Ways of Financing Government Consumption Intro: From Solow to Ramsey to Endogenous Growth A simple model of endogenous long-run growth Distortionary Taxes in the Ramsey Model Intro to Endogenous Growth Lecture 15 Topics in Macroeconomics December 3, 2007 Lecture 15 1/24 Topics in Macroeconomics
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Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Distortionary Taxes in the Ramsey Model
Intro to Endogenous GrowthLecture 15
Topics in Macroeconomics
December 3, 2007
Lecture 15 1/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Outline
Ways of Financing Government ConsumptionReview: Ricardian equivalence: lump-sum taxes vs. debtToday: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Intro: From Solow to Ramsey to Endogenous GrowthReview: Solow ModelReview: Ramsey Model
A simple model of endogenous long-run growthGrowth rates are constantGrowth rates of c, k and y are the sameBalanced growth path theorem
Lecture 15 2/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Distortionary taxes on capital income 3
Tax rate on capital income, τk , with receipts being used tofinance government consumption, g.
The main modifications to the basic model are:
◮ Government’s budget constraint
τkt rtat = gt
◮ Household’s budget constraint
ct + at+1 = wt + (1 + (1 − τkt)rt )at
The tax rate influences the after-tax rate of return on savings.
Lecture 15 3/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Distortionary taxes on capital income 4
The Euler equation becomes:
EE :ct+1
ct= [β(1 + (1 − τkt+1)rt+1)]
1/σ
In equilibrium,
EE :ct+1
ct= [β(1 + (1 − τkt+1)(f
′(kt+1) − δ)]1/σ
RC : ct + gt + kt+1 = f (kt) + (1 − δ)kt
in addition to k0 > 0 and the transversality condition.
Notice that now the EE changes, hence the intertemporalconsumption-savings decision is distorted.
Lecture 15 4/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Switching from LS taxes to DT on K to finance g (*) 5
Recall: Permanent increase in g financed with LS taxesIf there is a permanent increase in g and HH did not expect itbefore but perceive it as permanent now
◮ Graphically, ∆k = 0 shifts down by the magnitude of ∆g
◮ The economy adjusts instaneously
through a downward jump of c
→ wealth effect
◮ No dynamic effect on capital accumulation
Hence no effect on output
Lecture 15 5/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Switching from LS taxes to DT on K to finance g (*) 6
Switching to distortionary taxes on K income to finance gA higher tax rate reduces steady-state capital per capita.*
k∗τk =
(
(1 − τk )α
ρ + (1 − τk )δ
)1
1−α
<
(
(1 − τk )α
(1 − τk )ρ + (1 − τk)δ
)1
1−α
=
(
α
ρ + δ
)1
1−α
= k∗τLS = k∗
k∗τk < k∗τLS = k∗
Lecture 15 6/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Switching from LS taxes to DT on K to finance g (*) 7
Switching to distortionary taxes on K income to finance g
◮ Graphically, the ∆c = 0 curve shifts to the left.*(k∗τk < k∗)
◮ Adjustment features an immediate upward jump ofconsumption to reach the path to the new steady state.
◮ This is because saving is less profitable.
◮ In the long-run, consumption will also be lower bec. outputfalls.
Lecture 15 7/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Taxes on labour income 8
Tax rate on labour income, τn, with receipts being used tofinance government consumption, g.
The main modifications to the basic model are:
◮ Government’s budget constraint
τntwt = gt
◮ Household’s budget constraint
ct + at+1 = (1 − τnt)wt + (1 + rt)at
The tax rate influences the after-tax wage.
Lecture 15 8/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Distortionary taxes on labour income 9
The Euler equation becomes:
EE :ct+1
ct= [β(1 + rt+1)]
1/σ
In equilibrium,EE :
ct+1
ct= [β(1 + f ′(kt+1) − δ)]1/σ
RC : ct + gt + kt+1 = f (kt) + (1 − δ)kt
in addition to k0 > 0 and the transversality condition.
Notice that now the EE and RC are the same as for LS taxes.This is because labour is supplied inelastically in this model.That means a change in the after-tax return to work (price)does not affect any decision.
Next: Example with elastic labour supply and labour incometaxes → distortion
Lecture 15 9/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Distortionary taxes on labour income 10
Static (one period) modelHousehold solves
maxc,l
u(c) + v(l)
s.t. c = (1 − τn)(1 − l)w
where c is consumption, l is leisure, w wage,τn is labour income tax, 1 is time endowment,(1 − l) is hours worked
Lecture 15 10/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Distortionary taxes on labour income 11
Static (one period) model: rewrittenAfter substituting for c in the utility function, household solves
maxl
u((1 − τn)(1 − l)w) + v(l)
The first order condition is:
u′(c)((1 − τn)w) = v ′(l)
Lecture 15 11/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Today: Distortionary taxes on capital incomeToday: Distortionary taxes on labour income
Distortionary taxes on labour income 12
Static (one period) model: solutionIn terms of Marginal rate of substitution:
MRSc,l =v ′(l)u′(c)
= (1 − τn)w
Thus if the labor tax goes up,
Substitution effect: leisure becomes cheaper relative toconsumption → work less
Income effect: less after-tax income even if work the same→ consume less, work more
Lecture 15 12/24 Topics in Macroeconomics
Ways of Financing Government ConsumptionIntro: From Solow to Ramsey to Endogenous Growth
A simple model of endogenous long-run growth
Review: Solow ModelReview: Ramsey Model
Review: Solow Model 13
Solow Model
◮ Exogenously given savings rate s◮ Production function