Supportive Quantitative Methods

The Cartesian product of two sets \(A\) and \(B\) is a set with elements pairs combining an element of \(A\) and an element of \(B\). Specifically, we write \[ A \times B = \left\{(a,b)\colon\, a\in A,\, b\in B\right\}. \] Elements of the Cartesian product are denoted by \((a,b)\in A\times B\). Cartesian products are analogously defined of any finite collection of \(n\) sets. For example, \[ \times_{i=1}^{n} X_{i} = \left\{(x_{1}, \dots, x_{n})\colon\, x_{1}\in X_{1},\, \dots,\, x_{n}\in X_{n}\right\}. \]

It is rare for economic choices to be made in isolation. When a person visits a retail store, she purchases a variety of products instead of a single one on most occasions. In production settings, entrepreneurs combine labor, capital, and other production factors to produce the desired output. In addition, entrepreneurs have to consider not only a particular pair of labor and capital but rather how this pair compares to other feasible pairs of capital and labor they could employ in production. A commonly used way (for reasons going beyond this introduction's scope) to mathematically describe such collections of choices is via their convex combinations. For any real number \(\alpha\in[0,1]\), and any two points \(x_{1}, x_{2} \in X\), we say that \(x = \alpha x_{1} + (1 - \alpha) x_{2}\) is a convex combination of \(x_{1}\) and \(x_{2}\).

We say that a set \(X\) is convex if it contains all the convex combinations of its elements. Namely, \(X\) is a convex set if for every \(\alpha\in[0,1]\), and every \(x_{1}, x_{2} \in X\), we have \(\alpha x_{1} + (1 - \alpha) x_{2} \in X\).

The functions' graphs are among the most common and easily communicated ways to represent functions and depict their properties. As a mathematical object, the graph of a function is defined as a set. The graph of a function \(f \colon X \to Y\) is the subset of the Cartesian product \(X \times Y\) defined by \[ \mathcal{G}(f) = \left\{(x, f(x)) \in X\times Y \colon\, x\in X \right\}. \] Graphs take familiar geometric representations of collections of points in the Cartesian coordinates for real-valued functions of one or two real variables.

The graphs of functions of two real variables are also called surfaces.

Although each \(x \in X\) is uniquely mapped to \(f(x)\), different elements of \(X\) can be mapped to the same element of \(Y\). A function that maps each element of \(X\) to a unique element of \(Y\) is called one to one or injection.

Not all elements of \(Y\) have to be necessarily used by the association rule of the function. The subset of the codomain \(Y\) used by the function's rule is called range, and it is denoted by \(f(X)\). In general, we have \(f(X)\subseteq Y\), and if the last relation holds with equality, i.e., \(f(X) = Y\), we say that \(f\) is onto or a surjection. A function that is simultaneously an injection and a surjection is called a bijection.

It is fruitful to define rules for associating things from multiple sets to a codomain set on some occasions. For example, sweet recipes use different combinations of essential ingredient quantities to prepare various desserts. Chocolate cakes use chocolate and sugar but no vanilla, while vanilla cakes use sugar and vanilla but not chocolate. If we let \(X_{1}\), \(X_{2}\), and \(X_{3}\) be the set of potential quantities for chocolate, sugar, and vanilla, and \(Y\) be the set of deserts, we could conceive recipes as functions of the form \(f: X_{1}\times X_{2}\times X_{3} \to Y\).

A mapping \(f\colon \times_{i=1}^{n} X_{i} \to Y\) is called a function of multiple variables (in this case \(n\) variables). We write \(f(x_{1}, x_{2}, \dots, x_{n})\in Y\) to denote the value of the an element \((x_{1}, x_{2}, \dots, x_{n}) \in \times_{i=1}^{n} X_{i}\). Sometimes, it is more convenient to think of the elements of the Cartesian product \(\times_{i=1}^{n} X_{i}\) as vectors and shorten the notation to \(f(x)\) with the implicit convention that \(x = (x_{1}, x_{2}, \dots, x_{n})\).

