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THE ECONOMICS OF TAXATION Static Ramsey Tax

School of Economics, Xiamen University

Fall 2015

The Economics of Taxation | Ye Chen, Xiamen University

1

Overview of Optimal Taxation

Combine lessons on incidence and efficiency costs to analyze optimal design of

commodity taxes.

What is the best way to design taxes given equity and efficiency concerns?


2

From an efficiency perspective, would finance government purely through lump-sum

taxation.

With redistributional concerns, would ideally levy individual-specific lump sum taxes.

Tax higher-ability individuals a larger lump sum.

Problem: cannot observe individuals’ types.

Therefore must tax economic outcomes such as income or consumption, which

leads to distortions.


3

Ramsey vs. Mirrleesian Approaches Two approaches to optimal taxation:

1. Ramsey: restrict attention linear (t x ) tax systems 2. Mirrleesian: non-linear ( ( )t x ) tax systems, with no restrictions on ( )t x

Ramsey approach: rule out possibility lump sum taxes by assumption and consider

linear taxes.

Mirrleesian approach: permit lump sum taxes, but model their costs in a model with

heterogeneity in agents’ skills.


4

Primal vs. Dual Approaches

Regardless of which approach is used, there are two ways of solving the optimal

taxation problem.

1. Primal approach: the government chooses allocations directly.

The optimal tax formulas are then typically expressed directly in terms of the

primitives of the model.

2. Dual approach: the government chooses the taxes directly.

The optimal tax formulas are easily expressed in terms of supply and demand

elasticities.


5

Four Central Results in Optimal Tax Theory

1. Ramsey (1927): inverse elasticity rule

2. Chamley (1985), Judd (1986): no capital taxation in infinite horizon Ramsey models

3. Diamond and Mirrlees (1971): production efficiency

4. Atkinson and Stiglitz (1976): no consumption taxation with optimal non-linear

income taxation


6

Ramsey Tax Problem

Government sets taxes on uses of income in order to accomplish two objectives:

1. Raise total revenue of amount R

2. Minimize utility loss for agents in economy

Key assumptions:

1. Lump sum taxation prohibited

2. Cannot tax all commodities (e.g., leisure untaxed)

3. Production prices fixed (and normalized to one): 1ip

1i iq


7

Ramsey Model: Setup

One individual (no redistributive concerns) As in efficiency analysis, assume that individual does not internalize effect of i

on government budget

Captures idea that any one individual accounts for a small fraction of economy

Individual maximizes utility

1 ,( , , )Nu x x l

subject to budget constraint

1 1 N Nq x q x wl Z

Z = non wage income, w = wage rate


8

Ramsey Model: Consumer Behavior

Lagrangian for individual’s maximization problem:

1 1 1( , , , ) ( ( ))N N NL u x x l wl Z q x q x

First order condition:

ix iu q

where V Z is marginal value of money for the individual

Yields demand function ( , )ix q Z and indirect utility function ( , )V q Z where

1( , , , )Nq w q q


9

Ramsey Model: Government’s Problem

Government solves either the maximization problem max ( , )V q Z

subject to the revenue requirement

1

( , )N

i ii

x x q Z R

Or, equivalently, minimize excess burden of the tax system min ( ) ( , ( , )) ( , ( , ))EB q e q V q Z e p V q Z E

subject to the same revenue requirement


10

For maximization problem, Lagrangian for government is:

( , ) ( , )G i ii

L V q Z x q Z E

..

jGi j

ji i imechanical effectpriv welfare behavioral responseloss to indiv

xL V xq q q

Using Roy’s identity ( ii

V xq

):

( ) 0i j j ij

x x q

Note connection to marginal excess burden formula, where 1 and 1 .


11

Ramsey Optimal Tax Formula

Optimal tax rates satisfy system of N equations and N unknowns:

( )j ij

j i

x xq

Same formula can be derived using a perturbation argument, which is more intuitive.


12

Ramsey Formula: Perturbation Argument Suppose government increases i by id

Effect of tax increase on social welfare is sum of effect on government revenue and

private surplus.

