Optimal fiscal policy with heterogeneous agents
and aggregate shocksú
François Le Grand
Xavier Ragot†
Abstract
We show that allocations in incomplete insurance market economies with capital and
aggregate shocks can be represented as the solution of the program of a constrained quasiplanner. This representation generates a finite-dimensional state space, which allows for
solving Ramsey programs. We apply this framework to derive optimal fiscal policy and public
debt dynamics after persistent technology shocks, when the planner can levy distorting taxes
on capital and labor and positive lump-sum transfers. Average capital taxation is proved
to be a simple function of the tightness of credit constraints. In a quantitative exercise, it
is shown that private savings increase too much after a technology shock and are absorbed
by an increase in public debt and a decrease in capital taxes. Simulations of these optimal
solutions can be obtained by simple perturbation methods.
Keywords: Incomplete markets, optimal policy, public debt.
JEL codes: E21, E44, D91, D31.
ú
We thank Edouard Challe, Wouter den Haan, Christos Koulovatianos, Felix Kubler, Stephanie Schmitt-Grohe,
Martin Uribe, Gianluca Violante, Pierre-Olivier Weill, and the seminar participants at the conference on Heterogeneous Agents at the University of Zurich, University of Luxembourg, New York University, National Bank of
Belgium, and at T2M and the EEA conference. We thank Sumudu Kankamange et Thomas Weitzenblum for
excellent research assistance. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement Integrated Macro-Financial
Modeling for Robust Policy Design (MACFINROBODS, grant no. 612796).
†
LeGrand: emlyon business school and ETH Zurich; [email protected]. Ragot: Sciences Po, OFCE and
CNRS; [email protected].
1
1
Introduction
Heterogeneous agent models provide a framework to think about many relevant aspects of inequalities and individual risk in general equilibrium. In these models, infinite-lived agents face
incomplete insurance markets and borrowing limits that prevent them from perfectly hedging
their idiosyncratic risk, in the tradition of the Bewley-Huggett-Aiyagari literature (Bewley, 1983;
Imrohoro lu, 1989; Huggett, 1993, and Aiyagari, 1994). These frameworks are becoming increasingly popular and are now widely used, since they fill a gap between micro- and macroeconomics
and enable the inclusion of aggregate shocks and a number of additional frictions on both the
goods and labor markets. However, considering normative analysis, little is known about optimal policies in these environments with capital, due to the difficulties generated by the large
and time-varying heterogeneity across agents. This is unfortunate, since a vast literature, reviewed below, suggests that the interaction between capital accumulation, income and wealth
inequalities have first-order implications for the optimal design of time-varying fiscal policies.
This paper presents a methodological contribution that offers a general and tractable representation of incomplete insurance-market economies. This representation allows us to easily
solve for Ramsey program in incomplete insurance-market economies with both capital and aggregate shocks. We apply our framework to provide a theoretical and quantitative analysis of
optimal fiscal policy. We derive new results about the optimal dynamics of public debt, distorting capital and labor taxes and transfers, considering rich trade-offs involving redistribution,
insurance, and incentives.
In these economies, heterogeneity is known to increase as time goes by, because agents differ
according to the whole history of the realization of their idiosyncratic risk. Huggett (1993),
using the results of Hopenhayn and Prescott (1992), and Aiyagari (1994) have shown that these
economies without aggregate risk have a recursive structure when the distribution of wealth is
introduced as a state variable. Unfortunately, the distribution of wealth has an infinite support,
which is the root of many analytical difficulties. Furthermore, in the presence of aggregate,
shocks as introduced in the seminal work of Krusell and Smith (1998), the nature of the state
vector may be a difficult question (see Miao 2006).
Our methodological contribution is to show that incomplete-market economies can be represented as the limit of economies with finite support. More precisely, we construct an environment
2
where the heterogeneity across agents depends only on a finite but arbitrarily large number, denoted N , of consecutive past realizations of the idiosyncratic risk. As a theoretical outcome,
agents having the same idiosyncratic risk history for the previous N periods choose the same
consumption and wealth levels. The interest of this truncated representation of incomplete
insurance-market economies lies in four properties. First, the allocation can be represented
as the solution of a family-head program, which ensures the existence of the equilibrium with
aggregate shocks. Second, the standard incomplete-market model is indeed the limit of our
truncated representation for large N , under general conditions. Third and more importantly, as
our representation has a finite state-space, we can use the tools derived in Marcet and Marimon
(2011) to study Ramsey programs. These tools rely on the extensive use of Lagrange coefficients. Finally, the finite state-space simplifies to a large extent the simulation of the model, as
standard perturbation methods can be used.
This limited-heterogeneity environment enables us to solve for the Ramsey policy in an
incomplete-market economy with technological shock. The planner has four instruments: linear
taxes on capital and on labor, public debt, and transfers. These instruments are standard in
the Ramsey literature. We derive three sets of results.
First, we show that the average optimal long-run capital tax is directly related to the equilibrium severity of credit constraints. More formally, the tax on capital is proportional to the sum
of the Lagrange multipliers on agents’ credit constraints. As a direct consequence, the capital
tax is always non-negative (as already found in Aiyagari 1995) and is positive if and only if credit
constraints bind for some agents in equilibrium. This result contributes to clarify the deviations
from the Chamley (1986) and Judd (1985a) no-capital tax result found in the literature. In
addition, pre-tax marginal return on capital is uniquely pinned down by the planner discount
factor, as originally found by Aiyagari (1995). Finally, labor taxes are shown to directly depend
on the elasticity of labor supply.
Second, the dynamics of the fiscal policy is mainly driven by the difference in two valuations
of liquidity. The first one is the planner valuation of government liquidity, which is measured
by the Lagrange multiplier of the government budget constraint. The second one is the social
valuation of the liquidity of private agents. Since the latter internalizes agents’ saving incentives,
this valuation differs from their own private valuation, which is simply equal to their marginal
utility. The difference in the social valuations of government and private liquidities is denoted
3
the liquidity valuation gap, and is key to understand optimal fiscal policy dynamics. A simple
example shows that capital taxes are used to transfer resources to low-income agents and are
likely to be more volatile than labor taxes, which distort labor supply.
