EC9012 - Economic Analysis: Macro
Irfan Qureshi
University of Warwick
October 17, 2015
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
1 / 17
Today
1
Housekeeping - 1
2
Housekeeping - II
3
Housekeeping - III
4
Problem Set- 1
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
2 / 17
About me
Irfan Qureshi, Ph.D. Candidate in Economics, area: Monetary
economics, empirical macroeconomics
Email: i.a.qureshi@warwick.ac.uk
Office:
I
Location: S2.94, Department of Economics
I
Hours
F
Friday 3 p.m - 4 p.m
F
Book online, https://irfanqureshi.youcanbook.me
Website: www.warwick.ac.uk/iqureshi
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
3 / 17
Course modalities
4 Lecture on growth, 4 on DSGE modeling
4 seminars with me, 4 with Lucio D’Aguanno
I revision seminar (TBA)
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
4 / 17
Organization of the seminars
Interactive
I
Go through problem sets before each class
Organized
I
Finish each problem set
Feedback
I
An opportunity for you to receive feedback on your own thoughts
about the course.
Revise
I
Any concepts you might have missed out in class
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
5 / 17
Question - (a)
Production: Yt = F (Kt , Lt ) = AKtα L1−α
t
Labour: Lt+1 = (1 + n)Lt and α ∈ (0, 1)
F (λKt , λLt ) = A(λKt )α (λLt )1−α
⇒ λA(Kt )α (Lt )1−α
⇒ λF (Kt , Lt )
Homogoneous of degree 1 ⇒ CRS
Remember: if Homogoneous of degree greater than 1 ⇒ IRS
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
6 / 17
Question - (b)
Production: Yt = F (Kt , Lt ) = AKtα L1−α
t
Labour: Lt+1 = (1 + n)Lt and α ∈ (0, 1)
Divide the production function by Lt
⇒
⇒
⇒
⇒
Ktα L1−α
t
Lt
AKtα L−α
t
Kt α
A( Lt )
yt = Aktα
Yt
Lt
=A
Why do we do this?
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
7 / 17
Question - (b)
Production: Yt = F (Kt , Lt ) = AKtα L1−α
t
Labour: Lt+1 = (1 + n)Lt and α ∈ (0, 1)
Divide the production function by Lt
⇒
⇒
⇒
⇒
Ktα L1−α
t
Lt
AKtα L−α
t
Kt α
A( Lt )
yt = Aktα
Yt
Lt
=A
Why do we do this?
To represent the production function (and therefore, the model) in terms
of output and capital per labour (or per effective worker, depending on the
definition of Lt
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
7 / 17
Question - (c)
Production: Yt = F (Kt , Lt ) = AKtα L1−α
t
Labour: Lt+1 = (1 + n)Lt and α ∈ (0, 1)
Take partial derivative w.r.t each factor of production
I
wt =
I
rt =
∂F (Kt ,Lt )
∂Lt
∂F (Kt ,Lt )
∂Kt
Irfan Qureshi (University of Warwick)
= (1 − α)AKtα L−α
= (1 − α)Aktα
t
= αAKtα−1 L1−α
= (α)Aktα−1
t
EC9012 - Economic Analysis: Macro
October 17, 2015
8 / 17
Question - (d)
Production: Yt = F (Kt , Lt ) = AKtα L1−α
t
Labour: Lt+1 = (1 + n)Lt and α ∈ (0, 1), s ∈ (0, 1), δ ∈ (0, 1]
Production can be written as: yt = f (kt ) = Aktα
Kt+1 = It + (1 − δ)Kt , where It is investment
Kt+1 = sYt + (1 − δ)Kt
This system describes the evolution of capital over time.
Capital is a result of last periods saved capital and left-over capital
(i.e. capital leftover after depreciation)
Divide this expression by Lt+1 :
⇒
Kt+1
Lt+1
=
sYt +(1−δ)Kt
Lt (1+n)
Irfan Qureshi (University of Warwick)
Kt+1
Lt+1
⇒ kt+1 =
=
Kt+1
Lt+1
sYt +(1−δ)Kt
Lt+1
=
sAktα +(1−δ)kt
1+n
EC9012 - Economic Analysis: Macro
October 17, 2015
9 / 17
Question - (e)
Find an expression for the growth rate of per worker capital
Production: Yt = F (Kt , Lt ) = AKtα L1−α
t
Labour: Lt+1 = (1 + n)Lt and α ∈ (0, 1), s ∈ (0, 1), δ ∈ (0, 1]
γk t =
kt+1 −kt
,
kt
⇒ γkt =
⇒
replace kt+1 from the previous expression
sAktα +(1−δ)kt
1+n
kt
sAktα−1 −(δ+n)
1+n
−kt
, simplify this to obtain:
(reference: class notes)
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
10 / 17
Question - (f)
Kaldor’s facts: six statements about economic growth (1957)
I
Statement 2: The rate of growth of the capital stock per worker is
roughly constant over long periods of time
I
In order to retain an unchanging level of capital per worker k over time,
we have to invest enough to create new capital to offset the loss in
capital per worker due to depreciation and population growth over
time.
