Econ 211 Math Review (Spring 2022-2023) PDF

Econ 211 (Intermediate Microeconomics) Math Review Spring 2022-2023 Fall 2022 This document gives an overview of some basic mathematical tools. Ple...

Econ 211 (Intermediate Microeconomics) Math Review Spring 2022-2023 Fall 2022 This document gives an overview of some basic mathematical tools. Please note it shall be complemented with Chapter 0, ‘Foundational Preliminaries’ from Nechyba. Functions Functions describe the relationship between input variables and outputs y = 𝑓(𝑥) Definition: “A function is a rule that associates with every member of a set, a single member of another set. The first set is called the domain of the function. The function with domain A is said to be defined on A.” Reference: Martin J. Osborne: Intervals and Functions Multivariate functions: function of more than one variable. A multivariable function is just a function whose input and/or output is made up of multiple numbers. For example consider the function, 𝑓(𝑥, 𝑦) = 4𝑥 + 𝑦 This function maps points from a two-dimensional real space onto real number line A rule that assigns values to points in N dimensions can then be written as: 𝑓: ℝ → ℝ This notation is read as follows: ‘The function 𝑓 takes points that lie in the space ℝ and assigns to these points a value from the real number line ℝ ’. Examples of Multivariate functions in economics include demand and supply functions, expenditure functions, etc. Reference: Nechyba, Chapter 0 Khan Academy: Multidimensional Graphs (to visualize multivariate functions) Examples of some single variable functions along with their general form are given below:  Linear functions 𝑓(𝑥) = 𝑎𝑥 + 𝑏  Quadratic functions 𝑓(𝑥) = 𝑎𝑥 + 𝑏𝑥 + 𝑐  Reciprocal functions 𝑓(𝑥) = 𝑐⁄𝑥  Exponential functions 𝑓(𝑥) = 𝑒 References: Martin J. Osborne Graphical representation of Functions Lines and Curves Line: an important function is a line graphed on an x-y plane, written in ‘slope-intercept form’ 𝑦 = 𝑚𝑥 + 𝑏 where ‘m’ is the slope of the curve and ‘b’ is the intercept.  The slope of a line describes the rate at which y changes as x varies. For a line, this value is constant and equal to: Δ𝑦 𝑦 − 𝑦 𝑚= = Δ𝑥 𝑥 − 𝑥 Practice: Write the slope-intercept form of a line that has a slop of -2 and intercept of 1 and plot the line on Cartesian plane. The slope of a line answer two questions: o How steep or flat is the line? o Is the relationship between x and y positive or negative? Curves, on the other hand can represent more complex relationships between y and x, Tangency The point at which a given line and curve just touch without crossing is the tangent point, and is characterized by both the line and curve having the exact same slope Understanding tangency is important in microeconomics Used to solve optimization problems such as profit or utility maximization, or cost minimization Level Curves Let 𝑓 be a function of two variables, and c a constant. The set of pairs (x, y) such that 𝑓(𝑥, 𝑦) = 𝑐 is called the level curve of 𝑓 for the value c. Let 𝑓(𝑥, 𝑦) = 𝑥 + 𝑦 for all (𝑥, 𝑦). The level curve of 𝑓 for the value 1 is the set of all points (𝑥, 𝑦) such that 𝑥 + 𝑦 = 1 i.e., a circle of radius 1. The figure on the right below depicts a collection of level curves of 𝑓 where the number on each curve is the value of the function to which it corresponds. The two-dimensional (2-D) graphs are often simply a slice of some higher dimensional graph whereby, all other variables are held constant. For instance, if a 2-D graph arises from some third variable held constant at some value then, changing that third variable is equivalent to shifting to a different ‘slice’ of the underlying 3-D graph. This shift on the 2-D graph appears as a shift in a line or curve. In consumer theory one of the commonly used level curves is the Indifference curve whereas in producer theory the most common level curve you will encounter is the Isoquant. Indifference curves are level curves of the utility function and isoquants are the level curves of firms’ production function. Reference: Martin J. Osborne: Graphical representation of functions Nechyba, Chapter 0 (A.1.3; B.1.1) Concavity and Convexity A set of points is said to be convex if the straight line that connects any two points in the set is fully contained in the set. For instance, consider the magenta set [0,1] below. No matter which two points in that set we pick, the line that connects the points lies on the real number line and within