Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66
Basic Math Concepts In this lecture we will review some basic mathematical concepts employed throughout the course: 1 Single Variable Functions 2 Limit of a Function and Continuity 3 Derivative, Higher Order Derivatives, Differential 4 Multi-Variable Functions 5 Partial Derivatives of a Multi Variable Functions 6 Total Differential, Chain Rule, Second Partial Derivatives 7 Simple Integration Van Essen (U of A) Basic Math Concepts 2 / 66
What is a Function? Definition y is a function of x if there is a relationship between y and x that defines, for each value of x, a corresponding value of y. Van Essen (U of A) Basic Math Concepts 3 / 66
Useful Single Variable Functions and their Graphs We describe a few special families of single variable functions: linear functions, general polynomial functions, the exponential function, and the natural logarithm function. Van Essen (U of A) Basic Math Concepts 4 / 66
Linear Functions A linear function is a function of the form f (x) = ax + b, where a and b are parameters. The parameter a is the slope of the function and the parameter b is the vertical intercept i.e., the value the function takes at x = 0. The main property of a linear function is that the dependent variable changes at a constant rate with respect to the independent variable i.e., their graph is just a straight line. Van Essen (U of A) Basic Math Concepts 5 / 66
Linear Functions Example The graph of the linear function f (x) = 2x + 1 is given below: y 4 2 5 4 3 2 1 1 2 3 4 5 2 x 4 Figure: Linear Function f (x) = 2x + 1 Van Essen (U of A) Basic Math Concepts 6 / 66
Polynomial Functions The linear function is a special case of a polynomial function. An n-th degree polynomial takes the form f (x) = a 0 + a 1 x + a 2 x 2 + + a n x n. The quadratic function i.e., a function of the form f (x) = a 0 + a 1 x + a 2 x 2. Van Essen (U of A) Basic Math Concepts 7 / 66
Polynomial Functions The graph of a quadratic is easy to remember it is either a mountain or a valley. It looks like a mountain if a 2 < 0, and it looks like a valley if a 2 > 0. A quadratic with a 2 > 0 is given below. y 4 2 5 4 3 2 1 1 2 3 4 5 2 x 4 Figure: Example of Quadratic with a 2 > 0. Van Essen (U of A) Basic Math Concepts 8 / 66
Inverse Functions Definition If y = f (x), then x = h(y) is the inverse function of f if f (h(y)) = y. Example If y = f (x) = 2x, then the function x = h(y) = 1 2 y is the inverse function since f (h(y)) = 2 1 2 y = y The exponential and logarithm functions are inverses of one another. Van Essen (U of A) Basic Math Concepts 9 / 66
Exponential and Natural Log Functions The exponential and logarithm functions are two useful functions for a variety of reasons. The exponential function is a function of the form f (x) = ae x and the natural logarithm function is of the form f (x) = ln x. y 4 2 5 4 3 2 1 1 2 3 4 5 2 x 4 Figure: The exponential and natural logartihm functions Van Essen (U of A) Basic Math Concepts 10 / 66
Exponential and Natural Log Functions Logarithms and exponential functions are inverses of one another. In other words, ln e x = x e ln x = x Van Essen (U of A) Basic Math Concepts 11 / 66
Exponential and Natural Log Functions In addition to the two inverse relationships, the following four properties of these functions are also useful ones to keep in mind. 1 ln(xy) = ln x + ln y 2 ln( x y ) = ln x ln y 3 ln x n = n ln x 4 e x +y = e x e y Example Suppose f (x, y) = x 2 y 3 where x, y > 0, then by properties (1) and (3) we have that ln f (x, y) = 2 ln x + 3 ln y. Van Essen (U of A) Basic Math Concepts 12 / 66