Linear functions are special functions that preserve addition and (vector) multiplication. This property makes linear functions very useful in sciences because they allow us to calculate results of addition and multiplication in the codomain from the corresponding operations in the domain and vice-versa, which can significantly reduce the calculation complexities.

A function \(f\colon X \to Y\) is linear if (and only if) \[ f(x_{1} + \alpha x_{2}) = f(x_{1}) + \alpha f(x_{2}) \] for all \(x_{1}, x_{2} \in X\) and \(\alpha \in \mathbb{R}\). Linear functions necessarily satisfy the condition \(f(0) = 0\). Geometrically, linear functions of a single variable have line graphs passing through the origin of the Cartesian coordinates. Formally, not all lines are linear functions (although this misconception becomes increasingly more mainstream). Functions with line graphs that do not pass through the origin of the Cartesian coordinates are called affine. A function is affine if (and only if) it preserves convex combinations, i.e. \[ f(\alpha x_{1} + (1 - \alpha) x_{2}) = \alpha f(x_{1}) + (1 - \alpha) f(x_{2}), \] for all \(x_{1}, x_{2} \in X\) and \(\alpha \in [0,1]\).

In general, the value of a function can erratically change when it is given different domain values. Monotonic functions constitute a special class of functions with more predictable value changes restricted in particular directions. There are two main types of monotonic functions, namely increasing and decreasing functions. Although each type can be specialized using stricter monotonicity concepts, these two types adequately (for this introduction) describe the basic idea of monotonicity. A real valued function \(f\) is increasing if for all \(x_{1}, x_{2}\in\mathbb{R}\) such that \(x_{1} \ge x_{2}\), we have \(f(x_{1}) \ge f(x_{2})\). A real valued function \(f\) is said to be decreasing if for all \(x_{1}, x_{2}\in\mathbb{R}\) such that \(x_{1} \ge x_{2}\), we have \(f(x_{1}) \le f(x_{2})\).

Functions are well-defined rules mapping elements of a set \(X\) to elements of a set \(Y\). What happens, however, if we are interested in examining how elements of \(Y\) are associated with elements of \(X\) according to a given function? For example, suppose that \(c\) is a consumption policy function (typically found in macroeconomics and finance) associating wealth (measured in Euros) with optimal consumption choices (measured in Euros). Thus, for each wealth level \(w\), \(c(w)\) gives the optimal spending allocated to consumption commodities and services. Sometimes it is also relevant to inquire about the required wealth for which a particular consumption level is optimal. Such an inquiry associates consumption spending with wealth levels, which goes in the inverse direction of the association that \(c\) describes.

This idea of inversion generalizes in mathematics via the concept of the inverse function. However, function inversion is not always possible. Suppose that we are given a function \(f\colon X \to Y\). To have a well-defined inverse function one has to be able to associate each element of \(Y\) with exactly one element of \(X\). If for some \(y\in Y\), there exist two \(x_{1}, x_{2} \in X\) such that \(f(x_{1}) = y = f(x_{2})\) one cannot unambiguously define an association from \(Y\) to \(X\) based on \(f\) (which value of \(X\) should be chosen? \(x_{1}\) or \(x_{2}\)?). Thankfully, we do not encounter such problems if we are given a one-to-one function because such functions guarantee that each \(x\in X\) is mapped to a unique element of \(Y\). The inverse function of a function \(f\colon X \to Y\), if it exists, is a function undoing the operation of \(f\). Namely, it is a function \(f^{-1}\colon f(X) \to X\) such that \(f^{-1}(f(x)) = x\) for all \(x \in X\).

A secant is a line with endpoints two distinct points of a function's graph. The secant's slope measures the rate of change of a function for a given increment. For a real function \(f\) and two points \(x_{1}, x_{2}\) in its domain, the secant's slope is given by the ratio of changes in the range and the domain of the function, i.e., \[ \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x} = \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}. \]

For highly curved functions, the slope of the secant can change significantly as the increment \(\Delta x\) changes.

