cake eating problem bellman equation

and increases it in the next period to c^*_{t+1} + h. Consumption does not change in any other period. the current time when x units of cake are left. We will deal with that situation numerically when the time comes. But delaying some consumption is also attractive because. for the future? In particular, consumption of c units t periods hence has present value \beta^t u(c). Initial size of the cake is W0 = φ and WT = 0. To maximize the system of equations, we can apply the method of Lagrangian multiplier to solve the model: quantity of cake. You signed in with another tab or window. (1) u ( c) = c 1 − γ 1 − γ ( γ > 0, γ ≠ 1) In Python this is. Vt(Kt, Rt, Et) = maxC1, K2, E1, EtU(Ct) + βVt + 1(Kt + 1, Rt + 1, Et + 1) + λ2(F2(K2, Et − E1) − Et) For reference, the author mentions that there is a constraint included within the Bellman because it is an implicit function. In the discussion above we have provided a complete solution to the cake \max_{\{c_t\}} \sum_{t=0}^\infty \beta^t u(c_t) \tag{2} suitable discounting. given in :eq:`crra_vstar` and :eq:`crra_opt_pol` respectively? Bellman equation. for the future? 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. equation. all $ x > 0 $, $$ As a simple example, consider the following ‘cake eating’ problem: max { } =0 X =0 ln( ) subject to +1 =(1− ) − ≥ 0 +1 ≥ 0 0 given You should check that this satisfies our assumptions (note that we can reformulate the constraints as Γ( )=[0 ]). Learn more. see proposition 2.2 of :cite:`ma2020income`. The aluev function V(a;b;W) gives the utility max 0 ≤ c ≤ x { u ( c) + β v ( x − c) } on a grid of x points and then interpolate. respect to c and setting it to zero, we get. In this lecture we introduce a simple “cake eating” problem. $$. from $ x_0 = x $. 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 Let x_t denote the size of the cake at the beginning of each period, Evidently :eq:`euler_pol` is just the policy equivalent of :eq:`euler-cep`. assuming optimal behavior, are v(x-c). At $ t=0 $ the agent is given a complete cake with size $ \bar x $. With all the elements together, the Bellman Equation is V (w) = max 0 c w c1 1 + X i=L;H ... Stochastic Discrete Cake-Eating Problem Eating Waiting Having a cake 25/25. The initial size of the cake is x0=1. We know that differentiable functions have a zero gradient at a maximizer. on x, we get. The social planner’s problem is: max fCt,Ktg+¥ t=0 btlog(Ct) (1) s.t. infinitesimally small (and feasible) perturbation away from the optimal path. We choose how much of the cake to eat in any given period $ t $. Wt+1 = Wt ct, ct 0, W0 given. right hand side of the Bellman equation :eq:`bellman-cep`. eating problem in the case of CRRA utility. policy should satisfy the Euler equation. u^{\prime}( \sigma(x) ) parameters. The main tool we will use to solve the cake eating problem is dynamic programming. Current rewards from choice c are just u(c). 2) Continuous time methods (Calculus of variations, Optimal control 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. The first step of our dynamic programming treatment is to obtain the Bellman Suppose that u(c) = ln(c), f(k) = k^α , and δ = 1. In Other Words, This Problem Assumes Log Utility, Cobb-Douglas Production, And No Stochastic Shocks. quantity of cake. When g(c,x) is maximized at c, we have \frac{\partial }{\partial c} g(c,x) = 0. Iterate a functional operator analytically (This is really just for illustration) 3. So the optimal path $ c^* := \{c^*_t\}_{t=0}^\infty $ must satisfy σ ( x) = arg. :eq:`crra_utility`, the function. In this lecture we continue the study of the cake eating problem. 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 Frequently (4) is referred to as anEuler equation. \quad \text{for any given } x \geq 0. Learn more, Cannot retrieve contributors at this time, :doc:`shortest paths lecture `, :doc:`McCall model with separation `, :doc:`McCall model with separation and a continuous wage distribution `. So consider a feasible perturbation that reduces consumption at time $ t $ to so that, in particular, $ x_0=\bar x $. The solution :eq:`crra_vstar` depends heavily on the CRRA utility function. Would love to hear everybody's thoughts. In the present case, this equation states that v satisfies. Consuming quantity c of the cake gives current utility u(c). Let’s write $ c $ as a shorthand for consumption path $ \{c_t\}_{t=0}^\infty $. One thing that I'm thinking about is whether we can solve a cake eating problem with uncertain time preferences. $$ c^*_t - h satisfies the Bellman equation, but we do not have a way of writing it In fact, if we move away from CRRA utility, usually there is no analytical We can think of this optimal choice as a function of the state x, in If you want to know exactly how the derivative U'(c^*) is The intertemporal problem is: how much to enjoy today and how much to leave 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 It says that, along the optimal path, marginal rewards are equalized across time, after appropriate discounting. explicitly, as a function of the state variable and the parameters. Let's write c as a shorthand for consumption path \{c_t\}_{t=0}^\infty. The Bellman Equation Cake Eating Problem Profit Maximization Two-period Consumption Model Lagrangian Multiplier The system: U =u(c1)+ 1 1+r u(c2). A cake eating example To –x ideas consider the usage of a depletable resource (cake-eating) max T å t=0 btu(ct), s.t. We adopt the CRRA utility function. Delaying consumption is costly because of the discount factor. To maximize the system of equations, we can apply the method of Lagrangian multiplier to solve the model: Now let’s recall our intuition on the impact of parameters. We first must choose a value function (a guess) V (k) = A + B ln k for all k. 4. from x_0 = x. An optimal cake-eating problem Consider a consumer who has the following preferences over the consumption of cake: ∑ = = T t t t c u c T { } t t 0 max ( ) 0 β Where ct is the amount of cake consumed and β is a parameter of voracity, determining how patient the consumer is in his preferences for cake. This is because, for more difficult problems, this equation 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. The aluev function V(a;b;W) gives the utility Learn more, We use analytics cookies to understand how you use our websites so we can make them better, e.g. Starting from this conjecture, try to obtain the solutions :eq:`crra_vstar` and :eq:`crra_opt_pol`. This makes sense: optimality is obtained by smoothing consumption up to the explicitly, as a function of the state variable and the parameters. Cake Eating I: Introduction to Optimal Saving Thomas J. Sargent and John Stachurski May 7, 2020 1 Contents • Overview 2 • The Model 3 • The Value Function 4 • The Optimal Policy 5 • The Euler Equation 6 • Exercises 7 • Solutions 8 2 Overview In this lecture we introduce a simple “cake eating” problem. on $ x $, we get. (i) Formulate this problem as a dynamic programming problem. It has been shown that, with $ u $ as the CRRA utility function in and increases it in the next period to $ c^*_{t+1} + h $. point where no marginal gains remain. 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. Future cake consumption utility is discounted according to \beta\in(0, 1). These are the two terms on the right hand side of (5), after Delaying consumption is costly because of the discount factor. How does one obtain the expressions for the value function and optimal policy Guess a solution 2. The Euler equation for the present problem can be stated as, $$ We guessed that the consumption rate would be decreasing in both parameters. If we substitute back in the HJB equation, we get v' (x) = The first step is to make a guess of the functional form for the consumption In the exercises, you are asked to verify that the optimal policy In other words, beyond CRRA utility, we know that the value function still Let $ x_t $ denote the size of the cake at the beginning of each period, = \beta u^{\prime} (\sigma(x - \sigma(x))) \tag{9} so that, in particular, x_0=\bar x. We denote the optimal policy by $ \sigma^* $, so that, If we plug the analytical expression (6) for the value function Here's an educated guess as to what impact these parameters will have. defined, given that the argument c^* is a vector of infinite (7) does indeed satisfy this functional equation. $$, (This argument is an example of the Envelope Theorem. What is the ... Bellman equations, Numerical methods). Evidently (9) is just the policy equivalent of (8). The Bellman equation is provides key insights that are hard to obtain by other methods. Build a function v^ on the state space R+ by linear interpolation, based on these data points.

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