Derivative of a Single Variable Function In economics, we are often interested in measuring the rate of change of a function. For instance, a firm is interested in knowing how much profit will change if it produces an additional unit of output. A manager may care about how production is changing when he adds an additional unit of labor to the production line. A buyer wants to know how much happier they will be if the consume another unit of a certain product. All of these questions are about marginal changes and it turns out that the set of tools provided by calculus are well suited for answering these types of questions. Van Essen (U of A) Basic Math Concepts 13 / 66
Derivative of a Single Variable Function A derivative tells us the instantaneous rate of change of a function. We first need to first define rate of change. Suppose we have a function f : R R and want to know how this function would change if we go from input x to a new input x + x, where x is the change in x. Van Essen (U of A) Basic Math Concepts 14 / 66
Derivative of a Single Variable Function In order to find the change in f, denoted f, we can simply take f (x + x), the value of the function at the new input x + x, and subtract f (x), the value the function takes at the old input x i.e., f = f (x + x) f (x). The rate of change is defined by f x or written out f x f (x + x) f (x) =. x Geometrically, this rate is the slope of a line that starts at the point (x, f (x)) and ends at the point (x + x, f (x + x)). Van Essen (U of A) Basic Math Concepts 15 / 66
Derivative of a Single Variable Function Van Essen (U of A) Figure: BasicRate Math Concepts of Change 16 / 66
Derivative of a Single Variable Function For example, suppose a firm was producing 10 units of output with 5 workers and that after they added 3 more workers to the production process output jumped to 12. If we let f (l) be the production function that tells us how much output we can produce when we use l workers, the change in output was f = f (8) f (5) = 2 and the rate of change f l = 2 3. The derivative is the instantaneous rate of change (or the marginal change) i.e., it is the rate of change of a function when we make a small change in the input. Specifically, the derivative is defined as d f (x) = lim dx x 0 f (x + x) f (x). x Van Essen (U of A) Basic Math Concepts 17 / 66
Derivative of a Single Variable Function Graphically, the derivative of the function f at the point x = x 0 is the slope of a line that is just tangent to f (x 0 ). This is illustrate below. Van Essen (U of A) Basic Math Concepts 18 / 66
Derivative of a Single Variable Function In summary, a derivative tells us how a function is changing at a particular point (i.e., the slope). Graphically, this is just the slope of a particular tangent line. We care about derivatives because they tell us how the function is changing. In particular, the sign of the derivative tells us whether the function increasing ( df dx > 0), decreasing ( df dx < 0), or constant ( df dx = 0). The actual derivative evaluated at a point is just a number which tells us the magnitude of the change i.e., how fast is the function changing. In other words, a function with a derivative equal to 10 at x = 1 is increasing faster than a function with a derivative of 2 at x = 1. Van Essen (U of A) Basic Math Concepts 19 / 66
Derivative of a Single Variable Function Now we come to the problem of how to find the derivative of a function. Of course, we have a definition of the derivative and we could always apply this definition to a particular function. The point you should take away from the following example is that this way of calculating a derivative is rather annoying and tedious. Van Essen (U of A) Basic Math Concepts 20 / 66
