Graduate Macroeconomics 2

Problem set 3. - Solutions

Question 1

The consumer’s utility function is given by:

U(Ct, 1− Lt) =

(Cφt (1− Lt)1−φ

)1−γ1− γ

In this non-separable case, consumption and leisure will be either complements or

substitutes, depending on the value of γ. Taking the first derivative with respect to

consumption, it is easy to see that when γ > 1, the marginal utility of consumption

is decreasing in leisure, (so consumption and leisure are substitutes), and when γ <

1 they become complements. When they are substitutes (γ < 1), when someone

works more, they want to be compensated by more consumption, whereas when

they are complements (γ > 1), then people need leisure to enjoy certain types of

consumption goods.

What this suggests is that when γ > 1, optimal response of the households will

induce a stronger positive co-movement between C and N , whereas with γ < 1,

this co-movement will be weaker.

Note that γ governs the inter-temporal elasticity of substitution over the whole

bundle of consumption and leisure, and that varying γ will have no effect on the

elasticity of labor supply. The elasticity of labor supply can be observed from the

intra-remporal optimization condition: Wt

Ct= (1−φ)

φ(1−N) .

Key equations of the model

Since only the utility function is different, it is only the consumer’s FOCs which

change. All other equations governing the model are the same as in Lecture 3.

These are: output, law of motion for capital, return on capital, wage rate, exoge-

nous process for technology.

The Euler equation:

βE

[Rt+1

(Ct+1

Ct

)φ(1−γ)−1(1− Lt+1

1− Lt

)(1−φ)(1−γ)]= 1

Intra-temporal FOCWt

Ct=

1− φφ(1− Lt)

1

The log-linearized model

Again, we only need to log-linearize the new equations, i.e. the two new FOCs.

Euler equation:

E

(rt+1 + (φ(1− γ)− 1) (ct+1 − ct)− (1− φ)(1− γ) L∗

1− L∗ (lt+1 − lt))

= 0

Intra-temporal FOC

wt − ct =L∗

1− L∗ lt

Steady state values that we need

From the intra-temporal FOC on the BGP we know that:

W ∗t

C∗t

=1− φ

φ(1− L∗)

and as before we can express W ∗t /C

∗t as a function of L∗, and parameters (α), and

steady state ratios (Y ∗t /C

∗t ) that we have calculated before. Just as in the lecture,

using that L∗ = 1/3, we can pin down the parameter value of φ.

Calibration

α δ R∗ G L∗ Y ∗t

K∗t

C∗t

K∗t

W∗t

C∗t

φ

1/3 0.025 1.015 1.005 1/3Gβ −(1−δ)

α 1− δ −G+Y ∗t

K∗t

1−αL∗

Y ∗tK∗t

C∗t

K∗t

1

1+(1−L∗)W∗t

C∗t

We will vary the remaining two parameters: ρ ∈ [0, 1] and γ ∈ [0,∞).

Response to technological shock

As usual, when the production process is not persistent, there is no persistence

in the deviation of other variables.

Observe the role of γ. As discussed earlier, γ determines whether consumption

and leisure are complements (γ < 1), or substitutes (γ > 1). The usual problem

in business cycle models is that labor is not pro-cyclical enough, and wages are

too pro-cyclical. This can be alleviated when consumption and leisure are substi-

tutes, as in this case as consumption increases, people are more willing to decrease

leisure, i.e. increase labor supply. We can see this on the graphs, though it is

masked by the differential response of consumption to the technological shock.

Notice that for smaller γ, the amplification is stronger (output reacts more), and

the path of consumption is more hump-shaped. That is when the shock hits, con-

sumption hardly increases, and therefore even though consumption and leisure

2

10 20 30 400

0.005

0.01

0.015y

10 20 30 400

0.5

1

1.5x 10

−3 k

10 20 30 400

2

4

6

8x 10

−4 c

10 20 30 40−5

0

5

10x 10

−3 l

10 20 30 40−2

0

2

4

6x 10

−4 r

10 20 30 400

2

4

6x 10

−3 w

10 20 30 40−5

0

5

10

15x 10

−3 a

10 20 30 400

0.005

0.01y

10 20 30 400

2

4

6

8x 10

−3 k

10 20 30 400

2

4

6x 10

−3 c

10 20 30 40−2

0

2

4

6x 10

−3 l

10 20 30 40−2

0

2

4x 10

−4 r

10 20 30 400

2

4

6x 10

−3 w

10 20 30 400

0.005

0.01

0.015a

Figure 1: Response to a technological shock: influence of ρ (left: 0.05, right: 0.95),γ = 1

are complements, there is not a lot of pressure on increasing leisure. Hence, actu-

ally for smaller values of γ labor supply increases more, which leads to the larger

amplification effect. However, a larger γ leads to more endogenous persistence.