Third, we simulate the model using standard values for preferences, household income process, and public spending. Our simulated equilibrium features, at the steady-state, an optimal
public debt-to-GDP ratio of 148%, a labor tax rate of 36%, and a capital tax rate of 11%, while
3% of households are credit-constrained. The equilibrium dynamics feature capital taxes that
are much more volatile than labor taxes, which confirms earlier results in complete insurancemarket economies (Aiyagari, Marcet, Sargent, and Seppälä 2002; or Farhi, 2010, for a recent
discussion). A final property of the equilibrium dynamics is that public debt is pro-cyclical,
while capital taxes are counter-cyclical. The dynamic behavior of public debt comes from the
fact that after a positive technology shock, household savings are too high compared to the
constrained efficient allocation. The planner absorbs these excess savings by a higher public
debt, and redistributes wealth by cutting capital taxes.
Related literature.
This paper first contributes to the literature on the theory of incom-
plete insurance-market economies with aggregate shocks. Some environments already provide a
tractable framework. This is the case of no-trade equilibria with permanent idiosyncratic shocks
(Constantinides and Duffie, 1996), used for instance in Heathcote, Storesletten, and Violante
(2016). More recently, Krusell, Mukoyama and Smith (2011) study a class of no-trade equilibria
in an economy without capital and with a tight-enough credit constraint. This environment
is used by Ravn and Sterk (2015) in a model with endogenous idiosyncratic risk. Departing
from no-trade, a class of “small trade” equilibria, featuring “reduced heterogeneity” with a finite number of wealth levels, has been studied. This reduced heterogeneity relies on a specific
assumption, such as quasi-linearity in utility functions in Challe and Ragot (2016) or Le Grand
and Ragot, (2016), or partial insurance mechanisms in Challe, Matheron, Ragot and RubioRamirez (2015). This framework allows one to study optimal policies, as in Ragot (2016) with
limited financial-market participation, or in Bilbiie and Ragot (2017) with nominal frictions.
The current paper extends these previous works and provides a general theory of truncated
representations of Bewley economies. In addition, it derives new tools to study optimal policies,
based on the dynamic structure of Lagrange coefficients.
4
Second, our paper contributes to the literature on distortions and optimal policies in incomplete insurance-market models. Many contributions identify a number of relevant trade-offs,
but the general case with capital accumulation and aggregate shocks has not been studied yet,
to the best of our knowledge. In economies without aggregate shocks, Aiyagari (1995) shows
that the capital tax is non-negative. Aiyagari and McGrattan (1998) compute the optimal
steady-state level of debt. Dávila, Hong, Krusell, and Ríos-Rull (2012) identify constrained
inefficiencies in incomplete-insurance market models with production, showing that the capital
stock can be too low. Açikgöz (2015) solves for the Ramsey program to obtain the steady-state
optimal level of public debt. Dyrda and Pedroni (2016) solve numerically for the optimal policies
along the transition between an initial and a final steady-state. Gottardi, Kajii, and Nakajima
(2014) solve for the Ramsey allocation in an incomplete-market model with human capital accumulation. Nuño and Moll (2017) use a continuous-time approach and mean-field games to
characterize differences in inequalities in both constrained-efficient and constrained-inefficient
economies. Bhandari, Evans, Golosov, and Sargent (2013) characterize optimal taxation, in an
environment without capital, but with agents having heterogeneous skills and facing aggregate
risks and incomplete insurance markets. In a similar framework, Bhandari, Evans, Golosov, and
Sargent (2016b) provide a set of indeterminacy results for the public debt, which extends the
result of Barro (1974) to incomplete market economies.1 Shin (2006) studies a two-agent economy to derive additional results. Some papers have introduced incomplete insurance markets in
overlapping generation models to quantitatively investigate optimal fiscal policies (Imrohoro lu
1998, and Conesa, Kitao, and Krueger 2009). Finally, many papers have considered incompleteinsurance markets to analyze the effect of fiscal policies. The positive effects are studied in
Heathcote (2005), who considers aggregate shocks. A recent contribution is Kaplan and Violante (2014), who introduce transaction costs for some assets. Our truncated approach allows
us to derive a new set of results.
Third, this paper is also related to the vast literature on optimal fiscal policy with aggregate
shocks. Seminal contributions consider a complete-market economy with a representative agent
(Barro 1979; Lucas and Stokey 1983; surveyed in Chari and Kehoe 1999). More recent contributions consider incomplete markets for the aggregate risk, introducing non state-contingent
1
There is a large literature on debt irrelevance, for instance explained by generational accounting: Auerbach,
Gokhale, and Kotlikoff (1991, 1994) for seminal contributions and Bassetto and Kocherlakota (2004) for an
extension to discretionary taxes.
5
public debt (Aiyagari, Marcet, Sargent, and Seppälä 2002; Farhi 2010; Bhandari, Evans, Golosov,
and Sargent 2016a). Several papers have additionally introduced ex ante heterogeneity among
agents. Bassetto (2014) considers tax payers and “rentiers”, who do not pay any tax. Azzimonti,
de Francisco, and Krusell (2008a, 2008b), Azzimonti and Yared (2017) and Correia (2010) consider agents ex ante differing in initial wealth or earnings. Greulichy, Laczo, and Marcet (2016)
characterize Pareto-optimal fiscal policy in a model where agents differ according to their productivity and wealth. The New Dynamic Public Finance literature focuses on optimal fiscal policy
in environments with heterogeneous and private information. This approach was pioneered in
Mirrlees (1971) and applied to optimal taxation problems by Golosov and Tsyvinski (2007) and
Werning (2007). Here, we use a Ramsey approach, limiting the number of instruments (see
Farhi and Werning 2013, and Golosov, Tsyvinski, and Werquin 2016, for recent contributions
and discussions of the difference in methods).
Fourth, this paper is related to the computational literature studying incomplete insurance
markets with perturbation methods. Reiter (2009) uses a perturbation method to solve for
aggregate dynamics around a steady-state equilibrium of a Bewley model, with idiosyncratic
shock but no aggregate shock. Mertens and Judd (2012) use perturbation methods around a
steady-state with neither aggregate nor idiosyncratic shock. They use a penalty function to pin
down equilibrium saving decisions. Other papers using perturbation methods but restricting
exogenously the state-space are Preston and Roca (2007) and Kim, Kollmann, and Kim (2010).
In our model, we use perturbation methods around a steady-state with idiosyncratic shocks
and without aggregate shocks, in an economy which delivers a finite-dimensional state-space
as a theoretical outcome. This last property is key to be able to derive optimal policies using
Lagrangian techniques.
