Problem Set 1
EC2450A
Fall 2016
Problem 1
An economy is populated by individuals with preferences over consumption and labor. They
have utility ui c, y where y is income, uc c, y > 0 and uy c, y < 0. Suppose the tax schedule
in place has a constant marginal tax rate τ above a fixed threshold y ∗ . The government wants to
choose τ to maximize the tax revenue raised from top earners.
(a) As we saw in class, the tax rate that maximizes revenues depends on a Pareto parameter a and
the elasticity of total income of the top earners who are in the top bracket, ε. Provide intuition
about why ε is a mix of substitution and income effects.
Increasing τ by dτ generates a negative substitution effect (less slope) leading to less work, and
a negative income effect leading to more work. Hence, ε is a mix of substitution and income effects.
(b) The individual solves the following utility maximization problem:
max ui c, y
c,y
subject to:
c = (1 − τ )y + I
Denote by yi 1 − τ, I the Marshallian income supply. The uncompensated elasticity of labor
supply with respect to 1−τ is εui = ∂yi /∂(1−τ ) (1−τ )/yi . We denote by ηi = (1−τ )∂yi /∂I
the income parameter.
Suppose a government advisor suggests to run an experiment where the top tax rate τ (above
y ∗ ) is raised by dτ . The advisor claims that the response dyi can be rewritten as a function of
εui and ηi . Is the advisor right? If yes, show how dyi depends on εui and ηi
Let zi (1, I) be the earnings supply function obtained from solving the individual utility maximization problem under a linear tax:
max ui c, y
c,y
subject to:
c = (1 − τ )y + I
We denote by εui the uncompensated elasticity of yi with respect to (1 − τ ) and by ηi =
(1 − τ )∂yi /∂I the income effect parameter. As a simple graph shows (see Saez Restud01 Section
3), the reform changes (1 − τ ) by −dτ and changes I by dI = y ∗ dτ . Note, that the virtual income
is defined as Iy ∗ τ (in Saez 2001 it is written as I = ŷτ ), and as we assume that y ∗ is fixed it follows
immediately that dI = y ∗ dτ . Hence, we have:
dyi = −dτ
∂yi
∂yi
dτ
1 − τ ∂yi
dτ ∗
∂yi
dτ i u
dτ ∗
+y ∗ dτ
=−
yi
+
y (1−τ )
=−
y εi +
y ηi
∂(1 − τ )
∂I
1−τ
yi ∂(1 − τ ) 1 − τ
∂I
1−τ
1−τ
(c) Using the expression derived in point b) write ε as a function of the Pareto parameter a =
y m /(y m − y ∗ ) and a weighted average of the uncompensated elasticities and income effect
parameters. Why are uncompensated elasticities weighted by incomes yi , while the ηi s are not?
Recall that the elasticity ε is defined as
1−τ
ε= P
yi ≥y ∗
P
yi ≥y ∗
dyi
yi d(1 − τ )
Hence we have
P
P
u
X 1
1
z ∗ yi ≥y∗ ηi
yi ≥y ∗ yi εi
u
∗
yi ε −
y ηi = P
− m
1−τ i
1−τ
z
N
yi ≥y ∗ yi
yi ≥y ∗ yi
∗
1−τ
ε= P
yi ≥y
with N number of top bracket taxpayers. Hence ε = ε̂u − a−1
a η̂ the income weighted average of
εui and η̂ the straight average of ηi among top bracket taxpayers.
The uncompensated elasticity is income weighted because those with higher income should count
more in the response. In contrast, the income effect parameter is not an elasticity and hence should
not be income weighted.
(d) Now suppose the utility is logarithmic in consumption and exponential in income. It takes the
following form:
1
ui (c, y) = log c − φi y 1+ ε
where φi can vary across individuals and captures heterogeneity in the disutility from labor.
Derive the uncompensated elasticity, income, and compensated elasticity parameters (i.e., εui ,
ηi , εci ) by solving the utility maximization problem of the individual under the linear tax and
the same budget constraint as above.
