Problems

1. Study the four accompanying diagrams. Consider the following statements and indicate which diagram matches each statement. Which variable would appear on the horizontal and which on the vertical axis? In each of these statements, is the slope positive, negative, zero, or infinity?
   

   • Question

     LBVc+dF2n7H+4+ZPVVpXFlQcliMVQNGvTJTiFM+6t0sn9Cn2zO+aAfQ8UzJhp31cNhCtACPp4CjpZFUX+7FDA2a4Da3yi/Cym0f1bbMeB++pHi5h

     Prob 100 1a. If the price of movies increases, fewer consumers go to see movies.

   • Question

     m4VmXuv70qUYN9hStU1qqGrtY5is5oIoW8Q52kOnVu9sn2Wjg2HcpM7lF4dIIOcb5YeHRSHlUMtHUJvSEpCv1XJmyb1+CupytpVQiQgrmiyOYJQxKN5PX6NsPN/MyPoSG7RlFrIGkSQ=

     Prob 100 1b. More experienced workers typically have higher incomes than less experienced workers.

   • Question

     xBXJHuwrPVRpML6EIC1DaF0eO46LDT+dAeGoTp0phMMuCSKU2iiL0LQF4LG0H7k9tIzO42Q17V/ZehqGHR4LfI6S82RhQ5JgYRJpQLn/xIl+qUXqrh1ilJ98hJdolqVxIuawreC4B88=

     Prob 100 1c. Whatever the temperature outside, Americans consume the same number of hot dogs per day.

   • Question

     im/2NW1/KleXsTTdMD+JhBcPPXUHJH5LXgNEi6438hKdXlzre2qhln66B2xyWrBtQtaCymSvOGp8yqgkQVkMpttZY54tlr1NhYSM8utjj0YolLdKOv/49w==

     Prob 100 1d. Consumers buy more frozen yogurt when the price of ice cream goes up.

   • Question

     SOr20rQRrLUN8p2Y7o8BnSf0qxhA5axwwXL3Sn2gKeh6N3AdXRY3HHesJC/LAYNj4034LI7LzZ51Kfg/2eGVcY2Sg7XBnUPAgzmfw61gEGThPa8w9kcH/MyBGEfiuX5kODa+YH7HAz0+vNEI4FDz1PjefHkb+M66Fm4zAFMXc3yn0FvvrwBA9NnKqMI53QYZ

     Prob 100 1e. Research finds no relationship between the number of diet books purchased and the number of pounds lost by the average dieter.

   • Question

     tM88sDSwP7tXUzgB313fdTRel0nUawmYIuqKqmc1PZ0ZxK49Ry+alx7rFF7RfoiRf2Zfs66p/8JD6ObXfV5BdtPPz9KB/Zapd4YMt9QB5q0CCkrP

     Prob 100 1f. Regardless of its price, Americans buy the same quantity of salt.
2. During the Reagan administration, economist Arthur Laffer argued in favor of lowering income tax rates in order to increase tax revenues. Like most economists, he believed that at tax rates above a certain level, tax revenue would fall because high taxes would discourage some people from working and that people would refuse to work at all if they received no income after paying taxes. This relationship between tax rates and tax revenue is graphically summarized in what is widely known as the Laffer curve. Plot the Laffer curve relationship assuming that it has the shape of a nonlinear curve. The following questions will help you construct the graph.

   • Question

     hSO2UILC43wEdTquEVjCkhbpIQF2GrQxrBiuJRXgIkegeYsy7nferl2QfsKlO0WTKQNImLFH0kiANXvLn1n9p3CQcnIxCMU/S6r2pHFII4j0djpItgy0S2l0aV/mtMHPjgqniK/cEWKaW8e/zniLmXxqEcDl0Cz1rVkMIkMldpR1J4wTz2nifWlVcLK5/OAdZLaiij5HNiaYu9WiSTdhcxbBJhlDH4j7h6cFtfR9ijHcvfgnF1nL4QfjJsiVMqdJ

     Prob 100 2a. Which is the independent variable? Which is the dependent variable? On which axis do you therefore measure the income tax rate? On which axis do you measure income tax revenue?

