Consider the one-period RBC model studied in the lecture notes under this new utility function. How, if at all, does labour supply depend on the real wage?

Math/Physic/Economic/Statistic Problems

Seminar Class 1: Real Business Cycle Theory (week 3)

Q1. Consider the following Cobb-Douglas production function (0< α<1): 𝑌𝑡 =𝐾𝑡𝛼(𝐴𝑡𝐿𝑡)1−𝛼 which shows that output depends upon capital input, labour input and ‘technology’ (A).

i) Show that the marginal product of capital is positive (for K,L,A>0) but diminishing in the level of capital employed.

ii) Show that the marginal product of labour is positive for (K,L,A>0) but
diminishing in the level of labour employed.

iii) Provide an economic interpretation for the parameter α.

Q2.

For the two-period RBC model, show that a logarithmic instantaneous utility function
of the form (where b>0 is a leisure preference parameter):

𝑢𝑡 =ln𝑐𝑡 +𝑏ln(1−ℓ𝑡)

with arguments in consumption (𝑐𝑡) and leisure time (1−ℓ𝑡), produces an intertemporal
elasticity of substitution of 1 between period 1 and period 2 leisure time. Explain how
this is relevant to the labour supply in the two-period RBC model.

Q3. For a ‘simplified’ RBC model with no government and full depreciation in each period,
we can present the following expression for Ỹt , which you can interpret as the deviation
of (log) output from its long-run path, i.e. a short-run fluctuation (Romer, 2019, p.206):

𝑌̃𝑡 =(𝛼+𝜌𝐴)𝑌̃𝑡−1 −𝛼𝜌𝐴𝑌̃𝑡−2 +(1−𝛼)𝜀𝐴,𝑡

n.b. you do not need to derive this expression.

Set α=1/3, ρA=0.9 and consider a temporary shock of 1/(1–α) to εA. Assume that the shock
hits the economy in period t and that it lasts for only a single time period. Show what
happens to Ỹ between period t and period t+8.

Q4. Now suppose the period-t utility function for households is:

𝑢𝑡 =ln𝑐𝑡 +𝑏(1−ℓ𝑡)1−𝛾 (1−𝛾)⁄

where b>0 and γ>0.

Consider the one-period RBC model studied in the lecture notes under this new utility
function. How, if at all, does labour supply depend on the real wage?

Q2.
We need to use the intertemporal elasticity of substitution between period 1 and period
2 leisure time to answer this question. This measures the percentage change in the ratio
of leisure time, (1−ℓ1)/(1−ℓ2), for a given percentage change in relative wages, w2/w1.
Intuitively, an increase in relative wages will make it more attractive to work in period 2,
i.e. households want less leisure time in period

2. However, because this elasticity is calculated for a given indifference curve, the household will need to increase leisure time in period 1 to compensate for the reduced leisure time in period 2.

Mathematically, we need to calculate the following elasticity:

If this elasticity is large, households are willing to substitute a relatively large amount of
current leisure for future leisure in response to a rise in the wage ratio w2/w1. This
response is clearly relevant to labour supply in the model because households allocate
4 their time endowment between labour and leisure time in each period. An increase in
leisure time in period 1, say, implies a decrease in labour time in that same time period.

Recall that the first order conditions for current and future leisure time can be combined
to obtain (see lecture slides):

1−ℓ1
1−ℓ2
= 1
𝑒−𝜌(1+𝑟)
𝑤2
𝑤1

Therefore:

𝜕[(1−ℓ1) (1−ℓ2⁄ )]
𝜕[𝑤2/𝑤1] = 1
𝑒−𝜌(1+𝑟)

Using this result in the formula for the intertemporal elasticity of substitution for leisure
time given above:

𝐼𝐸𝑆𝐿𝑒𝑖𝑠𝑢𝑟𝑒 = 1
𝑒−𝜌(1+𝑟) 𝑤2 𝑤1⁄
(1−ℓ1)/(1−ℓ2)

Using the first order condition presented at the top of this page to substitute out for
relative leisure time, we can write this as:

= 1
𝑒−𝜌(1+𝑟) 𝑤2 𝑤1⁄
1
𝑒−𝜌(1+𝑟)
𝑤2
𝑤1
=1

We interpret this result as follows: an x% increase in relative wages, w2/w1, leads to an
equal % increase in relative leisure time, (1−ℓ1)/(1−ℓ2).

Q3.
To repeat the expression given in the question:

𝑌̃𝑡 =(𝛼+𝜌𝐴)𝑌̃𝑡−1 −𝛼𝜌𝐴𝑌̃𝑡−2 +(1−𝛼)𝜀𝐴,𝑡

This is an AR(2), i.e. a second order autoregressive, process to an econometrician.

The positive coefficient on the first lag of Ỹ and the negative coefficient on the second lag
of Ỹ can generate ‘hump-shaped’ responses to shocks. This is needed to replicate
empirical evidence according to a well-known paper by Cogley and Nason (1995).

Using the parameter values given in the question, we can construct the following table:

Time period Impact on 𝒀̃ Calculation
t 1 (1–α)*shock
t+1 1.23 (α+ρA)*1
t+2 1.22 (α+ρA)*1.23 – (αρA)*1
5

t+3 1.14 (α+ρA)*1.22 – (αρA)*1.23
t+4 1.03 (α+ρA)*1.14 – (αρA)*1.22
t+5 0.94 (α+ρA)*1.03 – (αρA)*1.14
t+6 0.84 (α+ρA)*0.94 – (αρA)*1.03
t+7 0.76 (α+ρA)*0.84 – (αρA)*0.94
t+8 0.68 (α+ρA)*0.76 – (αρA)*0.84
t+… … …

This replicates the ‘hump-shaped response’ in (log) output that we typically observe in
the data. However, we rely heavily on the assumed persistence of the technology shock
to generate this response.

The internal propagation mechanisms of standard RBC models
are not particularly strong. See lecture slides and chapter 5 of Romer (2019) for a full
discussion.

Q4.
The utility function is as follows:

𝑢𝑡 =ln𝑐𝑡 +𝑏(1−ℓ𝑡)1−𝛾 (1−𝛾)⁄

Recall from the lecture slides that the budget constraint in the one-period model is simply
𝑐 =𝑤ℓ. The Lagrangian problem is therefore (drop time subscripts from this point on
since there is only one time period):

ℒ =ln𝑐+𝑏(1−ℓ)1−𝛾 (1−𝛾)+𝜆(𝑤ℓ−𝑐)⁄

With first order conditions:

𝜕ℒ
𝜕𝑐 =1
𝑐−𝜆=0

𝜕ℒ
𝜕ℓ =−𝑏(1−ℓ)−𝛾 +𝜆𝑤 =0

Using the budget constraint, the FOC for c can be expressed as:

𝜆=1
𝑐 = 1
𝑤ℓ

Substitute this result into the FOC for ℓ to obtain:

1
ℓ =𝑏(1−ℓ)−𝛾

This is only an implicit function for ℓ but it is sufficient to show that ℓ does not depend
upon the real wage.

This gives us the same results as the lecture slides – the income and
substitution effects of a change in w exactly offset each other in this one-period model,
even though we have used a different utility function in the seminar class.