The rest of the paper is organized as follows. In Section 2, we present the environment. We
describe the central planner problem and derive the associated allocation in Section 3. We then
show in Section 4 how the planner allocation can be decentralized, and we provide convergence
properties. We then take advantage in Section 5 of the finite equilibrium structure to solve
the Ramsey program. We discuss in Section 6 the implications of the Ramsey program for
fiscal policy. In Section 7, we provide a numerical application illustrating our findings. Finally,
conclusions are given in Section 8.
6
2
The environment
Time is discrete, indexed by t Ø 0. The economy is populated by a continuum of agents of size
1, distributed on a segment J following a non-atomic measure ¸: ¸(J) = 1.2
2.1
Risk
Aggregate risk.
The aggregate risk is represented by a probability space (S Π, F, P). In any
period t, the aggregate state, denoted st , takes values in the state space S µ R+ and follows
a first-order Markov process. The history of aggregate shocks up to time t is summarized by
the sequence of aggregates shocks from date 0 to date t and denoted st = {s0 , . . . ., st } œ S t+1 .
Finally, the period-0 probability density function of any history st is denoted mt (st ).
Idiosyncratic risk.
At the beginning of each period, agents face an uninsurable idiosyncratic
labor productivity shock et that can take E values in the set E = {1, . . . , E} œ RE
+ . Households
in state e œ E have a labor productivity ◊e > 0, which is assumed to be increasing in e, as
a normalization. The productivity shock et follows a discrete first-order Markov process with
transition matrix M (st ) œ [0, 1]E◊E . The probability Me,eÕ (st ) is the probability for an agent to
switch from state e at date t to state eÕ at date t + 1, when the aggregate state is st in period t.
The history of idiosyncratic shocks up to date t is denoted et = {e0 , . . . , et } œ E t+1 .
Remark 1 (Notations) For the sake of clarity, for any random variable Xt : S t æ R, we
will denote Xt , instead of Xt (st ), its realization in state st , and for any random variable Yt :
S t ◊ E t æ R, we will denote Yt,et its realization in state (st , et ).
2.2
Preferences
In each period, the economy has two goods: a consumption-capital good and labor. Households
rank consumption c and labor l according to a smooth periodic utility function U (c, l), satisfying
standard regularity properties. As standard in this class of models, we consider a Greenwood2
We assume that the law of large numbers holds. See Green (1994) for a proper construction of J and ¸. See
also Feldman and Gilles (1985), Judd (1985b), and Uhlig (1996) for other solutions.
7
Hercowitz-Huffman (GHH) utility function, exhibiting no wealth effect for the labor supply:3
A
U (c, l) = u c ≠ ‰
≠1
B
l1+1/Ï
,
1 + 1/Ï
(1)
where Ï > 0 is the Frisch elasticity of labor supply, ‰ > 0 scales labor disutility, and u : R+ æ R
is twice continuously derivable, increasing, and concave, with uÕ (0) = Œ. Each household ranks
consumption and labor streams, denoted respectively as (ct )tØ0 and (lt )tØ0 , according to the
intertemporal criterion
2.3
qŒ
t=0 —
t U (c , l ),
t t
where — œ (0, 1) is the discount factor.
Production and assets
In any period t, a production technology with constant returns to scale (CRS) transforms capital
Kt≠1 and labor Lt into F (Kt≠1 , Lt , st ) units of output. Capital must be installed one period
before production, and the state of the world possibly affects productivity through a technology
shock. This formulation allows for capital depreciation, which is subsumed by the production
function F as in Farhi (2010) for instance. Labor Lt is measured in efficient units, and is equal
´
to the sum of the individual labor efforts expressed in efficient units: Lt = iœJ ◊ei li,t ¸(di). The
t
production function is smooth in K and L and satisfies the standard Inada conditions. The
good is produced by a unique profit-maximizing representative firm. We denote as w̃t the real
before-tax wage rate in period t and as r̃t the real before-tax rental rate of capital in period t.
Profit maximization yields in each period t Ø 1:
r̃t = FK (Kt≠1 , Lt , st ) and w̃t = FL (Kt≠1 , Lt , st ).
(2)
Finally, agents save using two assets, which are claims on the capital stock and public debt.
In addition, agents cannot borrow more than an exogenous borrowing limit ≠ā Æ 0.
2.4
Government, fiscal tools and resource constraints
In each period t, the government has to finance an exogenous public good expenditure G and
a positive lump-sum transfer Tt > 0 paid to all agents. The government can levy distorting
3
All our results can be derived with a general utility function U (c, l). A GHH utility function simplifies slightly
the algebra, especially when deriving the Ramsey program in Section 5. Admittedly, and as shown by Marcet,
Obiols-Homs, and Weil (2007), considering alternative utility function would affect the optimal tax schedule, as
aggregate labor supply would depend on the wealth distribution.
8
taxes on capital income ·tK or on labor income ·tL or issue an amount Bt of a riskless one-period
public bond.4 As in Heathcote (2005), we assume that the public debt pays the economy-wide
interest rate r̃t for any aggregate history st œ S t . The same tax rate ·tK applies to public bonds
and capital shares. In consequence, both assets are perfectly substitute for households. Positive
lump-sum transfers Tt > 0 are allowed because Heathcote, Storesletten, and Violante (2016)
show that they are needed to properly approximate the current US fiscal system. Following the
tradition of Lucas and Stokey (1983), lump-sum taxes (or negative Tt ) are not available.5
As is standard, we also assume that the date-0 capital tax rate, bearing on initial capital, is
exogenously set. Indeed, taxing capital in the first period is non-distorting and the government
would heavily tax the initial capital stock (see Farhi 2010, or Sargent and Ljungqvist 2012,
Section 16.7). We can thus deduce that the period-t budget constraint of the government is:
G + (1 + r̃t )Bt≠1 + Tt Æ ·tL w̃t Lt + ·tK r̃t At≠1 + Bt .
(3)
We denote the after-tax real interest and wage rates respectively as:
rt = (1 ≠ ·tK )r̃t
and wt = (1 ≠ ·tL )w̃t .
(4)
Using the CRS property of the production function, the budget constraint (3) becomes:
G + rt Kt≠1 + wt Lt + (1 + rt )Bt≠1 + Tt Æ F (Kt≠1 , Lt , st ) + Bt .
(5)
Finally, if Cttot denotes the total consumption in period t, the economy-wide resource constraint
is G + Cttot + Kt Æ F (Kt≠1 , Lt , st ) + Kt≠1 .