   • Question

     98kI2jwEJo6u6oTWne0ZUV8yibJchHlx/7A8AOb3Nlms2tDdUsJx9CMU0OCYwMHUI0QVgZ1xLBATDMWVG9NBtUhs2D0=

     Prob 100 2b. What would tax revenue be at a 0% income tax rate?

   • Question

     tYF028A7Tl8yvqXYBevR7q0js0LwwmVHNj06tnbCnZyKx0VJiVlmbFIAPB0+jJpQKFMUEP1oUGlb82qHA8J+yE3Jq32qDd6tZwqhlfOoL89iFwUjiT7AzzAq+DEqhaE8hdLK8Ynjfun5EQKw1OQiZbXAWPg=

     Prob 100 2c. The maximum possible income tax rate is 100%. What would tax revenue be at a 100% income tax rate?

   • Question

     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

     Prob 100 2d. Estimates now show that the maximum point on the Laffer curve is (approximately) at a tax rate of 80%. For tax rates less than 80%, how would you describe the relationship between the tax rate and tax revenue, and how is this relationship reflected in the slope? For tax rates higher than 80%, how would you describe the relationship between the tax rate and tax revenue, and how is this relationship reflected in the slope?
3. In the accompanying figures, the numbers on the axes have been lost. All you know is that the units shown on the vertical axis are the same as the units on the horizontal axis.
   

   • Question

     mPeuxEoggw4hfH2KQvQ58saQQ5eN6vceqcCh+/FxJBDCoxyUeOXBx+tRYQkcpXgIcChQTfqp39rUaVNornZ9qH/Vt6sAvUvvIPLGRtr5Mf6z6HW0PNm2C6P541c7LXkKcqHuLQNjSc1JHw6V

     Prob 100 3a. In panel (a), what is the slope of the line? Show that the slope is constant along the line.

   • Question

     F22z3A86TBspESnSqoSz/vthdPNm7eicAm/CSe/9KrwBft6+7VpOUdu83n12U9R/9M+V2e8Ywc7PQh7gze8hmMZuuEzOOgfYUCmM5MZgRO5IL69/pqK37F1hJGNSOeoXj3KqPO2wmsdQ+nsk

     Prob 100 3b. In panel (b), what is the slope of the line? Show that the slope is constant along the line.
4. Answer each of the following questions by drawing a schematic diagram.

   • Question

     Mb3p3bSzoA2xhUEor3eMS7Jnkzll159LHd4Bm7C9BbZGZt+SCLU16yruMLLNzP8NDN00giJx1mDdvrmDynkAMULR9aW+aW1TGaOH0EMN+f8BD5JcKJRZU3cP5ldW0OwjbTcT9QdFWLulCa9rHvnc2xprZL7K3P12pYOjBGvb0OYFuMcT88vQ9+Hyei2IZgjgkzYU3dQnHpSSZ4trV0yaajK1z22TLIkOFlYYdrXOSYtkP5AimJZ6URDthkpd4Ux6E7rKoZiPZvo/qbdlcXT9f3FBDHo3MCJwRsV+yY5Cy3HKSIj8pPhR9ASVcFxhhMW2+UhDg2uSDPWlwq3mOy9xtKqxOEfOs6tKrniYraan9ywrs9T1LGtToMINmZMcSzvVgmZj0PoNqvBDwgZ3JHE5ctENJVXRfXV++mAMIfDSESu47Np8FNkrM7vKLGT6gW/NIB+nhA==

     Prob 100 4a. Taking measurements of the slope of a curve at three points farther and farther to the right along the horizontal axis, the slope of the curve changes from −0.3, to −0.8, to −2.5, measured by the point method. Draw a schematic diagram of this curve. How would you describe the relationship illustrated in your diagram?

