This Document Contains Appendix A to E APPENDIX A MATH ESSENTIALS: UNDERSTANDING GRAPHS AND SLOPE Learning Objectives LO A.1: Create four quadrants using x- and y-axes and plot points on graph. LO A.2: Use data to calculate slope. LO A.3: Interpret what direction and steepness of slope indicate about a line. Appendix Outline Creating a Graph (LO A.1) Graphs of one variable Graphs of two variables Slope Calculating slope (LO A.2) The direction of a slope (LO A.3) The steepness of a slope Problems and Applications Review Questions 1. Create four quadrants using x- and y-axes. Use your graph to plot the following points. [LO A.1] a. (1,4) b. (-2,1) c. (-3, -3) d. (3, -2) Answer: Each point is an (x,y) coordinate. For example, for point a, we move up to a positive 1 on the horizontal, or x, axis, and then move up to positive 4 on the vertical, or y, axis. 2. Create four quadrants using x- and y-axes. Use your graph to plot the following points. [LO A.1] a. (0,4) b. (0, -2) c. (1,0) d. (-3, 0) Answer: Each point is an (x,y) coordinate. For example, for point a, we stay on 0 on the horizontal, or x, axis and then move up to positive 4 on the vertical, or y, axis. 3. Use the curve labeled “Demand” in Figure AP-1 to create a table (schedule) that shows Price in one column and Quantity in another. What is the slope of the curve labeled “Demand”? [LO A.2] Answer: Price Quantity Demanded 70 0 50 10 30 20 10 30 0 35 Each point is an (x,y) coordinate. For example, for point a, we stay on 0 on the horizontal, or x, axis and then move up to positive 4 on the vertical, or y, axis. The slope of the curve labeled “Demand” is -0.5. 4. Use the curve labeled “Demand” in Figure AP-2 to create a table (schedule) that shows Price in one column and Quantity in another. What is the slope of the curve labeled “Demand”? [LO A.2] Answer: Price Quantity Demanded 70 0 50 10 30 20 10 30 0 35 The slope of the curve is the rise over the run, or the change in the price divided by the change in the quantity: (−70/35) = −2. 5. Use the information about price and quantity in Table AP-1 to create a graph, with Price on the y-axis and Quantity on the x-axis. Label the resulting curve “Demand”. What is the slope of that curve? [LO A.2] Answer: The slope of the line is the rise over the run, or the change in the price divided by the change in the quantity: (-12/120) = -0.1. 6. Use the information about price and quantity in Table AP-2 to create a graph, with Price on the y-axis and Quantity on the x-axis. Label the resulting curve “Demand”. What is the slope of that curve? [LO A.2] Answer: The slope of the curve is the rise over the run, or the change in the price divided by the change in the quantity: (-25/5) = -5. 7. Use the curve labeled “Supply” in figure Ap-3 to create a table (schedule) that shows Price in one column and Quantity in another. What is the slope of the curve labeled “Supply”? [LO A.2] Answer: The slope of the curve labeled “Supply” is the rise over the run, or the change in the price divided by the change in the quantity: (30/10) = 3. Price Quantity Demanded 0 0 30 10 60 20 90 30 120 40 150 50 8. Use the curve labeled “Supply” in figure Ap-4 to create a table (schedule) that shows Price in one column and Quantity in another. What is the slope of the curve labeled “Supply”? [LO A.2] Answer: The slope of the curve is the rise over the run, or the change in the price divided by the change in the quantity: (40/160) = 0.25. Price Quantity Supplied 0 0 10 40 20 80 30 120 40 160 9. Use the information about price and quantity in Table AP-3 to create a graph, with Price on the y-axis and Quantity on the x-axis. Label the resulting curve “Supply”. What is the slope of that curve? [LO A.2] Answer: The slope of the curve is the rise over the run, or the change in the price divided by the change in the quantity: (125/25) = 5. 10. Use the information about price and quantity in Table AP-4 to create a graph, with Price on the y-axis and Quantity on the x-axis. Label the resulting curve “Supply”. What is the slope of that curve? [LO A.2] Answer: The slope of the curve is the rise over the run, or the change in the price divided by the change in the quantity: (48/12) = 4. 