Choices Under Certainty

Financial Markets

Department of Mathematical Sciences, Aalborg University

About the course

This course serves as an “introduction” to financial theory.
Both mathematical and economic aspects are discussed.

Among others, the following topics will be covered:

* choices under certainty

* expected utility and risk taking

* mean-variance diagram

* minimum variance portfolio

* Capital Asset Pricing Model (CAPM) and evidence

* Arbitrage Pricing Theory (APT) and evidence

About the course, practicalities

Evaluation is pass/fail.
It is a “running evaluation”, so working through the semester is required to pass.
We will use GitHub for handling the (5) self-study sessions.
You have to “commit” your solutions as proof of participation.

Choices under certainty

Finance is the science of managing resources and investments, including the creation and analysis of derivative products.
At is core, finance deals with the notion of uncertainty: returns on investments are unknown a-priori.
So the question arises on how/why trading is done.
Which brings us to talk about how to model decisions under different choices.

Agent’s choice problem

We start by abstracting from randomness.
To model consumers’ choices, we define a set of choices \(X\subset \mathbb{R}^n\), and a weak preference relation \(\succsim\).
\(X\) is typically composed of \(n\) “goods”.
For \(x,y\in X\), we say that \(x\) is at least as preferred as \(y\) if \(x\succsim y\).
The pair \((X,\succsim)\) characterizes the agent’s choice problem.

Preference relation

From the weak preference relation we can define:

the strong preference relation, \(\succ\), as \[x\succ y\ \Leftrightarrow x\succsim y \text{ but not } y\succsim x\]
the indifference relation, \(\sim\), as \[x\sim y\ \Leftrightarrow x\succsim y \text{ and } y\succsim x\]

Moreover, the indifference curve for \(x\) by \(\{y\in X: x\sim y\}\).

Analogously, the lower and upper contour sets.

Preference relation \((ii)\)

The fact that we consider the components of \(X\) to be “goods” can be used to show that the preference relation is strongly monotone.

Strong monotonicity

The preference relation \(\succsim\) on \(X\) is strongly monotone if for \(x,y\in X\) with \(y\geq x\) and \(y\neq x\), then \(y\succ x\).

Strong monotonicity is colloquially stated as “more is better”.

Preference relation \((iii)\)

We assume that the preference relation satisfies the rationality assumption.

Rationality assumption

The preference relation \(\succsim\) on the set of choices \(X\) is said to be rational if it satisfies:

Completeness: \(\forall x,y\in X\), it holds that \(x\succsim y\) or \(y\succsim x\).

Transitivity: \(\forall x,y,z\in X\), if \(x\succsim y\) and \(y\succsim z\), then \(x\succsim z\)

Preference relation \((iv)\)

Failings of rationality:

Utility functions

Utility functions are defined so that we can use our mathematical tools for the analysis.

Definition

We say that \(u:X\rightarrow\mathbb{R}\) is a utility function representing the preference relation \(\succsim\) if \[x\succsim y \ \ \Leftrightarrow \ \ u(x)\geq u(y), \ \ \ \ \forall x,y\in X.\]

In short, utility functions take us from the “abstract” space of bundles and preferences to the real line.

Utility functions \((ii)\)

The existence of an utility function is not guaranteed unless the preference relation satisfies the continuity assumption.

Continuity assumption

The preference relation \(\succsim\) on the set of choices \(X\) is said to be continuous if \(\forall x\in X\), the sets \(\{y\in X:y\succsim x\}\) and \(\{y\in X:x\succsim y\}\) are closed.

That is the lower and upper contour sets are closed.

Utility functions \((iii)\)

In fact, for continuous preference relations (plus extra conditions), we can show that there exists a continuous utility function that represents them.

Theorem (Existence of utility function)

Let \(\succsim\) be a strongly monotone, rational, and continuous preference relation on \(X\). Then, there is a continuous utility function \(u(\cdot)\) that represents \(\succsim\).

Utility functions \((iv)\)

Proof for the existence of utility function theorem.

Construction:

We consider the positive quadrant given that we are dealing with “goods” and strong monotonicity.
For \(x\in X\), consider the indifference curve given by \(\{y\in X: x\sim y\}\).
We will show that the ray (line section) given by \(\{\alpha \mathbb{e}: \alpha\in\mathbb{R}\}\) with \(\mathbb{e}=(1,\cdots,1)\) intersects the indifference curve at a single point.
Define \(u(x):=\alpha_x\), where \(\alpha_x\) is the implied unique value above.