   • Question

     3Lt7z6KLYcdOuU8XurKSLg1E/0sR6cDvfYoSEInykS74kpMq2wpQGu0DDM1W6z6obygFKDq2xkdA0xPVjuoShmVkHq9VreZEGyKkSt3OwzqnsVZMKi/YzTTbBxJjKoKWOgL3IfY6aGxwmFcC9jRpG17csH8eRTWlYzs2epw1AV2RBkFvbhXWTXwtHm/jXdJiByhEhI6PcuAvGuxnwQJyc6qKtS8+hGnf4MYnBNB1DanSF20dZlb2CBwC8vvVizt9xAiaoYJyxaScZUqPeiYCqf4U07vZUFhUy3MwxNJeLOk6DycY3IsKJKFgdh06LoaJMmCS6y0ZQEeMTBJh73AwDhWHyN0dMN7IsjHizvP+mOz7aYU58U/BPRqnUqBsZUfCVvaAFuOO/1o5robVNBTf5Oa4ceXN59594tBl2LcKkuQ=

     Prob 100 4b. Taking measurements of the slope of a curve at five points farther and farther to the right along the horizontal axis, the slope of the curve changes from 1.5, to 0.5, to 0, to −0.5, to −1.5, measured by the point method. Draw a schematic diagram of this curve. Does it have a maximum or a minimum?

5. Question

   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

   Prob 100 5. For each of the accompanying diagrams, calculate the area of the shaded right triangle.

6. Question

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   Prob 100 6. The base of a right triangle is 10, and its area is 20. What is the height of this right triangle?
7. The accompanying table shows the relationship between workers’ hours of work per week and their hourly wage rate. Apart from the fact that they receive a different hourly wage rate and work different hours, these five workers are otherwise identical.

   Name	Quantity of labor (hours per week)	Wage rate (per hour)
   Athena	30	$15
   Boris	35	30
   Curt	37	45
   Diego	36	60
   Emily	32	75

   • Question

     iAn+WQ082KWCo7BsC2e/EmV82d8K8P+jCP0d6mXjOfpU5w5XPqrmmgjlDI37GMszcdRwZa3MTaM5ZZs5VAWmQbZ7WJfXvbtoppuPAjVw8bo7qemE/0hmWyVrsUM=

     Prob 100 7a. Which variable is the independent variable? Which is the dependent variable?

   • Question

     uoaq52qyXmqMtxcejFB4M4YmKCQ7mVyAYy4wJBFmfQuAbTGu1Hmi2HjkPlQIw/AFS+nZ0ahzxsicluiovLuRDXckGfpeOudxzvvvg30974GH0PAWzHThj0PaFyTB9k5Ij4510u4OxdTYfp9lDr5iA9O2B+bgckCgB4ON27GbFYKTuCo99/JKzVAsc86Vp2DQU5vfkWgJVOWxTkJQ7fkwOx3LMLoDwI1bAMAUxQ==

     Prob 100 7b. Draw a scatter diagram illustrating this relationship. Draw a (non-linear) curve that connects the points. Put the hourly wage rate on the vertical axis.

   • Question

     IN9tr+5z6U19BJaMyywAg3c9UL4YZUPnnX2xtCq0Wy1R5QGLacUH5ps+lCAkIXI+hS+9tmumjX2AdngxiCFUUn/7jNz7NPy8kS1tsIAWhwQ5eFNQH2Ze2n84yDkqYfBu0GKwoqTiNepYHwtPaL1YdgVskEaDODlP8DO8Oj4oPCMJ/6LpJocQVNijdTSeG6+ZUqt2c59QgrwoG2UbGBYo6PMyTljw+3UiPm0+DOPOsiouLQAz5uj2ZsYhgFO0NNDhzdjJA2xj+EvhITeV0i4uvRcWjaGSn1UynoaienbD+TZ9/2tr6KM4Ug+ivl38/qBfxq45MROpgh4rI3zbO3J7qA==

     Prob 100 7c. As the wage rate increases from $15 to $30, how does the number of hours worked respond according to the relationship depicted here? What is the average slope of the curve between Athena’s and Boris’s data points using the arc method?

   • Question

     kJtqKE3E8Ix1y2EzH7uFuuqyCYp6D7eExHlpiQ9x74BRf0hNkk9yirMt7QVFt8vnwZSvktZjvicGk9svZWqkspJfxQ9vCn/YXDCLUZzZ61yujtvSmIjsNBzSOeGhwuckstCCJcry0pYBwg1G3gWZu/CuNQGBKUpkJs0pXmNy0LE1ClShD48Kc88haXm7jT+NID4tA4r8O9TV1ZMCH7QBAD2GnPxTwZjcHV6oBfSv150fpGM3+n7WXfwR8eEbGOF+PNsyMQucuks9x8HMwh5tUKiiKQ/CTfdyjCmDq4e0cCUcQAmA1pN42qa5RmKHNk8DhcrniOeyYL6v4NSwp9GZiw==

     Prob 100 7d. As the wage rate increases from $60 to $75, how does the number of hours worked respond according to the relationship depicted here? What is the average slope of the curve between Diego’s and Emily’s data points using the arc method?
8. Studies have found a relationship between a country’s yearly rate of economic growth and the yearly rate of increase in airborne pollutants. It is believed that a higher rate of economic growth allows a country’s residents to have more cars and travel more, thereby releasing more air-borne pollutants.

