EC9012 - Economic Analysis: Macro
Irfan Qureshi
University of Warwick
November 9, 2015
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
1 / 15
Today
1
Question - 1
2
Question - 2
3
Stable systems
4
OLG revision Q-2:hints
5
Galor-Zeira revision:hints
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
2 / 15
Setting out the problem
Income groups:
I
1
3
of the population have income x i = H
I
1
3
of the population have income x i = M
I
1
3
of the population have income x i = L
H>M>L
Tax rate: τ ∈ (0, 1]
θ ∈ (0, 1] of tax revenue equally distributed.
What is a distortion? A distortion is departure from the allocation of
economic resources from the state in which each agent maximizes his/her
own welfare. In this model this is equal to 1 − θ
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
3 / 15
part(a)
Each group pays τ x i of their income, and keep (1 − τ )x i of their
pre-tax income.
They receive τ θx̄ of tax revenue, and x̄ is average income, which can
be calculated: x̄ = H+M+L
3
If the M-income group supports the tax, then it will be the L-income
group will support the tax as well.
I
Why? Because they pay less tax(τ L < τ M ) but receive the same tax
revenue, as defined before.
For the other case, assume that if society members of M-income
group object the tax then high income (H) will also object the tax.
I
Why? They pay more taxes (τ M < τ H) to enjoy the same benefits.
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
4 / 15
The condition for the majority support therefore is:
I
I
x M ≤ x̃ M (income of middle income at least as good as their post-tax
income)
M ≤ (1 − τ )M + τ θ H+M+L
(post tax income + revenue from transfer)
3
I simplify this expression to get:
I
τ M ≤ τ θ H+M+L
3
I
M ≤ θ H+M+L
3
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
5 / 15
part (b)
For θ = 1, the condition in (a) becomes:
M≤
H+M+L
.
3
What does this mean?
I
Income of the two lowest income groups must be below the average
income for the proposal to receive support, so that the transfer can
bring them to the point with less than or equal to their actual income.
I
As the M-income group contains the median income, this amounts to
saying that the median income must be lower than the mean income.
If they are higher then they wont support the tax bill.
For θ = 1 the condition turns out to be the ratio between
median income and average income.
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
6 / 15
Question - 2, part (a)
Q: Find tax rate that maximizes output:
A: Replace output equation with factor inputs:
e = τ λb
(1)
h(e) = e γ
(2)
h(e) = (τ λb)γ
(3)
K = (1 − τ )λb
(4)
Y
Irfan Qureshi (University of Warwick)
= K α h1−α
EC9012 - Economic Analysis: Macro
November 9, 2015
7 / 15
Question - 2, part (a)
Y
=
K α h1−α = [(1 − τ )λb]α [τ λb](1−α)γ
=
(1 − τ )α τ (1−α)γ λb α+(1−α)γ
max(1 − τ )α τ (1−α)γ λb α+(1−α)γ
τ
→ τ∗ =
Irfan Qureshi (University of Warwick)
(1 − α) γ
α + (1 − α) γ
EC9012 - Economic Analysis: Macro
November 9, 2015
8 / 15
part (b)
Find the optimal tax rate from the viewpoint of the workers
wh
=
→
τ
∗
=
dY
h = (1 − α) K α h1−α = (1 − α) Y
dh
arg max wh
Maximise the income w.r.t τ
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
9 / 15
part (c)
Find the optimal tax rate from the viewpoint of the capitalists
R(1 − τ )b
=
→
τ
∗
=
dY
(1 − τ )b = αY
dK
arg max R(1 − τ )b
Maximise the income w.r.t τ
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
10 / 15
part (d)
What is going on here?
Poor (own labour), Rich (own capital), CD production function.
Labour could be augmented with education, but who finances the
education? The rich. How much would the rich want to pay for
education? How much would the poor want the rich to pay for
education and what are the efficiency levels?
The interesting thing is when you have a CD production function, all
three generate the same result (no conflict of interest).
Not obvious exante. You have owners of capital and owners of labour.
You could augment labour by human capital financed by the wealthy.
Now what is the trade-off from the point of the rich? If they give as a
group part of their capital they pay taxes so its from capital. The
more they transfer to education they have less physical capital but
more human capital in their economy.
What is interesting in the CD is that the share of labour is constant
and the share of capital is constant, so everyone wants to drive to
that point.
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
11 / 15
Question - (d) - problem set 2
OLG Model:
Since individuals only consume in the second year of life - when old:
kt+1 = st = wt
⇒
kt+1 =
kt2 for
4 for
kt ≤ 2
≡ φ(kt )
kt > 2
Set kt+1 = k̄ & kt = k̄. Then I can solve to obtain the steady states.
Steady states are 0, 1 and 4.
I
I
For stability |φ0 (k̄)| < 1
Stable steady states are are 0 and 4. 0 is trivial. So the answer is
k̄ = 4.
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
12 / 15
(a) (b) and (c) discussed in detail in problem set 1.
(d) We want ’unstable’ system so set |φ0 (kt )| > 1 and solve for s.
(e) Use condition from (d) and s ≤ 1 to find B
(f) maximize production function w.r.t labour.
(g) and (c) use dynamical system (st = βwt ), replace savings and
wage to find ψ(kt )
(h) For dynamical system find an expression φ(kt ) = kt+1
(i) Set ψ(k̄) = k̄ to find steady state
(j) Ans:1 why?
(k) Indifferent implies what? (Hint:compare to condition found in (d))
(l) Mechanical, as individuals save their entire income, so you can find
k̄
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
13 / 15
1
Refer to problem set 3.
2
Conditions for poverty trap in the long run.
3
Plot, similar to problem set.
4
Intuition and use plot.
5
Plot prob. dist. function and a 45 degree line.
6
Discussed in seminar. Different from what happens to inequality.
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
14 / 15
Thank you and good luck!
Irfan Qureshi (University of Warwick)
EC9012 - Economic Analysis: Macro
November 9, 2015
15 / 15