Mouse-in-Hole Problem and Recursion Concepts

<p>To decrease the value of future utility. (D)</p> Signup and view all the answers

<p>State variable and control variable. (C)</p> Signup and view all the answers

<p>It breaks down the problem into smaller, self-similar subproblems. (D)</p> Signup and view all the answers

<p>Closed-form solutions for the value function $V(k)$ are only possible for a limited set of payoff functions $u(c_t)$ and dynamic constraints $k_{t+1} = g(k_t, c_t)$. (C)</p> Signup and view all the answers

<p>It can help identify characteristics of the solution without needing to find the exact analytical form of $V(k)$. (D)</p> Signup and view all the answers

<p>The value function must be differentiable. (C)</p> Signup and view all the answers

<p>The partial derivative of the function $g$ (dynamic constraint) with respect to $k_0$, evaluated at the optimal consumption $c_0^*$. (D)</p> Signup and view all the answers

<p>Constant. (D)</p> Signup and view all the answers

<p>The steady-state level of capital. (D)</p> Signup and view all the answers

<p>Welfare increases with $k_0$. (D)</p> Signup and view all the answers

<p>It influences the productivity of capital. (A)</p> Signup and view all the answers

<p>The optimal value function starting from period 1 with initial state $k_1$. (B)</p> Signup and view all the answers

<p>An optimal policy has the property that, whatever the initial state and initial decision, the remaining decisions must constitute an optimal policy with respect to the state resulting from the first decision. (D)</p> Signup and view all the answers

<p>The discount factor, reflecting the decision-maker's preference for present versus future utility. (B)</p> Signup and view all the answers

<p>It represents the production function, determining how capital and consumption in period t determine the capital stock in period t+1. (D)</p> Signup and view all the answers

<p>Convert a multi-period optimization problem into a recursive form, simplifying the solution process. (D)</p> Signup and view all the answers

<p>An increase in $c_0$ decreases $k_1$, and thereby influences $V_1(k_1)$. (C)</p> Signup and view all the answers

<p>The problems in period 0 and period 1 become identical except for their initial conditions. (B)</p> Signup and view all the answers

<p>$U_0$ is the instantaneous utility derived from consumption at time 0, while $V_0(k_0)$ represents the total value from time 0 onwards. (B)</p> Signup and view all the answers

<p>Applying the operator T to V(k) leaves V(k) unchanged. (D)</p> Signup and view all the answers

<p>It confirms that the solution, V(k), is optimal and stable. (C)</p> Signup and view all the answers

<p>$y = \alpha \beta k_t^\alpha$ (B)</p> Signup and view all the answers

<p>It must be independently and identically distributed with a zero mean conditional on the past. (B)</p> Signup and view all the answers

<p>The value function $V(k_t)$. (B)</p> Signup and view all the answers

<p>The sensitivity of the value function to changes in capital stock. (B)</p> Signup and view all the answers

<p>It updates the value function estimate in each iteration. (B)</p> Signup and view all the answers

<p>It mathematically expresses that applying the operator T does not change the value function at the fixed point. (A)</p> Signup and view all the answers

<p>Capital stock in the next period must be non-negative and cannot exceed the total output of the current period. (D)</p> Signup and view all the answers

<p>As a constant fraction of current capital stock and technology. (D)</p> Signup and view all the answers

<p>0.387 (A)</p> Signup and view all the answers

<p>The Euler equation, representing an interior solution for optimal consumption. (C)</p> Signup and view all the answers

<p>It converges to $\frac{1}{1 - \beta}$. (D)</p> Signup and view all the answers

<p>To reflect that the functional form of the value function may change over time as the horizon approaches. (A)</p> Signup and view all the answers

<p>The present value of lifetime utility. (B)</p> Signup and view all the answers

<p>It governs the persistence of technology shocks. (A)</p> Signup and view all the answers

<p>It equates the marginal utility of consumption at time 0 to the discounted marginal utility of consumption at time 1, adjusted by the capital's marginal product. (D)</p> Signup and view all the answers

<p>The present value of the marginal utility of consuming one less unit today, invested to increase future capital. (A)</p> Signup and view all the answers

<p>$u'(c_0^<em>) = \beta (1 + r) u'(c_1^</em>)$ (B)</p> Signup and view all the answers

<p>The Euler equation is a first-order condition that avoids dealing directly with the value function. (D)</p> Signup and view all the answers

<p>The marginal product of capital, indicating how much output increases with an additional unit of capital. (B)</p> Signup and view all the answers

<p>It represents the discount factor, reflecting the preference for present versus future consumption. (C)</p> Signup and view all the answers

<p>It decreases current consumption ($c_0^<em>$) and increases future consumption ($c_1^</em>$). (C)</p> Signup and view all the answers

<p>The wage rate or labor income. (A)</p> Signup and view all the answers

<p>Current consumption becomes relatively more valuable than future consumption. (C)</p> Signup and view all the answers

<p>An increase in savings and investment. (C)</p> Signup and view all the answers

The mouse chooses to enter each hole with a probability of 1/3.

The average time the mouse takes to reach its goal based on the probabilities of choosing holes.

Expected time to goal given the mouse entered hole i.

A problem that can be defined in terms of itself, using previous results to solve for future ones.

A function that reflects the satisfaction or value gained from consumption at time t.

A variable that represents decisions made at each time step in a dynamic model.

A variable that describes the state of a system at time t, impacting future decisions.

The maximum overall utility obtainable at time t given the initial state variable.

The formulation for maximizing utility over a time span, incorporating current and future utility.

The initial decision that maximizes utility based on current state and future expectations.

Describes how the state changes based on the decision and the previous state.

An optimal policy will remain optimal even after the first decision is made.

A recursive formula representing the relationship between the value of the current state and future states.

Function that gives the maximum utility achievable from a given state in the next period.

In log utility, the saving rate remains steady over time.

In steady state, capital k equals (αβ) raised to the power of (1 - α).

Welfare increases as capital k0 increases, indicating positive effects.

A method for analyzing value functions without needing closed-form solutions.

The Envelope Theorem assumes the value function is differentiable.

V(k0) combines utility from current consumption and the future value function.

Via the Envelope Theorem, V′(k0) simplifies involving β and gk.

The relationship kt+1 = g(kt, ct) shows how current decisions affect future capital.

An equation connecting current and future utility based on consumption.

A necessary condition for optimality in a dynamic program relating to utility derivatives.

The additional satisfaction gained from consuming an extra unit of good c.

Describes how current capital and consumption affect future state variables.

Decision making where outcomes depend on consumption over different time periods.

A method for solving complex problems by breaking them into simpler subproblems.

An equation describing how the state variable evolves over time.

A technique employed to find approximate solutions to equations, especially in dynamic contexts.

A mathematical operator applied to a function to generate a new function.

A point where T V(k_t) equals V(k_t), indicating stability.

Finding the highest value or maximum of an expression under defined constraints.

Repeating steps to approach a desired result, often used in functions.

The process where a sequence approaches a specific value as steps increase.

Rules governing logarithmic functions, such as ln(a*b) = ln(a) + ln(b).

A system that changes in response to decisions over time.

The optimal consumption level that maximizes utility.

Future capital level after consumption decisions are made.

The value function that represents the maximum utility obtainable.

Utility function representing satisfaction from consumption.

A stochastic process with the Markov property, dependent only on the current state.

A method of solving dynamic programming problems starting from final conditions.

Function that determines future capital based on current capital, productivity, and consumption.

A discount factor indicating the value of future utility.