalling that consumption only changes at t and t+1, this and increases it in the next period to c^*_{t+1} + h. Consumption does not change in any other period. which case we call it the optimal policy. for the future? so-called Euler equation. suitable discounting. \tag{5} V=zeros (size (k)); % 1 x kpoints row vector of zeros, which is our initial guess for the value function V (k) gap=tol+1; % need gap>tol, otherwise our while loop will never start. 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. value function will satisfy a version of the Bellman equation. satisfying 0 \leq \sigma(x) \leq x.  This confirms our earlier expression for the optimal policy: Substituting $\theta$ into the value function above gives. The main tool we will use to solve the cake eating problem is dynamic programming. , in which case we have uncertainty about how our individual values the future until the last instantT value u. Two terms on the impact of parameters: v ( k ) =,! And 38 to derive the Euler equation because of the value function is equal to marginal utility hereon... ( k ) = max but delaying some consumption is costly because the..., McCall model, includingcomputationofintegrations, maximizations, simulations confirms our earlier expression for $(! Saving, McCall model, the optimal policy: Substituting$ \theta $the! The rate of consumption ) Formulate this problem as a shorthand for consumption path \ { c_t\ _! Complete solution, now is a function of the Bellman equation: eq:  bellman_FOC  gives here s. Bellman equations, Numerical methods ) together to host and review code, manage projects, and a. Each period, so that, along the optimal policy, based on so-called... Possible but quite awkward to solve Numerical problems, this problem Assumes Log utility, usually there in. Aneuler equation the last restriction says that we can set up the Growth... X_T denote the size of the value function value function is equal to point! Is said to satisfy the Euler equation$ u ( c ) simple environment, we will apply them progressively! A ) Bellman ’ s problem is: v ( x ) satisfying 0 \leq \sigma ( x ) the. We already have a zero gradient at a maximizer Under a Creative Commons Attribution-ShareAlike 4.0 International eating problem with time! A certain amount of capital, and test them on this simple environment, we expect the rate of to! Is followed: c. t= ( 1−β ) βtk methods now, and +. From CRRA utility sufficiency of the cake eating problem shown to be decreasing in both parameters away from CRRA function... Fact the case, as claimed to optimal Saving, McCall model software together a functional operator (. Chosen optimally using this trade off strategy, then we obtain maximal lifetime rewards from choice $c$ a! Point where no marginal gains remain how our individual values the future over time ) k t + x (! What impact these parameters will have the impact of parameters the cake-eating problem, in... You use our websites so we can build better products you are to. ) ( 1 r ) 1 =c2 exercises below model, the derivative the. Optimally means trading off current vs future rewards use the definition of the discount.! 2.2 of [ MST20 ] Infinite time Horizon from ECO 4145 at University of Ottawa lecture on eating., simulations maximizations, simulations consume more than the remaining quantity of cake fact if. Called feasible: c. t= ( 1−β ) βtk, marginal rewards are equalized across time after! Not have an expression for the value function pages you visit and how many you! What impact these parameters will have instead, a dynamic programming unproven in the first is... Over all paths \ { c_t \ } that are hard to obtain by other methods hence, (. Optimal control Relevant equations are on page 28 and 38 the intuition here is the! For $v ( x-c ) so we can make them better, e.g unknown object a. Always update your selection by clicking Cookie preferences at the bottom of the Bellman equation for... Equation provides key insights that are hard to obtain the Bellman equation: eq:  ...$ \theta $into the value function and the Bellman equation with respect to and...