I
Remember that in our model capital stock per worker is denoted by kt
To maintain a steady state where capital per worker is constant over
time, we must have that the change in capital stock must be
zero(γkt = 0).
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
11 / 17
Question - (f) - continued
γk t =
sAktα−1 −(δ+n)
1+n
Denote steady state as k̄, and set γkt = 0
1
sA 1−α
)
sAk̄ α−1 = δ + n ⇒ k̄ = ( δ+n
1
α
s
ȳ = f (k̄) = Ak̄ α ⇒ A 1−α ( δ+n
) 1−α
For consumption, remember that c̄ + i¯ = ȳ
But i¯ = s ȳ
1
α
s
c̄ = (1 − s)ȳ ⇒ (1 − s)A 1−α ( δ+n
) 1−α
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
12 / 17
Question - (g)
Find an expression for the golden rate (find k such that consumption in
the steady state is maximised)
ct = f (kt ) − sf (kt ), sf (k) = (n + δ)k
Now take first order conditions, which maximise c at the steady state
f 0 (kt ) = n + δ from ct . Lets call the desired rate of capital k GR
f 0 (k GR ) = n + δ
For y = Ak α , find the first derivative and replace in equation:
1
αA 1−α
αAk α−1 = n + δ ⇒ k GR = ( δ+n
)
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
13 / 17
Question - (h)
s GR = α
f 0 (k GR ) = n + δ ⇒ For k = k GR ,
f 0 (k)k
f (k)
f 0 (k)k
f (k)
=
(n+δ)k
f (k)
= share of capital (α), derived in part (b)
Remember, in the steady state:
(n+δ)k
f (k)
=
f 0 (k)k
f (k)
= s.
s GR = share of capital
What is the inuition here?
Can I set s = 0?
Can I set s =1?
Note: You can take the derivative of consumption at steady state w.r.t
saving to show that s GR = α holds
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
14 / 17
Question - (i)
1
α
s
ȳ = f (k̄) = Ak̄ α = A 1−α ( δ+n
) 1−α
Definition: Elasticity is the measurement of how responsive an economic
variable is to a change in another.
Take logs:
∂lny
∂lns
α
α
α
)lnA+ 1−α
lns− 1−α
ln(δ+n))
∂((1+ 1−α
∂lns
Then calculate: ey ,s =
ey ,s =
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
15 / 17
Question - (i)
1
α
s
ȳ = f (k̄) = Ak̄ α = A 1−α ( δ+n
) 1−α
Definition: Elasticity is the measurement of how responsive an economic
variable is to a change in another.
Take logs:
∂lny
∂lns
α
α
α
)lnA+ 1−α
lns− 1−α
ln(δ+n))
∂((1+ 1−α
∂lns
α
1−α
Then calculate: ey ,s =
ey ,s =
ey ,s =
Application: For α = 13 , and assuming a difference in savings rate of 3
times across countries, the explainable gap of GDP per capita is about
150% accroding to the Solow Model, with Cobb-Douglas production.
However, in a model with a poverty trap, a small rise in savings may
release an economy from the poverty trap and imply very large differences
in income.
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
15 / 17
Question - (j)
Plug in δ = 1 into equation describing dynamics of capital:
kt+1 =
syt
1+n
α
Remember that: ⇒ yt+1 = Akt+1
α . This yields the following expression:
Now replace kt+1
syt α
yt+1 = A( 1+n
) is the dynamic expression
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
16 / 17
Question - (k)
yt+1
yt
= from previous part
Taking logs
ln
i
yt+1
yti
= ln(1 + gi ) = lnA + αlns i − (1 − α)lnyti − αln(1 + ni )
and we know from kindergarten that ln(1 + x) ≈ x, so:
ln(1 + gti ) = gti
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
17 / 17
Question - (k)
yt+1
yt
= from previous part
Taking logs
ln
i
yt+1
yti
= ln(1 + gi ) = lnA + αlns i − (1 − α)lnyti − αln(1 + ni )
and we know from kindergarten that ln(1 + x) ≈ x, so:
ln(1 + gti ) = gti
Sources of growth?
Beta convergence?
I
Coefficient on log output per person (=-1) describes convergence
(Baumol 1986, Romer page 33)
I
Perfect convergence → higher initial income on average lowers
subsequent growth one-for-one.
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
October 17, 2015
17 / 17