Principles of Microeconomics
Principles of Microeconomics ECON 10011
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This 0 page Class Notes was uploaded by Dr. Tracy Kulas on Sunday November 1, 2015. The Class Notes belongs to ECON 10011 at University of Notre Dame taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/232729/econ-10011-university-of-notre-dame in Economcs at University of Notre Dame.
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Date Created: 11/01/15
Econ 1001 1 Principles of Microeconomics Math Review Graphs and Algebra Microeconomic problems generally fall into one of two categories individual choice and market equilibrium Analyzing choice problems focuses on how individuals balance cost and bene t tradeoffs I would like to buy a new car but in order to afford a new car I have to go out to dinner less often and take fewer vacations Equilibrium analysis focuses on how the competing interests of individuals balance Producers would like to sell their goods at high prices but if the price is too high there will not be enough willing buyers To study choice problems and equilibrium problems at the introductory level we will use two mathematical skills graphical analysis and algebra If you go on to major in economics you can also expect to use calculus and statistics in advanced courses Professional economists who work on theoretical issues often use even more advanced mathematics including differential equations real and complex analysis topology and differential geometry 1 Graphs Graphs help us visualize the relationship between several variables Figure 1 provides an example of a 2dimensional graph Figure 2 provides an example of a 3dimensional graph Figure 1 mfurmatmn yuu put m a graph x n m 1 andy The axis unentatmntells us Lhatxmmeases m value as unemuves m thenght vanebles x y snaz The variable mnges 39nm u m 4H and 15 platted unthe axis wnh dxfferent ms nut essenual m use the same scale un each sens O en umes usmg dxfferent scales makes n 32st tn msushze me relatmnshlp between vanables n Cuurdmates Everypumt un a gaphxs desenbea by avalue fur each cuurdmate Fur example un ageugaphmmap each leesan esn be desenbea by alungxtude and a latde m Curve Acume can be lmear urnunrlmear A curve 15 away m msuallyrepresentthe In urzfxy In 3 dimensmns a curve Wm luukhke a surface as m gure 2 ecunumcpmblems m the real wurldmvulve much mure Lhanthree vmbles Ofcuurse drawing graphs m mure Lhanthree amensmns 15 pretty challengng Furtunately we Wm 3 can study most economic problems effectively by looking at 2 and 3dimensional cases Graphs help us visualize how several variables relate to each other In Figure 1 increases in x first coincide with increases in y then increases in x coincide with decreases in y and nally further increases in x coincide with increases in y Notice that I used the term coincide and not cause All we can tell from a graph is how variables change in relation to each other It is important not to always view the variable plotted on the vertical aXis as a dependent variable or to view the variable plotted on the horizontal aXis as an independent variable In economics it will be common for both variables to be jointly determined by some other variables that are not being plotted In Figure 2 increases in x and y coincide with increases in 2 Changes in x appear to coincide with smaller changes in 2 than do changes in y We will learn more about how to read graphs like this when we study consumer and producer theory Many of the graphs found in economics exhibit a simple but important type of curvature Consider the following two curves in Figure 3 rX Figure 3 Both curves have positive slopes higher values of x correspond to higher values of y The key difference is that the curve on the left gets atter as x increases while the curve on the right gets steeper as x increases For the curve on the left as x increases further increases in x coincide with smaller increases in y For the curve on the right as x increases further increases in x coincide with larger increases in y Now consider two different curves in Figure 4 y y X X Figure 4 Unlike their counterparts in Figure 3 both of the curves in Figure 4 are downward sloping higher values of x correspond to lower values of y For the graph on the left as x increases the curve gets steeper As x increases further increases in x coincide with larger decreases in y For the curve on the right as x increases the curve is getting atter Further increases in x coincide with smaller decreases in y This comparison suggests that the two curves on left of Figures 3 and 4 and the two curves on the right of Figures 3 and 4 have similar curvature properties In fact the curves on the left are called concave and the curves on the right are called convex There is a simple way to tell whether a curve is concave or convex Draw a straight line connecting two points on the curve If the line lies below the curve it is concave If the line lies above the curve it is convex Figure 5 illustrates this procedure Since the red line in the left graph lies below the curve and this will be true regardless of which two points on the curve we connect the curve is concave Likewise because the red line in the right graph lies above the curve the curve is convex Figure 5 In economics most bene t functions are concave while most cost functions are convex A concave bene t function corresponds to the concept of diminishing marginal benef1t For example suppose spending 1 million on advertising increases your company39s sales from 250000 units per month to 350000 units per month The concept of diminishing marginal benefit or concavity suggests that spending a second 1 million on advertising will increase sales by less than 100000 units that is monthly sales will be less than 450000 units A convex cost function corresponds to increasing marginal costs Again suppose you can increase monthly sales by 100000 units to 350000 units by spending 1 million on advertising The concept of increasing marginal cost means that to increase monthly sales by another 100000 units to 450000 units you would need to increase advertising expenditures by more than 1 million In total you will need to buy more than 2 million of advertising to increase monthly sales to 450000 units 2 Algebra a Understanding equations Looking at Figures 1 and 2 allows us to ask two questions When do variables take on positive values and when do they take on negative values And how do two variables change in relationship to each other While graphs give us general answers to these questions we can use algebra to come up with precise answers Rather than saying the two variables increase together or one decreases when the other increases we can actually measure the rate of change by analyzing the equation used to generate a graph When the relationship between variables is linear the equation that describes the relationship can be