Consider a world in which there are only two dates: 0 and 1. At date 1 there are three possible states of nature: a good weather state (G), a fair weather state (F), and a bad weather state (B). Denote S₁ as the set of these states, i.e., 8₁ € S₁ = {G, F, B}. The state at date zero is known. Denote probabilities of the three states as π = (0.4, 0.3, 0.3). There is one non-storable consumption good, apple. There are three consumers in this economy. Their preferences over apples are exactly the same and are given by the following expected utility function & + β Σ πou (chi), 81 ES1 where subscript k = 1, 2, 3 denotes each consumer. In period 0, the three consumers have a linear utility and, in period 1, the three consumers have the same instantaneous utility function: cl-7 u (c) = where y = 0.2 (the coefficient of relative risk aversion). The consumers' time discount factor, ß, is 0.98. The consumers differ in their endowments, which are given in the table below: Consumer 1 Consumer 2 Consumer 3 Endowments t = 1 G F B 3.2 1.8 0.9 1.6 1.2 0.4 1.2 0.6 0.2 t=0 80 0.4 1.2 2.0 Assume that atomic (Arrow-Debreu) securities are traded in this economy. One unit of 'G security' sells at time 0 at a price qc and pays one unit of consumption at time 1 if state 'G' occurs and nothing otherwise. One unit of 'F security' sells at time 0 at a price qF and pays one unit of consumption at time 1 if state 'F' occurs and nothing otherwise. One unit of 'B security' sells at time 0 at a price qв and pays one unit of consumption in state 'B' only. 1. Write down the consumer's budget constraint for all times and states, and define a Market Equilibrium in this economy. Is there any trade of atomic (Arrow-Debreu) securities possible in this economy? 2. Write down the Lagrangian for the consumer's optimisation problem, find the first order necessary conditions, and characterise the equilibrium (i.e., compute the op- timal allocations and prices defined in the equilibrium). ( 3. At the equilibrium, calculate the forward price and risk premium for each atomic
Consider a world in which there are only two dates: 0 and 1. At date 1 there are three possible states of nature: a good weather state (G), a fair weather state (F), and a bad weather state (B). Denote S₁ as the set of these states, i.e., 8₁ € S₁ = {G, F, B}. The state at date zero is known. Denote probabilities of the three states as π = (0.4, 0.3, 0.3). There is one non-storable consumption good, apple. There are three consumers in this economy. Their preferences over apples are exactly the same and are given by the following expected utility function & + β Σ πou (chi), 81 ES1 where subscript k = 1, 2, 3 denotes each consumer. In period 0, the three consumers have a linear utility and, in period 1, the three consumers have the same instantaneous utility function: cl-7 u (c) = where y = 0.2 (the coefficient of relative risk aversion). The consumers' time discount factor, ß, is 0.98. The consumers differ in their endowments, which are given in the table below: Consumer 1 Consumer 2 Consumer 3 Endowments t = 1 G F B 3.2 1.8 0.9 1.6 1.2 0.4 1.2 0.6 0.2 t=0 80 0.4 1.2 2.0 Assume that atomic (Arrow-Debreu) securities are traded in this economy. One unit of 'G security' sells at time 0 at a price qc and pays one unit of consumption at time 1 if state 'G' occurs and nothing otherwise. One unit of 'F security' sells at time 0 at a price qF and pays one unit of consumption at time 1 if state 'F' occurs and nothing otherwise. One unit of 'B security' sells at time 0 at a price qв and pays one unit of consumption in state 'B' only. 1. Write down the consumer's budget constraint for all times and states, and define a Market Equilibrium in this economy. Is there any trade of atomic (Arrow-Debreu) securities possible in this economy? 2. Write down the Lagrangian for the consumer's optimisation problem, find the first order necessary conditions, and characterise the equilibrium (i.e., compute the op- timal allocations and prices defined in the equilibrium). ( 3. At the equilibrium, calculate the forward price and risk premium for each atomic
Chapter1: Making Economics Decisions
Section: Chapter Questions
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Transcribed Image Text:Question 4. Arrow-Debreu
Economy
Consider a world in which there are only two dates: 0 and 1. At date 1 there are three
possible states of nature: a good weather state (G), a fair weather state (F), and a bad
weather state (B). Denote S₁ as the set of these states, i.e., 8₁ € $₁ = {G, F, B}. The
state at date zero is known. Denote probabilities of the three states as π = (0.4, 0.3, 0.3).
There is one non-storable consumption good, apple. There are three consumers in
this economy. Their preferences over apples are exactly the same and are given by the
following expected utility function
& + β Σ Tsu (c),
$1 ES1
where subscript k = 1, 2, 3 denotes each consumer. In period 0, the three consumers have
a linear utility and, in period 1, the three consumers have the same instantaneous utility
function:
c²-y
u (c)
=
where y = 0.2 (the coefficient of relative risk aversion). The consumers' time discount
factor, 3, is 0.98.
The consumers differ in their endowments, which are given in the table below:
Endowments
t=0
t = 1
So
G F B
0.4
3.2 1.8 0.9
Consumer 1
Consumer 2
1.2
1.6 1.2 0.4
Consumer 3 2.0 1.2 0.6 0.2
Assume that atomic (Arrow-Debreu) securities are traded in this economy. One unit
of 'G security' sells at time 0 at a price qc and pays one unit of consumption at time 1 if
state 'G' occurs and nothing otherwise. One unit of 'F security' sells at time 0 at a price
qF and pays one unit of consumption at time 1 if state 'F' occurs and othing otherwise.
One unit of 'B security' sells at time 0 at a price qв and pays one unit of consumption in
state 'B' only.
1. Write down the consumer's budget constraint for all times and states, and define a
Market Equilibrium in this economy. Is there any trade of atomic (Arrow-Debreu)
securities possible in this economy?
2. Write down the Lagrangian for the consumer's optimisation problem, find the first
order necessary conditions, and characterise the equilibrium (i.e., compute the op-
timal allocations and prices defined in the equilibrium). (
3. At the equilibrium, calculate the forward price and risk premium for each atomic
security. What do your results suggest about the consumers' preference? (
Transcribed Image Text:Suppose that instead of atomic (Arrow-Debreu) securities there are three linearly
independent securities, a riskless bond, a stock, and a one-period put option on this stock
available for trade in this economy. The riskless bond pays 1 apple in every state, the
stock pays 2, 1 and 0 apples in G, F and B, respectively. The put option has a strike
price of 1.
4. Write down the budget constraint for each consumer using the newly available
securities.
5. Write down the Lagrangian for the consumer's optimisation problem, find the first
order necessary conditions, and characterise the equilibrium (i.e., compute equilib-
rium allocations and prices of the newly available securities).
6. Now, price the newly available securities using the atomic prices from part 2. Com-
ment on your results in light of the arbitrage-free markets.
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