(This follows Diamond, 1982 .)^49 Consider an island...

(This follows Diamond, $1982 .)^{49}$ Consider an island consisting of $N$ people and many palm trees. Each person is in one of two states, not carrying a coconut and looking for palm trees (state $P$ ) or carrying a coconut and looking for other people with coconuts (state $C$ ). If a person without a coconut finds a palm tree, he or she can climb the tree and pick a coconut; this has a cost (in utility units) of $c .$ If a person with a coconut meets another person with a coconut, they trade and eat each other's coconuts; this yields $\bar{u}$ units of utility for each of them. (People cannot eat coconuts that they have picked themselves.) A person looking for coconuts finds palm trees at rate $b$ per unit time. A person carrying a coconut finds trading partners at rate $a L$ per unit time, where $L$ is the total number of people carrying coconuts. $a$ and $b$ are exogenous. Individuals' discount rate is $r$. Focus on steady states; that is, assume that $L$ is constant. (a) Explain why, if everyone in state $P$ climbs a palm tree whenever he or she finds one, then $r V_{P}=b\left(V_{C}-V_{P}-c\right),$ where $V_{P}$ and $V_{C}$ are the values of being in the two states. (b) Find the analogous expression for $V_{C}$. (c) Solve for $V_{C}-V_{P}, V_{C},$ and $V_{P}$ in terms of $r, b, c, \bar{u}, a,$ and $L$. (d) What is $L$, still assuming that anyone in state $P$ climbs a palm tree whenever he or she finds one? Assume for simplicity that $a N=2 b$. (e) For what values of $c$ is it a steady-state equilibrium for anyone in state $P$ to climb a palm tree whenever he or she finds one? (Continue to assume $a N=2 b$. (f) For what values of $c$ is it a steady-state equilibrium for no one who finds a tree to climb it? Are there values of $c$ for which there is more than one steady-state equilibrium? If there are multiple equilibria, does one involve higher welfare than the other? Explain intuitively.

(This follows Diamond, $1982 .)^{49}$ Consider an island consisting of $N$ people and many palm trees. Each person is in one of two states, not carrying a coconut and looking for palm trees (state $P$ ) or carrying a coconut and looking for other people with coconuts (state $C$ ). If a person without a coconut finds a palm tree, he or she can climb the tree and pick a coconut; this has a cost (in utility units) of $c .$ If a person with a coconut meets another person with a coconut, they trade and eat each other's coconuts; this yields $\bar{u}$ units of utility for each of them. (People cannot eat coconuts that they have picked themselves.)
A person looking for coconuts finds palm trees at rate $b$ per unit time. A person carrying a coconut finds trading partners at rate $a L$ per unit time, where
$L$ is the total number of people carrying coconuts. $a$ and $b$ are exogenous. Individuals' discount rate is $r$. Focus on steady states; that is, assume that
$L$ is constant.
(a) Explain why, if everyone in state $P$ climbs a palm tree whenever he or she finds one, then $r V_{P}=b\left(V_{C}-V_{P}-c\right),$ where $V_{P}$ and $V_{C}$ are the values of being in the two states.
(b) Find the analogous expression for $V_{C}$.
(c) Solve for $V_{C}-V_{P}, V_{C},$ and $V_{P}$ in terms of $r, b, c, \bar{u}, a,$ and $L$.
(d) What is $L$, still assuming that anyone in state $P$ climbs a palm tree whenever he or she finds one? Assume for simplicity that $a N=2 b$.
(e) For what values of $c$ is it a steady-state equilibrium for anyone in state $P$ to climb a palm tree whenever he or she finds one? (Continue to assume $a N=2 b$.
(f) For what values of $c$ is it a steady-state equilibrium for no one who finds a tree to climb it? Are there values of $c$ for which there is more than one steady-state equilibrium? If there are multiple equilibria, does one involve higher welfare than the other? Explain intuitively.

