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Lecture 2. Economic Models, Functions, Logs, Exponents, e. Variables, Constant, Parameters. Variables: magnitude can change Price, profit, revenue… Represented by symbols Can be ‘frozen’ by setting value Try to setup models to obtain solutions to variables
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Lecture 2 Economic Models, Functions, Logs, Exponents, e
Variables, Constant, Parameters • Variables: magnitude can change • Price, profit, revenue… • Represented by symbols • Can be ‘frozen’ by setting value • Try to setup models to obtain solutions to variables • Endogenous variable: determined by model • Exogenous variable: determined outside model
Variables, Constant, Parameters • Constants, do not change, but can be joined with variables. • Called coefficients • Ex: 1000q, 1000 is constant, quantity is variable • When constants are not set? • Ex: βq, where β stands for the coefficient and q for quantity • Now, β can change! A ‘variable’ constant is called a parameter.
Equation vs. Identity • An identity, is a definition, like profit, revenue, and cost • where π is profit, R is revenue, C is cost • Behavioral equations • Specify how variables interact • Conditional equations • Optimization and Equilibrium conditions are examples
Real Numbers • Rational numbers: (‘ratio’) • Whole numbers: + Integers • Fractions • Irrational numbers • Can’t be expressed as a fraction • Nonrepeating, nonterminating decimals • Ex: • Real Numbers combine rational and irrational • Non-imaginary, i=
Set Properties and Set Notation • Definition: A set is any collection of objects specified in such a way that we can determine whether a given object is or is not in the collection. • Notation:e Ameans “e is an element of A”, or “e belongs to set A”. The notatione Ameans “e is not an element of A”.
Null Set • Example. What are the real number solutions of the equation x2 + 1 = 0? There is no answer to this… you can write this as { }, [], or .
Set Builder Notation • Sometimes it is convenient to represent sets using set builder notation. For example, instead of representing the set A (letters in the alphabet) by the roster method, we can useA = {x | x is a letter of the English alphabet} which means the same as A = {a , b, c, d, e, …, z} In statistics… {x|a} would be read x, given a. Here, we will use the vertical line in set notation • Example. {x | x2 = 9} = {3, -3}This is read as “the set of all x such that the square of x equals 9.” This set consists of the two numbers 3 and -3.
Union of Sets The union of two sets A and B is the set of all elements formed by combining all the elements of set A and all the elements of set B into one set. It is written A B. B A In the Venn diagram on the left, the union of A and B is the entire region shaded.
Intersection of Sets The intersection of two sets A and B is the set of all elements that are common to both A and B. It is written A B. In the Venn diagram on the left, the intersection of A and B is the shaded region. B A
The Complement of a Set The complement of a set A is defined as the set of elements that are contained in U, the universal set, but not contained in set A. The symbolism and notation for the complement of set A are In the Venn diagram on the left, the rectangle represents the universe. A is the shaded area outside the set A.
Application A marketing survey of 1,000 commuters found that 600 answered listen to the news, 500 listen to music, and 300 listen to both. Let N = set of commuters in the sample who listen to news and M = set of commuters in the sample who listen to music. Find the number of commuters in the set The number of elements in a set A is denoted by n(A), so in this case we are looking for
Solution The study is based on 1000 commuters, so n(U)=1000.The number of elements in the four sections in the Venn diagram need to add up to 1000.The red part represents the commuters who listen to both news and music. It has 300 elements. The set N (news listeners) consists of a green part and a red part. N has 600 elements, the red part has 300, so the green part must also be 300. Continue in this fashion.
Solution(continued) U 200 people listen to neither news nor music is the green part, which contains 300 commuters. M N 300 listen to news but not music. 200 listen to music but not news 300 listen to both music and news
Functions The previous graph is the graph of a function. The idea of a function is this: a correspondence between two sets D and R such that to each element of the first set, D, there corresponds one and only one element of the second set, R. The first set is called the domain, and the set of corresponding elements in the second set is called the range. For example, the cost of a pizza (C) is related to the size of the pizza. A 10 inch diameter pizza costs $9.00, while a 16 inch diameter pizza costs $12.00.
Function Definition 10 9.00 12 10.00 16 12.00 domain D or x range R or f(x) You can visualize a function by the following diagram which shows a correspondence between two sets: D, the domain of the function, gives the diameter of pizzas, and R, the range of the function gives the cost of the pizza.
Functions Specified by Equations • Consider the equation Input:x = -2 -2 Process: square (–2),then subtract 2 (-2,2) is an ordered pair of the function. Output: result is 2 2
Vertical Line Test for a Function If you have the graph of an equation, there is an easy way to determine if it is the graph of an function. It is called the vertical line test which states that: An equation defines a function if each vertical line in the coordinate system passes through at most one point on the graph of the equation. If any vertical line passes through two or more points on the graph of an equation, then the equation does not define a function.