Question 2

Consider a version of Campbell (1992) in which there are no technology shocks,

but government expenditure is stochastic.

Key equations of the model

Since we are only adding government expenditure, and shocks to government ex-

penditure to the model, we only need to change the equations, where this matters.

There are only two equations that change in the model: the capital accumulation

equation, and the nature of the shock. Both are given in the question:

Kt+1 = (1− δ)Kt + Yt − Ct −Xt

Where δ ∈ [0, 1] is the depreciation rate, Ct is private consumption, and Xt is gov-

ernment expenditure. Implicitly we are assuming that there is no public debt and

that lump-sum taxes are used to finance Xt. The assumption of no public debt

is not important since this economy satisfies all assumptions required for obtain-

ing Ricardian Equivalence. Government expenditure in log-deviations from the

nonstochastic BGP follows an AR(1) process with autoregressive coefficient of φ,

3

10 20 30 400

0.005

0.01

0.015y

10 20 30 400

2

4

6

8x 10

−3 k

10 20 30 400

2

4

6x 10

−3 c

10 20 30 40−5

0

5

10x 10

−3 l

10 20 30 40−2

0

2

4

6x 10

−4 r

10 20 30 400

2

4

6x 10

−3 w

10 20 30 400

0.005

0.01

0.015a

10 20 30 400

0.005

0.01

0.015y

10 20 30 400

2

4

6

8x 10

−3 k

10 20 30 400

2

4

6x 10

−3 c

10 20 30 40−2

0

2

4

6x 10

−3 l

10 20 30 40−2

0

2

4

6x 10

−4 r

10 20 30 400

2

4

6x 10

−3 w

10 20 30 400

0.005

0.01

0.015a

10 20 30 400

0.005

0.01y

10 20 30 400

2

4

6

8x 10

−3 k

10 20 30 400

2

4

6x 10

−3 c

10 20 30 40−2

0

2

4x 10

−3 l

10 20 30 40−2

0

2

4x 10

−4 r

10 20 30 400

2

4

6x 10

−3 w

10 20 30 400

0.005

0.01

0.015a

10 20 30 400

0.005

0.01y

10 20 30 400

2

4

6

8x 10

−3 k

10 20 30 400

2

4

6x 10

−3 c

10 20 30 400

1

2

3x 10

−3 l

10 20 30 40−2

0

2

4x 10

−4 r

10 20 30 400

2

4

6x 10

−3 w

10 20 30 400

0.005

0.01

0.015a

Figure 2: Response to a tech shock: influence of γ (top left: 0.1, top right: 0.5,bottom left1.5, bottom right: 5), ρ = 0.95

i.e.:

xt = φxt−1 + εt

Where εt is a zero-mean white noise process and xt ≡ logXt − logX∗t . Under

the assumption that X∗t /Y

∗t is constant, output, consumption, capital, and real

wages all grow at the same rate as technology,G, and bothR∗ andN∗ are constant.

Note that technology here is assumed to be non-stochastic, therefore, it is on its

balanced-growth path.

The log-linearized model

Again, we only have to log-linearize our one new equation, the capital accumula-

4

tion equation, as the other one is already in log-linear form.

Gkt+1 = (1− δ)kt +Y ∗t

K∗t

yt −C∗t

K∗t

ct −X∗t

Y ∗t

Y ∗t

K∗t

xt

Steady state values that we need

We are told thatX∗t /Y

∗t = 0.2. We can use this in the capital accumulation equation

(along the BGP) to express the consumption-capital ratio as:

C∗t

K∗t

= (1− δ)−G+

(1− X∗

t

Y ∗t

)Y ∗t

K∗t

Calibration

α δ R∗ G L∗ Y ∗t

K∗t

X∗t

Y ∗t

C∗t

K∗t

W∗t

C∗t

θ

1/3 0.025 1.015 1.005 1/3Gβ −(1−δ)

α 0.2 1− δ −G+(1− X∗

t

Y ∗t

)Y ∗t

K∗t

1−αL∗

Y ∗tK∗t

C∗t

K∗t

(1− L∗)γ 1−αL∗

Y ∗tK∗t

C∗t

K∗t

We will vary the remaining two parameters: φ ∈ [0, 1] and γ ∈ [0, 10].

