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Question 1 (12 points)

  1. (a)  Write down the Euler equation for the two-period consumption-saving problem in general terms (as derived in class) and describe its economic intuition.
  2. (b)  Consider the Euler equation for the two-period consumption-saving problem. Suppose β(1 + rt) < 1. Is the growth rate of consumption positive, negative, or zero? Explain.
  3. (c)  Graphically depict the solution to the consumption-saving problem. Clearly state why you know it is the solution.

Question 2 (16 points)

Consider the following consumption-savings problem. The consumer solves max Cα+βCα

Ct ,Ct+1 ,St t t+1 subject to the lifetime budget constraint

Ct+ Ct+1 =Yt+ Yt+1 1+rt 1+rt

where we assume that β ∈ (0,1) and α > 0.

  1. (a)  Under which assumption on α are marginal utilities from consumption positive but diminishing?
  2. (b)  Derive the Euler equation from the household’s constrained maximization problem.
  3. (c)  Using the Euler equation and the lifetime budget constraint, solve for the period t consumption function (i.e. Ct∗).
  4. (d)  What happens to the household’s optimal consumption in period t when β increases? Provide some intuition.

Question 3 (15 points)

Suppose that a household only lives for one period. The household’s optimization problem is:

max U=lnCt+θtln(1−Nt) Ct ,Nt

subject to

Ct = wtNt + Yt
where Nt denotes labor supply, Ct denotes consumption, wt is the wage rate, and Yt is

exogenous non-labor income. θ > 0 is a parameter.

  1. (a)  Solve for the household’s optimal labor supply and consumption.
  2. (b)  What can you say about the effect of wt on the optimal Nt? Provide some economic intuition.
  3. (c)  What can you say about the effect of Yt on the optimal Nt? Provide some economic intuition.

Question 4 (15 points)

Consider the following data for average hours of work per week across countries.

Figure 1: Average hours of work per week across countries. Source: Bick et al. (2018). Reproduced from Kurlat (2020).

Assume we want to write down a model of household labor supply that allows us to under- stand this pattern. To do so, we assume that household preferences are given by:

u(c, l) = ln(c) + θl

where c is consumption, l is leisure and θ is a parameter. Households have a total of one unit of time available and can supply labor at a wage rate w. Note that the household budget constraint is then given by c = w · (1 − l).

  1. (a)  Find an expression for the fraction of their time that households optimally spend in market work.
  2. (b)  If this was the right model and one looked at households in different countries, how would hours of work correlate with wage levels?
  3. (c)  Considering the empirical evidence above, do you think this is a good model of house- hold labor supply?

Question 5 (18 points)

Suppose these are the prices (in US dollars) and quantities of goods A and B produced in the US in 2017 and 2018:

pA qA pB qB 2017 4 5 3 3 2018 1 10 4 2

  1. (a)  Compute nominal GDP in 2017.
  2. (b)  Compute nominal GDP in 2018.
  3. (c)  What was real GDP in 2018 at 2017 prices (computed using fixed 2017 prices)? Using this measure, how much did GDP grow between 2017 and 2018?
  4. (d)  What was real GDP in 2017 at 2018 prices (computed using fixed 2018 prices)? Using this measure, how much did GDP grow between 2017 and 2018?
  5. (e)  Why is there a difference between your answer in (c) compared to (d)?
  6. (f)  Statisticians use a technique called ’chain-weighting’ in order to avoid the issues arising from different base years. It works as follows
    1. (a)  Calculate real GDP at 2017 prices.
    2. (b)  Calculate real GDP at 2018 prices.
    3. (c)  Calculate growth rate of real GDP between 2017 and 2018 using both base years and take the geometric average of the two growth rates. [Hint: The geometric

√

average of a and b is given by:
How much did GDP grow between 2017 and 2018 using the chain-weighted method?

Question 6 (24 points)

[Excel Problem] Download quarterly, seasonal adjusted data on US real GDP, personal consumption expenditures, and gross private domestic investment for the period 1960Q1- 2021Q4. You can find these data in the BEA NIPA Table 1.1.6, ”Real Gross Domestic Product, Chained Dollars”.

(a) c

ab.]

  1. (b)  The growth rate of a random variable x, between dates t − 1 and t is defined as g tx = x t − x t − 1 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?
  2. (c)  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.
  3. (d)  In this subquestion we try to get a sense of the cost of the COVID-19 recession that started in 2020Q1. Compute the average growth rate of real GDP for the period 2015Q1-2019Q4. Compute a counterfactual time path of the level of real GDP if it had grown at that rate between 2020Q1 and 2021Q4. Visually compare that counterfactual time path of GDP to actual GDP, and come up with a rough estimate of the cost of the COVID-19 recession (in dollars).

References

[1] Alexander Bick, Nicola Fuchs-Schu ̈ndeln, and David Lagakos. How do hours worked vary with income? cross-country evidence and implications. American Economic Review, 108(1):170–99, 2018.

[2] Pablo Kurlat. A Course in Modern Macroeconomics. 2020.

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