11. What is the direction of slope indicated by the following examples? [LO A.3] a. As price of rice increases, consumers want less of it. b. As the temperature increases, the amount of people who use the town pool also increases. c. As farmers use more fertilizer, their output of tomatoes increases. Answer: a. As price of rice increases, consumers want less of it. The direction of the slope is negative. b. As the temperature increases, the amount of people who use the town pool also increases. The direction of the slope is positive. c. As farmers use more fertilizer, their output of tomatoes increases. The direction of the slope is positive. When the slope of a line is positive, we know that y increases as x increases, and y decreases as x decreases. If a line leans upward, then its slope is positive. When the slope of a line is negative, we know that y decreases as x increases, and y increases as x decreases. If a line leans downward, then its slope is negative. As the price of rice increases, consumers want less of it. The direction of the slope is negative. As the temperature increases, the amount of people who use the town pool also increases. The direction of the slope is positive. As farmers use more fertilizer, their output of tomatoes increases. The direction of the slope is positive. 12. Rank the following equations by the steepness of their slope from lowest to highest. [LO A.3] a. y = -3x + 9 b. y = 4x + 2 c. y = -0.5x + 4 Answer: The slope of the curve is the rise over the run or the change in the price divided by the change in the quantity. The steepness rank is determined by the absolute value of the slope. a. Slope is -3 Rank is 2. b. Slope is +4 Rank is 3. c. Slope is -0.5 Rank is 1. APPENDIX B MATH ESSENTIALS: WORKING WITH LINEAR EQUATIONS Learning Objectives LO B.1: Use linear equations to interpret the equation of a line. LO B.2: Use linear equations to explain shifts and pivots. LO B.3: Use linear equations to solve for equilibrium. Appendix Outline Interpreting the Equation of a Line (LO B.1) Turning the graph into an equation Turning the equation into a graph Equations with x and y reversed Shifts and Pivots (LO B.2) Solving for Equilibrium (LO B.3) Problems and Applications 1. Use the demand curve in Figure BP-1 to derive a demand equation. [LO B.1] Answer: First, use the endpoints to calculate the slope. The slope is the rise over the run, or the change in the Y, or price, direction (120 - 0), divided by the change in the X, or quantity, direction (0 - 40). The slope is 120/-40 or -3. To find the Y-intercept, look at the graph and determine where the line intersects the Y-axis. This occurs at $120, so the Y-intercept is 120. The equation is then: P = 120 - 3Q. 2. Use the demand schedule in Table BP-1 to derive a demand equation. [LO B.1] Answer: The y-intercept occurs where quantity demanded = 0, or 80. The slope is the change in price (10) divided by the change in quantity demanded (-40), which is (-10/40) or 0.25. The equation is P = 80 - 0.25Q. 3. Use the supply curve in Figure BP-2 to derive a supply equation. [LO B.1] Answer: The y-intercept occurs where the line crosses the y-axis, which in this case is 0. The slope is the rise over the run, or the change in price (2) divided by the change in quantity (40), which is (2/40) or 0.05. The equation is P = 0.05Q. 4. Use the supply schedule in Table BP-2 to derive a supply equation. [ [LO B.1] Answer: The y-intercept occurs where quantity supplied = 0, or 100. The slope is the change in price (100) divided by the change in quantity supplied (25), which is (100/25) or 4. The equation is P = 100 + 4Q. 5. Graph the equation P = 2Q + 3. Is this the supply curve or demand curve? [LO B.1] Answer: This line slopes upward, so it is a supply curve. The slope is the rise over the run or the change in price (13 - 3) divided by the change in quantity (5 - 0). This curve has a slope of (10/5) or 2. 6. Graph the equation P = -8Q + 10. Is this the supply curve or demand curve? [LO B.1] Answer: This line slopes downward, so it is a demand curve. The slope is the rise over the run or the change in price (10 - 0) divided by the change in quantity (0 - 1.25), which is (10/1.25) or -8. 7. Rearrange the equation Q = 5 – 0.25P and sketch the graph. Is this a supply curve or a demand curve? [LO B.1] Answer: Start with Q = 5 - 0.25P. Subtract 5 from both sides: Q – 5 = - 0.25P. Divide both sides by - 0.25: -4Q + 20 = P. Re-arrange: P = 20 - 4Q. This is a downward-sloping demand curve. 