ECOS3021 Business Cycles and Asset Markets University of Sydney
2021 Semester 1
Tutorial #2
1. Study the business cycle facts reported for the image labeled “Table 9” (from Jorda, Schularick, and Taylor (2017), Macrofinancial History and the New Business Cycle Facts). In this table sd refers to the standard deviation of a variable, and corr refers to the correlation between two variables. Output=y; Consumption=c; Investment=i; Government expenditure=g; Net exports=nx. Statistics are reported for the United States and then all other countries (“Pooled”). The full sample period covers the years 1870-2013. There are also three sub-sample periods: pre-WWII (1870-1939), post-WWII (1939-2013), and the more recent period of floating exchange rates (1972-2013).
1.a) Over the full sample period, which variables are more volatile than GDP? Which variables are less volatile? Do these patterns vary across the United States and other countries?
ANSWER:
Investment and government spending is more volatile than GDP in both sets of countries. Consumption is more volatile than GDP in the other countries, but less volatile in the US. Net exports is more volatile than GDP in the other countries, but less volatile in the US.
1.b) For the United States, which variable experiences the largest decline in volatility over time? For the other countries, which variable experiences the largest decline in volatility over time?
ANSWER:
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For the US, relative investment volatility falls from 5.54 to 2.68 (a 52% decline) from pre-WWII to the Floating period.
For the other countries, relative government spending volatility falls from 2.94 to 1.73 (a 41% decline) from the pre-WWII to the Floating period. [Net exports falls from 2.01 to 1.37, a 32% decline].
1.c) Describe the correlation between government spending and GDP. Is this correlation consistent across time? Why might the Full Sample period provide a misleading picture of this correlation?
ANSWER:
Generally, GDP is weakly correlated with government spending. E.g. corr=-0.10, 0.00, -0.03, -0.10, etc
However, for the US government was moderately negatively correlated with spending pre-WWII, and then moderately positively correlated with spending post-WWII
In the US, the pattern of correlation clearly flips from moderately negative to moderately positive over time. However, the Full Sample period suggests there is only a weak negative correlation.
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2. Study the business cycle facts reported for the image labeled “Table 1” (from Aguiar and Gopinath (2007), Emerging Market Business Cycles: The Cycle Is the Trend). In this table σ refers to the standard deviation of a variable, and ρ refers to the correlation between two variables. Output=Y ; Consumption=C; Investment=I; Trade Balance=TB. Statistics are reported for emerging markets countries and developed markets countries (i.e. wealthy countries).
2.a) In which group of countries is output more volatile? In which group of countries is relative consumption more volatile? In which group of countries is consumption absolutely more volatile (justify your answer using simple mathematical relationships)?
ANSWER:
σ(Y)forEM=2.74¿1.34forDM
σ(C)/σ(Y)forEM=1.45¿0.94forDM
To figure out which country has greater absolute volatility:
σ(CEM = σ(CEM) ×σ(YEM)=1.45×2.74=3.973 σ(YEM)
σ(CDM = σ(CDM) ×σ(YDM)=0.94×1.34=1.25 σ(YDM)
2.b) Are trade balances pro-cyclical or counter-cyclical? This suggests that exports are -cyclical, and imports are -cyclical.
ANSWER:
Tradebalances are counter-cyclical (EM=-0.51, DM=-0.17). 3
Since the trade balance is = X – M, this suggests that exports are counter-cyclical Since the trade balance is = X – M, this suggests that imports are pro-cyclical
2.c) Suppose a fall in output causes a fall in consumption.1 If a recession results in a decline in output of 10%, how much do we expect consumption to decline? Provide an answer for each group of countries. Note: There is an easy way to answer the question (a good answer) and a hard way to answer the question (an excellent answer)!
ANSWER:
Easy answer: The correlation coefficients are ρE M (C, Y ) = 0.72 and ρE M (C, Y ) = 0.66. The 10% decline in output is associated/correlated with a 7.2% decline in consumption in EM countries, and a 6.6% decline in consumption in DM contries.
Hard answer: Suppose the relationship between output and consumption follows a linear regression equation: c = βy + ε. The regression coefficient is given by β = cov(C,Y ) . The table does not
σ(Y )2
give us cov(C, Y ). However, from statistics we know that the correlation coefficient is given by:
ρ(C, Y ) = cov(C,Y ) . Therefore, we can compute: σ(Y )σ(C)
β = cov(C, Y ) σ(Y )2
= cov(C,Y) × σ(C) σ(Y )σ(Y ) σ(C) = cov(C,Y) × σ(C) σ(Y )σ(C) σ(Y )
= ρ(C, Y ) × σ(C) σ(Y)
where the table gives us both of these components. Therefore:
βEM =ρEM(C,Y)× σEM(C) =0.72×1.45=1.044
σEM(Y)
βDM =ρDM(C,Y)× σDM(C) =0.66×0.94=0.62
σDM(Y)
Finally, this means that a 10% fall in output leads to a 10.44% fall in consumption in EM countries,
and leads to a 6.2% fall in consumption in DM countries.