the magenta set [0,1]. Thus, [0,1] is a convex set. Now consider a set that made up of the segment [0,1] and [2,3], i.e., a set that is the union of [0,1] and [2,3] denoted by [0,1] ∪ [2,3]. Since the set has a “hole” in it we can pick a pair of points such as 0.5 and 2.5 whereby the line connecting the points is not fully contained in the set [0,1] ∪ [2,3]. Thus, the set [0,1] ∪ [2,3] is not convex, i.e., it is a non-convex set. Note that, for a set to be non-convex, all we need is to have one such pair of points whereby the line connecting the points lies at least partially outside the set. For the set to be convex, all lines connecting any pair of points from the set must fully lie within the set. Reference: Nechyba, Chapter 0 (Part A: A.1.1) Concave and Convex functions of single variable A function of a single variable is:  Concave if every line segment joining two points on its graph is never above the graph  Convex if every line segment joining two points on its graph is never below the graph The figure below illustrates this concept. Precisely, Consider a function ′𝑓 defined on the interval [x1, x2]. This function is concave if, for every pair of points a and b given that x1 ≤ a ≤ x2 and x1 ≤ b ≤ x2, the line segment from (a, f(a)) to (b, f(b)) lies on or below the graph of the function. That is: 𝑓 (1 − 𝜆)𝑎 + 𝜆𝑏 ≥ (1 − 𝜆)𝑓(𝑎) + 𝜆𝑓(𝑏) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜆 𝑤𝑖𝑡ℎ 0 ≤ 𝜆 ≤ 1 This is illustrated in the graph below The inequality above holds for any pair of points a and b on the graph above. Different points on the line segment can be analyzed by choosing different values of 𝜆. Likewise, the definition of a convex function can be written as follows: 𝑓 (1 − 𝜆)𝑎 + 𝜆𝑏 ≤ (1 − 𝜆)𝑓(𝑎) + 𝜆𝑓(𝑏) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜆 𝑤𝑖𝑡ℎ 0 ≤ 𝜆 ≤ 1 A function can be both concave and convex if and only if it is linear (affine), of the form 𝑓(𝑥) = 𝛼 + 𝛽𝑥 for all x, and some constants α and β. Note that the strictly concave and strictly convex functions can be defined just as above except that the inequality symbols are written without an ‘equality’ sign. In economics, production function for a firm is often assumed to be increasing and concave. An example of a single input function is depicted in the figure on the right below:  Its increasing as more input results in more output  Its concave as it depicts ‘diminishing returns’ i.e., each additional unit of input results in a smaller increase in output. Reference: Martin J. Osborne: Concave and Convex functions of a single variable Convexity- for functions of more than one variable A set S of n-vectors is convex if (1−λ)x + λx' ∈ S whenever x ∈ S, x' ∈ S, and λ ∈ [0,1]. We call (1 − λ) x + λx' a convex combination of x and x'. Geometrically, the set of all convex combinations of two points x and x' is the line segment connecting x and x'. The figure below illustrates the case for n=2. The set in the first figure is convex, as every line segment joining a pair of points in the set lies entirely in the set. The set in second figure is not convex as the line segment joining the points x and x' does not lie entirely in the set. Reference: Martin J. Osborne: Concave and Convex functions of many variables Rules for Working with Exponents and Logarithms Exponential Rules: 1. 𝑥 = 1 2. 𝑥 = 1 3. 𝑥 𝑥 = 𝑥 4. =𝑥 5. (𝑥 ) = 𝑥 6. 𝑥 = 7. = √𝑥 𝑎𝑛𝑑 𝑥 = ( √𝑥) 8. (𝑥𝑦) = 𝑥 𝑦 and ( ) = Logarithm Rules: 1. ln(𝑥𝑦) = ln(𝑥) + ln(𝑦) 2. ln (𝑥 𝑦) = ln(𝑥) − ln (𝑦) 3. 𝑒 = 𝑥 𝑎𝑛𝑑 ln(𝑒 ) = 𝑥 4. 𝑙𝑛1 = 0 5. 𝑙𝑛𝑥 > 0 𝑓𝑜𝑟 𝑥 > 1 𝑎𝑛𝑑 𝑙𝑛𝑥 < 0 𝑓𝑜𝑟 𝑥 < 1 Reference: Nechyba, Chapter 0 (Part B) Calculus Review First Derivatives First derivative is simply the slope of the function. A single-variable function & its derivative can be represented as: 𝑦 = 𝑓(𝑥) 𝑑𝑓(𝑥) 𝑑𝑦 𝑓 (𝑥) = = 𝑑𝑥 𝑑𝑥 A positive first derivative indicates that a function is increasing and a negative first derivative indicates that a function is decreasing over that region Differentiation Rules: 1. f(x) f’(x) 2. k 0 3. kxn knxn−1 4. ln x 1/x 5. ex ex 6. ax axln a 7. cos x −sin x 8. sin x cos x 9. tan x 1 + (tan x)2 Some General rules 1. Sum rule: If F(x) = f(x) + g(x) then F'(x) = f'(x) + g'(x) 2. Product rule: If F(x) = f(x)g(x) then F'(x) = f'(x)g(x) + f(x)g'(x) 3. Quotient rule: If F(x) = f(x)/g(x) then F'(x) = [f'(x)g(x) − f(x)g'(x)]/(g(x))2 4. Chain rule: If F(X)=f(g(x)) then F’(X)= f’(g(x)g’(x) Practice: Take the derivatives of the following functions a) f(x)= 5x2+4 