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.
(e) Suppose in part (d) the planner uses Pareto weights a = b = 1/2. Solve for the ecient allocation.
Why does this allocation defer from your answers to parts (b) and (c)?
Risk Sharing with a Continuum of States
There are two periods t = 0, 1. There are two agents, a and b, who consume only at date 1. At date 1, there
is a continuum of states. In particular, the state s is drawn from the U [0, 1] distribution.1 The endowment of
each agent is given by
ya (s) = Gs
yb (s) = G Gs
where G > 0 is a positive constant. Agents have utility over consumption in date 1 given by the CRRA form
u (c) =
c1
1
(1)
(a) For any state s, what is the aggregate endowment?
(b) Set-up the planners problem with Pareto weights a = 1/2 and b = 1/2. Derive the conditions which
characterize the ecient allocation in this economy and solve for the ecient allocation.
(c) In the ecient allocation, what is the correlation between agent as consumption and his own endowment
ya (s)? Give intuition for your answer.
A Risk Neutral and a Risk averse agent
There are two periods t = 0, 1. There are two agents, a and b, who consume only at date 1. Agent a is risk
averse and has CRRA utility function given by (1) with = 2. On the other hand, agent b is risk neutral with
linear utility function given by u (c) = c.
There are two states, high and low, denoted by s = h, `. In the high state, endowments of each agent are
given by
yah = 4, and ybh = 2
In the low state, endowments of each agent are given by
ya` = 1,
and yb` = 2
Each agent believes the high state will occur with probability 1/4.
(a) What is the aggregate endowment in state h? What is the aggregate endowment in state `?
(b) Set-up the planners problem with Pareto weights a = 1/2 and b = 1/2. Derive the conditions which
characterize the ecient allocation in this economy and solve for the ecient allocation.
(c) In the ecient allocation, what is the correlation between agent as consumption and his own endowment?
What is the correlation between agent as consumption and the aggregate endowment? Give intuition for
your answer.
(d) In the ecient allocation, what is the correlation between agent bs consumption and his own endowment?
What is the correlation between agent bs consumption and the aggregate endowment? Give intuition for
your answer.
1 The
Uniform distribution on [0, 1].
The Standard Banking Problem
The set-up of this problem is exactly the same as the banking setup presented in Lecture Notes 4 and 5.
Suppose that there are three dates, indexed by t = 0, 1, 2. There is a single good which can be used either
for consumption or investment.
There is a unit-mass continuum of ex-ante identical agents. Each consumer has an endowment of one unit
at date 0 and nothing at the future dates. All consumption however takes place at dates 1 and 2.
Assets. There are two assets that consumers can use in order to provide for future consumption: there is a
short-term liquid asset and a long-term illiquid asset.
The liquid, short asset is represented by a storage technology that allows one unit of the good at date t to
be converted into one unit of the good at date t + 1, for either t = 0, 1.
The illiquid, long asset is represented by an investment technology that allows one unit of the good invested
at date 0 to be converted into 1 + r units of the good at date 2, with r > 0. If the long asset is liquidated
prematurely at date 1, then it pays ` where 0 < ` < 1.
Preferences. At date 1, each consumer learns his or her type. There are two possible types: early consumers
who only want to consume at date 1 and late consumers who only want to consume at date 2. Initially, in
period 0, each consumer does not know his own typehe only knows the probability of being an early or a late
consumer. Let denote the probability of being an early consumer and 1 be the probability of being a late
consumer. The consumer only learns whether he is an early or late consumer at the beginning of date 1.
More specically, the agents utility is given by
u (c1 )
u (c2 )
with probability
with probability
(a) Consider the consumers problem under autarky. Derive the conditions that characterize the autarkic
allocation of (c1 , c2 ).
(b) Suppose there exists a market at date 1 in which agents can buy and sell holdings of the long asset after
learning their true types. Thus, early types will sell their holdings of the long asset and late types will
buy these assets. Let p be the equilibrium price in this market. Solve for the market allocation (c1 , c2 )
when this market exists.
(c) Consider the planners problem. Derive the conditions that characterize the planners solution for the
ecient allocation.
(d) In one graph with c1 on the x axis and c2 on the y-axis, plot three things: (i) the feasible consumption
set under autarky, (ii) the planners feasible consumption set, and (iii) the market allocation.
(e) Show that if u (c) = ln c, the ecient allocation is the same as the market solution. In addition, in a graph
as in part (d), plot the ecient allocation and show how it coincides with the market solution. (You do
not need to plot the feasible set under autarky.)
(f) If utility is CRRA as in equation (1), with > 1, then the ecient allocation has greater early consumption
c1 and lower late consumption c2 than in the market allocation. Show this in a graph as in part (e).
(g) If > 1 does the planner provide more or less liquidity insurance compared to the market allocation?
Give intuition for your answer.
The Banking Problem with Uncertainty
The set up for this problem is the same as in the previous problem, except with one change. Assume now that
the liquidation value ` for the long asset can take two values, ` = 0 or ` = `h , where 1 < `h < 1 + r. The value
of ` is not known at date 0, but consumers know that at date 1, ` will be drawn from the following distribution
0
with probability
`=
`h with probability 1
At date 1, consumers learn the value of ` (at the same time they learn their own type).
(a) Set up the consumers problem under autarky, letting denote the amount the consumer invests in the
short asset. Derive the consumers optimality condition for .
(b) Let a denote the consumers optimal level of under autarky. Is a increasing or decreasing in ? Explain
why. No need to do math to answer this, just provide economic intuition.
(c) What is the value of a if = 0?
Nash Equilibria
(a) Give the Nash equilibrium (or equilibria)2 for the following 2-player game with two actions.
Ballet
Boxing
Ballet
2, 1
0, 0
Boxing
0, 0
1, 2
(b) Give the Nash equilibrium (or equilibria)3 for the following 2-player game with three actions.
Top
Middle
Bottom
Left Center Right
0, 4
4, 0
5, 3
4, 0
0, 4
5, 3
3, 5
3, 5
6, 6
2 If youve taken game theory, please report only the pure-strategy Nash equilibrium (equilibria). You do not need to report any
mixed-strategy Nash equilibrium (equilibria).
3 Again please report only the pure-strategy Nash equilibrium (equilibria).
Bank Runs
The setup for this problem is the same as in Question 5, but assume that agents have CRRA utility as in
equation (1) with > 1. Suppose there is a bank which can pool and invest the endowments so as to implement the planners allocation.
(a) Describe the banking contract that implements the ecient allocation.
(b) Under the banking solution, show that there exists two Nash Equilibria: one in which all of the late
consumers withdraw late, and another in which all of the late consumers withdraw early.
(Hint: to show that an equilibrium is a Nash equilibrium, you must check that each agent is playing a
best response to all other agents strategies.)
(c) We call the Nash equilibrium in which all of the late consumers withdraw early the Bank run equilibrium.
Is this equilibrium ecient? Explain why or why not.
(d) One way to eliminate bank runs is to have deposit insurance. Explain what deposit insurance is and how it eliminates the Bank run equilibrium.
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