Abstract
In this chapter, we look at financial markets. With complete and perfect markets, the models in this chapter are within the Arrow-Debreu world.
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- 1.
This is implied by the fact that, for any matrix, the rank of its column vectors is the same as the rank of its row vectors.
- 2.
Suppose that there are two assets and one asset is money and the other is bonds. If I issue 10 $100 bonds, I get $1000 but lose 10 bond units. My budget for this transaction is ($1000 × 1) − ($1000 × 1) = 0. In this example, qm = $1000, qb = $100, and \( \theta_{b} = 10, \) where \( m \) indicates money and \( b \) indicates bonds.
- 3.
This is implied from (3). Intuitively, since the equilibrium price vector is unique up to a positive multiplier and since there is only one good, the price can be simply \( 1 \).
- 4.
This means that the good \( x_{0} \) is used as the numeraire for \( p \) and \( q. \) As we know, for CME prices, one of them can be set to \( 1. \)
- 5.
The MRS \( v_{i}^{\prime} \left( {\tilde{x}^{i} } \right)/v_{i}^{\prime} \left( {x_{0} } \right) \) is there because of risk aversion. Without it, the price of the security will be the discounted expected income.
- 6.
Let \( v_{i} \left( {p,I_{i} } \right) \equiv \mathop { \hbox{max} }\nolimits_{{x_{i} \in {\mathcal{C}}}} \{ u_{i} \left( {x_{i} } \right) | p \cdot x_{i} \le I_{i} \} . \) Then, \( \lambda_{i} \equiv \partial v_{i} \left( {p,I_{i} } \right)/\partial I_{i} . \)
- 7.
If there are only finite states \( s = 1, \ldots ,S, \) then we write \( p \) as a vector \( p = \left( {p^{1} , \ldots ,p^{S} } \right). \) For any \( x = \left( {x^{1} , \ldots ,x^{S} } \right), \) we have \( p \cdot \,x = \mathop \sum \nolimits_{s} p^{s} x^{s} . \)
- 8.
Conversely, these two conditions actually imply \( \alpha_{i} \) and \( \beta_{i} \) as defined in the proposition.
- 9.
\( \mathop \sum \nolimits_{n} d_{n} \left( {X_{t + 1} } \right) \) is the same as \( M \) in (19).
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Wang, S. (2018). Micro-foundation of Finance. In: Microeconomic Theory. Springer Texts in Business and Economics. Springer, Singapore. https://doi.org/10.1007/978-981-13-0041-7_5
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DOI: https://doi.org/10.1007/978-981-13-0041-7_5
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