/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Solve the problem $$ \max E\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 8

Solve the problem $$ \max E\left[\sum_{1 \leq t \leq T-1} u_{t}^{1 / 2}+a X_{t}^{1 / 2}\right] \quad \text { subject to } X_{t+1}=\left(X_{t}-u_{t}\right) V_{t+1} $$ where \(a\) and \(T\) are given positive numbers, and where \(V_{t+1}=0\) with probability \(1 / 2\), \(V_{t+1}=1\) with probability 1/2. (Hint: Try \(J(t, x)=2 a_{t} x^{1 / 2}, a_{t}>0\).)

Short Answer

Expert verified
Assume \(J(t,x) = 2a_t x^{1/2}\); solve \(u_t^{1/2}=a_t (X_t-u_t)^{1/2}\) to maximize.

Step by step solution

01

Define the Value Function

Assume the value function is of the form given in the hint: \( J(t, x) = 2a_t x^{1/2} \) where \(a_t\) are coefficients to be determined.
02

Bellman Equation

Substitute the value function into the Bellman equation: \( J(t, x) = \text{max}_u \, E \, \left[ u^{1/2} + a (X - u) V_{t+1}^{1/2} + J(t+1, X - u) \right] \)
03

Simplify Using Probabilities

Account for the probabilities of \(V_{t+1}\): \[ E \left[ J(t+1, X_{t+1}) \right] = \frac{1}{2} J(t+1, 0) + \frac{1}{2}J(t+1, X - u) \]
04

Substitution and Simplification

Substitute \(J(t+1, x) = 2a_{t} x^{1/2}\) into the expectation: \[ \frac{1}{2}(2a_{t+1}(0)^{1/2}) + \frac{1}{2} (2a_{t+1}(X - u)^{1/2}) \]
05

Solve the Optimization Problem

Maximize the expression with respect to \(u\): \[ \text{max}_u \, \left[ u_t^{1/2} + \frac{1}{2}(2a_{t+1}(X_t - u_t)^{1/2}) \right] \]
06

Determine \(u_t\)

Find \(u_t\) by solving: \( u_t^{1/2} = a_t (X_t - u_t)^{1/2} \)
07

Substitute Back to Find \(a_t\)

Determine the coefficients \(a_t\) by comparing coefficients on both sides of the equation from the hint.
08

Verifying the Solution

Verify the solution by substituting \(a_t\) back into the original Bellman equation and ensuring consistency.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Value Function
The **value function** is a key concept in stochastic dynamic programming. It represents the maximum expected utility that can be obtained from a given state and time. In our problem, we assumed a value function of the form: \ \( J(t, x) = 2a_t x^{1/2} \) This helps to simplify the complex optimization problem. Here, the coefficients \(a_t\) need to be determined through further steps. The value function encodes all future expectations and choices, guiding decisions from one state to another to maximize the overall benefit.
Bellman Equation
The **Bellman equation** is an essential tool for solving stochastic dynamic programming problems. It breaks down an optimization problem into simpler sub-problems represented by recursive relationships. The Bellman equation for our problem is: \ \( J(t, x) = \max_u \, E \, \left[ u^{1/2} + a (X - u) V_{t+1}^{1/2} + J(t+1, X - u) \right] \) This equation involves finding the decision \(u\) which maximizes the expected utility. By substituting the value function and simplifying using probabilities, the problem transforms into a more manageable form. Hence, the Bellman equation helps iteratively determine the optimal strategy at each stage.
Optimization Problem
An **optimization problem** involves finding the best solution from a set of possible choices. In our context, it means finding the consumption \(u_t\) that maximizes the utility: \ \( \max_u \, \left[ u_t^{1/2} + \frac{1}{2}(2a_{t+1}(X_t - u_t)^{1/2}) \right] \) This optimization takes into account the trade-off between immediate utility from consumption and future utility, represented by the expression involving \(a_{t+1}\). Solving the optimization problem often requires calculus and algebra to derive the best value of \(u_t\). Here, solving \( u_t^{1/2} = a_t (X_t - u_t)^{1/2} \) helps in finding the optimal consumption path.
Expected Utility
The concept of **expected utility** is central to decision-making under uncertainty. It represents the average utility over different possible outcomes, weighted by their probabilities. In this problem, expected utility comes into play when considering the different values of \(V_{t+1}\): \ \[ E \left[ J(t+1, X_{t+1}) \right] = \frac{1}{2} J(t+1, 0) + \frac{1}{2}J(t+1, X - u) \] Here, the future value function is averaged over the probabilities of \(V_{t+1} = 0\) and \(V_{t+1} = 1\). Expected utility helps in making decisions that are optimal on average, considering the inherent uncertainties in future states.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the following problem with \(\beta \in(0,1):\) $$ \max _{u_{t} \in(-\infty, \infty)} \sum_{t=0}^{\infty} \beta^{t}\left(-\frac{2}{3} x_{t}^{2}-u_{t}^{2}\right), \quad x_{t+1}=x_{t}+u_{t}, \quad t=0,1, \ldots, \quad x_{0} \text { given } $$ (a) Suppose that \(J(x)=-\alpha x^{2} .\) Find a third degree equation for \(\alpha .\) Find the associated value of \(u^{*} .\) (Disregard condition (2).) (b) Given a start value \(x_{0}\). By looking at the objective function, show that it is reasonable to assume that \(\left|x_{t}\right| \leq\left|x_{t-1}\right|\) and that \(u_{t} \leq\left|x_{t-1}\right|\). Does \((2)\) then apply?