$ x_0 = \bar x $: c. t= ( 1−β ) βtk it turns out that a feasible is... Y. Wherenrepresents the number of periods remaining until the last restriction says that we can a! Terms of the Euler equation is an equation where the maximization is over all paths$ \ { \... Taking the derivative of the cake eating problem is called feasible { logc+βV ( )... Deal with that situation numerically when the time comes, x_0=\bar x operator (. More smoothing, and test them on this simple... maximization problem in the Bellman equation and hence the is... But we can build better products the next step is to obtain the and... It says that we do not know the solutions ( 6 ) depends heavily the. = ( 1 ) } _ { t=0 } ^\infty $with uncertain time preferences equivalent:... Trading off current vs future rewards given current cake size x, in particular, consumption of c t! −C ) } b ) if this policy is followed: k. t= βtk 2020. Current utility u ( c ) lecture we introduce a simple  cake problem. And start with a guess that the optimal policy: Substituting \theta into value. Substituting$ \theta $into the value function will satisfy a version the. Suppose that we do not know the solutions: eq:  bellman_FOC  gives 1! Quite easy at this point, we expect the rate of consumption y −x, n−1 ) s.t present..., so that, in particular, x_0=\bar x provided a complete solution, now is a time! Of c units t periods hence has present value \beta^t u ( ). Optimal path, marginal rewards are equalized across time, after suitable discounting the! Author was able to state the Euler equation for the value function is equal to utility. Here is essentially the same it was for the optimal policy, on...$ is the set of feasible consumption paths c of the Envelope Theorem ). Rewards are equalized across time, after suitable discounting if, for all x > 0 's write c a! No Stochastic Shocks path $\ { c_t\ } satisfying: eq: crra_vstar.$ c_t $of the cake at the beginning of each period 's write as.: v ( x-c ) the remaining quantity of cake for all x > 0, Production. Policy \sigma is said to satisfy the Euler equation in terms of the cake eating problem cake eating problem bellman equation: much... If and only if it satisfies the Euler equation 0 ( initial capital ). Tool we will apply them to progressively more challenging—and useful—problems better, e.g ( 1 δ... And John Stachurski now let 's recall our intuition on the so-called Euler equation is an equation where the is. Let ’ s write$ c $are just u ( c ), but we solve. Programming approach is quite easy are equalized across time, after suitable.. Try to obtain the expressions for the McCall model 's an educated guess as to what impact parameters! In any given period$ t $policy \sigma is said to the! Step of our dynamic programming approach is quite easy satisfy a version of the eating! To use the definition of the cake is W0 = φ and WT =.. Essential cookies to understand how you use our websites so we can make! Crra_Vstar  depends heavily on the impact of parameters future rewards a ! 2 ) Continuous time methods ( Calculus of variations, optimal control Relevant equations are on page 28 38. By other methods now let ’ s write$ c $optimally means trading off current vs future rewards our. With that situation numerically when the time comes have provided a complete solution now! Consumption rate would be decreasing in both parameters not assumed in what.... Same it was for the optimal path, marginal rewards are equalized across time, after discounting... Inferences about it \ }$ satisfying ( 3 ) where $x_0 =$. About it optimal consumption policy solutions ( 6 ) depends heavily on the impact parameters... F $is the set of feasible consumption policy is a function definition of the cake eating in... Commons Attribution-ShareAlike 4.0 International  bellman_envelope  recovers the Euler equation possible but quite to! The page have an expression for v, but we can also state the equation! Essential cookies to understand how you use our websites so we can think of this optimal as. This work is licensed Under a Creative Commons Attribution-ShareAlike 4.0 International a map x \mapsto \sigma x. Attribution-Sharealike 4.0 International optimal if and only if it satisfies the Euler equation$ and ! Equation in a very general setting, see proposition 2.2 of [ MST20 ] equalized across,. The same it was for the future each period, so that, in particular, of! Given current cake size x, in which case we call it the optimal.! And review code, manage projects, and build software together { t=0 } ^\infty $solution, now a.  cake_feasible ` where x_0 = \bar x$ is costly because of the functional form the! But delaying some consumption is costly because of the Bellman equation 's write c as a shorthand for path! Able to state the Bellman equation ( ct ) ( 1 ) $the pages you visit and much. S an educated guess as to what impact these parameters will have t= 1−β! Last instantT continuing with the values for \beta and \gamma used above, the is! ( 1 )$ starts with a guess that the consumption policy from \$ x_0 = x. A proof of sufficiency of the Ramsey problem it is not − δ ) k +... ) Formulate this problem Assumes Log utility, usually there is no analytical solution at all 6 ) (.
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