written as y mlx1 m22 max3 mnxn b The n1 variables arey and x1 x2xn Each ml is the slope parameter that describes the relationship between the two variables y and xl and b is the yintercept or the value of the variable y when the all the xl equal 0 The interpretation of the value of m is that for every oneunit change in x1 y changes by m units By varying the value of the index 139 between 1 and 71 you can focus on the effect of a different variable Using the index notation saves time and space and helps us see that the general concept of a slope does not depend on which variable we look at Example Supposey 40x1 b and x1 increases by 3 How much doesy change Answer The variable y changes by 403 or 12 units A positive slope means that the two variables y and x1 increase together or decrease together A negative slope means the two variables change in opposite directions Example If y 2x1 b then for every unit decrease in x1 y increases by 2 units and for every unit increase in x1 y decreases by 2 units b Solving systems of equations Most of the problems covered in this class will frequently require solving two equations with two unknowns There are two general methods for solving systems of equations It is a good idea to know how to use both as there will be some problems for which one method is easier and some problems for which the other method is easier i The Substitution Method Consider the following set of equations 2x 3y 40 and 3x 5y 100 To use the Substitution Method you need to pick one equation and one variable the choice is arbitrary but different choices can result in different amounts of arithmetic and then solve for the chosen variable in the chosen equation For example let39s choose the first equation and the variable x Solving for x results in x2015y l The next step is to substitute this expression into the other equation and then solve for the remaining variable For the example this step generates the following expressions 320 l5y 5y 100 or 60 45y 5y 100 or 95y 160 or y 16095 1684 2 The nal step is to substitute the value from 2 into the equation in 1 Doing so implies x 20 151684 526 Thus the solution to the above set of equations is xy 5261684 ii The Linear Combination Method This method involves adding the equations together in a way that causes one of the variables to drop out Let s consider the same example used to illustrate the Substitution Method 2x3y40 and 3 3x 5y 100 4 The rst step is to choose a variable to eliminate Again the choice will not affect the answer Let s choose to eliminate x The next step is to multiply the rst equation by the coef cient on x in the second equation and to multiply the second equation by the coef cient on x in the rst equation Doing so results in 6x 9y 120 and 4 6x 10y 200 5 Now subtract the equation 5 from equation 4 This implies 6x 6x 9y 10y 120 200 or 19y 320 or y 320191684 The nal step is to take this value for y and substitute it into either equation 3 or 4 The choice will not matter If one chooses equation 3 then you get 2x 31684 40 or 2x 1052 or x 526 If instead you choose equation 4 then you get 3x 51684 100 or 3x 1580 or x 526 As a nal illustration let39s solve 3 and 4 again but now let39s start by eliminating y Now you want to multiply 3 by the coefficient on y in 4 and you want to multiply 4 by the coefficient ony in 3 Doing so results in 10x 15y 200 and 9x 15y 300 Subtracting the second equation from the first yields 19x 100 or x 10019 526 Substituting this value of x into 3 implies 2526 3y 40 or 3y 5052 or y 1684 Often times in general models the coefficients do not have specific values This allows us to solve our model for a wide range of parameter values all at once Rather than the solution being a pair of numbers we will get a pair of expressions that depend only on our coefficients Despite the more abstract nature of our system of equations the Substitution Method and the Linear Combination Method still work and they work the same way The most general way to write two equations and two unknowns is xx By y and 6 6x ey A 7 The Greek letters alpha cc beta 5 gamma y delta 6 epsilon e and lambda A represent numbers we can plug in later while x and y still represent the variables for which we wish to solve Using the Linear Combination method we first get 056x 56y yo and 056x eocy cc Subtracting the second equation from the first yields 56 eocy yo 05A or y Y5 cab66 ea 8 Finally substituting the expression in 8 into 6 or I could have chosen 7 implies xx 5y6 och56 eoc y or xx y 5y6 och56 60 or putting the terms on the right under a common denominator xx y56 eoc By6 000 56 eoc BA OCE YB 60 or x BA EYl55 ea 9 Please do not spend time memorizing the expressions in 8 and 9 For homework quiz and exam questions you will need to show me that you know how to solve problems like this c Quadratic Equations A quadratic equation is an equation of the form ax2bxc0 10 where x is the unknown variable and a b and c are parameters As in the previous section we may or may not know their values or we may wish to assign values to these parameters after we have solved for x The quadratic formula is used to generate all possible solutions to quadratic equations of the form in 10 The quadratic formula is x btb2 4ac 2a If the radical term b2 4ac equals 0 then the only solution to 10 is x b2a If the radical term is positive then there are two real solutions Often times in economics one of the solutions can be ruled out on feasibility grounds For instance one solution might imply a negative quantity or price If the radical term is negative then the expression ax2 bx c is either always positive or always negative In this case 10 has no real solutions d Exponents Many examples will involve exponential terms Here is a summary of the main rules for working with exponents i xk xm xklm ii xkxm xk39m iii 6 x iv x39k lxk One way to remember rule iv is to first recall that any number raised to the zero power equals 1 Thus x39kx 39kx xk Econ 1001 1 Topic Review Economic Tradeoffs 1 List and de ne the 3 key concepts economists use to describe production tradeoffs within an economy 2 Explain the economic signi cance ofa concave PPF 3 List and de ne the two concepts used to describe productivity differences between countries 4 Explain how each concept is used to determine the set of goods a country will import and the set of goods it will export 5 To what does the phrase quotterms of tradequot refer How is it measured 6 Draw a PPF graph consistent with the following information Explain how each piece of information is re ected in your graph a There are two countries Each country is capable of producing the same two goods b Each country39s marginal cost of producing each good is strictly increasing c Country 1 has an absolute advantage in producing good 1 Country 2 has an absolute advantage in producing good 2 d Country 1 is a net exporter of good 2
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