2
1
Suppose each individual utility takes the form ui (c, y) = log c−φi y 1+ ε where phii is a parameter
(that can vary across individuals). The individual solves
1
max log((1 − τ )y + I) − φi y 1+ ε
c,y
The individual FOC in y is:
1−τ
1 1
= φi (1 + )y ε
(1 − τ )y + I
ε
or
log(1 − τ ) − log((1 − τ )y + I)) =
1
1
log(y) + log(φi (1 + ))
ε
ε
We now derive elasticities at the limit where y → ∞ since we know from the FOC that when
φ → 0 the agent supplies and infinite amount of labor. This defines implicitly y i (1 − τ, I) with the
following comparative statics. A small change dI leads to dzi such that:
dyi
h1
i
1−τ
dI
+
=−
εy (1 − τ )y + I
(1 − τ )y + I
Hence
ηi = (1 − τ )
dyi
=−
dI
1+
1
y+I/(1−τ )
εy
→y→∞ −
ε
ε+1
A small change d(1 − τ ) leads to dyi such that:
dyi
h1
i d(1 − τ )
1−τ
yd(1 − τ )
+
=
−
εy (1 − τ )y + I
1−τ
(1 − τ )y + I
Hence
(1 − τ )
i
dzi h 1
1
y
+
=1−
d(1 − τ ) εy (1 − τ )y + I
(1 − τ )y + I
εui =
1 − (1−τz)y+I
1 − τ dzi
=−1
→y→∞ 0
y
y d(1 − τ )
ε + (1−τ )y+I
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(e) Study what happens to ηi and εui when φi becomes small (find their limits). Using the relation
found previously, write the optimal top tax rate formula as a function of ε and the Pareto
parameter a when φi is small.
The parameter ε is the Frisch elasticity of labor supply for this class of utility functions. Suppose
we calibrate the parameters a and ε such that a = 1.5 and ε = 1. What is the optimal τ ?
What is the optimal τ when ε is very large? Discuss why the optimal tax rate is high even
with a large Frisch elasticity.
Use c) to obtain an optimal top tax rate formula as a function of ε and the Pareto parameter
a in that case. We have ε = ε̂u −
τ=
a−1
a η̂
'
a−1 ε
a ε+1
and hence:
ε+1
1
1
=
ε =
1 + aε
1 + (a − 1) 1+ε
1 + aε
If a = 1.5 and ε = 1, we have τ = 1/(1 + 0.5 × 0.5) = 80%.
With ε = ∞, we get τ = 1/(1 + 0.5) = 66.6%. This utility specification has zero uncompensated
elasticity and hence large income effects when the compensated elasticity is large. As a result, the
tax rate is substantial, even with a very large Frisch elasticity.
Problem 2
Suppose that utility is quasi-linear and takes the form: u c, l = c −
l1+
1+
with > 0. Each
individual earns income y = wl and consumes c = y − T y). The wage rate w can be interpreted
as a measure of skills and is distributed with density f w > 0 over 0, ∞ . The total population
R∞
is normalized to one so that 0 f w dw = 1
(a) Suppose the tax schedule is linear with a flat tax rate τ . The tax is hence T y = −S + τ y
where S > 0 is the transfer that the individual receives when labor supply is zero (T 0 = −S).
Find the optimal labor supply choice as a function of the parameters S and w 1 − τ . Also,
derive the uncompensated and compensated elasticities of labor supply as a function of and
find the income effect parameter.
h
i1/
maxl wl(1 − τ ) + S − l1+ /(1 + ) =⇒ w(1 − τ ) = l =⇒ l = w(1 − τ )
so εu = εc = 1/ and
η = 0. Let us denote by ε = 1/ the common compensated and uncompensated elasticity.
(b) Assume that taxes are entirely rebated to the individuals in the economy. We have that S = τ Y ,
where Y is average earnings in the economy. Find the optimal tax rate τ in the case where
the government only cares about the worst-off individual (i.e. the government is Rawlsian)
and in the case where the government maximizes the sum of utilities (i.e. the government is
utilitarian). Always explain the intuition behind your results.
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max wl(1 − τ )
1
Z
τ
1
w1+ f w dw =⇒ τ ∗ =
1
=
+1
1+ε
Worst off individual has w = 0 and hence l = 0 and utility u = S = τ Y so Rawlsian optimal rate
maximizes tax revenue (to maximize S) and is set at τ ∗ =
+1
from (b). Given that all utilities are
linear, there is no concern for redistribution and hence the optimal utilitarian tax rate is zero.
(c) Do points (a)-(b) again using utility function u c, l = log c −l. If exact analytical expressions
are not possible to derive, just provide implicit formulas with economic explanation. Is this
utility function more or less realistic than the one used in questions (a)-(b)?
Go back to utility function u c, l = c −
such that:
(
T y =
l1+
1+ .
We now study an economy with two tax brakets
−S + τ1 y
if y ≤ ŷ
−S + τ1 ŷ + τ2 y − ŷ
if y > ŷ
−S is the transfer to non-working individuals.
