Economics 101Spring 2001Section 4 - HallamProblem Set #5Due date: March 2, 20011. Consider the following data on quantities of q1 and q2 and utility. In the table q2 is held fixed at 3 units. Computemarginal utility for each change in q1. Rather than placing it between each value of q1 and the subsequent one, put itnext to the subsequent one.q1q2u, q2 = 3 MU1, q2 = 30 3 0.00 NA1 3 6.93 6.932 3 8.243 3 9.124 3 9.805 3 10.376 3 10.857 3 11.278 3 11.66 0.399 3 12.0010 3 12.3311 3 12.6212 3 12.90 0.282. Consider the following hypothetical data on cloth and wine production in England and Portugal. Cloth is measuredin 1,000 yards, wine is measured in 1,000 liters. Assume that capital is freely mobile so only labor costs matter. Alsoassume that real wages will tend to equalize so that only labor quantities matter. The data below gives the number ofworkers required per unit of output.Cloth WineEngland 100 workers 75 workersPortugal 60 workers 30 workersa. Who has a comparative advantage in cloth? (Show work)b. Who has a comparative advantage in wine? (Show work)c. Convert the data to output per worker and make a new table.d. Who has the absolute advantage in cloth?e. Using the new data, show who has the comparative advantage in cloth.3. In the table below are listed a number of combinations of q1 and q2. Based on this data draw this individual’sindifference curves for u = 12, u = 14, u = 16 and u = 18. You can use just 5 or 6 points for each indifferencecurve. Use graph paper.q1q2costu bar41.569 0.75 143.89 12.0027.000 1 135.00 12.009.546 2 181.09 12.005.196 3 253.39 12.006.831 2.5 216.16 12.005.196 3 253.39 12.004.123 3.5 291.75 12.003.375 4 330.75 12.002.415 5 409.83 12.001.837 6 489.67 12.001.458 7 569.92 12.0042.875 1 166.75 14.0023.338 1.5 168.18 14.0015.159 2 192.32 14.0010.847 2.5 224.19 14.008.251 3 259.50 14.006.548 3.5 296.60 14.005.359 4 334.72 14.003.835 5 412.67 14.002.917 6 491.83 14.002.315 7 571.63 14.0064.000 1 209.00 16.0034.837 1.5 191.17 16.0022.627 2 207.25 16.0016.191 2.5 234.88 16.0012.317 3 267.63 16.009.774 3.5 303.05 16.008.000 4 340.00 16.005.724 5 416.45 16.004.355 6 494.71 16.003.456 7 573.91 16.0091.125 1 263.25 18.0049.602 1.5 220.70 18.0032.218 2 226.44 18.0023.053 2.5 248.61 18.0017.537 3 278.07 18.0013.917 3.5 311.33 18.0011.391 4 346.78 18.008.150 5 421.30 18.006.200 6 498.40 18.004.920 7 576.84 18.004. Assume that p1 = 4, p2 = 12, and I = 80. Draw in a budget line for Problem 3. What are the optimum levels of q1and q2?5. Now assume that p1 = 2 and p2 = 81 and I = 135. Draw in a budget line for Problem 3. What are the optimumlevels of q1 and q2? (You may guess).6. For this problem p1 = 16, p2 = 20 and I = 240. Below is a table of alternative consumption choices and the utilityand the marginal utility they provide. Which is the optimal choice?q2q1u MU1MU20 15 0.000 0.000 1 13.75 19.257 0.351 9.6292 12.5 26.592 0.532 6.6484 10 35.566 0.890 4.4466 7.5 40.537 1.352 3.3798 5 42.295 2.115 2.64410 2.5 39.764 3.977 1.989Show and explain the answer using the three following equilibrium conditions.p2p1MUq2MUq1 p2p1 MRSq1q2∆ q1∆ q2MUq1p1MUq2p27. For this problem p1 = 6, p2 = 4 and I = 58. Below is a table of alternative consumption choices and the utility andthe marginal utility they provide. Which is the optimal choice?q1q2cost u MU1MU29.666667 0 58 5.369 0.151003 0.9395758.333333 2 58 6.851 0.220211 0.7992837 4 58 8 0.3 0.75.666667 6 58 8.855 0.398475 0.619854.333333 8 58 9.409 0.529256 0.5488583 10 58 9.615 0.721125 0.480751.666667 12 58 9.348 1.05165 0.4089750 14.5 58 7.71 2.313 0.29173Show and explain the answer using the three following equilibrium conditions.p2p1MUq2MUq1 p2p1 MRSq1q2∆ q1∆ q2MUq1p1MUq2p2u(q1, q2) 20 q1 14 q2 q21 q228. Consider the following utility functionwhich is valid for low levels for q1 and q2. Complete the following table where MU1 is computed holding q2constant.q1q2uMU10113—113219214917316415517102 —122232527203 —132333537305 —1564258117355575u0 20 q1 14 q2 q21 q22u0 20 q1 14 q2 q21 q22< q21 q22 u0 20q1 14q2< q21 20 q1 u0 q22 14 q2 0< q21 20 q1 (u0 q22 14 q2) 0a 1, b 20, and c u0 q22 14 q2q120 ± ( 202) (4)(1)(u0 q22 14 q2)220 ± 400 4(u0 q22 14 q2)2q120 ± 400 4(120 9 42)220 ± 52220 ± 7.2112< q1 13.6055, or q1 6.39459. An indifference curve shows combinations of goods that all lead to the same utility level. For the two good case, wecan find the indifference curve from the utility function by specifying the level of utility and then solving the utilityfunction equation for q1 as a function of q2. Consider then the indifference curve for the utility function in problem8. Letting the given level of utility be denoted u0, the function is given byWe can write this as a function of q1 as followsThis is a quadratic equation in q1 that can be solved using the quadratic formula whereSolving we obtainFor example, if q = 120, and q2 = 3, then q1 = 6.3945. This is clear fromWe chose the more efficient level of q1 = 6.3945. For a utility level of 120 and q2 = 2, find the level of q1 that puts the consumer on the indifference curve.10. For a utility level of 120 and levels of q2 of 1, 2, 3, 4, 5, 6, 6.5 and 7, find the level of q1 puts the consumer on theindifference curve. Make a table with these points.q1q2Utility1 1202 1206.3944 3 1204 1205 1204.7085 6 1206.5 1204.6148 7 12011. Graph these points in a diagram with q1 on the vertical axis. Use graph paper.12. Assume the price of q1 = 10 and the price of q2 = 4 for problem 11.a. Show on the same graph all combinations of q1 and q2 that cost $50.00.b. Show on the same graph all combinations of q1 and q2 that cost $80.00.c. By shifting this line in a parallel fashion, find the combination of q1 and q2 that has the lowest cost amongthose that allow for 120 units of utility.How many units of q1 and q2 are chosen?How much income do you think is spent to obtain this utility level?13. Work question 3 from Skills and Tools in Chapter 5.14. Work question 4 from Skills and Tools in Chapter 5.15. Work question 6 from Skills and Tools in Chapter
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