8. Rearrange the equation Q = 0.2P and sketch the graph. Is this a supply curve or a demand curve? [LO B.1] Answer: Start with Q = 0.2P. Divide both sides by 0.2: 5Q = P. Re-arrange: P = 5Q. This is an upward-sloping line, so it is a supply curve. 9. The entrance fee at your local amusement park is $20 for the day. The entrance fee includes all rides except roller coasters. Roller coasters cost an extra $2 per ride. [LO B.2] a. Write an equation that represents how much money you will spend on rides as a function of the number of rides you on: S = total spending on rides; Q = the quantity of roller coasters rides. b. What is your total spend on rides if you ride 4 roller coasters? c. Draw a graph of the relationship between total spending on rides and the number of roller coaster rides. d. Redraw the graph from part (c) to show what changes if the entrance fee increases to $25. e. Rewrite the equation from part (a) to incorporate the increased entrance fee of $25. f. After the entrance fee increases to $25, what is your total spending on rides if you ride 4 roller coasters? Answer: a. The y-intercept is 20, which is the cost of the entrance fee when no roller coasters are ridden. Every additional roller coaster ride costs $2. So, the equation of the line is S = 20 + 2Q. b. If you ride 4 roller coasters, you will pay $20 for entry plus $2 × 4 for the four rides. The total is $28. c. See the correct graph below. The line should start at a price of $20 and a quantity of 0 roller coaster rides and end at a price of $40 and a quantity of 10 roller coaster rides. d. If the entrance fee increases to $25, the line will shift up by $5, but will be parallel to the original line. It will start at $25 and 0 roller coaster rides and end at $45 and 10 roller coaster rides. e. Since the entrance fee is all that has changed, only the y-intercept changes in the equation. The new equation is S = 25 + 2Q. f. If you ride 4 roller coasters, you will pay $25 for entry plus $2 × 4 for the four rides. The total is $33. 10. Use the following two equations: [LO B.3] (1) P = 12 – 2Q (2) P = 3 +Q a. Find the equilibrium price and quantity. b. Graph the demand and supply equations. Illustrate the equilibrium point. Answer: a. Setting the equations equal to each other: 12 – 2Q = 3+ Q. Solving for Q: 9 = 3Q or Q = 3. Substituting into either equation: P = 12 – 2(3) = $6. The x-intercept for the demand line occurs where P = 0. So 0 = 12 - 2Q. Solving, -12 = -2Q, so the x-intercept is Q = 6. The y-intercept for the demand line occurs where Q = 0. So the y-intercept is P = 120. The y-intercept for the supply line occurs where Q = 0. So the y-intercept is P = 3. The endpoint for the supply line is determined where Q = 6. So when Q = 6, P = 3 + 6 or $9. 11. With reference to Table BP-3: [LO B.3] a. Use the information from the table to create the demand and supply equations. b. Use your demand and supply equations to solve for equilibrium. c. Graph supply and demand curves. Illustrate the equilibrium point. Answer: a. The y-intercept of the demand equation occurs where the quantity demanded is 0 at 120. The slope of the demand equation is the change in P (20) divided by the change in quantity demand (-2) or -10. The demand equation is P = 120 – 10Q. The y-intercept of the supply equation occurs where the quantity supplied is 0 at 0. The slope of the supply equation is the change in P (20) divided by the change in quantity supplied (4) or 5. The supply equation is P = 5Q. b. Setting the equations equal to each other: 120 – 10Q = 5Q. Solving for Q: 120 = 15Q Q = 8. Substituting into either equation: P = 120 – 10(8) = $40. The x-intercept for the demand line occurs where P = 0. So 0 = 120 - 10Q. Solving: -120 = 10Q, so the x-intercept is Q = 12. The y-intercept for the demand line occurs where Q = 0. So the y-intercept is P = 120. The y-intercept for the supply line occurs where Q = 0. So the y-intercept is P = 0. The endpoint for the supply line is determined where Q = 24. So when Q = 24, P = 5 (24) or $120. This is also shown in the table. c. APPENDIX C MATH ESSENTIALS: CALCULATING PERCENTAGE CHANGE, SLOPE, AND ELASTICITY