Consider the following special case of Problem 2, where \(r=0\) : $$ \max _{u_{t} \in[0,1]} \sum_{t=0}^{T} \sqrt{u_{t} x_{i}}, \quad x_{t+1}=\rho\left(1-u_{t}\right) x_{t}, \quad t=0, \ldots, T-1, \quad x_{0}>0 $$ (a) Compute \(J_{T}(x), J_{T-1}(x), J_{T-2}(x) .\) (Hint: Prove that \(\max _{u \in[0,1]}[\sqrt{u}+A \sqrt{1-u}]=\) \(\sqrt{1+A^{2}}\) with \(\left.u=1 /\left(1+A^{2}\right) .\right)\) (b) Show that the optimal control function is \(u_{s}(x)=1 /\left(1+\rho+\rho^{2}+\cdots+\rho^{T-s}\right)\), and find the corresponding \(J_{s}(x), s=1,2, \ldots, T\)

Consider the problem $$ \begin{gathered} \max E \sum_{t=0}^{\infty} \beta^{t}\left(-u_{f}^{2}-X_{t}^{2}\right), \quad \beta \in(0,1), \quad u_{t} \in \mathbb{R} \\ X_{t+1}=X_{t}+u_{t}+V_{t}, \quad E\left(V_{t+1}\right)=0, \quad E\left(V_{t+1}^{2}\right)=d \end{gathered} $$ (a) Guess that \(J(x)\) is of the form \(a x^{2}+b\), and insert it into (5) to determine \(a\) and \(b\). (b) Solve the corresponding finite horizon problem assuming \(J(t, x)=J(t, x, T)=\) \(\beta^{t}\left(a_{1} x^{2}+b_{t}\right) .\) (We now sum only up to time \(T\).) Find \(J\left(0, x_{0}, T\right)\), let \(T \rightarrow \infty\) and prove that the solution in (a) is optimal (we are in case B).

Consider the problem $$ \max _{u_{\mathrm{f}} \in \mathbb{R}}\left[\sum_{t=0}^{T-1}\left(-e^{-\gamma u_{t}}\right)-\alpha e^{-\gamma x_{T}}\right], x_{l+1}=2 x_{t}-u_{t}, t=0,1, \ldots, T-1, x_{0} \text { given } $$ where \(\alpha\) and \(\gamma\) are positive constants. (a) Compute \(J_{T}(x), J_{T-1}(x)\), and \(J_{T-2}(x)\). (b) Prove that \(J_{l}(x)\) can be written in the form $$ J_{t}(x)=-\alpha_{t} e^{-\gamma x} $$ and find a difference equation for \(\alpha_{f}\).

Use the stochastic Euler equation to solve the problem $$ \max E \sum_{t=0}^{2}\left[1-\left(v_{t+1}+X_{t+1}-X_{t}\right)^{2}+\left(1+v_{3}+X_{3}\right)\right], \quad X_{0}=0, X_{1}, X_{2}, X_{3} \in \mathbb{R} $$ where all \(v_{l}\) are identically and independently distributed, with \(E v_{t}=1 / 2\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.