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UW-Madison ECON 102 - Homework #1

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Economics 102Summer 2010Answers to Homework #1This homework is due at the beginning of the class lecture. Please staple your homework and make sure that your homework includes your name and the time of your discussion section. There will be no stapler atthe class lecture. Homework should be neat, legible and professional looking. Please show all necessary work and please make sure that it is easy to identify your final answer for each problem. 1. Review of math:a. You are told that there are two linear relationships between Y and X where Y is the variable measured on the vertical axis and X is the variable measured on the horizontal axis. The first linear relationship is given by the equation Y = 50 – 2X. You are told that the second linear relationship goes through the origin and that for every 1 unit increase in the X variable, the Y variable increases by 8 units. What is the equation for the second line and what is the solution (X, Y) for your two equations?Answer:For the second equation you know that the definition of slope is rise/run. The rise, or the change in Y, is 8 units; the run, or the change in X, is 1 unit. Thus, the slope of this equation is8. Since the equation goes through the origin, the y-intercept must be equal to 0. So, the equation for the second line is Y = 8X.Using these two equations, you can find the values of X and Y where the two lines intersect. This will occur at (5, 40).b. You are told that there are two linear relationships between Y and X where Y is the variable measured on the vertical axis and X is the variable measured on the horizontal axis. The first linear relationship contains the points (125, 75) and (50, 150). The second linear relationship contains the points (50, 100) and (150, 150). Find the equations for thetwo lines and then calculate the solution (X, Y) for these two equations. Answer:The slope of line 1 is equal to (75 – 150)/(125 – 50) = -75/75 = -1. Then, to find the y-intercept use the general y-intercept form (y = mx + b) and substitute in the slope measure you calculated and one of the points you were given. Thus, the equation can be written as Y = 200 – X.The slope of line 2 is equal to (100 – 150)/(50 – 150) = -50/-100 = ½. Then, to find the y-intercept use the general y-intercept form (y = mx + b) and substitute in the slope measure you calculated and one of the points you were given. Thus, the equation can be written as Y = (1/2)X + 75. The solution to the two equations is (83.3, 116.7).c. Hint: this one is a bit more challenging!) You are told that there are two linear relationships between Y and X where Y is the variable measured on the vertical axis and X is the variable measured on the horizontal axis. The first linear relationship is describedas follows: the Y variable is equal to 5 more than twice the X variable. The second linear relationship is described as follows: the X variable is equal to 5 less than twice the Y variable. Find the equations for the two lines and then calculate the solution (X, Y) for these two equations. Answer:The first line can be written as Y = 5 + 2X while the second line can be written as X = 2Y – 5.You can rewrite the second equation, solving for Y, in order to have this equation in slope-intercept form. Thus, Y = (1/2) X = (5/2). Take a moment to verify that each equation actually fits the description you were given in the problem. Use these two equations to find the solution: (X, Y) = (-5/3, 5/3).If you are having difficulties with the fractions you might want to take some time to review the rules about adding, subtracting, multiplying and dividing fractions. I will assume that you know these rules and that you can use them accurately.2. More math review:a. You have three hundred books and you decide to purchase an additional one hundred books. What is the percentage increase in books?Answer:To find a simple percent you want to use the following formula:Percentage change = [(Current Value – Initial Value)/(Initial Value)] * 100In this case, the percentage change in books is equal to [(400 – 300)/300] * 100 = 33%.b. You have four hundred books and you decide to decrease your personal library by selling one hundred books. What is the percentage reduction in books?Answer:To find a simple percent you want to use the following formula:Percentage change = [(Current Value – Initial Value)/(Initial Value)] * 100In this case, the percentage change in books is equal to [(300 – 400)/400] * 100 = a decrease of 25%.c. Look at your answers in parts (a) and (b). Are your answers the same in absolute value terms or are they different? Explain this similarity or this difference. Answers:The absolute values of the two answers are different (the signage of your answer tells you if the percentage change is an increase or a decrease) because the two calculations are based upon different base, or initial value, measures. Even though both examples use the same numbers the base in part (a) is 300 while the base in part (b) is 400. If this is unclear to you, you will want to spend some time making sure that you review how to calculate simple percentages. 3. Absolute and Comparative Advantage:Suppose Joe and Mary both have 1 hour available to spend in some combination of cleaning baths and washing dishes. You are told that Joe can clean one bath in 10 minutes or wash one set of dishes in 20 minutes. Mary can clean one bath in 5 minutes or wash one set of dishes in 15 minutes. a. Draw two graphs representing Joe’s PPF and Mary’s PPF. In your graph measure clean baths on the horizontal axis. Label your graphs carefully and completely. Assume in drawing the two PPFs that they are linear. Answer:b. Fill in the blank:___________ has the absolute advantage in cleaning baths.___________ has the absolute advantage in washing dishes.___________ has the comparative advantage in cleaning baths.___________ has the comparative advantage in washing dishes. Answers:___Mary____ has the absolute advantage in cleaning baths.___Mary____ has the absolute advantage in washing dishes.___Mary____ has the comparative advantage in cleaning baths.___Joe _____ has the comparative advantage in washing dishes. c. Joe’s apartment has three dirty baths and 2 sets of dirty dishes while Mary’s apartment has 6 dirty baths and 2 sets of dirty dishes. Assume that Joe and Mary both have 1 hour they can devote to washing dishes and cleaning baths. If they do not specialize and trade, is it possible for Joe to get


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UW-Madison ECON 102 - Homework #1

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