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ECON 212: Intermediate Macroeconomics

Fall 2021

Problem Set 2

Instructions: In this problem set you are allowed to turn in the solution in groups of 3 people at most, although you can choose to work alone. Please make sure to legibly write your name(s) on the first page of your solutions. Also: i) be neat, ii), for the data part, if the question asks for a graph, copy the graph from Excel to Word and write your answer next to or below it, iii) when graphing, label all plots, series, and axes accordingly, iv) if a graph has a time dimension, always plot it on the horizontal axis, v) write down your name, vi) scan your solutions and upload them to Canvas. Have fun!

Measuring the Economy

1. Warming up. In years 1 and 2, there are two products produced in an economy, computers and bread. In year 1, 300 computers are produced and sold at $2,000 each, and in year 2, 400 computers are sold at $2,300 each. In year 1, 15,000 breads are sold for $2.80 each, and in year 2, 20,000 breads are sold at $3.20.

(a) Calculate nominal GDP in each year.

(b) Calculate real GDP in both years using year 1 as the base year. Back out the implicit price deflator for both years. What are the implied growth rates of real output and prices (inflation)?

(c) Now calculate real GDP in both years using year 2 as the base year. Back out the implicit price deflator for both years. What are the implied growth rates of real output and prices (inflation)?

(d) Are your answers for 1b and 1c the same or different? Why?

2. Labor market. Suppose unemployment rate is 8%, total working age population is 206 million, and the number of unemployed is 10 million. Determine:

(a) Labor force.

(b) Participation rate.

(c) Number of employed workers.

(d) Employment-Population rate.

3. Williamson, 6th edition, problem 2.3. Assume an economy with two firms. Firm A produces wheat and firm B produces bread. In a given year, firm A produces 50,000 bushels of wheat, sells 20,000 bushels to firm B at $3 per bushel, exports 25,000 bushels at $3 per bushel, and stores 5,000 bushels as inventory. Firm A pays $50,000 in wages to consumers. Firm B produces 50,000 loaves of bread, and sells all of it to domestic consumers at $2 per loaf. Firm B pays consumers $20,000 in wages. In addition to the 50,000 loaves of bread consumers buy from firm B, consumers import and consume 15,000 loaves of bread, and they pay $1 per loaf for this imported bread. Calculate the Gross Domestic Product for the year using:

(a) The production approach.

(b) The expenditure approach.

(c) The income approach.

Empirical Section

Download quarterly, seasonal adjusted data on US real GDP, personal consumption expenditures, and gross private domestic investment for the period 1960Q1–2021Q2. You can find these data in the BEA NIPA Table 1.1.6, “Real Gross Domestic Product, Chained Dollars”.

1. Take the natural logarithm of each series (“=ln(series)”) and plot each against time. Which series appears to move around the most? Which series appears to move the least?

2. The growth rate of a random variable x, between dates t − 1 and t is defined as

Calculate the growth rate of each of the three series (using the raw series, not the logged series) and write down the average growth rate of each series over the entire sample period. Are the average growth rates of each series approximately the same?

3. We argued in class that the first difference of the log is approximately equal to the growth rate:

Compute the approximate growth rate of each series this way. Comment on the quality of the approximation.

4. The standard deviation of a series of random variables is a measure of how much the variable jumps around about its mean (“=stdev(series)”). Take the time series standard deviations of the growth rates of the three series mentioned above and rank them in terms of magnitude.

5. The National Bureau of Economic Research (NBER) declares business cycle peaks and troughs (i.e. recessions and expansions) through a subjective assessment of overall economic conditions. A popular definition of a recession (not the one used by the NBER) is a period of time in which real GDP declines for at least two consecutive quarters. Use this consecutive quarter decline definition to come up with your own recession dates for the entire post-war period. Compare the dates to those given by the NBER.

6. One of most recent recessions is dated by the NBER to have begun in the fourth quarter of 2007, and officially ended after the second quarter of 2009, though the recovery in the last years has been weak. Compute the average growth rate of real GDP for the period 2003Q1–2007Q3. Compute a counterfactual time path of the level of real GDP if it had grown at that rate, over the period 2007Q4-2019Q4. Visually compare that counterfactual time path of GDP, and comment on the cost of the recent recession.

Anti-Stiffness Math

1. Solve the following constrained maximization problem:

subject to

Explain what the values of and mean.

2. Let the with :

(a) Take the first, second, and cross derivative of . Explain what the sign of each one means.

(b) Divide the function by  and show that the function can be written as .

(c) Letting , express  as a function of just and let that be .

(d) Take the first and second derivative of . Explain what the sign of each one means.

(e) Based on those derivatives, sketch a graphical representation of the function.