*Advanced course « Economics of
Inequality » (Master APE, M2 year)*

Thomas Piketty

Année universitaire 2009-2010

Academic year 2009-2010

**Course Notes E : **

**Optimal redistributive taxation of
labor income**

__The
optimal labor income tax problem__

Mirrlees
(1971) : basic labor supply model used to analyse optimal labor income
taxes:

- each
agent i is characterized by an exogeneous wage rate w_{i}
(=productivity),

- labor
supply l_{i}

- pre-tax
labor income y_{i} = w_{i}l_{i}

- income tax t = t(y_{i})

(t(y_{i})
can be >0 or <0 ; if <0, then this is an income transfer, or
negative income tax)

- after-tax labor income z_{i} = y_{i}
– t(y_{i})

- agents
choose labor supply l_{i} by
maximizing U(z_{i},l_{i})

- social
welfare function W = ∫ W(U(z_{i},l_{i}))
f(y_{i})dy_{i} subject
to budgetary constraint: ∫ t(y_{i})
f(y_{i})dy_{i} > 0 (or >G, with G = exogenous public
spendings)

- if
individual productivities w_{i} were fully observable, then the
first-best efficient tax system would be t=t(w_{i}), i.e. would not
depend at all on labor supply behaviour, so that there would be no distorsion =
fully efficient redistribution

- however
if the tax system can only depend on income, i.e. t = t(y_{i}), e.g.
because of unobservable productivites w_{i} (adverse selection), then
we have an equity/efficiency trade-off

>>>
Mirrlees 1971 provides analytical solutions for the second-best efficient tax
system in presence of such an adverse selection pb

But
problems with the Mirrlees 1971 formula:

(i) very
complicated and unintuitive formulas, hard to apply empirically

(ii) only
robust conclusion: with finite number of productivity types w_{i} ,…, w_{n}, then zero
marginal rate on the top group = completely off-the-mark

>>>
Diamond (1998), Saez (2001), Piketty (1997): continuous distribution of types
(no upper bound, so that the artificial zero-top-rate result disappears),
first-order derivation of the optimal tax formulas, very intuitive and
easy-to-calibrate formulas

__First-order
derivation of linear optimal labor income tax formulas__

Linear tax schemes: t(y) = ty – t_{0}

I.e. t =
constant marginal tax rate

t_{0}
>0 = transfer to individuals with zero labor income

Define e
= labor supply elasticity

I.e. if
the net wage (1-t)w_{i} increases by 1%, labor supply l_{i}
(and therefore labor income y_{i}) increases by e%

E.g. if
U(z_{i},l_{i}) = z_{i} - V(l_{i}) (separable
utility, no income effect), with V(l)=l^{1+µ}/(1+µ), then e=1/µ

More
generally, whatever the random labor income generating process y_{i} =
y(productivity w_{i}, labor supply l_{i}, effort e_{i}
, luck l_{i}), one can define e = generalized labor supply elasticity =
if the net-of-tax rate (1-t) increases by 1%, labor income y increases by e%

Assume
we’re looking for the tax rate t* maximizing tax revenues R = ty

(revenue-maximizing
tax rate t* = top of the Laffer curve)

(revenue-maximizing
tax rate t* = social optimum if W = Rawlsian, i.e. W=0 for all U>U_{min},
i.e. social objective = maximizing transfer t_{0})

(= useful
reference point: by definition, socially optimal tax rates for non-rawlsian
welfare functions will be below revenue-maximizing tax levels)

First-order
condition: if the tax rate goes from t to t+dt, then tax revenues go from R to
R+dR, with:

dR = y dt + t dy

with dy/y = - e dt/(1-t)

I.e. dR = y dt – t ey dt/(1-t)

dR = 0 if
and only if t/(1-t) = 1/e

**I.e. t*
= 1/(1+e) **

I.e. pure
elasticity effect : if the elasticity e is higher, then the optimal tax t*
is lower

I.e. if
e=1 then t*=50%, if e=0,1 then t*=90%, etc.

**= the basic principle of optimal taxation
theory: other things equal, don’t tax what’s elastic**

(other example:
Ramsey formulas on optimal indirect taxation: tax more the commodities with a
less elastic demand, and conversely)

__First-order
derivation of non-linear optimal labor income tax formulas__

General
non-linear tax schedule t(y)

I.e.
marginal tax rates t’(y) can vary with y

Note f(y)
the density function for labor income, and F(y) the distribution function

Assume
one wants to increase the marginal tax rate from t’ to t’+dt’ over some income
bracket [y; y+dy]. Then tax revenues go from R to R+dR, with:

dR =
(1-F(y)) dt’ dy – f(y)dy t’ey dt’/(1-t’)

dR = 0 if
and only if **t’*/(1-t’*) = (1-F(y))/yf(y) 1/e**

I.e. two effects:

Elasticity
effect: higher elasticities e imply lower marginal tax rates t’*

Distribution
effect: higher (1-F)/yf ratios imply higher marginal rates t’*

Intuition
: (1-F)/yf = ratio between the mass of people above y (=mass of people paying
more tax) and the mass of people right at y (=mass of people hit by adverse
incentives effects)

For low
y, the ratio (1-F)/yf is declining: other things equal, marginal rates should
fall

But for
high y, the ratio (1-F)/yf is increasing: other things equal, marginal rates
should rise

>>>
for constant elasticity profiles, U-shaped pattern of marginal tax rates

__Asymptotic
optimal marginal rates for top incomes__

With a Pareto distribution 1-F(y) = (k/y)^{a} and f(y)=ak^{a}/y^{(1+a)},
then (1-F)/yf converges towards 1/a, i.e. t’* converges towards:

**t’* = 1/(1+ae)**

with e= elasticity, a = Pareto coefficient

Intuition: higher a (i.e. lower coefficient b=a/(a-1),
i.e. less fat upper tail) imply lower tax rates, and conversely

Exemple:
if e=0,5 and a=2, t’* = 50%