To show that \(\alpha_x\) exists:

Consider the upper and lower contour sets of \(x\) so that \(\{\alpha\in\mathbb{R}_+:\alpha\mathbb{e}\sim x\}\) is nonempty and closed.
By monotonicity, both sets are non-empty.
By completeness their union is equal to all \(\mathbb{R}_+\).
By continuity, both are closed.
Hence given that \(\mathbb{R}_+\) can be written as the union of two closed sets, their intersection is non-empty.
Which shows that there is a value \(\alpha_x\) such that \(\alpha_x \mathbb{e} \sim x\), given that it lives in the intersection.

Second, \(\alpha_x\) is unique:

By monotonicity \(\alpha_1\mathbb{e}\succ \alpha_2\mathbb{e}\) whenever \(\alpha_1>\alpha_2\).

Third, it represents \(\succsim\):

\(\Rightarrow]\) If \(\alpha_x\geq \alpha_y\), then \(x\sim \alpha_x \mathbb{e}\succsim \alpha_y\mathbb{e}\sim y\), so that \(x\succsim y\), where we used monotonicity.
\(\Leftarrow]\) If \(x\succsim y\), then \(x\sim \alpha_x \mathbb{e}\succsim \alpha_y\mathbb{e}\sim y\), so that \(\alpha_x\geq\alpha_y\), where we used monotonicity.

Finally, \(u(x)\) is continuous:

We show that for any sequence \(\{x_n\}_{n=1}^{\infty}\) such that \(\lim_{n\to\infty} x_n=x\), we have that \(\lim_{n\to\infty} \alpha_{x_n}=\alpha_x\).
Let \(\epsilon>0\) and \(\{x_n\}_{n=1}\) be a sequence that converges to \(x\), by monotonicity and continuity, there exist an \(N\) such that \(|x_n-x|<\epsilon\) \(\forall n>N\).
That can be used to show that \(\alpha_{x_n}\) converges to \(\alpha_x\).

Utility functions \((v)\)

Note that the utility function from the Theorem is only guaranteed to be continuous.
Nonetheless, in practice we usually require it to be (twice) differentiable for the analysis.
Additional conditions as homotheticity, (quasi)-linearity, and quasiconcavity are sometimes required.

Utility functions (example)

Cobb-Douglas utility function: \(u(x,y) = K x^\alpha y^{1-\alpha}\)

Code

#Julia code
using Pkg; Pkg.activate(pwd())
using Plots
X = 0.1:0.005:0.35
K = 0.2:0.1:0.5
α = 0.5

f(x,k) = (k/x^α)^(1/(1-α))

y=[f(x,k) for x in X, k in K]

plot(X, y, label=["K=0.2" "K=0.3" "K=0.4" "K=0.5"])

Utility functions (example)

Cobb-Douglas utility function: \(u(x,y) = K x^\alpha y^{1-\alpha}\)

Code

#Julia code
using Plots
X = 0.1:0.005:0.35     
α = 0.5

f(x,k) = (k/x^α)^(1/(1-α))

y=[f(x,0.5) for x in X]

plot(X,y, fillrange = zeros(size(X,1),1), 
fillalpha = 0.3, label="{y∈R²:y≺x}")

plot!(X,y,line = :dash, color = :red, 
linewidth = 3, label="{y∈R²:y~x}")

Utility functions (properties)

Monotonic transformations of utility functions represent the same preference relation.
Utility functions are downward sloping.
Each indifference curve is a level set of the utility function.
Each bundle is part of an indifference curve.
Two indifference curves cannot intersect.

Budget constraint

With the utility function defined, the two extra components required to solve the agent’s choice problem are:

the agent has some (initial) wealth, \(w\in\mathbb{R}_+\).
she is price taker with price vector given by \(p\in \mathbb{R}^n_{+}\).

Under these conditions, the (Walrasian) budget set is given by \[B_{p,w} := \{x\in X: p\cdot x\leq w\}.\]

Agent’s choice problem (at last)

The choice problem of a rational agent with utility function \(u(\cdot)\), wealth \(w\), and prices \(p\) is given by:

\[\max u(x) \text{ s.t. } p\cdot x \leq w.\]

Theorem (Solution of Agent’s choice problem)

If \(p>0\) and \(u(\cdot)\) is continuous, then the utility maximization problem has a solution.