3
The island economy
In general, the previous economy features a growing heterogeneity in wealth levels over time,
because agents with different idiosyncratic histories will choose different savings. This hetero4
The question of the optimal mix of these financing tools will be the focus of the second part of the paper and
in particular of the Ramsey program studied in Section 5.
5
The absence of non-distorting taxes (Tt < 0) is easy to rationalize, as it is sufficient to introduce a positive
–but small– mass of households with very low market income (for instance living out of home production). These
households do not affect the Ramsey program below, except by preventing lump-sum taxes. Allowing for lumpsum taxes would be straightforward.
9
geneity can be represented by a time-varying distribution of wealth levels with infinite support,
which raises considerable theoretical and computational challenges. We now present an environment in which the savings of each agent depend on the realizations of the idiosyncratic risk for
only a given number of consecutive past periods, and not on the whole history. As an endogenous outcome, the heterogeneity among the population is summarized by a finite (but possibly
large) number of agent types.
To simplify the exposition, we present this economy using the family and island metaphor
(see Lucas 1975 and 1990, or Heathcote, Storesletten, and Violante 2016 for a recent reference).
The gain of this presentation strategy is that equilibrium existence can be proved using standard
techniques. In Section 4 below, we show that the island allocation can be decentralized.
3.1
The islands
We denote by N Ø 0 the length of the truncation for idiosyncratic histories.
Island description.
There are E N different islands, where we recall that the cardinal of
the set E of idiosyncratic risk realizations is E. Agents with the same idiosyncratic history
for the last N periods are located on the same island. Any island is represented by a vector
eN = (e≠N +1 , . . . , e0 ) œ E N summarizing the previous N -period idiosyncratic history of all island
inhabitants. At the beginning of each period, agents face a new idiosyncratic shock. Agents
with history êN œ E N in the previous period are endowed in the current period with history
eN , and we denote eN ≤ êN when eN is a possible continuation of êN . The specification N = 0
corresponds to the full insurance case (only one island and one agent type), and thus to the
standard representative-agent assumption. Symmetrically, the case N æ Œ corresponds to a
standard incomplete-market economy with aggregate shocks, à la Krusell and Smith (1997).
The family head.
The family head maximizes the welfare of the whole family, attributing
an identical weight to all agents and being price-taker.6 The family head can freely transfer
resources among agents within the same island, but cannot do so across islands. All agents
belonging to the same island are treated identically and will therefore receive the same allocation,
6
As the family head does not internalize the effect of its choice on prices, the allocation is not constrainedefficient and the distortions identified by Davila et al. (2012) are present in the equilibrium allocation. The
planner will reduce them with its instruments, defined in Section 5.
10
as is consistent with welfare maximization. For agents in each island eN œ E N , the family head
will choose the per-capita consumption level ct,eN , the labor supply lt,eN , and the end-of-period
savings at,eN (remember that capital and public debt are substitute).
Agents face borrowing constraints, and their asset holdings must be higher than ≠ā.7 Fur-
thermore, some proofs below require that households cannot save more than amax . This maximal
amount can be chosen to be arbitrarily large, in particular such that it is never a binding constraint.8 Finally, we assume that all households enter the economy with an initial wealth denoted
(a≠1,eN )eN œE N .
Island sizes.
The probability
t,êN ,eN
that a household with history êN = (ê≠N +1 , . . . , ê0 ) in
period t experiences history eN = (e≠N +1 , . . . , e0 ) in period t+1 is the probability to switch from
state ê0 at t to state e0 at t + 1, provided that histories êN and eN are compatible. Formally:
t,êN ,eN
= 1eN ≤êN Mê0 ,e0 (st ) ,
(6)
where 1eN ≤êN = 1 if eN is a possible continuation of history êN and 0 otherwise. From the
expression (6), we can deduce the low of motion of island sizes (St,eN )tØ0,eN œE N :
St+1,eN =
ÿ
St,êN
(7)
t,êN ,eN ,
êN œE N
where the initial size of each island (S≠1,eN )eN œE N , with
of motion (7) is thus valid from period 0 onwards.
Timing.
q
eN œE N
S≠1,eN = 1, is given. The law
At the beginning of each period, agents move from an island êN to another island
eN , by taking with them their wealth, equal to the per-capita saving at≠1,êN . On island eN ,
the wealth of all agents coming from island êN (equal to St≠1,êN
t≠1,êN ,eN at≠1,êN )
–and for all
islands êN – are pooled together and then equally divided among the St,eN agents of island eN .
7
See Aiyagari (1994) for a discussion of the relevant values for of ā, called the natural borrowing limit in an
economy without aggregate shocks. See Shin (2006) for a discussion with aggregate shocks. A standard value in
the literature is ā = 0, which ensures that consumption is positive in all states of the world.
8
As for instance in Szeidl (2013), the assumption on the maximal bound amax enables us to consider a general
utility function. An alternative option is to assume a bounded periodic utility function u, as in Miao (2006).
11
Therefore, each of these agents holds, in the beginning of period t, the wealth ãt,eN equal to:
ãt,eN =
ÿ
êN œE N
3.2
St≠1,êN
St,eN
(8)
t≠1,êN ,eN at≠1,êN .
Program of the family head
The program of the family head can now be expressed as follows:9
!
max "
at,eN ,ct,eN ,lt,eN
E0
tØ0,eN œE N
Œ
ÿ
t=0
S
—t U
ÿ
eN œE N
1
2
T
(9)
St,eN U ct,eN , lt,eN V ,
at,eN + ct,eN = wt ◊eN lt,eN + (1 + rt )ãt,eN + Tt , for all eN œ E N ,
(10)
ct,eN , lt,eN Ø 0, at,eN Ø ≠ā, for all eN œ E N ,
(11)
(S≠1,eN )eN œE N and (a≠1,eN )eN œE N are given,
(12)
and subject to the law of motion (7) for (St,eN )etØ0œE , and to the definition (8) of (ãt,eN )etØ0œE .10
N
N
N
N
The family head maximizes the aggregate welfare (9) subject to the budget constraints (10)
on all islands, to positivity and borrowing constraints (11), and to initial conditions (12). As
the objective function is concave, constraints are linear (i.e., the admissible set is convex), and
allocations are bounded (amax guarantees a compact admissible set), the existence of the equilibrium can be proved using standard techniques –see Stokey and Lucas (1989).11 If — t ‹t,eN m(st )
denotes the Lagrange multiplier of the credit constraint of island eN , first-order conditions are:
S
Uc (ct,eN , lt,eN ) + ‹t,eN = —Et U
ÿ
t,eN ,ẽN Uc (ct+1,ẽN , lt+1,ẽN )(1
ẽN ≤eN
lt,eN = (wt ◊eN )Ï ,
‹t,eN (at,eN + ā) = 0 and ‹t,eN Ø 0.
