04 Jun Question Arrow-Debreu Markets
Question
Arrow-Debreu Markets
There are two periods t = 0, 1. There are two states in period 1 denoted by s = h, `, where h denotes the high state, ` denotes the low state.
Consider an agent who consumes at both dates: date 0 and date 1, and can trade in Arrow-Debreu (contingent commodities) markets at date 0. Because we have two dates, and two states in period 1, there are three contingent commodities. An agent can purchase units of the consumption good at date 0, consumption in the high state at date 1, and consumption in the low state at date 1. Let p0 , p1h , and p1` denote the prices of these consumption goods, respectively.
Let u (c) denote the agents utility of consuming c units of the consumption good. Let denote the probability of state h occurring at date 1, and let 1 denote the probability of state ` occurring. The agents expected utility is given by
u (c0 ) + Eu (c1 ) = u (c0 ) + [u (c1h ) + (1 ) u (c1` )]
where < 1 is the agents discount factor.
Finally, the consumer is endowed with y0 units of the consumption good at date 0, y1h units of the consumption good in the high state at date 1, and y1` units of the consumption good in the low state at date
1. (a) Write down the consumers budget constraint.
(b) Write down the decision problem of the consumer and take First Order Conditions (FOCs) with respect
to c0, c1h , and c1` . Using these FOCs, show that:
u0 (c0 )
(u0 (c1h ) + (1 ) u0 (c1` ))
=
p0
p1h + p1`
This is a new version of the Euler equation, but now with Eu0 (c1 ) on the right-hand side.
Agents with Heterogeneous Beliefs
There are two periods t = 0, 1. There are two states in period 1 denoted by s = h, `, where h denotes the
high state, ` denotes the low state. There are two agents, a and b, who consume only at date 1 but can
trade in Arrow-Debreu markets at date 0. Suppose agents have dierent beliefs about the high and low states.
Specically, agent a has the following beliefs
ah = .75 and a` = .25
and agent b has the following beliefs
bh = .25 and b` = .75
Each agent has constant endowment across states: yah = ya` = 1 and ybh = yb` = 1. Agents have utility over
consumption in date 1 given by u (c) = ln c.
(a) What is the aggregate endowment in state h? What is the aggregate endowment in state `?
(b) Derive the conditions which characterize the equilibrium allocation in this economy with complete ArrowDebreu markets. Solve for the equilibrium allocation and price ratio.
(c) Consider the planners problem using Pareto weights a and b , where the planner respects the beliefs of
these agents. That is, when maximizing the weighted sum of expected utilities, the planner uses the agents
own beliefs of the probabilities of each state. Set-up this planners problem and derive the conditions which
characterize the ecient allocation in this economy. Show that their exists Pareto weights a and b which
implement the equilibrium allocation.
This implies that the rst and second welfare theorems hold, even if the agents have dierent beliefs!
(d) Now suppose the planner has beliefs over states which dier from that of the agents. The planners beliefs
are given by
ph = .6 and p` = .4
Set-up the planners problem where the planner now uses his own beliefs over states. Derive the conditions
which characterize the ecient allocation in this economy.
Our website has a team of professional writers who can help you write any of your homework. They will write your papers from scratch. We also have a team of editors just to make sure all papers are of HIGH QUALITY & PLAGIARISM FREE. To make an Order you only need to click Ask A Question and we will direct you to our Order Page at WriteDemy. Then fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Fill in all the assignment paper details that are required in the order form with the standard information being the page count, deadline, academic level and type of paper. It is advisable to have this information at hand so that you can quickly fill in the necessary information needed in the form for the essay writer to be immediately assigned to your writing project. Make payment for the custom essay order to enable us to assign a suitable writer to your order. Payments are made through Paypal on a secured billing page. Finally, sit back and relax.
About Writedemy
We are a professional paper writing website. If you have searched a question and bumped into our website just know you are in the right place to get help in your coursework. We offer HIGH QUALITY & PLAGIARISM FREE Papers.
How It Works
To make an Order you only need to click on “Order Now” and we will direct you to our Order Page. Fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Are there Discounts?
All new clients are eligible for 20% off in their first Order. Our payment method is safe and secure.