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Problem 1 Chapter 1-Exercise 2

Suppose the unemployment rate is 6%, the total working-age population is 120 million, and the number of unemployed is 3.5 million. Determine:

(a) The participation rate.
(b) The size of the labor force.

(c) The number of employed workers. (d) The Employment-Population rate.

Problem 2 Chapter 1-Exercise 4

This question demonstrates why the CPI may be a misleading measure of inflation. Go back to Micro Theory. A consumer chooses two goods x and y to minimize expenditure subject to achieving some target level of utility, u ̄. Formally, the consumer’s problem is

min E = px x + py y x,y

s.t. u ̄=xαyβ

Total expenditure equals the price of good x times the number of units of x purchased plus the price of good y times the number of units of y purchased. α and β are parameters between 0 and 1. px and py are the dollar prices of the two goods. All the math required for this problem is contained in Appendix A.

(a) Using the constraint, solve for x as a function of y and u ̄. Substitute your solution into the objective function. Now you are choosing only one variable, y, to minimize expenditure.

(b) Take the first order necessary condition for y.

(c) Show that the second order condition is satisfied. Note, this is a one

variable problem.

(d) Use your answer from part b to solve for the optimal quantity of y, y∗. y∗ should be a function of the parameters α and β and the exogenous variables, px,py and u ̄. Next, use this answer for y∗ and your answer from part a to solve for the optimal level of x, x∗. Note, the solutions of endogenous variables, x∗ and y∗ in this case, only depend on parameters and exogenous variables, not endogenous variables.

(e) Assumeα=β=0.5andu ̄=5. Intheyear2000,px =py =$10. Calculate x∗2000,y2∗000 and total expenditure, E2000. We will use these quantities as our “consumption basket” and the year 2000 as our base year.

(f) In 2001, suppose py increases to $20. Using the consumption basket from part e, calculate the cost of the consumption basket in 2001. What is the inflation rate?

(g) Now use your results from part d to calculate the 2001 optimal quantities x∗2001 and y2∗001 and total expenditures, E2001. Calculate the percent change between expenditures in 2000 and 2001.

(h) Why is the percent change in expenditures less than the percent change in the CPI? Use this to explain why the CPI may be a misleading measure of the cost of living.

Problem 3 Chapter 1-Exercise 5

[Excel Problem] Download quarterly, seasonal adjusted data on US real GDP, personal consumption expenditures, and gross private domestic invest- ment for the period 1960Q1-2016Q2. You can find these data in the BEA NIPA Table 1.1.6, “Real Gross Domestic Product, Chained Dollars”.

(a) 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?

(b) The growth rate of a random variable x, between dates t − 1 and t is

defined as

xt−1
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?

(c) In Appendix A we show that that the first difference of the log is

x xt − xt−1 gt= .

approximately equal to the growth rate:
gtx ≈lnxt −lnxt−1.

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

(d) 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.

(e) The National Bureau of Economic Research (NBER) declares business cycle peaks and troughs (i.e. recessions and expansions) through a sub- jective 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.

(f) The most recent recession 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 three 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 2007Q4-2010Q2. Visually compare that counterfactual time path of GDP, and comment (intelligently) on the cost of the recent recession.

Problem 4 Chapter 5-Exercise 5

Excel Problem. Suppose that you have a standard Solow model with a Cobb-Douglas production function. The central equation of the model can be written:

kt+1 = sAktα + (1 − δ)kt. Output per worker is given by:

α yt =Akt .

Consumption per worker is given by:
ct =(1−s)yt.

(a) Suppose that A is constant at 1. Solve for an expression for the steady state capital per worker, steady state output per worker, and steady state consumption per worker.

(b) Suppose that α = 1/3 and δ = 0.1. Create an Excel sheet with a grid of values of s ranging from 0.01 to 0.5, with a gap of 0.01 between entries (i.e. you should have a column of values 0.01, 0.02, 0.03, and so on). For

each value of s, numerically solve for the steady state values of capital, output, and consumption per worker. Produce a graph plotting these values against the different values of s. Comment on how the steady state values of capital, output, and consumption per worker vary with s.

(c) Approximately, what is the value of s which results in the highest steady state consumption per worker? Does this answer coincide with your analytical result on the previous question?

Problem 5 Chapter 6-Exercise 3

[Excel Problem] Suppose that you have the standard Solow model with both labor augmenting productivity growth and population growth. The production function is Cobb-Douglas. The central equation of the Solow model, expressed in per efficiency units of labor, is given by:

(1+z)(1+n) 130

̂ 1 ̂α ̂

kt+1 =

[sAkt + (1 − δ)kt] .

α

The other variables of the model are governed by Equations (6.23)–(6.27).

(a) Create an Excel file. Suppose that the level of productivity is fixed at A=1. Supposethats=0.2andδ=0.1. Supposethatα=1/3. Let z = 0.02 and n = 0.01. Solve for a numeric value of the steady state capital stock per efficiency unit of labor.

(b) Suppose that the capital stock per worker initially sits in period 1 in

steady state. Create a column of periods, ranging from period 1 to

period 100. Use the central equation of the model to get the value

̂̂

Continue to iterate on this, finding values of k in successive periods up through period 9. What is true about the capital stock per efficiency unit of labor in periods 2 through 9?

(c) In period 10, suppose that there is an increase in the population growth rate, from n = 0.01 to n = 0.02. Note that the capital stock per efficiency unit of labor in period 10 depends on variables from period 9 (i.e. the old, smaller value of n), though it will depend on the new value of n in period 11 and on. Use this new value of n, the existing value of the capital stock per efficiency unit of labor you found for period 9, and the central equation of the model to compute values of the capital stock per efficiency unit of labor in periods 10 through 100. Produce a plot showing the path of the capital stock per efficiency unit of labor from period 1 to period 100.

(d) Assume that the initial levels of N and Z in period 1 are both 1. This means that subsequent levels of Z and N are governed by Equations (6.7) and (6.9). Create columns in your Excel sheet to measure the levels

of N and Z in periods 1 through 100.

(e) Use these levels of Z and N, and the series for k you created above, to

̂
create a series of the capital stock per work, i.e. kt = ktZt. Take the

natural log of the resulting series, and plot it across time.

(f) How does the increase in the population growth rate affect the dynamic path of the capital stock per worker?

Problem 6 Chapter 7-Exercise 2

Excel Problem. Suppose that you have many countries, indexed by i, who are identical in all margins except they have different levels of A, which are assumed constant across time but which differ across countries. We denote these levels of productivity by Ai. The central equation governing the dynamics of capital in a country i is given by:

k =sAkα +(1−δ)k i,t+1 i i,t i,t

Output in each country is given by:

y =Akα i,t i i,t

(a) Solve for expressions for steady state capital and output in a particular country i as functions of its Ai and other parameters.

(b) Create an Excel sheet. Create a column with different values of A, each corresponding to a different level of productivity in a different country. Have these values of Ai run from 0.1 to 1, with a gap of 0.01 between entries (i.e. create a column going from 0.1, 0.11, 0.12, and so on to 1). For each level of Ai, numerically solve for steady state output. Create a scatter plot of steady state output against Ai. How does your scatter plot compare to what we presented for the data, shown in Figure

)

7.6?