Book contents
- Frontmatter
- Contents
- List of illustrations
- Introduction to the second edition
- Preface to the second edition
- Preface to the first edition
- Table of notation
- Table of assumptions
- A General equilibrium theory: Getting acquainted
- B Mathematics
- C An economy with bounded production technology and supply and demand functions
- D An economy with unbounded production technology and supply and demand functions
- E Welfare economics and the scope of markets
- F Bargaining and equilibrium: The core
- G An economy with supply and demand correspondences
- 23 Mathematics: Analysis of point-to-set mappings
- 24 General equilibrium of the market economy with an excess demand correspondence
- 25 U-shaped cost curves and concentrated preferences
- H Standing on the shoulders of giants
- Bibliography
- Index
23 - Mathematics: Analysis of point-to-set mappings
from G - An economy with supply and demand correspondences
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- List of illustrations
- Introduction to the second edition
- Preface to the second edition
- Preface to the first edition
- Table of notation
- Table of assumptions
- A General equilibrium theory: Getting acquainted
- B Mathematics
- C An economy with bounded production technology and supply and demand functions
- D An economy with unbounded production technology and supply and demand functions
- E Welfare economics and the scope of markets
- F Bargaining and equilibrium: The core
- G An economy with supply and demand correspondences
- 23 Mathematics: Analysis of point-to-set mappings
- 24 General equilibrium of the market economy with an excess demand correspondence
- 25 U-shaped cost curves and concentrated preferences
- H Standing on the shoulders of giants
- Bibliography
- Index
Summary
Correspondences
We will call a point-to-set mapping a correspondence. A function maps points into points. A correspondence (or point-to-set mapping) maps points into sets of points. Let A and B be sets. We would like to describe a correspondence from A to B. For each x ∈ A we associate a nonempty set β ⊂ B by a rule ϕ. Then we say β = ϕ(x), and ϕ is a correspondence. The notation to designate this mapping is ϕ : A → B. For example, suppose A and B are both the set of human population. Then we could let ϕ be the cousin correspondence ϕ(x) = {y | y is x's cousin}. Note that if x ∈ A and y ∈ B, it is meaningless or false to say y = ϕ(x), rather we say y ∈ ϕ(x). The graph of the correspondence is a subset of A × B : {(x, y) | x ∈ A, y ∈ B and y ∈ ϕ(x)}.
For example, let A = B = R. We might consider ϕ(x) = {y | x - 1 ≤ y ≤ x + 1}. The graph of ϕ(·) appears in Figure 23.1.
Upper hemicontinuity (also known as upper semicontinuity)
In the balance of this chapter and the next, we concentrate on mappings from one real Euclidean space into another, from RN into RK, for N ≥ 1 and K ≥ 1.
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- Information
- General Equilibrium TheoryAn Introduction, pp. 279 - 292Publisher: Cambridge University PressPrint publication year: 2011