With utility log c − l, we have maxl log wl(1 − τ ) + S − l =⇒ w(1 − τ )/ w(1 − τ )l + S = 1
so that l = 1 − S/ w(1 − τ ) . Note that l = 0 when w(1 − τ ) ≤ S. Income effect η = −1,
εu = S/ S − w(1 − τ ) , εc = εu − η = w(1 − τ )/ S − w(1 − τ ) . τ ∗ = 1/(1 + ε̂u ) where ε̂u is the
(income weighted) average uncompensated elasticity. ε̂u does not have a simple analytic expression.
Worst-off individual has l = 0 and utility u = log(S) so government wants to maximize S which is
done by maximizing tax revenue with τ ∗ = 1/(1 + ε̂u ). In utilitarian case, the optimal τ is given
by τ = (1 − ĝ)/(1 − ĝ + ε) as seen in class notes with ε a mix of uncompensated and income effects
(see Piketty-Saez handbook chapter for details). There is no simple analytic expression.
(d) Plot the budget constraint on a graph with axes l, c .
(e) Suppose that τ1 < τ2 . Find the optimal labor supply and earnings for an individual with wage
w. Consider the three cases where the individual is in the bottom bracket, the top bracket, or
exactly at ŷ.
The budget has a kink generating bunching at ŷ.
ε
Case 1 (first bracket): w ≤ w: l = w(1 − τ1 ) with w s.t. w1+ε (1 − τ1 )ε = ŷ
Case 2 (bunching at ŷ): w ≤ w ≤ ŵ: l = ŷ/w
ε
Case 3 (top bracket): w ≤ w: l = w(1 − τ2 ) with w s.t. w1+ε (1 − τ2 )ε = ŷ
Suppose that there are 3 types of individuals: disabled individuals unable to work w0 = 0, low
skilled individuals with wage rate w1 , and skilled individuals with wage rate w2 . We assume that
w1 < w2 . The fractions of disabled, low skilled, and high skilled in the population are respectively
λ0 , λ1 and λ2 such that λ0 + λ1 + λ2 = 1. Further assume that low skilled workers are always in
the bottom bracket and that high skilled workers are always in the top bracket.
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(f) Find the tax rate τ2∗ that maximizes taxes collected from the high skilled, assuming that S, τ1 ,
and ŷ are given. Express it as a function of and ŷ.
Amount of bunching is proportional to ε (see Saez AEJ:EP10 for details):
w1+ε 1 − τ1 ε
=
w1+ε
1 − τ2
n
h
io
T = λ1 τ1 (1 − τ1 )ε w11+ε + λ2 τ1 ŷ + τ2 (1 − τ2 )ε w21+ε − ŷ
Take the FOC wrt to τ2 to get:
τ2∗ /(1 − τ2∗ ) = (1/ε) y2 − y /y2
(g) Compute the tax rate τ1 that maximizes total taxes collected taking S and ŷ as given and setting
τ2 = τ2∗ (the optimal tax rate you found in the previous question). Explain why (intuitively)
τ2∗ < τ ∗ < τ1∗ , where τ ∗ is the one computed in question (b).
Take the FOC wrt to τ1 to get:
τ1∗ /(1 − τ1∗ ) = (1/ε) 1 + λ2 y/(λ1 y1 )
In order to explain τ2∗ < τ ∗ < τ1∗ :
1. Increasing the flat tax rate τ creates a mechanical increase in revenue proportional to average
earnings and creates a negative behavioral response proportional to average earnings as well.
2. Increasing the tax rate τ2 in the top bracket creates a mechanical increase in revenue proportional to (y2 − y) but creates a negative behavioral response proportional to y2 .
3. Increasing the tax rate τ1 in the bottom bracket creates a mechanical increase in revenue
proportional to y1 and creates a negative behavioral response proportional to y1 . However,
the tax rate increase also raises more tax from high skilled worker with no negative behavioral
response (inframarginal tax).
(h) (Bonus question:) Finally, the government introduces a third tax bracket with rate τ3 above
ȳ. Suppose τ3 > τ2 , ȳ > ŷ and that there is a continuous population with utility defined as
in (a). You have access to 5 years of income data before the reform and 5 years of data after
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the reform. Suggest a time series graph that you would draw to visually test whether creating
the 3rd tax bracket had an impact on reported incomes? How could you use the graph to
estimate the elasticity of earnings with respect to 1 − τ using this reform. Be precise about the
assumptions needed for the estimate to be unbiased. Do you have ideas about how to test the
robustness of your results with some alternative method of estimation?
