Econ 101 - Problem set 2 Problems 1-4 (Chapter 3: Krugman and Obstfeld) 1. The sector-specific model predicts that a fall in the relative price of the good will lower the real returns to the factor that is specific to the production of this good. Suppose a country called Texiana (Texas and Louisiana) that produces manufacturing and petroleum products.1 There are three inputs: labor (mobile), oil (specific to petroleum products), and capital (specific to manufacturing). Suppose further that P represents the relative price of petroleum products (with respect to the price of manufacturing), and it has declined. When you graph the production possibilities frontier of Texiana, you will find out that Texiana is producing less petroleum products and more manufacturing. Additionally, the marginal product of oil has declined. Therefore, the real return to oil-owners has declined. 2. An economy can produce Good 1 using labor and capital and Good 2 using labor and land. The total supply of labor is 100 units. Given the supply of capital and land, the output of the two goods depends on the amount of labor used:
Labor Input to Good 2 Output of Good 1 Labor Input to Good 2 Output of Good 2 0 0 0 0 10 25.1 10 39.8 20 38.1 20 52.5 30 48.6 30 61.8 40 57.7 40 69.3 50 66 50 75.8 60 73.6 60 81.5 70 80.7 70 86.7 80 87.4 80 91.4 90 93.9 90 95.9 100 100 100 100
a. In order to graph the production function, simply plot the points on the Output-Labor space. The line that connects these points is the production function.
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You should not worry about the distinction between oil and petroleum products.
Production Fuction good 1 100 93.9 90 87.4 80.7 75
73.6 66
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48.6 45 38.1 30 25.1
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Production function good 2 100 95.9 91.4
90 86.7 81.5
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75.8 69.3 61.8
60 52.5
45 39.8
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b. Recall that the production possibility frontier shows the maximum amount of Good 1 that can be produced once the decision has been made to produce any given amount of Good 2, and vice versa. In other words, the PPF represents all possible, technically efficient (i.e., all labor is used) combinations of labor that is used to produce some combinations of Good 1 and Good 2.
To graph the PPF you can choose a production plan for Good 1 and find the amount of Good 2 that you can produce using the quantity of labor not used in the production of Good 1.
Output of Good 1 0 25.1 38.1 48.6 57.7 66 73.6 80.7 87.4 93.9 100
Labor available for good 2 production 100 90 80 70 60 50 40 30 20 10 0
Output of Good 2 100 95.9 91.4 86.7 81.5 75.8 69.3 61.8 52.5 39.8 0
If we put Good 1 on the X-axis and Good 2 on the Y-Axis, we have the PPF shown below. Production Possibility Frontier 100 95.9 91.4 90
86.7 81.5 75.8
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69.3 61.8
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52.5 45 39.8
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The curvature of the PPF reflects diminishing returns to labor in each sector.
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3. a. We know that the demand for labor in Good 1 is given by _
w1 = p1 MPL1 ( L1 , K ) , and in Good 2 _
w2 = p 2 MPL 2 ( L2 , T ) p2 = 2 . Without loss of generality, we can assume that p2 =2 and p1 =1. p1 Using the table below, we can plot the demand of labor in both sector and determine the wage rate and allocation of labor. We know that
Workers Employed 10 20 30 40 50 60 70 80 90 100
Demand of labor: p1*MPL1 1.51 1.14 0.97 0.87 0.79 0.74 0.69 0.66 0.63 0.6
MPL in Sector 1 MPL in Sector 2 1.51 1.14 0.97 0.87 0.79 0.74 0.69 0.66 0.63 0.6
1.59 1.05 0.82 0.69 0.61 0.54 0.5 0.46 0.43 0.4
Demand of labor: p2*MPL2 3.18 2.1 1.64 1.38 1.22 1.08 1 0.92 0.86 0.8
Demand of labor 3.5
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Demand of labor good 2
As you can see, the wage rate is near to one. At this wage rate, the demand for labor in Sector 1 is approximately thirty units and in Sector 2 seventy units.
b. Given the wage rate and the allocation of workers, the production of Good 1 is 48.6 and the production of Good 2 is 86.7. Production Possibility Frontier 100
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The point (48.6, 86.7) is on the PPF, and, in equilibrium, the slop of the PPF is equal to the relative price, i.e. 2. c. If the price of Good 2 relative to that of Good 1 is 1, then the demand of labor is equal to the marginal productivity of labor. Works Employed MPL in Sector 1 MPL in Sector 2 10 20 30 40 50 60 70 80 90 100
1.51 1.14 0.97 0.87 0.79 0.74 0.69 0.66 0.63 0.6
1.59 1.05 0.82 0.69 0.61 0.54 0.5 0.46 0.43 0.4
Demand of labor: p1*MPL1
Demand of labor: p2*MPL1
1.51 1.14 0.97 0.87 0.79 0.74 0.69 0.66 0.63 0.6
1.59 1.05 0.82 0.69 0.61 0.54 0.5 0.46 0.43 0.4
Demand of labor 1.8
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1 Demand of labor good 1 Demand of labor good 2 0.8
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The wage rate is now between 0.8 and 0.6. Sector 1 demands 70 units of labor, Sector 2 30 units. Given the wage rate and the allocation of workers, the production of Good 1 is 80.7 and production of Good 2 is 61.8. Once again, this point is on the PPF and the slope of the tangent line at this point is 1. d. The effects of the fall in the relative price of Good 2 on the specific factors (capital and land) can be described as follows: Land is the specific factor in the production of Good 2. Landowners’ real return falls in terms of both goods. Capital owners, on the other hand, are better off. Recall that capital is specific to Good 1. The fact that the relative price of Good 2 has declined means that the relative price of Good 1 has increased. Capital owners’ real return increases in terms of both goods. 4. a. (You should have the answer in your class notes.) First of all, you can graph the addition to the labor force by shifting the PPF at every point so that the country produces more Good 1 and Good 2 at each point on the PPF. We are able to do this, because labor is used in both Good 1 and Good 2. Note, however, that there has been no change in the relative price of Good 2. Therefore, when you mark the new production point on the PPF, be sure that the slope of the new tangent line is exactly the same as before. Second, you can look at the labor allocation diagram. An increase in the size of the labor force necessitates an enlargement of the X-axis in the labor allocation diagram (You can achieve this by shifting the Y-axis (W) on the right-hand side to the right.) Marginal product of labor in both sectors will decline, so does the nominal wage rate. Note, however, that each sector is employing more workers than before. b. The results of the numerical example will prove the answer to Part (a) of this question.
