Question

Taylor's theorem shows that any function can be approximated in the vicinity of any convenient point by a series of terms involving the function and its derivatives, Here we look at some applications of the theorem for functions of one and two variables. a Any continuous and differentiable function of a single variable, \(f(x),\) can be approximated near the point \(a\) by the formula \\[ \begin{aligned} f(x)=& f(a)+f(a)(x-a)+0.5 f^{\prime \prime}(a)(x-a)^{2}+\\\ &+\text { terms in } f^{\prime \prime \prime}, f^{\prime \prime \prime}, \ldots . \end{aligned} \\] Using only the first three of these terms results in a quadratic Taylor approximation. Use this approximation together with the definition of concavity to show that any concave function must lie on or below the tangent to the function at point \(a\) b. The quadratic Taylor approximation for any function of two variables, \(f(x, y),\) near the point \((a, b)\) is given by \\[ \begin{aligned} f(x, y)=& f(a, b)+f_{1}(a, b)(x-a)+f_{2}(a, b)(y-b) \\ &+0.5\left[f_{11}(a, b)(x-a)^{2}\right.\\\ &\left.+2 f_{12}(a, b)(x-a)(y-b)+f_{22}(y-b)^{2}\right] \end{aligned} \\] Use this approximation to show that any concave function (as defined by Equation 2.84 ) must lie on or below its tangent plane at \((a, b)\)

Short answer

Question: Show that any concave function lies on or below its tangent line for a single-variable function and its tangent plane for a two-variable function. Answer: Based on our analysis, for both single-variable and two-variable functions, due to the properties of concavity, all terms containing the second derivative for single-variable or the Hessian matrix for two-variable functions resulted in values less than or equal to zero. Consequently, the actual function will be less than or equal to its quadratic Taylor approximation. Since the quadratic Taylor approximation includes the tangent line or tangent plane equation, it means that the concave function lies on or below its tangent line or tangent plane, respectively.

Step-by-step solution

Write down the quadratic Taylor approximation formula for f(x)

The quadratic Taylor approximation for f(x) near a point a is \\[f(x)=f(a)+f'(a)(x-a)+0.5f''(a)(x-a)^2+[\text{higher-order terms}]\\]

Define what it means for a function to be concave

A function is concave if its second derivative \(f''(x)\) is less than or equal to zero for all x, that is, \(f''(x) \le 0\).

Compare the approximation with the tangent to f(x) at point a

The tangent to the function f(x) at point a is given by the equation \\[y=f(a)+f'(a)(x-a)\\] Now we need to compare the values of the actual function \(f(x)\) and its quadratic Taylor approximation. For a concave function, we already know that \(f''(x) \le 0\). Hence, all the terms containing the second derivative will either be zero or negative, so the value of actual function \(f(x)\) will be less than or equal to the value of its quadratic Taylor approximation near point a. Since the quadratic Taylor approximation includes the tangent line equation, it means that the actual function \(f(x)\) lies on or below its tangent at point a. b) Two-variable function:

Write down the quadratic Taylor approximation formula for f(x, y)

The quadratic Taylor approximation for f(x, y) near point (a, b) is \\[\begin{aligned}f(x, y)=& f(a, b)+f_{1}(a, b)(x-a)+f_{2}(a, b)(y-b) \\ &+0.5\left[f_{11}(a, b)(x-a)^{2}+2 f_{12}(a, b)(x-a)(y-b)+f_{22}(y-b)^{2}\right] \end{aligned}\\]

Define the concavity for a function of two variables

A function is concave in two variables if its Hessian matrix is negative semi-definite, that is, the matrix \\[\begin{bmatrix}f_{11}(x, y) & f_{12}(x, y) \\ f_{21}(x, y) & f_{22}(x, y) \end{bmatrix} \\] satisfies the conditions: \\[f_{11}(x, y) \le 0, \quad f_{11}(x, y)f_{22}(x, y)-f_{12}(x, y)f_{21}(x, y) \ge 0\\]

Compare the approximation with the tangent plane to f(x, y) at point (a, b)

The tangent plane to the function f(x, y) at point (a, b) is given by the equation \\[z=f(a, b)+f_{1}(a, b)(x-a)+f_{2}(a, b)(y-b)\\] Now we need to compare the values of the actual function \(f(x, y)\) and its quadratic Taylor approximation. For a concave function in two variables, we already know that the Hessian matrix is negative semi-definite, which implies that the terms containing the second-order partial derivatives of the function will result in values less than or equal to zero. Similar to the single-variable case, since the quadratic Taylor approximation includes the tangent plane equation, it means that the actual function \(f(x, y)\) lies on or below its tangent plane at the point \((a, b)\).

Key concepts

Quadratic Taylor Approximation

Quadratic Taylor approximation is a powerful tool in microeconomics that helps to understand how a function behaves near a specific point. Imagine you are on a hike, and you want to predict the path ahead based on your immediate surroundings. Just like this, the quadratic Taylor approximation predicts the behavior of functions based on the information around a point called a.

The formula looks like this:
\[ f(x) = f(a) + f'(a)(x-a) + 0.5f''(a)(x-a)^2+\text{[higher-order terms]}\]

In essence, this is creating a parabolic shape that best fits the curve of the original function at point a, using the function's value, slope, and curvature (second derivative). The beauty of this approximation is its simplicity and usefulness in analysis, allowing us to create a 'best guess' scenario of a function's behavior in a local area without complex calculations.

Concave Functions

A concave function is shaped like an upside-down bowl, think of a satellite dish or a skateboard half-pipe. This shape has a critical property that if you pick any two points on the curve and connect them with a straight line, the line will always lie above the curve. In mathematical terms, a function is concave if its second derivative is less than or equal to zero across its domain:\[ f''(x) \le 0 \].

This characteristic is particularly meaningful in microeconomics, reflecting decreasing marginal returns or preferences with diminishing marginal utility. For example, the more you consume a good, the less additional utility you get from each new unit consumed. When we map this to a graph, we get our concave function, showcasing at each point along the graph how utility changes with consumption.

Hessian Matrix Negative Semi-definite

In the realm of functions with two variables, concavity is detected through the Hessian matrix. The Hessian is a square matrix of second-order partial derivatives of a function. It's like our multidimensional satellite dish curvature test. If a function is concave, the Hessian matrix is negative semi-definite. This sounds complex, but it simply means that for any vector v, when multiplied by the Hessian, the resulting value is non-positive. Here’s the Hessian for a two-variable function:\[\begin{bmatrix}f_{11}(x, y) & f_{12}(x, y) \ f_{21}(x, y) & f_{22}(x, y)\end{bmatrix}\].

The conditions \(f_{11}(x, y) \le 0\), and \(f_{11}(x, y)f_{22}(x, y) - f_{12}(x, y)f_{21}(x, y) \ge 0\) must hold for the function to be concave. In other words, this measure tells us about the 'bowl shape' of the graph. Applying this in an economic context helps to determine the nature of returns to scale in production or the level of risk aversion in utility functions.