Economics 102Summer 2011Answers to Homework #1Due 6/23/11Directions: The homework will be collected in a box before the lecture. Please placeyour name, TA name and section number on top of the homework (legibly). Make sureyou write your name as it appears on your ID so that you can receive the correct grade.Please remember the section number for the section you are registered, because you willneed that number when you submit exams and homework. Late homework will not beaccepted so make plans ahead of time. Please show your work. Good luck!1. For each of the following sets of information please write an equation that represents accurately this information. Assume that all relationships are linear. a. The line contains the points (x, y) = (10, 15) and (6, 11).b. The line contains the point (x, y) = (10, 15) and has a slope of -2.c. The line contains the point (x, y) = (20, 10) and you also know that each time the variable on the horizontal axis increases by two units the variable on the vertical axis increases by four units. d. The line contains the y-intercept 30 and you also know that each time the variable on the vertical axis decreases by two units the variable on the horizontal axis decreases by four units. e. The line contains the x-intercept 40 and you also know that each time the variable on the vertical axis increases by two units the variable on the horizontal axis decreases by four units. Answers:a. From the two points you can calculate the slope of the line as (15 -11)/(10 – 6) = 1. Then, use one of the points given to find the y-intercept: y = mx + b or y = x + b and substituting (10, 15) into the equation you get 15 = 10 + b or b = 5. The equation for the line is y = x + 5. b. From the information given you can write the equation as y = -2x + b and then substitute the given point into the equation to find the y-intercept, b. Thus, 15 = (-2)(10) + b or b = 35 and the equation is y = 35 – 2x.c. From the given information we can find the slope of the line as the change in y divided by the change in x or the rise/run. Using the information provided, this gives us a slope measure of 2. The general equation for the line is y = mx + b or y = 2x +b and substituting in the point (20, 10) we can find the y-intercept b. Thus, 10 – (2)(20) + b or b = -30. The equation is y = 2x – 30.d. From the information given we can calculate the slope of the line as the rise/run which in this case is equal to (-2)/(-4) or ½. We already know the y-intercept so the equation of the line is easily written as y = (1/2)x + 30.1e. From the information given we can calculate the slope of the line as the rise/run which in this case is equal to (2)/(-4) or (-1/2). We know the x-intercept which tells usone point (40,0) on the line. Plugging this point into the general equation for the line we can find the y-intercept of the line: thus, y = (-1/2)x + b or 0 = (-1/2)(40) + b or b = 20. The equation of the line is y = 20 – (1/2)x. 2. In each part of this question you are given information about two linear relationships. Use this information to write the two equations and then use these two equations to solve for a solution that is true simultaneously for both equations. a. The first line contains the points (x, y) = (10, 10) and (30, 10) and the second line contains the points (x, y) = (0, 15) and (30, 0). b. The first line contains the point (x, y) = (10, 20) and has a slope of 5. The second line contains the point (x, y) = (5, 5) and has a slope of -2.c. The first line contains the point (30, 0) and has a slope of -1. The second line contains the point (0, 15) and has a slope of 1.d. The first line contains the point (6, 2) and you also know that each time the variable on the vertical axis decreases by two units the variable on the horizontal axis increases by 1 unit. The second line contains the point (3,3)and you also know that each time the variable on the vertical axis increases by two units the variable on the horizontal axis also increases by two units. Answers:a. The first line if you graph it is a horizontal line where no matter what x’s value is, the y value is always 10. The equation for this line is y = 10. The second line has a y-intercept of 15 and a slope of -1/2. The equation for the second line is y = 15 – (1/2)x.The solution for this set of equations is (x, y) = (10, 10). b. The first line can be written as y = 5x – 30 and the second line can be written as y =15 – 2x. The solution for this set of equations is (x, y) = (45/7,15/7).c. The first line can be written as y = 30 – x and the second line can be written as y = x + 15. The solution for this set of equations is (x, y) = (7.5, 22.5).d. The first line can be written as y = 14 – 2x and the second line can be written as y =x. The solution for this set of equation is (x, y) = (14/3, 14/3).3. Joe, a baker, makes cookies and cakes. The following table provides five production combinations of cookies and cakes that Joe can make. Each combination lies on Joe’s production possibility frontier (PPF). Furthermore, Joe’sPPF is linear between each given point in the table: e.g., his PPF is linear betweenpoints A and B, B and C, C and D, etc. Use this information to answer this set of questions. ProductionCombinationCookies Cakes2A 1000 0B 800 70C 600 120D 300 160E 0 200a. What is the opportunity cost of producing one more cake if Joe is currently producing at combination A?b. What is the opportunity cost of producing one more cake if Joe is currently producing at combination B?c. What is the opportunity cost of producing one more cake if Joe is currently producing at combination C?d. What is the opportunity cost of producing one more cake if Joe is currently producing at combination D?e. What is the opportunity cost of producing one more cookie if Joe is currently producing at combination E?f. What is the opportunity cost of producing one more cookie if Joe is currently producing at combination D?g. What is the opportunity cost of producing one more cookie if Joe is currently producing at combination C?h. What is the opportunity cost of producing one more cookie if Joe is currently producing at combination B?Answers:a. The opportunity cost of producing one more cake is 20/7 cookies or approximately …
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