T
+ rt+1 )V ,
(13)
(14)
(15)
The Euler equations (13) and (14), with respect to consumption and labor respectively, have
the same flavor as in a Aiyagari-Bewley-Huggett economy, while equation (15) is a standard
complementary slackness condition. We discuss these equations more formally in Section 4.
9
To simplify notations, we denote ◊eN the productivity on island eN = (e≠N +1 , . . . , e0 ) instead of ◊e0 .
Note that Et [·] in (9) is the expectation operator at date t Ø 0 over all future aggregate histories.
11
Due to the finite heterogeneity representation, we could also prove the existence of a recursive equilibrium. To
save some space, we do not present this recursive formulation, as it is not necessary to derive first-order conditions.
10
12
3.3
Market clearing conditions
Labor market.
On island eN , the labor supply in efficient units at date t amounts to
◊eN St,eN lt,eN . Summing across all islands yields the total labor supply
ÿ
Lt =
(16)
◊eN St,eN lt,eN .
eN œE N
Financial market.
The total end-of-period savings of all agents, denoted At at date t is:
At =
ÿ
St,eN at,eN =
eN œE N
ÿ
(17)
St+1,eN ãt+1,eN ,
eN œE N
where the last equality stems from the pooling equation (8). The clearing of the financial market
at date t implies that at any date t, the following equality holds:
At = Bt + Kt .
3.4
(18)
Sequential equilibrium definition
Definition 1 (Sequential equilibrium) A sequential competitive equilibrium is a collection
1
of individual allocations ct,eN , lt,eN , ãt,eN , at,eN
2
tØ0,eN œE
1
, of island population sizes St,eN
N
2
tØ0,eN œE N
of aggregate quantities (Lt , At , Bt , Kt )tØ0 , and of price processes (wt , rt , r̃t , w̃t )tØ0 such that, for a
K , · L, B )
given fiscal policy (Tt , ·t+1
t tØ0 , for an initial distribution of island population and wealth
t
1
S≠1,eN , a≠1,eN
2
eN œE N
, for initial values of the capital stock K≠1 =
q
eN œE N
S≠1,eN a≠1,eN , of
the public debt B≠1 , of the capital tax ·0 , and of the initial aggregate shock s≠1 , we have:
1
1. given prices, individual strategies at,eN , ct,eN , lt,eN
tion program in equations (9)–(12);
2
tØ0,eN œE N
1
solve the agents’ optimiza-
2. island sizes and beginning of period individual wealth ãt,eN , St,eN
with law of motions (7) and (8);
2
tØ0,eN œE N
are consistent
3. labor and financial markets clear at all dates: for any t Ø 0, equations (16)–(18) hold;
4. the government budget constraint (5) holds at any date;
5. factor prices (wt , rt , r̃t , w̃t )tØ0 are consistent with (2) and (4).
13
,
The equilibrium is a finite-state equilibrium defined at each date by 6E N + 8 variables and
K , · L, B )
6E N + 8 equations for a given fiscal policy (Tt , ·t+1
t tØ0 , which is endogenized below.
t
4
Decentralization and convergence properties
We now show that the previous allocation can be decentralized and prove that it converges to the
allocation of a Bewley economy as N increases and under general conditions. We start with given
factor prices and without aggregate shocks. Two main reasons motivate these restrictions. First,
dropping aggregate shocks implies that we have existence proof of a recursive representation in
this case (see Huggett, 1993). Second, fixing factor prices avoids issues related to equilibrium
multiplicity that may otherwise emerge as shown in Açikgöz (2016).12
The economy is similar to the one of Section 2, except for the following differences. First,
we consider as given a constant after-tax interest rate r –with —(1 + r) < 1– an after-tax wage
w, and a constant transfer T . Second, no family head imposes allocations, and agents are
expected-utility maximizers taking fiscal policy as given. Finally, each agent receives at each
date a lump-sum transfer
N +1 ),
N +1 (e
which is contingent on her individual history eN +1 over
the previous N + 1 periods. This is the actual difference with a standard incomplete-market
framework. Using standard techniques, the agents’ program can be written recursively as:13
S
VN +1 (a, eN +1 ) = max
U (c, l) + —E U
Õ
a ,c,l
ÿ
(eN +1 )Õ ≤eN +1
aÕ + c = w◊eN +1 l + (1 + r)a + T +
Õ
eN +1 ,(eN +1 )Õ VN +1 (a ,
N +1
),
N +1 (e
c, l Ø 0, aÕ Ø ≠ā,
1
2Õ
T
eN +1 )V ,
(19)
(20)
(21)
where VN +1 : [≠a, amax ] ◊ E N +1 æ R is the value function of a given agent. Compared to the
economies studied by Huggett (1993) and Aiyagari (1994), the individual history eN +1 is a state
variable, as it determines the transfer
N +1 ).
N +1 (e
The Lagrange coefficient of the credit con-
straint aÕ Ø ≠ā is denoted as ‹, and the solution of this program consists of the policy rules gcN +1 ,
gaNÕ +1 , glN +1 and g‹N +1 –defined over [≠a, amax ] ◊ E N +1 – determining respectively consumption,
12
This section can be skipped if the reader is convinced by the relevance of the island economy and wants to
directly consider Ramsey policy in this environment.
13
Following the literature, we denote as aÕ the savings choice in the current period. The value a is thus the
beginning-of-period wealth.
14
savings, labor supply, and the Lagrange multiplier of the individual budget constraint.
Note that the standard Bewley economy is simply defined as the previous program where
we further impose
N +1 )
N +1 (e
= 0, for all periods. In this case, we need only the current
idiosyncratic state as a state variable (instead of the whole history eN +1 ). The value function
is then denoted V Bewley : [≠a, amax ] ◊ E æ R. We can now state our characterization result.