Derivative of a Single Variable Function Suppose we want to compute the derivative of the function f (x) = 25x 1 2 x 2. Using the definition of the derivative we know we have to compute f x and then take a limit of this expression as x 0. First, we compute the pieces of f x. The value of the function f evaluated at the new point x + x is f (x + x) = 25 (x + ) 1 (x + )2 2 = 1 2 x 2 x ( x) + 25x 1 2 ( x)2 + 25 ( x). Van Essen (U of A) Basic Math Concepts 21 / 66
Derivative of a Single Variable Function The value of the function f at the original point x is f (x) = 25x 1 2 x 2. Therefore the change in f is f = f (x + x) f (x) = 1 2 ( x)2 ( x) x + 25 ( x) The change in x is just x. Therefore the rate of change of f is f (x + x) f (x) x (25 (x + x) 1 2 (x + x)2) ( 25x 1 2 x 2) = x = 1 2 ( x)2 ( x) x + 25 ( x) x = 25 x 1 2 x Van Essen (U of A) Basic Math Concepts 22 / 66
Derivative of a Single Variable Function The derivative of f is the instantaneous rate of change so d f (x) = lim dx 25 x 1 x = 25 x. x 0 2 Finally, it is useful to evaluate this derivative at several points in order to see how the function is changing at different points. At the point x = 5, the derivative of f is 20 so the function is increasing. At the point 25 the derivative is zero so the function is constant. Last, at the point x = 30, the derivative is negative so the function is decreasing. Van Essen (U of A) Basic Math Concepts 23 / 66
Rules for Taking Derivatives There are simple rules for calculating most derivatives. Van Essen (U of A) Basic Math Concepts 24 / 66
Constant Rule Theorem Suppose f (x) = a, where a is a constant real number, then the derivative of f with respect to x is df (x) = 0. dx Van Essen (U of A) Basic Math Concepts 25 / 66
Adding Rule Theorem Suppose f (x) = g(x) + h(x), then the derivative of f with respect to x is df dg dh (x) = (x) + dx dx dx (x). Van Essen (U of A) Basic Math Concepts 26 / 66
Power Rule The power rule for taking a derivative applies to polynomial functions. Theorem Suppose f (x) = ax b, where a and b are constant real numbers, then the derivative of f with respect to x is df dx (x) = abx b 1. Van Essen (U of A) Basic Math Concepts 27 / 66
More Examples Example Suppose f (x) = 2, then df dx (x) = 0. Example Suppose f (x) = 3x, then df dx (x) = 3. Recall that x 0 = 1. Example Suppose f (x) = 3x 2, then df dx (x) = 6x. Example Suppose f (x) = 3x 1 2, then df dx (x) = 3 2 x 3 2. Van Essen (U of A) Basic Math Concepts 28 / 66
Example: Power Rule, Constant Rule, and Adding Rule Example Suppose we have a polynomial function f (x) = 1 + x 1 2 x 2 + 5x 3, then df (x) dx = 2 0 + 1 x + 15x = 1 x + 15x 2. Van Essen (U of A) Basic Math Concepts 29 / 66
Product Rule Theorem Suppose f (x) = f 1 (x)f 2 (x), then df dx (x) = df 1 dx (x)f 2(x) + f 1 (x) df 2 dx (x). Van Essen (U of A) Basic Math Concepts 30 / 66
Product Rule Example Suppose f (x) = 3 ln x, then Example df dx (x) = 3 x. Suppose f (x) = (3x 2 )(x 4 x + 1), then df dx (x) = (6x) (x 4 x + 1) + (3x 2 )(4x 3 1) = 18x 5 9x 2 + 6x. Van Essen (U of A) Basic Math Concepts 31 / 66
Quotient Rule The quotient rule applies when the function we are interested in taking the derivative can be thought of as the ratio of two functions. Theorem Suppose f (x) = f 1(x ) f 2 (x ), then Example f (x) = f 1(x ) f 2 (x ) = x 2 x, then df dx (x) = df1 dx (x)f 2(x) f 1 (x) df 2 dx (x) (f 2 (x)) 2. df dx (x) = (2x)(x) (x 2 )(1) (x) 2 = 2x 2 x 2 x 2 = 1 Van Essen (U of A) Basic Math Concepts 32 / 66
Derivative of the Natural Logarithm Function and the Exponential Function The derivative of the natural logarithm function and its inverse, the exponential function have their own rules. Theorem Suppose f (x) = ln x, where x > 0, then df dx (x) = 1 x. Theorem Suppose f (x) = e x, then df dx (x) = ex. Van Essen (U of A) Basic Math Concepts 33 / 66