   • Question

     iAn+WQ082KWCo7BsC2e/EmV82d8K8P+jCP0d6mXjOfpU5w5XPqrmmgjlDI37GMszcdRwZa3MTaM5ZZs5VAWmQbZ7WJfXvbtoppuPAjVw8bo7qemE/0hmWyVrsUM=

     Prob 100 8a. Which variable is the independent variable? Which is the dependent variable?

   • Question

     VzuHbTsAE3jTBmAGwdyJ986HoiYWOKQSTMi20gOXiOcDnXjfv58/IDIygcVpy9t+L82GUlZEfE4mXN21SFRJWleOsJklKciNFMP3qrTbxSsuttTvXQbFwdJpuHy+fvu7GrJ5qEGeimmexm/jDvYc8iekgBcyLwFESZ7MznpwJ/291yCRPA3vzNQX7MIrvIdXFOkbaDXgwuuj1GZ4HkP7cjPqyUfAiFexwW5PI+IgWRvYd97pOErr6NvH+L5FnpjQAks1i2BmNWQPAJj0ghwgo9Mm0IIYeIqXViism/wPcTqgFEBXHED0nTDZ2lupnogQ8GHufyJi7zcRQNzOMWxhBT+XpWeXUUgm7uqvgBR9cqRS2bN351hjH5/vlcM=

     Prob 100 8b. Suppose that in the country of Sudland, when the yearly rate of economic growth fell from 3.0% to 1.5%, the yearly rate of increase in air-borne pollutants fell from 6% to 5%. What is the average slope of a non-linear curve between these points using the arc method?

   • Question

     q5AqbqMhGJtHDq30Ptp/f6FeQfXM2zTE9HBxI8oDTPOmLVzMHS1XU1L7Ma/CuLF9/desSw7H8+np8YQraRXJcCze8TEZO9ReJQDaRKsyNiQZYZwu3MhqC0HYUnWY/Xj0zMiDE9Czfg+U4aGJOD1L0TmflIQ8pDt2g9PvacNowS7xh0+A8ZYCEMBitCRDTXuVhtqTDFt6THUtYCU7IPaAE68ni7XoBb62Dd/YXCqYRTJvMzl3sLl9sdL66RMZbqJzRyFtWFvaJWXgpkpbAvUBCfcmD9ZRghj5TjCY8b6x3nyFWm6iig/vO78l+mbZOVEsvwzs1cCdHZTqVb+nrxx9MvzmrnzcuOSOwLtqXQ==

     Prob 100 8c. Now suppose that when the yearly rate of economic growth rose from 3.5% to 4.5%, the yearly rate of increase in air-borne pollutants rose from 5.5% to 7.5%. What is the average slope of a nonlinear curve between these two points using the arc method?

   • Question

     UYnrLUSP5zyaVwTbEeT6ortx6PYld4JjvLfgGdLrOlZ0XwrheejNr/VU725pU4H87eT+xsxXx497Mg/mu0j9cgNMoAWj9RiMTLuNCgAx3jXJlINiJAi9eg==

     Prob 100 8d. How would you describe the relationship between the two variables here?
9. An insurance company has found that the severity of property damage in a fire is positively related to the number of firefighters arriving at the scene.

