Suggestions:
•    Start with question 3 first, to make sure you can download data from FRED, the database at the St. Louis Fed. If you had any problem with the data, let me know and keep working on the remaining questions.
•    Please turn in a fair copy of any handwritten material. You may attach to that any graph or table you generate in excel or other program (i.e.,your answers to problem sets don’t have to be all typed or all handwritten).

1.Exponents and logarithms (30 points). A production function (next class) is amathematical model of the relation between a _rm or country’s output and itsinputs, which here will be capital and labor. We’ll use the function

Y = K??L???;     (1)

where Y is the quantity of output, K is the quantity of capital input (plant andequipment), L is the quantity of labor input (number of employees), and ?= 1/3is a parameter (take this as given, unless we give a different value). We’ll set K = 100 and L = 50 in this problem.
(a) Compute output Y (make sure you understand how the formula works). (5 points)
(b) What is the natural logarithm of Y? (5 points)
(c) Show that the production function can be written

ln Y = ? lnK + (1-?) lnL;

where ln x means the natural logarithm of the variable x. What propertiesof logarithms do we need to derive this from equation (1)? (10 points)
(d) Compute (ln Y) using the expression in (c). Verify that your answer is the same as the one you computed in (b). (5 points)
(e) Consider the following experiment: Double both the amount of K and the amount of L in the problem and compute Y using formula (1). Comment in 1-2 lines. (5 points)

2. How much capital (40 points)? Your mission,if you decide to take it, is to use the production functionand market prices to decide how much capital a firm should “rent” (note: if you’re like Ethan Hunt, the protagonist of Mission Impossible, you will decide to take the mission. More realistically, you do want the 40 points here).

Let us saythat a firm sells output at price p per unit, rents/hires capital K and labor L atprices r and w, respectively, and produces output Y according to the production function (1).Its profit is therefore revenue minus cost:

Profit = pY– (rK + wL) = pK??L??? – (rK + wL);    (2)

How much capital does the firm want? To make this concrete, let us say thatp = 1, w = 1/2, r = 3/16 = 0.1875 (18.75%), L = 27, and (always) ? = 1/3.

We’ll solve the problem two ways: using a spreadsheet, and using calculus.

(a) In column A of a spreadsheet, create a column of values for capital K runningfrom 1 to 100 in increments of 1. Then, in column B, compute revenuepY = pK??L???for each value of K (remember L=27). Does revenue increase or decrease asyou increase capital? (10 points)
(b) In column C, compute cost (rK+wL) for each value of K. Does cost increaseor decrease as you increase capital? (10 points)
(c) In column D, compute profit for each value of K. Turn in a graph withprofit on the y-axis and capital on the x-axis, with the axes clearly labeled.At which value of capital is profit highest? (10 points)
(d) Now we do the same thing using calculus. Think of profit – equation (2) – as a function of capital K. Compute its derivative and set it equal to zero.For what value of K is the derivative equal to zero? Why does this give usthe profit-maximizing value of K? (10 points)

3. Macroeconomic volatility (30 points). The term “business cycle” refers to theperiodic ups and downs of the economy, evident in GDP (a measure of the totaloutput of the economy) and many other things (employment, retail sales, stockprices, and so on). Years of experience tells us that many things go up and downtogether, but some components go up and down more. We say they’re morevolatile, in the sense that the standard deviations of their growth rates are larger.
Our mission is to verify both facts using quarterly data on GDP and two ofits expenditure components, consumption and investment. The first step is todownload the data from FRED: http://research.stlouisfed.org/fred2/.

Table 1. FRED Data Codes
Series or Variable    FRED Code
GDP    GDPC96
Consumption    PCECC96
Investment    GPDIC96

The FRED series codes are given in Table 1. Most people use FRED’s Excel add-in to download the data straight from FRED to a spreadsheet on their computer (there is a user’s guide for this add-in in PDF format that you can download).If you have difficulty with setting the Excel add-in, use the following method(I have just checked it and it works, but let me know if you find any problem):

•    Go to FRED (see URL above).
•    Type GDPC96 in the search box in the upper right corner. That will generate a graph of real GDP.
•    Below the graph, you’ll see graph options labeled GRAPH SETTINGS, ADD DATA SERIES, and EDIT SERIES. Click on ADD DATA SERIES.
•    Type the code for consumption in the search box directly below the words ADD DATA SERIES and press enter. Then click on the blue button labeled Add Series below the search box. You should now have two lines, one for GDP, the other for consumption.
•    Repeat the last two steps with investment.
•    Once you have all three series on the graph, click on the Export tab just below the graph. Click on Download as XLS to download the data in a spreadsheet.

Whatever method you use, you should end up with a spreadsheet that has dates in thefirst column and the three series in the next three columns.

Once you have the data, compute, for each series, discretely-compounded ratesof growth “year on year” (YOY), i.e. the percent change from a year ago, for the period 1950Q1 to 2015Q2 (data are quarterly). The formula is

gt = [(Xt–Xt-4)/Xt-4]*100 = [(Xt/Xt-4)-1]*100;

The 100 at the end converts the YOY growth rate to a percentage. We generate a quarterly series of YOY growth rates. Placeeach growth rate in its own column. Note that the growth rate for 1950Q1 requiresdata for 1949Q1 (which you do have in this case).

Now you’re ready to do some analysis:

(a) For each of the three growth rates, compute the mean and standard deviation. How do the standard deviations compare? (10 points)
(b) What are the correlations among the 3 variables? Do you agree thatthese variables move up and down together but have different volatilities”?(A graph would be useful here, too, but is not required.) (10 points)
(c) Compute the continuously-compounded YOY growth rate for each of the original series using the formula

?t = (ln Xt- ln Xt-4)*100;

How do the means and standard deviationsof these new series compare to those of the respective discretely-compounded growth rates? What about the correlations (continuous vs discrete growth rates)? (10points) 