Chain Rule The chain rule applies when the function we are interested in taking the derivative can be thought of as the composition of two functions. For example, the function h(x) = ln(2x 2 ) can be thought of as the composition of the functions f (x) = ln(x) and g(x) = 2x 2 i.e., h(x) = f (g(x)). Theorem Suppose i(x) = f (g(x)), then Example di df (x) = dx dx (g(x))dg dx (x). Suppose f (x) = ln ( 2x 2), then df dx (x) = ( 1 2x 2 ) (4x) = 2 x. Van Essen (U of A) Basic Math Concepts 34 / 66
Example: Derivative of an Inverse Example (Derivative of an Inverse Function): Suppose f 1 is the inverse function of f, then f (f 1 (x)) = x. If we take the derivative of both sides with respect to x we have df dx (f 1 df 1 (x)) (x) = 1. dx Thus, the derivative of the inverse function f 1 is df 1 dx (x) = 1 df dx (f 1 (x)). Van Essen (U of A) Basic Math Concepts 35 / 66
Example Example Suppose f (x) = e x and f 1 (x) = ln(x). Since the exponential and logarithm functions are inverses of one another we have that df 1 dx (x) = d dx ln(x) = 1 df dx (f 1 (x)) = 1 e ln(x ) = 1 x. Similarly, if we let f (x) = ln(x) and f 1 (x) = e x. Then df 1 dx (x) = d dx ex = 1 1 e x = e x. Van Essen (U of A) Basic Math Concepts 36 / 66
Higher Order Derivatives The derivative of a function is also a function which, in some circumstances, can also be differentiated. In particular, the derivative of df dx (x), which is denoted d 2 f (x) or dx 2 sometimes f (x), is called a second order derivative. This is the only higher order derivative we will need in this course. The second derivative of a function tells us how the first order derivative function is changing with x. This information is very useful, for example, when trying to determine the maximum of a function. Van Essen (U of A) Basic Math Concepts 37 / 66
Derivative of a Single Variable Function Example Suppose f (x) = x 2 + 10, then df dx (x) = 2x. The second order derivative is d 2 f (x) = 2. This tells us that the function df dx 2 dx (x) is decreasing at a constant rate. Van Essen (U of A) Basic Math Concepts 38 / 66
Example: Higher Order Derivatives Suppose f (x) = ln x, where x > 0. Then the first derivative of f is df dx (x) = 1 x > 0 for x > 0. Thus, the logarithm function is an increasing function. The second order derivative is d 2 f dx 2 (x) = 1 x 2 < 0. So the logarithm function is increasing, but at a decreasing rate. Van Essen (U of A) Basic Math Concepts 39 / 66
Example y 5 4 3 2 1 0 0 1 2 3 4 5 x Van Essen (U of A) Basic Math Concepts 40 / 66
Multi-variable Real Valued Functions Almost every function you work with in economics is a multi-variable real valued function f : R N R. Utility functions and production functions are examples of real valued multi-variable functions that will use frequently in this course. We will briefly discuss how to graph 2 variable functions using three popular functions: linear function, Cobb-Douglas function, and the min function. Van Essen (U of A) Basic Math Concepts 41 / 66
Multi-variable Real Valued Functions Consider the function f : R 2 + R that is defined by for all (x, y) R 2 +. f (x, y) = x + y This is a relatively simple linear function. It takes, as input, a point (x, y) and adds the x component and the y component together. Van Essen (U of A) Basic Math Concepts 42 / 66
Multi-variable Real Valued Functions Its graph, in three dimensions, is the plane illustrated in the figure below. 10 z 5 0 2 0 0 2 4 x y 4 Figure: 3-D Plot of f (x, y) = x + y Van Essen (U of A) Basic Math Concepts 43 / 66
Multi-variable Real Valued Functions This 3-D depiction of the function demands too much artistic ability to be useful. We will use level curves to graph a multi-variable function. A level curve is a 2-D way of plotting a 3-D object. For example, a map of a mountain uses contour lines to indicate all of the spots on the mountain that have the same elevation. A level curve is the same thing. In particular, it is a line that connects all of the (x, y) such that function f obtains the same value. Van Essen (U of A) Basic Math Concepts 44 / 66