Vertical Line Test for a Function(continued) This graph is not the graph of a function because you can draw a vertical line which crosses it twice. This is the graph of a function because any vertical line crosses only once.
Function Notation The following notation is used to describe functions. The variable y will now be called f (x). This is read as “ f of x” and simply means the y coordinate of the function corresponding to a given x value. Our previous equation can now be expressed as
Function Evaluation • Consider our function • What does f (-3) mean?
Function Evaluation • Consider our function • What does f (-3) mean? Replace x with the value –3 and evaluate the expression • The result is 11 . This means that the point (-3,11) is on the graph of the function.
Some Examples 1. KEEP THIS ‘h’ example in mind for derivatives
Domain of a Function Consider which is not a real number. Question: for what values of x is the function defined?
Domain of a Function is defined only when the radicand (3x-2) is equal to or greater than zero. This implies that Answer:
Domain of a Function(continued) Therefore, the domain of our function is the set of real numbers that are greater than or equal to 2/3. Example: Find the domain of the function
Domain of a Function(continued) Therefore, the domain of our function is the set of real numbers that are greater than or equal to 2/3. Example: Find the domain of the function Answer:
Domain of a Function:Another Example Find the domain of
Domain of a Function:Another Example Find the domain of In this case, the function is defined for all values of x except where the denominator of the fraction is zero. This means all real numbers x except5/3.
Mathematical Modeling The price-demand function for a company is given bywhere x (variable) represents the number of items and p(x) represents the price of the item. 1000 and -5 are constants. Determine the revenue function and find the revenue generated if 50 items are sold.
Solution Revenue = Price ∙ Quantity, so R(x)= p(x) ∙ x = (1000 – 5x) ∙ x When 50 items are sold, x = 50, so we will evaluate the revenue function at x = 50: The domain of the function has already been specified. What is the ‘range’ over this domain?
Production Problem Either coal (C) or gas (G) can be used to produce steel. If Pc=100 and Pg=500, draw an isocost curve to limit expenditures to $10,000
Production Problem If the price of gas declines by 20%? What happens to the budget line? What if the price of coal rises by 25%? Expenditures rise 50%?
Production Problem All together?
Break-Even and Profit-Loss Analysis Any manufacturing company has costsC and revenuesR. The company will have a loss if R < C, will break evenif R = C, and will have a profit if R > C. Costs include fixed costs such as plant overhead, etc. and variable costs, which are dependent on the number of items produced. C = a + bx(x is the number of items produced) a and b are parameters
Break-Even and Profit-Loss Analysis(continued) Price-demand functions, usually determined by financial departments, play an important role in profit-loss analysis.p = m – nx (x is the number of items than can be sold at $p per item.) [Note: m and n are PARAMETERS] The revenue function is R = (number of items sold) ∙ (price per item) = xp = x(m - nx) The profit function is P = R - C = x(m - nx) - (a + bx)
Example of Profit-Loss Analysis A company manufactures notebook computers. Its marketing research department has determined that the data is modeled by the price-demand function p(x) = 2,000 - 60x, when 1 <x< 25, (x is in thousands). What is the company’s revenue function and what is its domain?
Answer to Revenue Problem Since Revenue = Price ∙ Quantity, The domain of this function is the same as the domain of the price-demand function, which is 1 ≤x ≤ 25 (in thousands.)
First, is the demand function, and the plot of it. Below, you can see that x*p(x) is our revenue function… But, firms don’t maximize revenue, they maximize profits, so we have to consider our cost function
Profit Problem The financial department for the company in the preceding problem has established the following cost function for producing and selling x thousand notebook computers: C(x) = 4,000 + 500x (x is in thousand dollars). Write a profit function for producing and selling x thousand notebook computers, and indicate the domain of this function.
Answer to Profit Problem Since Profit = Revenue - Cost, and our revenue function from the preceding problem was R(x) = 2000x - 60x2, P(x) = R(x) - C(x) = 2000x - 60x2 - (4000 + 500x) = -60x2 + 1500x – 4000. The domain of this function is the same as the domain of the original price-demand function, 1< x < 25 (in thousands.) Now, to get this profit function
First, is the cost function, and the plot of it. Below, you can see that the revenue function plotted with our cost function gives us a visual representation of profit… Where these two lines cross, profit is ZERO, since R=C. If But, firms don’t maximize revenue, they maximize profits, so we have to consider our cost function. We want to maximize the difference between the two functions.
This, is R-C, or our profit function Below, I have solved for this using Mathematica’s Maximize function
Types of functions • Polynomial functions
Polynomials or Quadratics can shift • Now, sketch the related graph given by the equation below and explain, in words, how it is related to the first function you graphed.