Select the top x% share where x% is the fractile in the 3rd bracket affected by the reform. Plot
the share of income going to the top x% in all years. If the graph looks like a step function, thats
convincing evidence of a tax effect. Key assumption is no change in income concentration from
before to after the reform [absent any tax change]. This can be checked (imperfectly) by seeing
whether the share of income going to a control group not affected by the reform [but fairly close
in income level to the affected group] is stable. An entirely different method would be a bunching
method in the years after the method.
Problem 3
In the standard Mirrlees model, the production function is implicitly additively linear: aggregate
output is simply the sum of individual outputs, i.e., the substitution between different agents’
outputs is infinite.
Consider instead the case where outputs are not perfectly substitutable.
We focus on a two-type model. There are n1 individuals of type 1 (low productivity) and n2
individuals of type 2 (high productivity). Let L1 be the labor effort of individuals of type 1 and
L2 the labor effort of individuals of type 2. The production function has constant returns to scale:
Y = F (n1 L1 , n2 L2 )
Individuals have quasilinear preferences of the form
ui = ci − v(Li )
where ci is their consumption and Li is their labor effort. If type i is paid a wage wi and pays a
tax Ti , his utility is:
ui = wi Li − Ti − v(Li )
The government’s objective is to maximize social welfare equal to:
SW F = n1 u1 + µn2 u2
with 0 < µ < 1.
7
1) Show that aggregate output can be rewritten as:
Y = n1 L1 f (l)
for l to be defined and where f is i) increasing and ii) concave.
Exploiting the CRS (homogeneity of degree 1) feature of the production function we can divide
F by n1 L1 to get:
Q
n2 L2 = F 1,
= f (l)
n1 L1
n1 L1
Where l = (n2 L2 )/(n1 L1 ). f (·) is increasing and concave.
2) What is the social marginal welfare weight on agents of type 2? What is the social marginal
welfare weight on agents of type 1? Are their absolute levels meaningful? Which type of agent is
implicitly valued more by the government?
The government social welfare function can be written as:
W = n1 U1 + µn2 U2
The marginal value of public funds λ is n1 + µn2 . Social marginal welfare weights are g1 = 1/λ
and g2 = µ/λ. Given the normalization by λ they are expressed in government money. Since µ < 1
the government is implicitly weigthing individual 1 more.
3) The government chooses a tax schedule T (y) specifying for each income level the tax to be
paid. Explain why the government can restrict itself to set a menu of contracts (T1 , Y1 ) and (T2 , Y2 )
that specifies for each type an output level and a tax level.
The government may rely on the revelation principle and use a direct revealing mechanism,
that is, two pairs (T1 , Y1 ) and (T2 , Y2 ) such that type 1 prefers the first pair and type 2 prefers the
second pair.This is equivalent to setting an optimal tax for the two levels of Y .
4) What are the two incentive compatibility constraints facing the government when setting
this menu of contracts? What is the resource constraint (government budget constraint) if the
government has an exogenous revenue requirement R?
Incentive compatibility constraints are:
Y1 − T1 − v
Y1 Y2 ≥ Y2 − T2 − v
w1
w1
Y2 − T2 − v
Y2 Y1 ≥ Y1 − T1 − v
w2
w2
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The resource constraint is:
n1 T1 + n2 T2 ≥ R
5) Among the two incentives constraints, which one will be binding if µ is very low? Explain
the intuition.
The incentive constraint binds for type 2. A government with redistribute motives will try to
keep the high type at the lowest possible utility, making her indifferent between the two contracts.
If instead the high type had some utility gain from choosing her designed contract, the government
could always tax her more and transfer money to the low type generating an increase in the social
welfare (µ < 1).
6) Suppose the wage is determined by the marginal product of labor given the aggregate production function. Write out an expression for w1 and w2 .
Wages are:
w1 = f l − lf 0 l
w2 = f 0 l
7) Rather than solving the constrained program with respect to Ti and Yi , replace Yi by the
auxiliary variable Li = Yi /wi (labor supplies of each type at the optimum). Eliminate T1 and T2
by using i) the government’s budget constraint and ii) the incentive constraint of type 2. Give an
expression for T1 and T2 as a function of the wages, labor supplies, and the government revenue
requirement.