The first derivative of a function gives us the rate of change of a function by formalizing what is meant by small changes in a way that avoids arbitrary choices of increments. The first derivative (or simply derivative if first is understood from context) of a function \(f\), if exists, is the limit of the secant of \(f\) as the domain change increments go to zero, i.e., \[ \frac{\mathrm{d} f(x)}{\mathrm{d} x} = \lim_{x_{2} \to x_{1}} \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}. \] The derivative is a (new) function giving the rate of change of \(f\) for infinitesimal changes in its domain variable. The derivative of \(f\) is often denoted by \(f'\).

If a function’s derivative is negative, the function is decreasing. If it is positive, the function is increasing.

Nothing prevents us from applying the same differentiation process to a first derivative of a function. Suppose that we are given a function \(f\), for which we can calculate the first derivative \(f'\). We can then think of \(f'\) as a function, say, \(g=f'\), and calculate the first derivative of \(g\). This gives us the derivative \(g'\) of \(g\), which can be associated back to the original function \(f\) by \(g' = (f')'\).

Why would we be interested in something like this? The derivative of the derivative of \(f\) reveals information about the curvature of \(f\). The curvature of a function can be geometrically thought of as the function's bending. In economics and finance, the curvatures are pivotal in calculating the risk preferences of individuals (see the relative and absolute risk aversion measures).

The second derivative of a function \(f\), if exists, is the limit of second-order finite differences of \(f\) as the domain change increments go to zero, i.e.,

\begin{align*} \frac{\mathrm{d}^{2} f(x)}{\mathrm{d} x^{2}} &= \lim_{\Delta x \to 0} \frac{f(x_{1} + \Delta x) - 2f(x_{1}) + f(x_{1} - \Delta x)}{(\Delta x)^{2}} \\ &= \lim_{\Delta x \to 0} \frac{\frac{f(x_{1} + \Delta x) - f(x_{1})}{\Delta x} - \frac{f(x_{1}) - f(x_{1} - \Delta x)}{\Delta x}}{\Delta x} \\ &= \lim_{\Delta x \to 0} \frac{f'(x_{1} + \Delta x) - f'(x_{1})}{\Delta x}. \end{align*}

If a function's second derivative is negative, the function is concave (curvature opening down). If its second derivative is positive, it is convex (curvature opening up).

We can recursively continue defining derivatives of higher order. The interpretations of higher-order derivatives become more obscure. A notable exception is the third derivative, which is used to measure the prudence of preferences (attitude towards savings for rainy days) in household finance and macroeconomics. Higher-order derivatives are, nevertheless, very useful in economics because they can be used to approximate many functions with arbitrary precision. See the discussion of the Taylor Expansion Section for an example.

For any \(k\ge 2\), the \(k\text{-th}\) order derivative of a function \(f\), if exists, is the limit of the secant of the \((k-1)\text{-th}\) derivative of \(f\) as the domain change increments go to zero, i.e.,

\begin{align*} \frac{\mathrm{d}^{k} f(x)}{\mathrm{d} x^{k}} &= \lim_{\Delta x \to 0} \frac{\frac{\mathrm{d}^{k-1} f (x_{1} + \Delta x)}{\mathrm{d} x^{k-1}} - \frac{\mathrm{d}^{k-1} f (x_{1})}{\mathrm{d} x^{k-1}}}{\Delta x}. \end{align*}

The \(k\text{-th}\) order derivative of a function \(f\) is more compactly denoted as \(f^{(k)}\). With this notation we can rewrite the definition for the \(k\text{-th}\) order derivative as \(f^{(k)}(x) = (f^{(k-1)})'(x)\).

A function is continuous if arbitrarily small changes in the domain result in arbitrarily small changes in the range of the function. Giving an exact definition or a good intuition of continuity requires introducing concepts that go way beyond the scope of the material. In business and economic studies, continuity is mostly treated as a technicality that is always present in the used functions. The good news is that whenever a function is differentiable, it is also continuous. Therefore, familiarity with the usual calculus toolbox can serve as a guide for continuity.

A stronger concept (that means more restrictive, i.e., fewer functions are satisfying it) is that of smoothness. We can define smoothness based on the ideas that we have already introduced. A function \(f\) is said to be smooth if its derivative \(f^{(k)}\) exists for any integer \(k\in\mathbb{N}\). The graphs of smooth functions do not exhibit any kinks or corners, which is from where these functions are named after.