Proposition 1 (Finite state space) There exists a balanced transfer
ú
N +1 ,
such that:
1. the solution of agent’s programs (19)–(21) for all eN +1 œ E N +1 is fully characterized by
only E N wealth levels, consumption levels, labor efforts, and Lagrange coefficients, that
are denoted (aÕeN , ceN , leN , ‹eN )eN œE N ;
2. for any eN œ E N , the solution (aÕeN , ceN , leN , ‹eN ) verifies the first-order conditions:
S
Uc (ceN , leN ) + ‹eN = —E U
ÿ
(eN )Õ œE N
eN ,(eN )Õ Uc
leN = (‰w◊eN )Ï ,
1
T
2
c(eN )Õ , l(eN )Õ (1 + r)V ,
aÕeN + ceN = w◊eN leN + (1 + r)âeN + T,
âÕeN =
ÿ
ẽN œE N
SẽN
SeÕ N
(22)
(23)
(24)
(25)
Õ
ẽN ,eN aẽN ,
‹eN (aÕeN + ā) = 0 and ‹eN Ø 0.
(26)
The previous proposition shows that there exists a transfer such that conditions (22)–(26)
determining the equilibrium allocation are exactly the same as the ones in the program of the
family head, given by (13)–(15) and the pooling equation (8) –when there is no aggregate shocks
and when the family head equations are written in a recursive form (Item 2 of the Proposition).
The wealth level âÕeN in equation (25) that results from the transfer
âÕ = a +
ú
N +1 ),
ú
N +1
(i.e., equal to
has the same expression as the after-pooling wealth in the island economy in
equation (8).14 In particular, although N + 1 periods are needed to construct the transfer
ú
N +1 ,
the after-transfer wealth depends only on the last N periods. The transfer consists in pooling
resources of all agents having the same idiosyncratic history for N + 1 periods, and redistributes
the same amount to agents having the same idiosyncratic history for N periods. Since the
14
The formal expression of
ú
N +1
is given in Appendix A.
15
transfer reduces wealth heterogeneity, there are only E N possible wealth levels (Item 1). The
corresponding value function is denoted VNú +1 . We can now state our convergence result.
Proposition 2 (Convergence) For given factor prices and for the transfer
ú
N +1 ,
if there
exists Ÿ œ (0, 1) and N Ø 1, such that for all N Ø N , such that for all (eN̄ ≠1 , . . . , e0 ) œ E N , and
¯
(fN , . . . , fN̄ ), (gN , . . . , gN̄ ) œ E N ≠N +1 and a1 , a2 œ [≠a, amax ]:
- N +1
-gaÕ (a1 , (fN , . . . , fN̄ , eN̄ ≠1 , . . . , e0 )) ≠ gaNÕ +1 (a2 , (gN , . . . , gN̄ , eN̄ ≠1 , . . . , e0 ))- < Ÿ |a1 ≠ a2 |
(27)
then limN æŒ |
ú
N +1 |
= 0, and for all a œ [≠a, amax ] and (eN , e) œ E N +1 :
1
2
lim VNú +1 a, (eN , e) = V Bewley (a, e).
N æŒ
Though involved, condition (27) has a simple meaning. It states that the marginal propensity
to save is always smaller than 1 for all agents, as soon as N is high enough. When this condition
is fulfilled, the transfer tends toward 0 as the length of idiosyncratic history N increases. Indeed,
if the saving propensity is strictly lower than one, initial differences in wealth vanish and agents
experiencing the same history of idiosyncratic shocks end up having the same wealth as time
goes by. As a consequence, the wealth pooling generated by the transfer
levels which tend to be closer to each other, and the transfer
ú
N +1
ú
N +1
concerns wealth
tends toward 0. In this case,
it is then easy to show that the value function of the truncated economy converges toward the
value function of the Bewley economy, which depends only on the current idiosyncratic shock.
By contradiction, it is easy to show that a necessary condition for inequality (27) to be fulfilled
is that the propensity to save is always strictly less than one for all agents in the corresponding
Bewley economy (i.e., without any transfers
ú
N +1 ).
To our knowledge, all calibrated Bewley
models found in the literature share this property, such that one can be confident about the
general relevance of this truncated representation of incomplete-market economies.
Introducing aggregate shocks.
In the economy with aggregate shocks, the limit of the
truncated economy can be proven to exist. However, it is difficult to compare this limit with other
models, such as Krusell and Smith (1998)’s for instance, as, to our knowledge, there is no proof
of the existence of a recursive representation for incomplete market economies with aggregate
16
shocks when the distribution of wealth is the only state variable (see Kubler and Schmedders,
2002 and 2003, and Miao, 2006, for example). The current construction of a truncated economy
could be the foundation of such a proof, but we leave this possibility for future research.15
5
Optimal fiscal policy: The Ramsey problem
5.1
The Ramsey problem
We now solve the Ramsey program in our incomplete-market island-economy with aggregate
shocks. The Ramsey program consists for the government to choose a sequence of lump-sum
taxes and of distorting taxes on capital and labor as well as a path of public debt levels that
maximize the aggregate welfare. This aggregate welfare computed using a utilitarian criterion
is simply the objective of the family head in equation (9).16 . The government assumes that
agents behave rationally. It also faces constraints on the economy-wide resources as well as on
its budget. The following definition formalizes this statement, using notations of Section 3.
Definition 2 (Ramsey program for a truncated economy) Let N > 0. Given initial con1
ditions about the wealth distribution S≠1,eN , a≠1,eN
2
eN œE N
, the initial public debt B≠1 , the ini-
tial capital tax ·0K , and the initial aggregate state s≠1 , the Ramsey program consists in choosing,
K , · L)
at date 0, a fiscal policy made of lump-sum, capital, and labor tax paths (Tt , ·t+1
t tØ0 , and of
public debt paths (Bt )tØ0 , that maximizes the aggregate welfare defined in (9) among the set of
competitive equilibria characterized in Definition 1.
Since the period-0 capital tax rate is given, the capital tax path starts at date 1. Equation
(4) implies that the government can equivalently decide the post-tax interest rate (rt )tØ1 and the
post-tax wage rate (wt )tØ0 instead of the distorting taxes (·tK )tØ1 and (·tL )tØ0 . As a consequence,
15
In this section, we achieved decentralization through a fiscal transfer úN +1 , but this is not the only option.
Indeed, following the constructions of Alvarez et al. (2009) and Khan and Thomas (2015), it is possible to provide
a sequential decentralization of the island economy. Indeed, islands are devices to pool income at each date t
(and as such provide insurance) for idiosyncratic risks occurring before date t ≠ (N + 1). As a consequence, if all
agents are ex ante identical, it would possible to achieve decentralization using insurance contracts, which hedge
at any date t the risks occurring before date t ≠ (N + 1) among agents having the same N -period history.
16
Alternative social welfare functions could be used, but we focus on the most standard case.