b) f(x)= √𝑥 (x-1)  (𝑥 − 2) c) 𝑓(𝑥) = (2𝑥 − 3) d) f(x) = √1 − 3𝑥 Reference: Martin J. Osborne: Calculus: one variable For implicit differentiation, you may refer to Martin J. Osborne: Derivatives of functions defined implicitly Second Derivatives These are the derivative of the derivative and describe the curvature of the curve i.e., if a curve is convex or concave For a single-variable function 𝑓(𝑥): 𝑑 𝑓(𝑥) 𝑑 𝑦 𝑓(𝑥) = = 𝑑𝑥 𝑑𝑥 Second derivative tells about the slope of the slope, i.e., whether the slope (measured by its first derivative) is increasing or decreasing over that region A curve is said to be concave if its second derivative is negative meaning that the slope decreases with higher values of x 𝑑 𝑓(𝑥) 0 𝑑𝑥 Partial Derivatives Often, economic equations are multi-variable (e.g., demand for a product may be a function of both price and income). For multivariate functions, partial derivative refers to the slope of the function at a point when we vary one variable while keeping others constant. For instance, consider the following two variable function ‘z’: z = 𝑓(𝑥, 𝑦) We can find two partial derivatives: 𝜕𝑓(𝑥. 𝑦) 𝜕𝑧 𝜕𝑓(𝑥. 𝑦) 𝜕𝑧 𝑓 (𝑥, 𝑦) = = , 𝑓 (𝑥, 𝑦) = = 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦 The rules for taking derivative (noted above), are applicable to partial derivatives- just that other variables are considered to be constant. Practice a) 𝑓(𝑥, 𝑦) = x3y2 b) 𝑓(𝑥, 𝑦) = 5 x2+ 4y3 c) 𝑓(𝑥, 𝑦) = xa y1-a d) Find the second partial derivatives of the function in part(c) above Homogeneous functions Homogeneous functions are often used in economic theory. For a given number k, a function is homogeneous of degree k if, when each of its arguments is multiplied by any number t > 0, the value of the function is multiplied by 𝑡. Precisely, Let f be a function of n variables defined on a set S for which (tx1,..., txn) ∈ S whenever t > 0 and (x1,..., xn) ∈ S, and let k be a number. Then f is homogeneous of degree k if: 𝑓(𝑡𝑥 , … , 𝑡𝑥 ) = 𝑡 𝑓(𝑥 , …. , 𝑥 ) 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝑥 , …. , 𝑥 ) ∈ 𝑆 and all 𝑡 > 0. Reference: Martin J. Osborne: Homogeneous functions Optimization One of the most important uses of calculus is optimization. We often need to optimize a function to find its maxima or minima. For instance, when solving for consumers utility maximization problem or when solving a firm’s expenditure or cost minimization problem. Unconstrained optimization Given the function 𝑦 = 𝑓(𝑥) First-order condition is equivalent to taking the derivative of a function and setting it equal to zero: ( ) 𝑓(𝑥) = =0 This gives the optimum points of the function- minima or maxima? Next? Find Second-order derivatives 𝑑 𝑓(𝑥) 𝑑 𝑦 𝑓′′(𝑥) = = 𝑑𝑥 𝑑𝑥 Remember if: 𝑑 𝑓(𝑥) < 0, the curve is 𝐶𝑜𝑛𝑐𝑎𝑣𝑒, the point is a 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑑𝑥 𝑑 𝑓(𝑥) > 0, the curve is 𝐶𝑜𝑛𝑣𝑒𝑥, the point is a 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑑𝑥 Practice a) Optimize the function y = 5 x2 – 100x and determine if the point is a maximum or a minimum. / / b) For the function, f(x)= 𝑥 (20 − 2𝑥) find the function value at the optimal point. Constrained optimization A convenient way of solving constrained optimization problems is by using the ‘Lagrangean’. The Lagrangean is a technique for transforming a constrained optimization problem into an unconstrained problem by combining the objective function and the constraint into one equation. The λ is a variable that multiplies the constraint. For instance, with the objective function, 𝑓(𝑥. 𝑦) and the constraint 𝑔(𝑥, 𝑦)=c, the maximization problem can be written as: 𝑚𝑎𝑥 , 𝑓(𝑥. 𝑦)𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑔(𝑥, 𝑦) = 𝑐 The Lagrange equation can then be written as: 𝐿(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) − 𝜆(𝑔(𝑥) − 𝑐) Now maximize by taking the partial derivatives of the above equation with respect to x, y, and λ, and set them equal to zero. These partial derivatives are known as the ‘first-order conditions’ or more commonly FOCs: 𝜕𝑓(𝑥. 𝑦) 𝜕𝑔(𝑥. 𝑦) 𝐿 (𝑥, 𝑦) = −𝜆 =0 𝜕𝑥 𝜕𝑥 𝜕𝑓(𝑥. 𝑦) 𝜕𝑔(𝑥. 𝑦) 𝐿 (𝑥, 𝑦) = −𝜆 =0 𝜕𝑦 𝜕𝑦 𝐿 (𝑥, 𝑦) = 𝑔(𝑥, 𝑦) − 𝑐 = 0 This system of equations with three equations and three unknowns can then be solved to find the optimal values of the variables i.e., 𝑥 ∗ and 𝑦 ∗. Note that the third first-order condition is simply the constraint. Reference: for more detailed discussion and examples refer to Martin J. Osborne: Optimization with an equality constraint