Multi-variable Real Valued Functions In the graph below, we plot three level curves for the function f (x, y) = x + y. Figure: Level Curves for f (x, y) = x + y Van Essen (U of A) Basic Math Concepts 45 / 66
Multi-variable Real Valued Functions The level curves f (x, y) = 1, f (x, y) = 3, and f (x, y) = 5. We can solve for the equation that defines the level curve. For instance, the level curve f (x, y) = 1 is defined by the equation if we solve for y we get y = 1 x. f (x, y) = x + y = 1 Van Essen (U of A) Basic Math Concepts 46 / 66
Multi-variable Real Valued Functions Cobb-Douglas functions are the family of real valued functions f : R 2 R of the form f (x, y) = Ax a y b with parameters A, a, b R +. For example, the function f (x, y) = x 2 y is a member of the Cobb-Douglas family with A = 1, a = 2, and b = 1. A 3-D plot of the Cobb-Douglas function f (x, y) = x 2 y is given below: z y 4 20 00 4 2 2 x 4 Van Essen (U of A) Basic Math Concepts 2 47 / 66
Multi-variable Real Valued Functions Again, to solve for a level curve f (x, y) = k we set the function equal to k. For the example, the level curves will be defined by the equation x 2 y = k or y = k x 2. A few of these level curves from this same function are: y 5 4 3 2 1 0 0 1 2 3 4 5 x Figure: Level Curves for the Cobb-Douglas Function f (x, y) = x 2 y Van Essen (U of A) Basic Math Concepts 48 / 66
Multi-variable Real Valued Functions Finally, the minimum function is a real valued function of the form f (x, y) = c min{ax, by}, where parameters a, b, c R +. The 3-D plot of this function looks like a side of a pyramid. 10 z y 5 5 0 0 0 2 4 x Figure: 3-D Plot of f (x, y) = min{2x, y} Van Essen (U of A) Basic Math Concepts 49 / 66
Multi-variable Real Valued Functions The level curves of the minimum function look like the letter L. The corners of these level curves are along the line y = a b x. This equation is found by setting the ax component in the min function equal to the by component of the min function. Figure: Level Curves of the Min Function f (x, y) = min{2x, y} Van Essen (U of A) Basic Math Concepts 50 / 66
Partial Derivatives If you know how to take derivatives of single variable functions then you know how to take partial derivatives of multi-variable functions. The trick is that we just pretend that all variables, other than one we are differentiating with respect to, are constants and then proceed as before. Example Suppose f (x, y) = x 2 y, then the partial derivatives with respect to x and y are f x f y = 2xy = x 2 respectively. Van Essen (U of A) Basic Math Concepts 51 / 66
Higher Order Partial Derivatives Suppose f : R N R, then the partial derivative n f x i is a function of N variables. Moreover, if this function is differentiable, then we can take partial derivatives of it as well. These are higher order derivatives. They tell us how the partial derivative function is changing with respect to different variables. Van Essen (U of A) Basic Math Concepts 52 / 66
Second Order Partial Derivatives We will mostly be concerned with first and second order derivatives of a function for a reason which we will learn about tomorrow-ish. A function of N variables has N first order derivatives and N 2 second order derivatives. Example Suppose f (l, k) = 4l 1 4 k 3 4, then the first order partial derivatives are f l = l 3 4 k 3 4 and f k = 3l 1 4 k 4 1 the second order partial derivatives of f are 2 f l 2 = 3 4 l 7 3 4 k 4 2 f = 2 f l k = 3 4 l 3 4 k 1 4 k l 2 f k 2 = 3 4 l 1 4 k 4 5 Van Essen (U of A) Basic Math Concepts 53 / 66
