Distortionary Taxes in the Ramsey Model eserved@d = *@let ... · taxes → distortion Lecture 15 9/24 Topics in Macroeconomics. Ways of Financing Government Consumption Intro: From

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



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




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.





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.





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






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∗






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.





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.





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






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






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)






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




Review: Solow ModelReview: Ramsey Model

Review: Solow Model 13

Solow Model

◮ Exogenously given savings rate s◮ Production function

◮ Assumptions (Lecture 3):

→ A1: Constant returns to scale (CRS)F (λK , λL) = λF (K , L)

→ A2: Marginal products positive and diminishingFK > 0, FL > 0, FKK < 0, FLL < 0

◮ From CRS (A1), we can write F in per capita termsf (k) = F (k , 1). Then (A2) becomes

FK > 0 ⇔ f ′(k) > 0FKK < 0 ⇔ f ′′(k) < 0

◮ For example, f (k) = Akα with α < 1





Review: Solow Model → Results 14

Solow Model

◮ Unless there is exogenous technological change(At+1 = (1 + g)At , g > 0), the economy converges to asteady state in per capita variables.

◮ No long-run growth except due to technological change.

◮ Golden Rule steady state level of consumption requiress = α

Investments result from choices: the Solow model has nothingto say about savings/investment decisions → Ramsey model





Review: Ramsey Model 15

Ramsey Model

◮ Households choose how much to save and how much toconsume → dynamics results from this choice.

◮ Production function: same as for the Solow model◮ Assumptions (Lecture 3):

→ A1: Constant returns to scale (CRS)→ A2: Marginal products positive and diminishing

◮ From CRS (A1), we can write F in per capita termsf (k) = F (k , 1). Then (A2) becomes

FK > 0 ⇔ f ′(k) > 0FKK < 0 ⇔ f ′′(k) < 0

◮ For example, f (k) = Akα with α < 1





Review: Ramsey Model → Results 16

Ramsey Model

◮ Unless there is exogenous technological change(At+1 = (1 + g)At , g > 0), the economy converges to asteady state in per capita variables (g = 0).

◮ No long-run growth except due to technological change.◮ Modified Golden Rule steady state level of consumption

and capital are smaller than Golden Rule. This is due toimpatience of households.

◮ If government consumption is financed with taxes oncapital gains, HH save less and the steady state (c∗τk ) and(k∗τk ) are lower than with LS taxes.

How does long-run growth occur endogenously?




Growth rates are constantGrowth rates of c, k and y are the sameBalanced growth path theorem

A simple model of endogenous long-run growth 17

Ak Model

◮ Take Ramsey Model but change 1 assumption: α = 1

◮ Thus the production function becomes: f (k) = Ak (orF (K , L) = AK , i.e. labor does not enter into the productionfunction)

◮ That is, f ′(k) = A > 0 → used in Euler equation◮ That is, f ′′(k) = 0. There are no more diminishing marginal

returns to capital. It is constant returns to capital alone!

How does long-run growth occur endogenously?





Growth rate of consumption and the Euler equation 18

Ak Model

◮ The Euler equation becomes:

ct+1

ct= [β(f ′(kt+1) + 1 − δ)]1/σ

ct+1

ct= [β(A + 1 − δ)]1/σ = γc

◮ There is no steady state in this model

◮ But clearly, consumption grows at a constant rate—

FOREVER!!!





Growth rate of consumption and the Euler equation 19

Ak Model: Assumption 1

ct+1

ct= [β(A + 1 − δ)]1/σ = 1 + γc

◮ Assumption 1: γc > 0

This requires: β(A + 1 − δ) > 1





Growth rate of capital and the resource constraint 20

Ak Model: The growth rate of capital is constant

◮ The resource constraint can be written as:

ct + kt+1 = (A + 1 − δ)kt

Dividing both sides by kt

ct

kt+

kt+1

kt= A + 1 − δ





Growth rate of capital and the resource constraint 21

Ak Model: The growth rate of capital is constant

ct

kt+

kt+1

kt= A + 1 − δ

◮ Suppose the growth rate of k is increasing over time

→ctkt

must be decreasing over time→ violates transversality condition

◮ Suppose the growth rate of k is decreasing over time

→ctkt

must be increasing over time→ drives k and in turn c to 0, cannot be optimal

◮ Thus, kt+1kt

= 1 + γk is constant





Equal growth rates 22

Ak Model: Equal growth rates γc = γk = γy

ct

kt+ (1 + γk ) = A + 1 − δ

◮ Thus the ratio of consumption to capital is constant

◮ Thus, capital and consumption must grow at the same rate

◮ Since yt = Akt , output per capita must be growing at thesame rate as well

Thus, γ = γc = γk = γy = [β(A + 1 − δ)]1/σ− 1





Conditions for a BGP to exist 23

Ak Model: Assumption 2

ct

kt+ (1 + γk ) = A + 1 − δ

ct

kt+ [β(A + 1 − δ)]1/σ = A + 1 − δ

ct

kt= (A + 1 − δ)(1 − β

1σ (A + 1 − δ)

1σ−1)

◮ Assumption 2:

β1σ (A + 1 − δ)

1σ−1 < 1





Conditions for a BGP to exist 24

Ak Model: TheoremConsider the social planner’s problem with linear technologyf (k) = Ak and CEIS preferences. Suppose (β, σ, A, δ) satisfy

β(A + 1 − δ) > 1 > β1σ (A + 1 − δ)

1σ−1

Then the economy exhibits a balanced growth path wherecapital, output and consumption all grow at a constant rategiven by

kt+1

kt=

yt+1

yt=

ct+1

ct= 1 + γ = [β(A + 1 − δ)]1/σ

The growth rate is increasing in A and β and it is decreasing inδ and σ.


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Distortionary Taxes in the Ramsey Model eserved@d = *@let ... · taxes → distortion Lecture 15 9/24 Topics in Macroeconomics. Ways of Financing Government Consumption Intro: From