Key Concepts

Equilibrium Multiplicity and Welfare Comparisons

Dynamic models can exhibit multiple equilibria where different sets of strategies and outcomes are self-enforcing. Analyzing these multiple equilibria involves comparing not just their stability but also the welfare outcomes they generate. This comparison helps in understanding which equilibria are socially desirable and the potential inefficiencies that arise from misaligned incentives.

Incentive Compatibility and Equilibrium Strategy

In dynamic models, equilibrium strategies must be incentive compatible, meaning that an agent’s policy is optimal given the expectations about other agents’ actions and the evolution of the system. This involves comparing the net benefits (or costs) of alternative actions and ensuring that no agent has an incentive to deviate, a concept that is essential to understanding policy stability in dynamic games.

Discounting in Continuous Time

Discounting is the process of valuing future payoffs less than immediate ones, and in continuous?time settings this is embodied through a constant discount rate. This rate affects the optimal timing of decisions by balancing immediate costs or benefits against those expected in the future, and is a fundamental aspect of intertemporal choice models.

Steady State Equilibrium

A steady state equilibrium refers to a situation where the system's aggregate variables — such as the distribution of agents across different states — remain constant over time. In these models, the value functions satisfy consistency conditions, meaning that the rates of entry into and exit from various states are balanced. This concept is crucial for understanding long?run outcomes in dynamic systems.

Dynamic Programming and the Bellman Equation

In continuous?time models, the Bellman equation is used to characterize the value functions for agents in different states. It equates the instantaneous return and the expected future gains (or losses) from switching states, taking into account the discount rate. This framework is fundamental for analyzing optimal decision?making over time and is used to derive equilibrium conditions in dynamic settings.

Search and Matching Models

These models analyze situations in which agents randomly encounter opportunities (or partners) according to a Poisson process or similar mechanisms. The encounter rates influence the transition between states and are central to understanding how individual decisions aggregate into overall market dynamics. They are widely used in labor market models and trading environments where search frictions play a role.

Transcript

00:01 So a peculiar tribe of natives in the south seas called the grads consume only coconuts.

00:08 They used the coconut for two purposes.

00:10 Either they consumed them for food or the bond them in the public religious sacrifice.

00:16 Now, suppose that each grant has an initial endowment of coconuts of giving an amount, and were to let that be the amount of coconuts that he consumes, and let the other one be the amount of coconut that it gives to the public offering.

00:36 The total amount of coconut contributed to the offering is given.

00:40 Now, the quality function is given by uixi -g as equal to xi plus a -i log of g.

00:54 Now, this is where a -i is greater than 1.

01:05 So now the first question is in determining this is gifts.

01:09 Each grad i assumes that the gifts of other grads remain constant and determines how much it would give on these basis.

01:20 Now, i'll let g minus i equals the summation of gj.

01:24 We note the gifts of other than grad i.

01:28 I want to write down a to the maximization problem that determines the grad i's gifts.

01:35 So we're going to start with our question first so giving this equation where a i is greater than one and then we're giving this equation to where g minus i is equal to the summation of g i and then i is equal to this detritory maximization is going to be equal to the max u i is going to be equal to ai log g so sub to xi plus g i is equal to w i so suppose that there are two groups one and two and then one is consumed and two is free ride now the person is utility max i utility max and it's going to be max ui um is equal to x i plus a i log g and then this is going to be x1i plus g i'm sorry equals to w i so now the person two is max ui is equal to x1 plus a 1 log g and then the sub 2 is equals x2 plus g2 is equal to x2 plus a 1 log g and then the sub 2 is equals x2 plus g2 is equals to w2.

03:57 So this is pressing 1 and this is pressing 2.

04:03 So now moving on to the second question.

04:09 This says recalling that g is equal to coming, recalling that g is equal to gi plus g minus 1, minus i for all segments i, what will be the equilibrium amount of the public good, good and not every agent to contribute a positive amount to the public good.

04:41 So given this, if the total amount of public good is xi, then xi plus x2 plus g1 is equal to w and g and g1 is equal to g1 plus g1 minus 1 and this is equation 2...

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