Learning Objectives LO C.1: Understand how to calculate percentage changes. LO C.2: Use slope to calculate elasticity. Appendix Outline Percentage Change (LO C.1) Slope and Elasticity (LO C.2) X over Y, or Y over X? Elasticity changes along with constant slope Problems and Applications 1. Calculate the percentage change in each of the following examples: [LO C.1] a. 8 to 12. b. 18 to 14. c. 130 to 120. d. 95 to 105. Answer: Using the mid-point method, percentage change in X = a. 8 to 12: b. 18 to 14: c. 130 to 120: d. 95 to 105: 2. Find the percentage change in price in each of the following examples using the mid-point method. [LO C.1] a. The price of a $4.50 sandwich increases to$5.50. b. The sale discounts the price of a sofa from $750 to $500. Answer: Using the mid-point method, percentage change in X = a. 8 to 10: [(10 - 8)/[(10 + 8)/2]] x 100 = 22.22% b. 8 to 6: [(6 - 8)/[(6 + 8)/2]] x 100 = -28.57% c. 140 to 130: [(130 - 140)/[(130 + 140)/2]] x 100 = -7.41% d. 90 to 110: [(110 - 90)/[(110 + 90)/2]] x 100 = 20.00% 3. Use the demand curve in CP-1 to answer the following questions. Use the mid-point method in your calculations. [LO C.2] a. What is the price elasticity of demand for a price change from $0 to $20? b. What is the price elasticity of demand for a price change from $20 to $40? c. What is the price elasticity of demand for a price change from $40 to $60? Answer: Price elasticity of demand = Price elasticity of demand = a. From $0 to $20: b. From $20 to $40: c. From $40 to $60: 4. Use the demand schedule in Table CP-1 to answer the following questions. Use the midpoint method in your calculations. [LO C.2] a. What is the price elasticity of demand for a price change from $4 to $8? b. What is the price elasticity of demand for a price change from $8 to $16? c. What is the price elasticity of demand for a price change from $20 to $24? Answer: Price elasticity of demand = % Change in QD/% Change in Price Price elasticity of demand = a. The price elasticity of demand for a price change from $4 to $8: b. The price elasticity of demand for a price change from $8 to $16: c. The price elasticity of demand for a price change from $20 to $24: 5. Use the demand schedule in Table CP-2 to answer the following questions. Use the midpoint method when calculating elasticity. [LO C.2] a. What is the price elasticity of demand for a price change from $2 to $3? What is the slope of the demand curve for a price change from $2 to $3? b. What is the price elasticity of demand for a price change from $3 to $5? What is the slope of the demand curve for a price change from $3 to $5? c. What is the price elasticity of demand for a price change from $6 to $7? What is the slope of the demand curve for a price change from $6 to $7? Answer: Price elasticity of demand = % Change in QD/% Change in Price Price elasticity of demand = Slope = a. The price elasticity of demand for a price change from $2 to $3: The slope of the demand curve for a price change from $2 to $3 is m = ($3 - $2) / (35 - 42) = -0.14. b. The price elasticity of demand for a price change from $3 to $5: The slope of the demand curve for a price change from $3 to $5 is m = ($5 - $3) / (21 - 35) = -0.14. c. The price elasticity of demand for a price change from $6 to $7: The slope of the demand curve for a price change from $6 to $7 is m = ($7 - $6) / (7 - 14) = 0.14. APPENDIX D MATH ESSENTIALS: THE AREA UNDER A LINEAR CURVE Learning Objectives LO D.1: Calculate surplus by finding the area under a linear curve. Appendix Outline The Area under a Linear Curve (LO D.1) Problems and Applications 1. Use the graph in Figure DP-1 to answer the following questions. [LO D.1] a. What is the amount of consumer surplus? b. What is the amount of producer surplus? c. What is the amount of total surplus? Answer: a. Consumer surplus is found by taking the area of the triangle above the market price and below the demand curve. In this instance, the amount of consumer surplus is 1/2(90 - 45) × 60 = $1,350. b. Measuring producer surplus is found by taking the area of the triangle below the market price and above the supply curve. In this instance, the amount of producer surplus is 1/2(45 × 60) = $1,350. c. Total surplus is the sum of consumer and producer surplus. In this instance, the amount of total surplus is $2,700. 