Homework #2
In your calculations, please round to the two places past the decimal wherever necessary.
1. Suppose that the production function for an economy is given by
Y = F(K, L) = AK1/2L1/2
Where Y is output, A is the level of technology, K is capital and L is labor.
a. Suppose that A has a value of 10, K has a value of 50, while L varies from 0 to 25 units. Using Excel prepare a spreadsheet showing the possible combinations of labor, capital, technology and output that are possible for this economy. Your table should include a column for labor, a column for capital, and a column for output. I have included a sample of what the table should look like below.
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Technology
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Labor
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Capital
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Output
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b. Now, use the data you generated in part (a) to draw the aggregate production function where you will illustrate the relationship between aggregate output on the vertical axis and labor on the horizontal axis. This graph will hold technology and capital constant. Execute this graph using Excel.
c. Now, suppose that this economy experiences a technological advance. A, the measure of technology, now has a value of 12. Redo your calculations from part (a) and with your new data, plot both the initial aggregate production function (where A = 10) and the new aggregate production function (where A' = 12).
d. Now, this economy finds that the value of A is 12, but that its level of capital has increased from 50 units to 60 units. Redo your calculations and generate a new graph illustrating two different production functions: one with A' = 12 and capital equal to 50 units; and the other with A' = 12 and capital equal to 60 units. Use Excel to do this work.
e. Summarize briefly the effect of a change in technology or a change in capital on the aggregate production function.
2. Let's revisit problem one from a different perspective. Suppose that we have the same production function and that the initial value of A is 10 and the initial value of K is 50. Labor ranges from 0 units to 25 units.
a. Using Excel calculate the marginal product of labor (MPL) for this production function. Insert a table with these values: your table should have a column for technology, labor, capital, output, and the MPL. (Hint: in your calculations just calculate to the hundredths place.) I have included a sample of what this table should look like below.
Technology Labor Capital Output MPL
b. Now, graph the MPL with output on the vertical axis and labor on the horizontal axis.
c. Describe in words what happens to the MPL as labor increases, holding everything else constant. Provide an intuitive explanation for your conclusion.
d. Now, redo your calculations from part (a), but with capital equal to 75 units. Provide a table with a column for technology, a column for labor, a column for capital, a column for output, and a column for MPL. I have included a sample of what this table should like below.
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Technology
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Labor
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Capital
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Output
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e. Provide a graph of both MPLs (on the same graph): the first MPL should have A = 10 and capital equal to 50 units; while the second MPL should have A = 10 and capital equal to 75 units. Put output on the vertical axis and labor on the horizontal axis.
f. Describe in words what happens to the MPL when capital increases while labor and technology are held constant. Provide an intuitive explanation to support your conclusion.
g. Let's do one more thing with this example. Consider the two scenarios we are working with: 1) A = 10, K = 50 and labor varies from 0 to 25 units; and 2) A = 10, K = 75, and labor varies from 0 to 25 units. Generate a table that displays the level of labor productivity for each scenario at each level of labor that is possible.
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First Scenario
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Second Scenario
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Labor
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Labor Productivity
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Labor
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Labor Productivity
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0
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XXX
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0
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XXX
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1
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1
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h. Summarize the effect of a change in capital on labor productivity, holding everything else constant.
3. You are given the following information about an economy. Its aggregate production function can be described with a Cobb-Douglas production function where labor's share of output is .75 and capital's share of output is .25. Furthermore, you know that this economy has 100 units of capital and its available technology measure has a value of 4. Currently, there are 350 people living in this economy who are 16 years old or older. Of this group of 350 people, 30 are voluntarily idle, institutionalized, or retired; there are also 20 people in the group of 350 who are looking for work, but have not yet been successful in finding employment. This economy has no military. (Hint: for this question assume that the amount of labor available in this economy equals the number of people working.)
a. What is the current employment rate in this economy?
b. What is the labor force participation rate in this economy?
c. What is the level of aggregate production equal to in this economy?
d. What is labor's income in this economy?
e. What is capital's income in this economy?
f. Holding everything else constant, if the available technology measure changes to 5, what will be the change in labor income in this economy?
4. You are the central banker for an economy. You know that the economy's production can be described using a Cobb-Douglas production function where the value of A is 10, the initial value of capital is 1000 units, and the initial value of labor is 500 units. You also know that labor's share of aggregate output is equal to 40%. In this economy k, the proportion of income that people want to hold as cash, is equal to .1.
a. What is the initial level of aggregate income in your economy? Assume that you are using a classical model, where labor, capital and technology are constant, to calculate this value. Show your work.
b. Suppose the initial value of the aggregate price level (the GDP deflator) is 100. What is the initial level of the money supply in your economy if the money market is in equilibrium where money supply equals money demand? Show your work.
c. Now, suppose that labor in your economy grows by 2% a year as does capital. Also, assume that k stays constant as does the level of technology. If you as the central banker want to keep the aggregate price level constant at 100 for each of the next five years, what adjustments will you need to make in the money supply? To answer this question first figure out what is happening to real GDP in each of the five years, and then calculate your nominal GDP in each of the five years. Then calculate what the level of the money supply must equal in order for the money market to be in equilibrium in each of the five years with the aggregate price level equaling 100. Summarize your findings in the following table.
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Year
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Money Supply
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Initial Level
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Year 1
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Year 2
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Year 3
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Year 4
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Year 5
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d. Using the data you calculated in part (c) and the information given in part (c), calculate what the rate of growth of the money supply needs to be in order for the aggregate price level to remain constant over the next five years in your economy.
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Year
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Money Supply
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growth rate of money supply
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Initial Level
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XXX
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Year 1
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Year 2
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Year 3
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Year 4
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Year 5
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e. Now, a junior economist at the central bank comes to you and tells you that the research unit at the central bank no longer anticipates that k will stay constant over the next four years. Instead you are told that k over this period will range from .08 to .12. How does this complicate your decisions with regard to the money supply if your goal is to maintain stable prices over the period? (Hint: you might find it helpful to use Excel and calculate what happens to the aggregate price level for your calculated values of nominal GDP, the money supply figures you calculated in part (d), and a k value of .08. Then, repeat this experiment using a k value of .12.)