LH Advanced Macroeconomics (07 33109)
Seminar Class 1: Real Business Cycle Theory (Week 3)
Q1. Consider the following Cobb-Douglas production function (0< α<1):
𝑌𝑡 = 𝐾𝑡𝛼 (𝐴𝑡 𝐿𝑡 )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. Consider the one-period RBC model studied in the lecture notes but now suppose the
utility function for households is:
𝑢𝑡 = ln 𝑐𝑡 + 𝑏(1 − ℓ𝑡 )1−𝛾 ⁄(1 − 𝛾)
where b>0 and γ>0.
How, if at all, does labour supply depend on the real wage under this new functional form
for utility?
Q3. 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 = ln 𝑐𝑡 + 𝑏 ln(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.
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SOLUTIONS
Q1. Recall the Cobb-Douglas production function given in the question:
𝑌𝑡 = 𝐾𝑡𝛼 (𝐴𝑡 𝐿𝑡 )1−𝛼
i) To find the marginal product of capital (MPK), take the first derivative of the production
function w.r.t. capital, K:
𝑀𝑃𝐾𝑡 = 𝛼𝐾𝑡𝛼−1 (𝐴𝑡 𝐿𝑡 )1−𝛼 > 0
Notice that under a Cobb-Douglas production function the MPK can be conveniently
expressed as:
𝑌𝑡
𝛼𝐾𝑡𝛼−1 (𝐴𝑡 𝐿𝑡 )1−𝛼 = 𝛼 > 0
𝐾𝑡
To show that the MPK diminishes as capital input increases, take the second derivative as
follows:
𝛼(𝛼 − 1)𝐾𝑡𝛼−2 (𝐴𝑡 𝐿𝑡 )1−𝛼 = 𝛼(𝛼 − 1)
𝑌𝑡
<0
𝐾𝑡2
The parameter restriction 0<α<1 is crucial to these results for MPK.
ii) To find the marginal product of labour (MPL), take the first derivative of the production
function w.r.t. labour, L (chain rule needed here):
𝑀𝑃𝐿𝑡 = (1 − 𝛼)𝐾𝑡𝛼 (𝐴𝑡 𝐿𝑡 )−𝛼 𝐴𝑡 > 0
Notice that under a Cobb-Douglas production function the MPL can be conveniently
expressed as:
𝑌𝑡
(1 − 𝛼)𝐾𝑡𝛼 (𝐴𝑡 𝐿𝑡 )−𝛼 𝐴𝑡 = (1 − 𝛼) > 0
𝐿𝑡
To show that the MPL diminishes as labour input increases, take the second derivative as
follows (chain rule needed again):
−𝛼(1 − 𝛼)𝐾𝑡𝛼 (𝐴𝑡 𝐿𝑡 )−𝛼−1 𝐴2𝑡 = −𝛼(1 − 𝛼)
𝑌𝑡
<0
𝐿2𝑡
Again, the parameter restriction 0<α<1 is crucial to these results.
iii) The parameter α represents capital’s share of output. To see this mathematically, note
that under perfect competition and profit maximisation each unit of capital (and labour)
is paid an amount equal to its marginal product. The total amount paid to those who own
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the capital is therefore MPK multiplied by the level of K. Let’s define the share of output
paid to those who own the capital as sK:
𝑠𝐾 ≡
(𝑀𝑃𝐾𝑡 )𝐾𝑡 𝛼𝐾𝑡𝛼−1 (𝐴𝑡 𝐿𝑡 )1−𝛼 𝐾𝑡
=
=𝛼
𝑌𝑡
𝐾𝑡𝛼 (𝐴𝑡 𝐿𝑡 )1−𝛼
It follows that (1–α) represents labour’s share of output (sL). You can easily check this
mathematically. Notice that these shares are constant in the Cobb-Douglas case, even
though this may not be consistent with empirical evidence. Also, the shares must sum to
1 because of the parameter restriction on α. In other words, we assume constant returns
to scale.
We have chosen to adopt a ‘Harrod-neutral’ (labour-augmenting technological progress)
form for our Cobb-Douglas production function. Alternatively, the Cobb-Douglas
production function could be written with ‘Hicks-neutral’ technological progress as
follows:
𝑌𝑡 = 𝐴𝑡 𝐾𝑡𝛼 𝐿𝑡 1−𝛼
Or with capital-augmenting (‘Solow-neutral’) technological progress:
𝑌𝑡 = (𝐴𝑡 𝐾𝑡 )𝛼 𝐿𝑡 1−𝛼
However, following Romer’s approach we shall adopt the labour-augmenting
specification given in the question.
Q2. The utility function is as follows:
𝑢𝑡 = ln 𝑐𝑡 + 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 (we can drop time subscripts from this
point on since there is only one time period):
ℒ = ln 𝑐 + 𝑏(1 − ℓ)1−𝛾 ⁄(1 − 𝛾) + 𝜆(𝑤ℓ − 𝑐)
With first order conditions:
𝜕ℒ 1
= −𝜆 =0
𝜕𝑐 𝑐
𝜕ℒ
= −𝑏(1 − ℓ)−𝛾 + 𝜆𝑤 = 0
𝜕ℓ
For completeness, take the F.O.C. with respect to λ:
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𝜕ℒ
= 𝑤ℓ − 𝑐 = 0
𝜕𝜆
Using the budget constraint, the FOC for c can be expressed as:
𝜆=
1
1
=
𝑐 𝑤ℓ
Substitute this result into the FOC for ℓ to obtain:
1
= 𝑏(1 − ℓ)−𝛾
ℓ
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.
Q3. 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 relevant to labour supply in the model because households allocate 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):
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1 − ℓ1
1
𝑤2
= −𝜌
1 − ℓ2 𝑒 (1 + 𝑟) 𝑤1
Therefore:
𝜕[(1 − ℓ1 )⁄(1 − ℓ2 )]
1
= −𝜌
𝜕[𝑤2 /𝑤1 ]
𝑒 (1 + 𝑟)
Using this result in the formula for the intertemporal elasticity of substitution for leisure
time given above:
𝐼𝐸𝑆𝐿𝑒𝑖𝑠𝑢𝑟𝑒 =
1
𝑒 −𝜌 (1 +
𝑤2 ⁄𝑤1
𝑟) (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
−𝜌
𝑒 (1 + 𝑟)
𝑤2 ⁄𝑤1
=1
1
𝑤2
𝑒 −𝜌 (1 + 𝑟) 𝑤1
We can 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 ).
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