Response to government expenditure shock

10 20 30 40−5

0

5

10

15x 10

−5 y

10 20 30 40−3

−2

−1

0

1x 10

−4 k

10 20 30 40−1.5

−1

−0.5

0x 10

−4 c

10 20 30 400

1

2x 10

−4 l

10 20 30 400

2

4

6

8x 10

−6 r

10 20 30 40−1

−0.5

0x 10

−4 w

10 20 30 400

0.005

0.01

0.015x

10 20 30 400

0.5

1x 10

−3 y

10 20 30 40−3

−2

−1

0

1x 10

−4 k

10 20 30 40−1.5

−1

−0.5

0x 10

−3 c

10 20 30 400

0.5

1

1.5x 10

−3 l

10 20 30 400

1

2

3

4x 10

−5 r

10 20 30 40−6

−4

−2

0x 10

−4 w

10 20 30 400

0.005

0.01

0.015x

Figure 3: Response to a government shock: influence of φ (left: 0, right: 0.95), γ = 1

γ = 1 and φ = 0. The results are roughly consistent with a negative wealth

effect: both consumption and leisure are reduced during the transition. The fluctu-

ations in government expenditure produce a business cycle which is not precisely

short-lived. However, during the transition, employment is countercyclical, which

5

is not consistent with the stylized facts of the US business cycle, as documented by

Stock and Watson (1999). The solution to the system can be characterized by:

γ = 1 and φ = 0.95. Note that when shocks to government expenditure

are very persistent, the business cycle becomes very long-lived (it takes roughly

100 quarters for the endogenous variables in the model to return to their steady-

state position). In this case, employment is pro-cyclical consistent with evidence

from filtered macro series for the US, but, unfortunately, wages are countercycli-

cal. Households supply more labor in spite of the lower real wage because there

is a large negative wealth effect on leisure, which is a normal good. The wealth

effect is stronger in this case, as the shock to government expenditure is close to

permanent.

10 20 30 400

0.5

1

1.5

2x 10

−3 y

10 20 30 400

1

2

3

4x 10

−4 k

10 20 30 40−1

−0.5

0x 10

−3 c

10 20 30 400

1

2

3x 10

−3 l

10 20 30 400

2

4

6

8x 10

−5 r

10 20 30 40−1

−0.5

0x 10

−3 w

10 20 30 400

0.005

0.01

0.015x

5 10 15 20 25 30 35 40−4

−2

0

2x 10

−4 y

5 10 15 20 25 30 35 40−1.5

−1

−0.5

0x 10

−3 k

5 10 15 20 25 30 35 40−2

−1.5

−1

−0.5

0x 10

−3 c

5 10 15 20 25 30 35 400

1

2

3x 10

−5 r

5 10 15 20 25 30 35 40−4

−3

−2

−1

0x 10

−4 w

5 10 15 20 25 30 35 400

0.005

0.01

0.015x

Figure 4: Response to a gov shock: influence of γ (left: 0, right: 1010), φ = 0.95

γ = 0 and φ = 0.95. The undesirable feature of countercyclical real wages is still

present is this case. Furthermore, capital is now hump-shaped and countercyclical

for most of the transition towards the non-stochastic BGP. Since labor supply is

more elastic in this case, the response of employment to the shock in xt is doubled

with respect to the previous case.

γ = 1010 and φ = 0.95. Although we obtain hump shape for output and

real wages are now pro-cyclical, labor input is unresponsive to the business cy-

cle, which is at odds with empirical evidence.

These results suggests that government spending is not the most plausible

source of business cycle, or at least, does not seem to be a better explanation than

technological shocks. Note that a little amount of persistence introduces a very

6

high persistence to the business cycle and the model has a hard time reconciling

the cyclical properties of total labor input and real wages that are documented in

Stock and Watson (1999), for example.

7