17
we can formalize the Ramsey program as follows:
!
max
Tt ,rt+1 ,wt ,Bt ,(at,eN ,ct,eN ,lt,eN )eN œE N
"
tØ0
S
T
Œ
ÿ
ÿ
E0 U — t
St,eN U (ct,eN , lt,eN )V ,
t=0
(28)
eN œE N
Bt + F (Kt≠1 , Lt , st ) Ø G + (1 + rt )Bt≠1 + rt Kt≠1 + wt Lt + Tt ,
(29)
for all eN œ E N :
at,eN + ct,eN = wt ◊eN lt,eN + (1 + rt )ãt,eN + Tt ,
1
S
2
Uc ct,eN , lt,eN + ‹t,eN = —Et U
ÿ
t+1,eN ,ẽN Uc
ẽN œE N
lt,eN = (‰wt ◊eN )Ï ,
1
2
At =
eN œE N
St,eN at,eN , Lt =
ÿ
eN œE N
(30)
ct+1,ẽN , lt+1,ẽN (1 + rt+1 )V, (31)
‹t,eN (at,eN + a) = 0,
ÿ
T
(32)
(33)
St,eN ◊eN lt,eN , Kt = At ≠ Bt ,
ct,eN , lt,eN , (at,eN + a) Ø 0,
(34)
(35)
with the law of motion (7) of (St,eN )tØ0,eN œE N , and the definition (8) of (ãt,eN )tØ0,eN œE N . All
constraints (29)–(35) should be understood, unless specified, for all st œ S t and all eN œ E N .17
Maximization devices in the Ramsey program are on the one hand individual quantities
–consumption level, labor effort, and asset holdings– and on the other hand fiscal instruments:
public debt, lump-sum taxes, and post-tax interest and wage rates. Equation (29) is the government budget constraint, while the individual budget constraint is given in equation (30).
The individual Euler equations for consumption and labor are provided in equations (31) and
(32), respectively. The complementary slackness condition is stated in equation (33). Equation
(34) gathers the aggregation for individual wealth and the labor supply, as well as the financial
market clearing. Finally, positivity and borrowing constraints appear in equation (35).
5.2
Simplification of the Ramsey program
We simplify the formulation of the Ramsey program exposed in equations (28)–(35), following
Marcet and Marimon (2011). We first denote — t mt (st )St,eN ⁄t,eN the (discounted) Lagrange
17
Again, Et [·] is the conditional expectation at date t with respect to aggregate shocks.
18
multiplier of the Euler equation of agent eN in state st . We also define for all eN œ E N :
t,eN
=
q
êN œE N
S t≠1,êN ⁄t≠1,êN
St,eN
t,êN ,eN
(36)
,
which can be interpreted as the average for agents of island eN of their previous period Lagrange
multipliers for the Euler equation. Finally, we can notice that we have ⁄t,eN = 0 if at,eN = ≠a:
⁄t,eN is zero when the credit constraint is binding. We can therefore drop the product ⁄t,eN ‹t,eN
(for any t and any eN ), which is actually always equal to 0. The following lemma summarizes
our simplification of the Ramsey program.
Lemma 1 (Simplified Ramsey program) The Ramsey program in equations (28)–(35) can
be simplified into:
!
max
rt+1 ,wt ,Bt ,Tt ,(at,eN ,ct,eN ,lt,eN )eN œE N
"
E0
Œ
ÿ
t=0
tØ0
—t
ÿ
1
St,eN U (ct,eN , lt,eN )
eN œE N
+ Uc (ct,eN , lt,eN )
1
t,eN (1
s.t. ⁄t,eN = 0 if at,eN = ≠a,
+ rt ) ≠ ⁄t,eN
(37)
22
,
(38)
and subject to equations (7), (8), (29)–(32), (34)–(35), and (36).
The proof is relegated to Appendix C.18 The simplification of the Ramsey program, which
eases the computation of the maximization problem, is based on a re-writing of the Lagrangian
to introduce Lagrange coefficients into the objective, as done by Marcet and Marimon (2011).
They also provide a recursive formulation of the Ramsey program that we do not need, as the
sequential representation allows us to derive first-order conditions.
6
Understanding fiscal policy
6.1
First-order conditions
An understanding of optimal fiscal policy can be obtained from the first-order conditions of the
program (37), which are necessary conditions. These conditions relate three different valuations
18
Açikgöz (2015) uses the same methodology to find a recursive formulation in a model without aggregate
shocks. Our proof carefully considers the effect of truncation on Lagrange multipliers.
19
of liquidity. The first one is the social valuation of government liquidity, defined as a marginal
valuation of an additional amount of goods for the government. It is measured by the (normalized) Lagrange multiplier of the government budget constraint (29), denoted as µt in period
t.19 The second valuation of liquidity is the marginal value of one unit of goods for agents eN
in period t when saving incentives are internalized. This value, called the social valuation of
liquidity for agents eN and denoted Ât,eN , is formally defined as:
1
Ât,eN = Uc (ct,eN , lt,eN ) ≠ Ucc (ct,eN , lt,eN ) ⁄t,eN ≠
t,eN
2
(1 + rt ) .
(39)
The valuation Ât,eN thus differs from the marginal utility of consumption Uc (ct,eN , lt,eN ), which
is the third valuation of liquidity called the private valuation of liquidity for agents eN . Indeed,
Ât,eN takes into consideration the Euler equations from periods t ≠ 1 to t and from periods t to
t + 1. An extra consumption unit makes the agent more willing to smooth out her consumption
between periods t to t + 1 and thus her Euler equation more “binding”. This more “binding”
constraint decreases the utility by the algebraic quantity Ucc (ct,eN , lt,eN )⁄t,eN , where ⁄t,eN is the
Lagrange multiplier of the agent’s Euler equation at date t. The extra consumption unit at t
also makes the agent less willing to smooth her consumption between periods t ≠ 1 to t and
therefore “relaxes” the constraint of date t ≠ 1. This is reflected by
t,eN
which is, for an agent
at date t on island eN , the average Lagrange multipliers of Euler equations at date t ≠ 1, when
the agent was on other islands.
It is easy to show that if the government could implement island-specific lump-sum transfers
(such as unconstrained Tt,eN ), it would implement µt = Ât,eN for all eN œ E N . As a consequence,
the difference µt ≠ Ât,eN is a measure of the cost for island eN of imperfect and distorting policy
tools. We will call the difference µt ≠Ât,eN the liquidity valuation gap for agents eN , as it is equal
to the marginal gain of transferring resources from the budget of island eN to the budget of the
government. The liquidity valuation gap µt ≠ Ât,eN can be either positive or negative depending
on the island, but, as shown below, the sum of social values over all islands is non-negative.
