<p>The endowment for individual 1 is <span class="math-container">$(4,0,1)$</span>.</p>
<p>The price for the asset is <span class="math-container">$q$</span> and assume individual 1 buys <span class="math-container">$\theta$</span> units of the asset at period 0.</p>
<p>Then the total amount of money left in period 0 is:
<span class="math-container">$$
x_1 = 4 - q \theta.
$$</span>
The 4 is the initial amount of money in period 0 and and the <span class="math-container">$- q \theta$</span> is the amount spend to buy the additional amount of assets.</p>
<p>The payout in state 1 is given by:
<span class="math-container">$$
x_1 = 0 + \theta
$$</span>
There is an endowment of 0 but the individual bought <span class="math-container">$\theta$</span> assets that pay out 1 unit in state 1.</p>
<p>The payout in state 2 is given by:
<span class="math-container">$$
x_2 = 1 + 2 \theta.
$$</span>
There is an endowment of 1 unit for the state 2 payout, and the asset contributes 2 euro per unit.</p>
<p>(remark: your optimization problem is missing an objective function).</p>

CC BY-SA 4.0
2023-07-30T08:17:41.143Z
2023-07-30T08:17:41.143Z
tdm
<p>Consider a 1-commodity, 2-consumers, 2-periods economy with
S = 2, J = 1. The asset pays one unit (of the commodity) in state 1 and 2
units in state 2. q denotes the price of the asset at time 0. The preferences of
both individuals are described by the utility function
<span class="math-container">$$ u(x0,x1,x2) = x0 · x1 +x2.$$</span>
The initial endowments are given by
<span class="math-container">$$ ω1 = (4,0,1); $$</span>
<span class="math-container">$$ ω2 = (1,1,1). $$</span>
Determine a competitive equilibrium and discuss its efficiency properties.</p>
<p>Solution: Individual 1:
<span class="math-container">$$ max. $$</span> <span class="math-container">$$ x10,x11,x12,θ1 $$</span>
<span class="math-container">$$ subject $$</span> <span class="math-container">$$ to $$</span>
<span class="math-container">$$qθ^1 = 4 - x0^1$$</span>
<span class="math-container">$$ x1^1 =  θ $$</span>
<span class="math-container">$$ x2^1 =  1 + 2 θ $$</span></p>
<p>How do you determine the subject to q0^1 = 4-x0 , x1^1 = 0, x2^1 = 1 +20 , I don't understand how you get these functions, that is my question</p>

<mathematical-economics><consumer-theory><markets><competitive-equilibrium>
Competitive Equilibrium how to determine subject to functions
jfnotk
# Competitive Equilibrium how to determine subject to functions

(Asked by: username_0 on 2023-07-15)

Consider a 1-commodity, 2-consumers, 2-periods economy with
S = 2, J = 1. The asset pays one unit (of the commodity) in state 1 and 2
units in state 2. q denotes the price of the asset at time 0. The preferences of
both individuals are described by the utility function
$$ u(x0,x1,x2) = x0 · x1 +x2.$$
The initial endowments are given by
$$ ω1 = (4,0,1); $$
$$ ω2 = (1,1,1). $$
Determine a competitive equilibrium and discuss its efficiency properties.

Solution: Individual 1:
$$ max. $$ $$ x10,x11,x12,θ1 $$
$$ subject $$ $$ to $$
$$qθ^1 = 4 - x0^1$$
$$ x1^1 = θ $$
$$ x2^1 = 1 + 2 θ $$

How do you determine the subject to q0^1 = 4-x0 , x1^1 = 0, x2^1 = 1 +20 , I don't understand how you get these functions, that is my question

**Tags:** mathematical-economics, consumer-theory, markets, competitive-equilibrium

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## 1 answers

### Answer by username_1 on 2023-07-30. Score: 1

The endowment for individual 1 is $(4,0,1)$.

The price for the asset is $q$ and assume individual 1 buys $\theta$ units of the asset at period 0.

Then the total amount of money left in period 0 is:
$$
x\_1 = 4 - q \theta.
$$
The 4 is the initial amount of money in period 0 and and the $- q \theta$ is the amount spend to buy the additional amount of assets.

The payout in state 1 is given by:
$$
x\_1 = 0 + \theta
$$
There is an endowment of 0 but the individual bought $\theta$ assets that pay out 1 unit in state 1.

The payout in state 2 is given by:
$$
x\_2 = 1 + 2 \theta.
$$
There is an endowment of 1 unit for the state 2 payout, and the asset contributes 2 euro per unit.

(remark: your optimization problem is missing an objective function).


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