Question

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 (GW), a fair weather state (FW), and a bad weather state (BW). Denote $S_1$ as the set of these states, i.e., $s_1 in S_1 = {GW, FW, BW}$. The state at date zero is known: call it $s_0$. Denote probability of the three states as $pi(cdot)$, $pi(S_1) = (0.4, 0.3, 0.3)$. There is one non-storable consumption good – say apples. There are three consumers in this economy. Their preferences over apples are exactly the same and are given by the following expected utility function $c_0^i(s_0) + eta sum_{s_1 in S_1} pi(s_1)u(c_1^i(s_1))$, where subscript $i = 1, 2, 3$ denotes consumers. In period 0, all agents have a linear utility while in period 1, the three consumers have the same CRRA instantaneous utility function: $u(c) = frac{c^{1-gamma}}{1-gamma}$, where the coefficient of RRA is $gamma = 0.2$. The consumers' time discount factor, $eta$, is 0.98. The consumers differ in their endowments which are given in the table below: Table 2 Endowments $t = 0$ $t = 1$ $s_0$ GW FW BW Consumer 1 0.4 3.2 1.8 0.9 Consumer 2 1.2 1.6 1.2 0.4 Consumer 3 2.0 1.2 0.6 0.2 Assume that Arrow-Debreu securities are traded in this economy. One unit of 'GW security' sells at time 0 at a price $q(GW)$ and pays one unit of consumption at time 1 if state 'GW' occurs and nothing otherwise. One unit of 'FW security' sells at time 0 at a price $q(FW)$ and pays one unit of consumption at time 1 if state 'FW' occurs and nothing otherwise. One unit of 'BW security' sells at time 0 at a price $q(BW)$ and pays one unit of consumption in state 'BW' only. 1. Write down the consumer's budget constraint for all times and states, and define a Sequential Market Equilibrium in this economy. Is there any trade of Arrow-Debreu securities possible in this economy? (1 mark) 2. Write down the Lagrangian for the consumer's optimisation problem, find the first order necessary conditions, and characterise the equilibrium (i.e., find the allocation and price defined in the equilibrium). (1 mark)

          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 (GW), a fair weather state (FW), and a bad weather state (BW). Denote $S_1$ as the set of these states, i.e., $s_1 in S_1 = {GW, FW, BW}$. The state at date zero is known: call it $s_0$. Denote probability of the three states as $pi(cdot)$, $pi(S_1) = (0.4, 0.3, 0.3)$.
There is one non-storable consumption good – say apples. There are three consumers in this economy. Their preferences over apples are exactly the same and are given by the following expected utility function

$c_0^i(s_0) + eta sum_{s_1 in S_1} pi(s_1)u(c_1^i(s_1))$,

where subscript $i = 1, 2, 3$ denotes consumers. In period 0, all agents have a linear utility while in period 1, the three consumers have the same CRRA instantaneous utility function: $u(c) = frac{c^{1-gamma}}{1-gamma}$, where the coefficient of RRA is $gamma = 0.2$. The consumers' time discount factor, $eta$, is 0.98.
The consumers differ in their endowments which are given in the table below:

Table 2
Endowments
$t = 0$ $t = 1$
$s_0$ GW FW BW
Consumer 1 0.4 3.2 1.8 0.9
Consumer 2 1.2 1.6 1.2 0.4
Consumer 3 2.0 1.2 0.6 0.2

Assume that Arrow-Debreu securities are traded in this economy. One unit of 'GW security' sells at time 0 at a price $q(GW)$ and pays one unit of consumption at time 1 if state 'GW' occurs and nothing otherwise. One unit of 'FW security' sells at time 0 at a price $q(FW)$ and pays one unit of consumption at time 1 if state 'FW' occurs and nothing otherwise. One unit of 'BW security' sells at time 0 at a price $q(BW)$ and pays one unit of consumption in state 'BW' only.

1. Write down the consumer's budget constraint for all times and states, and define a Sequential Market Equilibrium in this economy. Is there any trade of Arrow-Debreu securities possible in this economy? (1 mark)
2. Write down the Lagrangian for the consumer's optimisation problem, find the first order necessary conditions, and characterise the equilibrium (i.e., find the allocation and price defined in the equilibrium). (1 mark)

$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 (GW), a fair weather state (FW), and a bad weather state (BW). Denote S1 as the set of these states, i.e., s1 in S1 = GW, FW, BW. The state at date zero is known: call it s0. Denote probability of the three states as pi(cdot), pi(S1) = (0.4, 0.3, 0.3). There is one non-storable consumption good – say apples. There are three consumers in this economy. Their preferences over apples are exactly the same and are given by the following expected utility function c0^i(s0) + eta sums1 in S1 pi(s1)u(c1^i(s1)), where subscript i = 1, 2, 3 denotes consumers. In period 0, all agents have a linear utility while in period 1, the three consumers have the same CRRA instantaneous utility function: u(c) = fracc^1-gamma1-gamma, where the coefficient of RRA is gamma = 0.2. The consumers' time discount factor, eta, is 0.98. The consumers differ in their endowments which are given in the table below: Table 2 Endowments t = 0 t = 1 s0 GW FW BW Consumer 1 0.4 3.2 1.8 0.9 Consumer 2 1.2 1.6 1.2 0.4 Consumer 3 2.0 1.2 0.6 0.2 Assume that Arrow-Debreu securities are traded in this economy. One unit of 'GW security' sells at time 0 at a price q(GW) and pays one unit of consumption at time 1 if state 'GW' occurs and nothing otherwise. One unit of 'FW security' sells at time 0 at a price q(FW) and pays one unit of consumption at time 1 if state 'FW' occurs and nothing otherwise. One unit of 'BW security' sells at time 0 at a price q(BW) and pays one unit of consumption in state 'BW' only. 1. Write down the consumer's budget constraint for all times and states, and define a Sequential Market Equilibrium in this economy. Is there any trade of Arrow-Debreu securities possible in this economy? (1 mark) 2. Write down the Lagrangian for the consumer's optimisation problem, find the first order necessary conditions, and characterise the equilibrium (i.e., find the allocation and price defined in the equilibrium). (1 mark)$

Added by Jacob L.

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