Smooth functions are very useful in economics and finance because many of these functions can be approximated by expressions based on their derivatives. An example of such an approximation is the Campbell-Shiller decomposition, which provides a simple way to describe asset returns as functions of prices and dividends in finance.

For a smooth function \(f\) the Taylor series expansion of \(f\) at \(x_{0}\) is the function given by

\begin{align*} T_{f}(x) &= \sum_{k=0}^{\infty} \frac{f^{(k)}(x_{0})}{k!}(x - x_{0})^{k} \\ &= f(x_{0}) + f^{(1)}(x_{0})(x - x_{0}) + \frac{f^{(2)}(x_{0})}{2}(x - x_{0})^{2} + \dots \end{align*}

For many well-behaved functions, the Taylor series is convergent and it approximates \(f\). For such cases, we simply write

\begin{align*} f(x) &= f(x_{0}) + f^{(1)}(x_{0})(x - x_{0}) + \frac{f^{(2)}(x_{0})}{2}(x - x_{0})^{2} + \dots \end{align*}

The product rule is used to calculate the derivatives of products of functions. The derivative of the product of two functions \(f\) and \(g\) is given by \[ (fg)'(x) = f'(x)g(x) + f(x)g'(x). \]

The quotient rule is used to calculate the derivatives of ratios of functions. The derivative of the ratio of \(f\) to \(g\) is given by \[ \left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^{2}}. \]

A composite function is a function that combines the transformations of two functions. Suppose that we are given two functions \(f \colon X \to Y\) and \(g\colon Y \to Z\). We can define a composite function \(h\colon X \to Z\) by \(h(x) = g(f(x))\). The composition of \(g\) and \(f\) is sometimes denoted as \(g\circ f\).

The derivatives of composite functions are calculated according to the chain rule. The derivative of the composition of two functions \(f\) to \(g\) is given by \[ (g \circ f)'(x) = g'(f(x)) f'(x). \]

The idea of differentiation is not restricted to the functions of one variable. We can calculate the rate of change for functions with more variables by letting one variable variate while keeping all other variables fixed. This concept has many applications in economics, business, and finance because many commonly used functions have more than one variable.

Suppose that we are given a function \(f\colon X_{1} \times X_{2} \to Y\). The partial derivative of \(f\) with respect to the first variable is defined as \[ \frac{\partial f(x_{1}, x_{2})}{\partial x_{1}} = \lim_{\Delta x_{1} \to 0} \frac{f(x_{1} + \Delta x_{1}, x_{2}) - f(x_{1}, x_{2})}{\Delta x_{1}}. \] The partial derivative of \(f\) with respect to the second variable is defined as \[ \frac{\partial f(x_{1}, x_{2})}{\partial x_{2}} = \lim_{\Delta x_{2} \to 0} \frac{f(x_{1}, x_{2} + \Delta x_{2}) - f(x_{1}, x_{2})}{\Delta x_{2}}. \] Albeit a bit tedious, it is straightforward to generalize the concept for functions of more than two variables. The partial derivative of a function \(f\) of \(k\) variables with respect to the \(j\text{-th}\) variable is given by

\begin{align*} \frac{\partial f(x)}{\partial x_{j}} &= \frac{\partial f(x_{1}, \dots, x_{j}, \dots, x_{k})}{\partial x_{j}} \\ &= \lim_{\Delta x_{j} \to 0} \frac{f(x_{1}, \dots, x_{j} + \Delta x_{j}, \dots, x_{k}) - f(x)}{\Delta x_{j}}. \end{align*}

It is common to denote the partial derivatives using a shorthand notation based on the differentiation variable. In this notation, the partial derivative with respect to the first variable is written as \(f_{x_{1}}\), and the partial derivative with respect to the second variable as \(f_{x_{2}}\).