Second Order Partial Derivatives In the example we saw that 2 f k l = 2 f l k, this was not a coincidence. Theorem (Young s Theorem) Suppose that y = f (x 1,..., x N ) is C 2 on an open region J in R N. Then for all x J and for each i,j we have that 2 f x i x j = 2 f x j x i Van Essen (U of A) Basic Math Concepts 54 / 66
Total Differential The partial derivative tells us how a function changes when we vary one variable and hold the others fixed. Sometimes we wish to know how a function changes when more than one variable is changed. In particular, suppose z = f (x, y) and x goes from x 1 to x 2 and y goes from y 1 to y 2, then z = f (x 2, y 2 ) f (x 1, y 1 ). We can approximate z by considering a plane that is tangent to f at (x 1, y 1 ). The changes on this plane are denoted the by total differential formula dz = f f dx + x y dy The total differential is an approximation for z when x changes by x and y changes by y i.e., z f f x + x y y Van Essen (U of A) Basic Math Concepts 55 / 66
Total Differential: Example Approximate via differentials. Well let then (4.1) 3 (2.95) 3 (1.02) 3 f (x, y, z) = x 3 y 3 z 3 df = f f f dx + dy + x y z dz Van Essen (U of A) Basic Math Concepts 56 / 66
Total Differential: Example df = f f f dx + dy + x y z dz Choose (x 1, y 1, z 1 ) = (4, 3, 1), then f x (x 1, y 1, z 1 ) = 4 f y (x 1, y 1, z 1 ) = 9 4 f z (x 1, y 1, z 1 ) = 1 4 The differential is df = (4)(0.1) + ( 9 ( 4 )( 0.05) + 1 ) (0.02) 4 = 0.5075 Van Essen (U of A) Basic Math Concepts 57 / 66
Total Differential: Example The differential is df = (4)(0.1) + ( 9 ( 4 )( 0.05) + 1 ) (0.02) 4 Thus, = 0.5075 f = f (x 2, y 2, z 2 ) f (x 1, y 1, z 1 ) 0.5075 Since f (x 1, y 1, z 1 ) = 6, we have f (4.1, 2.95, 1.02) 6.5075 The actual value is 6. 495 2. Van Essen (U of A) Basic Math Concepts 58 / 66
Another Differential Example What is d ln y d ln x? Well, Thus, d ln y = 1 y dy d ln x = 1 x dx d ln y d ln x = dy x dx y which is the price elasticity formula. Van Essen (U of A) Basic Math Concepts 59 / 66
Total Derivative/ Chain Rule How do you find the change of the function f (x, y) with respect to y when x and y are related. Use the total derivative. Suppose z = f (x, y) and x = g(y) First, find the total differential Next, divide both sides by dy dz = f f dx + x y dy dz dy = f dx x dy + f y This decomposes the change in y into a direct effect f y indirect effect f dx x dy. and an Van Essen (U of A) Basic Math Concepts 60 / 66
Total Derivative/ Chain Rule Example Suppose f (x, y) = x 2 y, where y = g(x) = 3x 2 Then the total derivative of f with respect to x is df dx = f x + f dy y dx = 2xy + x 2 (6x) = 6x 3 + 6x 3 = 12x 3 Note: that if we replaced y with g(x) and tool the derivative we would get the same answer i.e., d ( 3x 4 ) = 12x 3. dx Van Essen (U of A) Basic Math Concepts 61 / 66
Antiderivatives Definition An antiderivative of a function f (x) is a function F (x) whose derivative is the original i.e., d dx F = f. Definition The function F is also called the indefinite integral of f and written F (x) = f (x)dx Van Essen (U of A) Basic Math Concepts 62 / 66
Common Indefinite Integrals af (x)dx = a f (x)dx (f (x) + g(x)) dx = f (x)dx + g(x)dx x n dx = x n+1 n+1 + C 1 x dx = ln x + C e f (x ) f (x)dx = e f (x ) + C f (x ) dx = ln f (x) + C f (x ) Van Essen (U of A) Basic Math Concepts 63 / 66
Example ( x 2 + x 3 + 2 ) dx = x 3 x 3 + x 4 4 + 2 ln(x) + C Van Essen (U of A) Basic Math Concepts 64 / 66
Definite Integral For fixed numbers a and b, the definite integral of f (x) from a to b is F (b) F (a) where F is the anti-derivative of f. b a f (x)dx = F (b) F (a) Graphical Interpretation of a Definite Integral: Area under a curve. Useful when we talk about Consumer and Producer Surplus. Van Essen (U of A) Basic Math Concepts 65 / 66
Example 1 0 ( x 2 + x 3) dx = = [ x 3 3 + x 4 4 [ 1 3 + 1 4 ] ] x =1 x =0 [ 0 3 + 0 4 ] = 7 12 Van Essen (U of A) Basic Math Concepts 66 / 66