1Note: this is not necessarily the case in reality!
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3. Consider a household choosing between consumption C and leisure L. They face the following decision problem:
max U(C,L)=CαL1−α C,L
s.t. L+N=1 C = wN
where N is labour hours and w is the wage.
3.a) Solve the household’s problem for the optimal choices of consumption, leisure, and labour.
ANSWER:
Substitute the time constraint and budget constraint into the problem: max(wN)α(1 − N)1−α
N
Take the first order condition with respect to labour N:
αw(wN)α−1(1 − N)1−α − (1 − α)(wN)α(1 − N)−α = 0
where we used both the chain rule and the product rule.
Simplifying with algebra:
αw(wN)−1(1 − N) − (1 − α) =0 αw(1−N)−(1−α)wN =0 αw−αwN −wN −αwN =0 αw − wN =0
⇒ N =α
Substituting back into the time constraint and the budget constraints, we get:
L=1−α, C=αw
3.b) What does the solution suggest about how the household allocates time between leisure and labour? How is this allocation affected by the parameter α? How is this allocation affected by the level of the wage, w?
ANSWER:
The household splits its time between leisure and labour according to the parameter α.
The higher is α, the less the household values leisure, and so the household allocates more time
towards labour: N = α.
The allocation of time is independent of the wage w. No matter how high the wage is, the household
always splits labour according to the rule: N = α, L = 1 − α. 5
4. Consider a household choosing between consumption C and leisure L. They face the following decision problem:
max U(C,L)=logC+b×L C,L
s.t. L+N=1
C = wN + Π
where N is labour hours and w is the wage, and Π is non-labour earnings (e.g. dividends from firms that households own shares in).
4.a) Solve the household’s problem for the optimal choices of consumption, leisure, and labour.
ANSWER:
Substitute the time constraint and budget constraint into the problem: maxlog(wN +Π)+b(1−N)
N
Take the first order condition with respect to labour N: w1−b=0
Simplifying with algebra:
wN + Π
N=1−Π bw
Substituting back into the time constraint we get: L=1−1+Π
bw
And substituting the labour demand equation into the budget constraint we get: C=w
b
4.b) Suppose Π = 0. How does an increase in the wage affect consumption, labour, and leisure? ANSWER:
When Π = 0, the demand functions are:
C=w, N=1, L=1−1
An increase in the wage leads to an increase in consumption: ∂C = 1
∂w b
But the increase in the wage has no effect on labor or leisure: ∂N=0, ∂L=0
bbb
∂w
∂w
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4.c) Suppose there is an increase in profits Π. How does this affect consumption, labour, and leisure? ANSWER:
The increase in profits has no effect on consumption: ∂C = 0
∂Π
But the increase in profits decreases labour, and increases leisure:
∂N=−1, ∂L=1 ∂Π w ∂w w
4.d) Suppose that profits are pro-cyclical, but that wages are sticky due to labour contracts agreed to between workers, unions, and firms. Now suppose that the economy is experiencing an expansion. In this economy, are consumption/labour/leisure: pro-cyclical, counter-cyclical, or acyclical? Can you provide economic intutition to explain these results?
ANSWER:
An expansion in this economy leads to an increase in profits, but not wages since they do not change.
From the previous answer (4.c), we saw that:
∂C=0, ∂N=−1<0, ∂L=1>0
∂Π ∂Π w ∂w w
So consumption is acyclical, labour is counter-cyclical, and leisure is pro-cyclical
Consumption is only affeted by labour income, but since wages are sticky during the expansion, consumption does not respond to the boom.
Labour is increasing in wages but decreasing in profit earnings. In contrast, leisure is decreasing in wages, but increasing in profit earnings.
An increase in the wage increases the price (i.e. opportunity cost) of leisure. Wages encourage the household to work more. However wages are sticky, and so do not provide any incentive to change the number of hours worked during the expansion.
The increase in pofits acts like an increase in wealth. When the household is wealthier, they would like to consume more leisure and work less. Hence the increase in profits during the expansion leads to an increase in leisure and a decrease in labour.
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