We can derive the following two conditions:
n
L1 w1 o
n1 + n2 T1 = R − n2 w2 L2 − w1 L1 − v L2 − v
w2
n
L1 w1 o
n1 + n2 T2 = R + n1 w2 L2 − w1 L1 − v L2 − v
w2
8) Use the expressions found for Ti to rewrite the objective of the government.
The Government’s objective can be rewritten as:
n1 w1 L1 − v L1
h
n1 n2 1 − µ n
L1 w1 io
+ n2 µ w2 L2 − v L2 +
w2 L2 − w1 L1 − v L2 − v
n1 + n2
w2
9
9) Suppose just for this question that, as in the Mirrlees model, wages are exogenous. Take the
first-order conditions with respect to L1 and L2 . Show that T10 > 0 and T20 = 0. Interpret this in
light of the binding incentive constraint.
First order conditions for the problem are:
0
v L1
n
n2 1 − µ h
1 0 L1 w1 io
= w1 1 −
v
1−
n1 + n2
w2
w2
v 0 L2 = w2
As you can see, the labor supply choice of type 2 is not distorted by the government: there is
no wedge on the first order condition. Therefore, T20 = 0. On the other hand, there is a wedge on
the choice of type 1. In particular, the monotonicity of allocations implies Y1 < Y2 , which means
L1 w1 /w2 < L2 . We therefore have:
v 0 L1 w1 /w2 < v 0 L2 = w2
by the convexity of v ·
and the FOC for L2 . It folows that the wedge on the labor supply
of type 1 is positive and therefore T10 > 0. The optimal marginal tax rate is zero for type 2 as it
is standard in this kind of models. The idea is that the government does not want to affect the
incentive to work of the high type to prevent her from imitating the low type.
10) Now suppose that wages are as determined above through the aggregate production function.
Give the derivatives of w1 and w2 with respect to L1 and L2 (note that both wages depend on the
labor supplies of both types). When employment of the more productive type increases, what
happens to w1 and w2 ?
Wages respond to labor supply according to the following:
∂w1
l2 00 =
f l
∂L1
L1
∂w1
l2
= − f 00 l
∂L2
L2
l
∂w2
= − f 00 l
∂L1
L1
∂w2
l 00 =
f l
∂L2
L2
10
Since f 00 l ≤ 0, these derivatives imply that when the employment of the more productive
increases, w2 decreases, while the wage rate of the less productive increases. This implies that
when agent 2 works more she has a positive ”externality” on type 1.
11) Differentiate the government’s objective function (into which you substituted the incentive
and resource constraints as in 8)) with respect to L2 . Can you show that T20 , the marginal tax rate
on type 2 has to be negative (if you need to make assumptions on some functions, do so)? What
does this mean? Can you provide the intuition for why this is the case?
As in point (9) we have a direct derivative of the SWF wrt L2 which is given by A w2 − v 0 L2 ,
where A is a positive coefficient. We then need to take into account the indirect effect of changes
in L2 on wages:
i
h ∂w2
∂w1
∂w2
n1 n2
∂w1
0 L1 w1 ∂ w1 /w2
n1
L1 + µn2
L2 +
1−µ ×
L2 −
L1 − L1 v
∂L2
∂L2
n1 + n2
∂L2
∂L2
w2
∂L2
That we can write as:
w
∂ 1
n1 + n2 µ ∂w1
∂w2
n1 n2
w2
0 L1 w1
n1 L1 +
n2 L2 +
1 − µ L1 v
n1 + n2 ∂L2
∂L2
n1 + n2
w2
∂L2
Notice that:
n1 L1
∂w1
∂w2
n2 L2 00 n1 L1
n1 L1 +
n2 L2 = −l2 f 00 l
+ lf 00 l n2 = −l
f l
+ lf 00 l n2 = 0
∂L2
∂L2
L2
n1 L1
L2
We know from the expressions derived in point (10) that ∂ w1 /w2 /∂L2 > 0. Given that µ < 1
we know that the indirect term is positive. Therefore the FOC for L2 reads:
0
v L2
L w ∂ w /w
1 n1 n2
1
2
1 1
= w2 +
1 − µ L1 v 0
A n1 + n2
w2
∂L2
Therefore, being the second term on the rhs positive, we have that T20 < 0. This is an application
of the so called ”additivity principle” of taxation. Whenever we have an income taxation problem
with some externality, we can always solve the externality and sum the tax rate that we find to the
tax rate we would derive being the externality absent. In this case, the tax rate with no externality
would be T20 = 0, while the ”Pigouvian” correction would imply T20 < 0, the result is a negatve
marginal tax rate on top incomes.
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