2. Use these two supply and demand equations to answer the following questions. [LO D.1] a. What is the equilibrium price? What is the equilibrium quantity? b. Draw a graph of supply and demand and illustrate the equilibrium. c. What is the amount of consumer surplus? d. What is the amount of producer surplus? e. What is the amount of total surplus? Answer: a. To find the equilibrium quantity, set demand equal to supply and solve for Q. 50 - 4Q = 2 + 2Q, so Q = 8. The equilibrium quantity is 8. To find the equilibrium price, put Q into either equation and solve for P. 50 - 4(8) = $18. The equilibrium price is $18. b. c. Consumer surplus is found by taking the area of the triangle above the market price and below the demand curve. The amount of consumer surplus is 1/2(50 - 18) × 8 = $128. d. Measuring producer surplus is found by taking the area of the triangle below the market price and above the supply curve. The amount of producer surplus is 1/2(18 - 2) × 8 = $64. e. Total surplus is the sum of consumer and producer surplus. The amount of total surplus is $128 + $64 = $192. APPENDIX E MATH ESSENTIALS: ALGEBRA AND AGGREGATE EXPENDITURE Learning Objectives LO E.1: Represent the components of planned aggregate expenditure algebraically and use these expressions to find equilibrium aggregate expenditure. LO E.2: Illustrate the multiplier effect of aggregate expenditure using the algebraic form of the aggregate expenditure equilibrium. Appendix Outline Using Algebra to Find Equilibrium Aggregate Expenditure (LO E.1) Using algebra to derive the expenditure multiplier (LO E.2) Problems and Applications 1. Consider the following components of the aggregate expenditure equilibrium model: C = 0.6(Y – 200) + 150 Iplanned = 175 G = 200 NX = 50 Assume all model parameters are in billions of dollars. [LO E.1, LO E.2] a. What is the marginal propensity to consume in this economy? b. What is the level of taxes in this economy? (You can assume that the functional forms above are consistent with those described earlier.) c. What is the equilibrium level of aggregate expenditure in this economy? Now suppose that planned investment decreases by $25 billion. d. Find the overall change in equilibrium aggregate expenditure that results from this initial change by finding the new level of equilibrium aggregate expenditure and comparing this new level to the initial level. e. Again, find the overall change in equilibrium aggregate expenditure that results from this initial change, but this time use the changes formulation of the equilibrium aggregate expenditure expression directly. Answer: a. The consumption function C = 0.6 (Y - 200) + 150 indicates consumption is $150 billion plus an amount that depends on disposable income. Disposable income is defined as income minus taxes, or Y - 200 in this case. The 0.6 is the marginal propensity to consume, indicating people spend 60% of their disposable income, and save the remaining 40%. b. Taxes are equal to $200 billion, given disposable income is Y - 200. c. The equilibrium level of aggregate expenditure is found by equating planned expenditure and actual expenditure, such that planned aggregate expenditure (PAE) is equal to output (Y). In other words, this means firms will want to produce a level of output equal to the total amount of spending in the economy. We know PAE = C + Iplanned + G + NX so: PAE = 0.6 (Y - 200) + 150 + 175 + 200 + 50 = 0.6 (Y - 200) + 575 = 0.6Y + 455. Now set PAE = Y so we have 0.6Y + 455 = Y. Rearrange terms to solve for Y = $1,137.5 billion. d. Planned investment is now equal to 150 and planned expenditure is PAE = 0.6Y + 430. The equilibrium level of aggregate expenditure is Y = $1,075 billion. e. The equilibrium level of aggregate expenditure declined from $1,137.5 billion to $1,075 billion, so the change was equal to $62.5 billion. The expenditure multiplier is equal to 1/(1- b) where b is the marginal propensity to consume, so we get 1/(1-0.6) or 2.5. Notice we can multiply the change in investment (-$25 billion) by the expenditure multiplier (2.5) to get the change in equilibrium output (-$62.5 billion). Solution Manual for Macroeconomics Dean Karlan, Jonathan Morduch 9781259813436
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