Suppose that we are given a decision problem described by a function \(f(c, p)\), where \(c\) is a decision variable under the agent's control and \(p\) is a parameter variable that the agent does not affect. The goal of the decision problem is to maximize \(f\). The unconstrained optimization problem corresponding to this decision setting is

\begin{align*} \max_{c} f(c, p). \end{align*}

The function \(f\) is known as the objective and the optimization variable \(c\) is the control variable of the optimization problem.

Using Fermat's theorem, we know that if \(f\) attains its maximum at \(c_{0}\) given \(p\), then its first (partial) derivative should necessarily be equal to zero, namely \(f'(c_{0}, p) = 0\). We can use this idea to locate any local maximizers. By imposing the condition

\begin{align*} f'(c, p) \overset{!}{=} 0, \end{align*}

and solving it for \(c\), we can locate potential local maximizers of the optimization problem. This condition is so commonly used that it is named. It is called the first-order or necessary condition. The candidate maximizers are called critical points.

Thus, we have a way of obtaining candidate maximizers, but how do we know which candidates are actually maximizers? Some of the candidates might as well be minimizers, as those too satisfy the first-order condition. Whether a candidate is a local maximizer or minimizer depends on the (local) curvature of the function. If the function is concave in a small neighborhood around the candidate (curvature opening down), then the candidate is a local maximizer. Instead, if the function is convex in a small neighborhood around the candidate (curvature opening up), then the candidate is a local minimizer.

Therefore, the problem's maximizers are all the candidates that satisfy the condition

\begin{align*} f''(c, p) \overset{!}{<} 0. \end{align*}

This condition is called the second-order or sufficient condition.

We have kept the parameter value \(p\) fixed in the preceding discussion. Thus, any maximizer that we have obtained depends on the fixed value of \(p\). We can change the value of \(p\) to \(p'\) and repeat the process. The new maximizer will be based on the new parameter value \(p'\) in this case. In this manner, we can define a function such that for each parameter value \(p\), we get an optimizer \(c(p)\). This mapping is called the optimal control function. Substituting the optimal control in the objective function we get

\begin{align*} v(p) = f(c(p), p) = \max_{c} f(c, p). \end{align*}

In this way, we have eliminated the dependence of the objective on the control variable and expressed the maximized values exclusively as a function of the parameter \(p\). The function \(v\) gives the maximum value of \(f\) for each parameter \(p\), and it is called the value function.

In some decision problems, agents do not have access to the full universe of alternatives. For instance, firms are subject to technological constraints, and they can only choose to produce technologically feasible quantities. Similarly, households can only buy commodities and services that are affordable for their budgets. How can we describe such decision problems?

Suppose that we are given a decision problem described again by a function \(f(c, p)\), where \(c\) is the control variable and \(p\) is a parameter. The goal of the decision problem is to maximize \(f\). However, the agent cannot freely choose \(c\). The agent's choices have to satisfy the constraint \(g(c, I) = 0\), where \(I\) is another parameter that the agent does not control. The constrained optimization problem corresponding to this decision setting is

\begin{align*} \max_{c} \, & f(c, p) \\ s.t. \quad &g(c, I) = 0. \end{align*}

The general approach with which one solves constraint optimization problems is by forming the Lagrangian function

\begin{align*} \mathcal{L}(c, \lambda, p, I) = f(c, p) - \lambda g(c, I). \end{align*}

The Lagrangian introduces an additional variable \(\lambda\), which is called the Lagrange multiplier. The Lagrange multiplier takes many useful interpretations in economics and finance decision problems. Specifically, the Lagrange multiplier gives us the shadow price of the constraint, i.e., the change in the value function when the constraint is infinitesimally relaxed.

Under some conditions, the maximizers or minimizers of the constrained maximization problem are obtained by solving the unconstrained optimization problem with \(\mathcal{L}(c, \lambda, p, I)\) as the objective and \(c\), \(\lambda\) as control variables. Therefore, we can use the Lagrangian to reduce a constrained optimization problem to the relatively simpler unconstrained problem

\begin{align*} \max_{c, \lambda} \mathcal{L}(c, \lambda, p, I), \end{align*}

We can get the maximizers or minimizers of the original constrained problem using the unconstrained problem's first and second-order conditions.