   • Question

     4HyZiAp89HY1qz6pEqoWE8YBL/Bdv2Llj1+wzDj2PH9ABosU1YA6zSx2oV6D4KqhHrOCsHMUcG0InAMbXvSnO8KzBd+ShzNAtqobejs8YAxbpUqyCk2dLzrdX6P7lB39AMdGiIm9MZ3XLijnNgzVuPSU3K4J3GP7AyFZu0cEHy7PQlxyZqVtCEskFnxORDxtJW5w3yEofdsbVzuBmARep/FlERsL+rPdUBAvcBecgJBKvC4i36I96H/o35EckvsU5IhlocRe7+y/SlnqxONDyXnSb6XkS89rV8uB44Q4ZwilS/ZchHGi8+1dpDTWkXdu5ZQ/YuhQzs3w0Jf/jiGIZXBJgii7LBzvEPCyHJtuuiV66ztXSTuLZydbSV+r36c0

     Prob 100 9a. Draw a diagram that depicts this finding with number of firefighters on the horizontal axis and amount of property damage on the vertical axis. What is the argument made by this diagram? Suppose you reverse what is measured on the two axes. What is the argument made then?

   • Question

     iIpTh0yWjW58FLhI/vHN+TnoC4NIu5HZLSAz4eJyvhduHWXbWvkN/tHHQHrRB8mKIE1Jr/3uwb0mdtUuJQGJPHP6mlGHEKuVeioG5/3po6DW5IEbqj1Y6BICcvcKWpQxqN/8BieeiQWwNUDk77HIHlQ0ayjUdRKzCDJ0J8x/5PF5YxqRAYLw7k+O2xc3xr2OFAV8Otbf4YKELUa6

     Prob 100 9b. In order to reduce its payouts to policyholders, should the insurance company therefore ask the city to send fewer firefighters to any fire?
10. The accompanying table illustrates annual salaries and income tax owed by five individuals. Apart from the fact that they receive different salaries and owe different amounts of income tax, these five individuals are otherwise identical.

    Name	Annual salary	Annual income tax owed
    Susan	$22,000	$3,304
    Eduardo	63,000	14,317
    John	3,000	454
    Camila	94,000	23,927
    Peter	37,000	7,020

    • Question

      o1Wc2xyVi8D04ap/fzrjEE9qraw40tONLyYaqNwc1O+gWvAUf5QUzddWxga9kKvQRu2YvjWGWuCj6zoHyOzefH1T1Yt25L6hOrldkqn/NQFqKyAO+eLTo2mgI1pPJORflG0yL8eD1omStPaRdEqzuTCJaM2mIvPUv8/pB6tdw+eA2NpzedF9jz5X1zokkiLGg1NlgFvzX5azvqJTX8wQe3uFxH5hLTQ0C8Iy5DssEqJls25GCkjsPdTp6Tt/b3pIa/+dr7FJWSfo+qUm8kZHzLxle9ZoUHMp4uAqi3Ie7pkTHYtsTiZH/zXqImDGP+u5ajIx6Q==

      Prob 100 10a. If you were to plot these points on a graph, what would be the average slope of the curve between the points for Eduardo’s and Camila’s salaries and taxes using the arc method? How would you interpret this value for slope?

    • Question

      InnPsWM7BWz2Tj4Wh4SgV107xYsMIsnGdk4uN41S87iCQyrxyWo+rD9muJChPkdd+TXmmjO39OfBSkfBRszZ4H+Muk5FAoY16b+bzFLTLx7txTsxW2bDzwWJ3/QskNaVucx5+WN1fxhWHeWIdoMq8aNbPQ5yN7l01l0qPa7+nMIJR+jbdw7xnU2rSxPPlTueuhYeGIpfmLmqx8OLWQS1RcGTNEcoE2TEmr37pb3b5f0H4U+sMce19HoAaK0=

      Prob 100 10b. What is the average slope of the curve between the points for John’s and Susan’s salaries and taxes using the arc method? How would you interpret that value for slope?

    • Question

      mtGpI03r2gugf9SdmQCRcr539QuxDVYArgKZ6+H91RBedwVtfvZslUOlycxsx1Z6Ba0+iPnKAY5tFKU7sjK7DCPHyHjITo9GJpegANhHnQB0ImTezwc5Eiv+pdVot3Jck6LACkKDj/ZEKH0kyuNbahKHsEMDvDgrO6NA+6CGdxrFHb2NgppLEjjUXV9Amspv48mXBi//oqjkH52Wzz4OLvMQNAS0gxHO1+N7AbICIx4zT3zV6SRyoC/Bx6w=

      Prob 100 10c. What happens to the slope as salary increases? What does this relationship imply about how the level of income taxes affects a person’s incentive to earn a higher salary?