Economics 102 Spring 2012 Homework #1 Due 2/8/12 Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on top of the homework (legibly). Make sure you 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 will need that number when you submit exams and homework. Late homework will not be accepted so make plans ahead of time. Please show your work. Good luck! 1. For the following pairs of equations below, find the point of intersection for each pair. a. 7y – 14x = 49 and -3x = y – 12 b. 13 + y = 3x and 3 – x = y c. 5y + 15x = 25 and 2x – y = 10 a. Set both equations in terms of y and solve for x, then you can plug that value of x into either equation and get the corresponding y. 7y – 14x = 49 (Divide both sides by 7) -3x = y – 12 (Add 12 to both sides) y – 2x = 7 (Add 2x to both sides) y = 12 – 3x y = 7 + 2x 7 + 2x = 12 – 3x (subtract 7 from both sides and add 3x to both sides) 5x = 5 (Divide both sides by 5) x = 1 => y = 7 + 2(1) = 12 – 3(1) = 9 b. 13 + y = 3x (Subtract 13 from both sides) 3 – x = y y = 3x – 13 3x – 13 = 3 – x (Add 13 to both sides and add x to both sides) 4x = 16 (Divide both sides by 4) x = 4 => y = 3 (4) – 13 = 3 – 4 = -1 c. 5y + 15x = 25 (Divide both sides by 5) 2x – y = 10 (Add y to both sides) y + 3x = 5 (Subtract 3x from both sides) 2x = 10 + y (Subtract 10 from both sides) y = 5 – 3x y = 2x – 10 5 – 3x = 2x – 10 (Add 3x to both sides and add 10 to both sides) 5x = 15 x = 3 => y = 5 – 3(3) = 2(3) – 10 = -4 2. Find the equation of a line described by the following: a. A line that goes through the points (7, 5) and (11, 3) b. A line that has a y-intercept of 8 and goes through the point (1, 2) c. A line with a slope of 3 and goes through the point (6, 5) a. Find the slope first. m = Δy/Δx = (3 – 5)/(11 – 7) = -2/4 = -1/2 y = mx + b. Find b. Plug one of the points into the line with the slope. 3 = (-1/2)(11) + b => 3 = -11/2 + b => b = 17/2 The line: y = (-1/2)x + 17/2b. The y-intercept has the point (0, 8). With two points we can find the line. m = Δy/Δx = (2 – 8)/(1 – 0) = -6. Since we already have the y-intercept as 8, b = 8. The line: y = -6x + 8 c. We have the slope and a point. We only need the y-intercept. Plug the point into the equation of the line and solve for b. 5 = 3(6) + b => 5 = 18 + b => b = -13 The line: y = 3x – 13 3. Assume Agrabah and Mypos are two countries. Both countries produce carpets and cups. Suppose that these two countries only use labor to produce these two goods (this is just a simplifying assumption to make our work easier). Agrabah is a larger country and has 400 hours of labor available while smaller Mypos only has 100 hours of labor. The following table tells you how many hours of labor are needed in each country to produce one carpet or one cup. Hours of Labor Needed to Produce One Carpet Hours of Labor Needed to Produce One Cup Agrabah 2 hours of labor 8 hours of labor Mypos 4 hours of labor 5 hours of labor a. Using the above table, draw the production-possibility frontiers (PPF) for both Agrabah and Mypos(on two separate charts). Put number of carpets produced on the Y axis, and cups produced on the X axis. b. What is Agrabah’s opportunity cost of producing one carpet? c. What is Agrabah’s opportunity cost of producing one cup? d. What is Mypos’s opportunity cost of producing one carpet? e. What is Mypos’s opportunity cost of producing one cup? f. Which country has the absolute advantage in producing carpets? g. Which country has the absolute advantage in producing cups? h. Which country has the comparative advantage in producing carpets? i. Which country has the comparative advantage in producing cups? j. What range of trading prices would be acceptable to both countries in terms of carpets for 1 cup? k. What range of trading prices would be acceptable to both countries in terms of cups for 4 carpets? l. Suppose now that the two countries agree to team up and combine their production. Draw the combined PPF for the two countries. a. To find the PPF of both countries, we want to find the maximum number carpets and cups each country would produce if it only produced one product. If Agrabah only produced carpets they could produce 400/2=200 carpets and if they only produced cups they could produce 400/8=50 cups. Mypos could produce 100/4=25 carpets and 100/5=20 cups. Plot these points on the vertical and horizontal axes and connect them to obtain the PPF for each country.b. In the time it takes Agrabah to produce one carpet, it can produce 2/8=1/4 of a cup. So the opportunity cost of a carpet is 1/4 cup. Or take maximum cups divided by maximum carpets to find 50/200=1/4. c. To find opportunity cost of cups, divide max number of carpets by max number of cups. This yields 200/50=4 d. Mypos opportunity cost for carpets is 20/25=4/5. e. Mypos opportunity cost for cups is 25/20=5/4. f. Agrabah has the absolute advantage in producing carpets because 200>25. g. Agrabah has the absolute advantage in producing cups because 50>20. h. To have comparative advantage means to have lower opportunity cost. Thus Agrabah has a comparative advantage in carpets since 1/4 is less than 4/5. i. Mypos has a comparative advantage in producing cups because 5/4 is less than 4. j. An acceptable trade price should lie between both countries opportunity cost for cups. So a trading price for one cup must be between 5/4 and 4 carpets. k. As in the previous problem, a trade price for one carpet must be between 1/4 and 4/5 cups. A price for 4 carpets must be 4 times this, so between 1 and 16/5 cups. l. To find the combined PPF, we want to find the Y intercept, the X intercept, and the location of the kink. If …
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