Marginal effect on government revenue:

i i j jj

dR x d dx

Marginal effect on private surplus:

i i ii

VdU d x dq

Optimum characterized by balancing the two marginal effects:

0dU dR


13

Ramsey Formula: Compensated Elasticity Representation

Rewrite in terms of Hicksian elasticities to obtain further intuition using Slutsky

equation:

j j ji

i i

x h xx

q q Z

Substitution into formula above yields:

( ) 0i j j i i jj

x h q x x Z

1 ij

ji j

hx q

where ( )j jZ jx

.


14

is independent of i and measures the value of the government of introducing a $1 lump sum tax

( )j jjx Z

Three effects of introducing a $1 lump sum tax:

1. Direct value for the government of

2. Loss in welfare for individual of

3. Behavioral effect → loss in tax revenue of ( )j jjx Z


15

Intuition for Ramsey Formula: Index of Discouragement 1 i

jji j

hx q

Suppose revenue requirement R is small so that all taxes are also small. Then tax j on good j reduces consumption of good i (holding utility constant) by

approximately ii j

j

hdhq

Numerator of LHS: total reduction in consumption of good i Dividing by ix yields % reduction in consumption of each good i = “index of

discouragement” of the tax system on good i Ramsey tax formula says that the indexes of discouragements must be equal across

goods at the optimum.


16

Inverse Elasticity Rule

Introducing elasticities, we can write Ramsey formula as:

1 1

Nj c

ijj j

Consider special case where 0ij if i j

Slutsky matrix is diagonal

Obtain classic inverse elasticity rule:

11

i

i ii


17

Example 1: Two Commodities

If N=2, then we can have:

1 21 2

1 21 1c ci i

This yields

If the goods are complementary ( 12 21 0c c ), the tax rates 1 and 2 will tend

to be closer to each other.

Intuitively, complementary goods look more alike, and there is less need to

differentiate between them.

1

1

2

2

1 22 12

11 211

c c

c c


18

Example 2: Corlett-Hague (1953)

We can reinterpret the results in the following way.

The compensated elasticities satisfy the adding-up property:

0 1 2 0c c ci i i

So, 2

2

1

1

1 11 22 20

11 22 101

( )( )

c c c

c c c

What matters for the relative tax rates is therefore the magnitude of 0ci .

If the good zero as leisure, a good i is more complementary with leisure than good j if 0 0

c ci j .

Goods that are more complementary with leisure should be taxed more heavily.


19

Example 3: Uniform Commodity Tax Rate

If utility function is weakly separable: ( ( ), ) ( ( )) ( )U G x l U G x v l

and the consumption component is homothetic, i.e. ( )G is a homothetic function,

then,

1 1ji

i j

or i j

Therefore, all goods should be taxed at the same rate.


20

Ramsey Formula: Limitations

Ramsey solution: tax inelastic goods to minimize efficiency costs

But does not take into account redistributive motives

Necessities likely to be less elastic than luxuries

Therefore, optimal Ramsey tax system is likely regressive

Diamond (1975) extends Ramsey model to take redistributive motives into account:

Basic intuition: replace multiplier with average marginal utility for consumers

of that good


21

Application of Ramsey Approach to Taxation of Savings

Standard lifecycle model of consumption

max ( )t tt

u c subject to t tq c W

where (1 )t t tq p and 0 0

Consumption in each period isomorphic to consumption of different goods

Can apply standard Ramsey formula to calculate t

Capital income tax is a constant tax on interest rate:

1

1 (1 )t tqr


22

Optimal Capital Income Tax Rate For any 0 , implied tax t approaches as t :

111 (1 )

lim

tt

tt

tt

q rp r

Ramsey formula implies that optimal t cannot be for any good

Therefore optimal capital income tax rate converges to 0 in long run (Judd 1985,

Chamley 1986)

Best policy is for gov’t to tax capital until it accumulates sufficient assets to fund

public goods and never tax capital again


23

Zero Capital Taxation in Ramsey Models

Fairly robust result in pure Ramsey framework (Bernheim 2002)

But not robust to

Allowing for progressive income taxation (Golosov, Kocherlakota, Tsyvinski

2003)

Allowing for credit market imperfections (Aiyagari 1995, Farhi and Werning 2011)

Finitely-lived agents with general utility function and age-dependent tax system

(Erosa and Gervais 2003) or finitely-lived agents with finite bequest elasticities

(Piketty and Saez, 2013)

the economics of taxationyechan.weebly.com/.../9/5/1/8951344/8_static_ramsey_tax.pdf · 2018. 9....

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