We now present and discuss the first-order conditions of the planner using these concepts.
We derive them formally in Appendix D.
19
Note that µt is the Lagrange coefficient on the period-t budget constraint of the government, and not on its
intertemporal budget constraint as in Lucas and Stokey (1983), that they call the ‘implementability condition’.
Our choice allows us to simplify the interpretation of first-order conditions in the Ramsey program below.
20
Social valuation of government liquidity, µt .
Its dynamics follows:
µt = —Et [µt+1 (1 + r̃t+1 )] .
(40)
Equation (40) sets equal the marginal benefit of one additional unit of debt at date t to the
marginal extra cost at date t + 1, using the before-tax return r̃t to value the next period. On the
one hand, the extra debt unit relaxes the government budget constraint at date t by one unit
and thus implies a benefit that amounts to the Lagrange multiplier of the government budget
constraint, µt . On the other hand, the extra debt unit implies debt reimbursement and interest
payment in the next payment, i.e., a total payment of 1 + r̃t+1 = 1 + FK (At ≠ Bt , Lt+1 ) that
makes the next-period government budget constraint stricter.
Liquidity valuation gaps, µt ≠ Ât,eN .
We begin with defining Ct as the set of islands on
which agents are credit-constrained at date t. Formally:
Ct = {eN œ E N , ⁄t,eN = 0}.
(41)
Then, for non credit-constrained islands, the dynamics of liquidity valuation gap is:
S
’eN œ E N \ Ct , µt ≠ Ât,eN = —Et U
ÿ
(1 + rt+1 )
ẽN œE N
t+1,eN ,ẽN
1
2
T
µt+1 ≠ Ât+1,ẽN V .
(42)
Equation (40) can be interpreted as a modified Euler equation for non credit-constrained agents.
It equalizes the current liquidity valuation gap µt ≠ Ât,eN to its discounted value tomorrow. The
Euler equation for the liquidity valuation gap is similar to the Euler equation for the private
valuation of liquidity for the same agents (equation 31). Both the agents and the planner
perceive that the marginal gain to transfer resources to the next period is rt .
Labor taxes.
The first-order condition for the post-tax real wage wt is:
ÿ
eN œE N
2
St,eN lt,eN ◊eN 1
·L
µt ≠ Ât,eN = µt Ï t L .
Lt
1 ≠ ·t
21
(43)
Equation (43) sets equal the social gain of financing the government budget using labor tax ·tL
(the left-hand side) to its cost (the right-hand side). More precisely, the left-hand side is the
marginal gain of transferring resources for all islands eN œ E N to the budget of the government
using an increase in labor tax ·tL . This implies a liquidity valuation gap µt ≠ Ât,eN , for every
island eN , which is weighted by its share in the total labor effort
St,eN lt,eN ◊eN
Lt
, expressed in
efficient units. This weight is thus proportional to the labor tax base. The right-hand side is
the cost of labor tax distortion, which reflects the reduction in the base of the labor tax. The
magnitude of the distortion depends positively on the fiscal wedge generated by labor tax ·tL ,
the Frisch elasticity of labor supply Ï which determines how agents adapt their labor effort to
the tax distortion, and the government liquidity valuation µt .
Capital taxes.
The first-order condition for the post-tax interest rate rt can be written as:
ÿ Ë
eN œE N
1
St,eN ãt,eN µt ≠ Ât,eN
2È
=
ÿ
St,eN Uc (ct,eN , lt,eN )
t,eN .
(44)
eN œE N
where ãt,eN is given by (8). Equation (44) sets equal the social gain of financing the government budget using the distorting capital tax ·tK (the left-hand side) to its cost (the right-hand
side). More precisely, the left-hand side is the average liquidity valuation gap weighted by the
before-tax wealth on each island, which is the tax base. The right-hand side is the sum of
individual distortions of a higher capital tax that affects individual Euler equations, and more
precisely, consumption smoothing between the previous period and today. Therefore, individual
distortions are measured by
t,eN ,
which assesses the tightness of Euler equations between t ≠ 1
and t, and thus the willingness to smooth out consumption between both periods.
Lump-sum transfer Tt .
ÿ
eN œE N
The first-order condition for the transfer Tt is:
1
2
St,eN µt ≠ Ât,eN Ø 0, with equality when Tt > 0.
(45)
This equation states that, when the positivity constraint on the lump-sum transfer is not binding,
the government sets the population-weighted sum of liquidity valuation gaps to 0. In particular,
this would also be the case in the absence of any positivity constraint. However, when the
constraint Tt Ø 0 is binding and when the government would actually like to tax some island
22
using the lump-sum instrument, the constraint Tt Ø 0 binds, and the social benefit of liquidity
for the government is higher than its average cost over all islands: µt >
6.2
q
eN œE N
St,eN Ât,eN .20
Steady-state fiscal policy
Using first-order conditions, we can derive theoretical implications about the steady-state optimal fiscal policy. The main results are summarized in the next proposition.21
Proposition 3 (steady-state) In the steady-state of the Ramsey equilibrium,
1. the marginal productivity of capital is pinned down by the discount factor —:
1 + FK (K, L) =
1
,
—
(46)
2. the capital tax is non-negative; it is positive if and only if credit-constraints bind for some
agents, or more formally:
·
K
=
(1 ≠ —)
q
S N‹ N
N
q e œC e e
eN œE N
SeN Uc (ceN , leN )
,
(47)
where we recall that C is the set of islands where credit constraints bind at the steady-state,
and ‹eN is the Lagrange multiplier of credit constraint for island eN ;
3. the labor tax is determined by the net average liquidity valuation gap:
µÏ
ÿ SeN leN ◊eN
·L
=
(µ ≠ ÂeN ) .
L
1≠·
L
N
N
(48)
e œE
The first item in equation (46) of Proposition 3 is a direct implication of the government
Euler equation (40). As a consequence, the marginal productivity of capital is determined by
the discount factor — only, as originally explained by Aiyagari (1995). This important restriction
is only possible in an economy with capital.
20
Allowing for negative transfers does not imply that (45) holds with equality. Indeed, the ability of the lowest
income agents to pay lump-sum taxes may provide a binding bound on negative taxes. If this bound does not
bind, it is easy to show that full insurance can be implemented, using only lump-sum taxes to pay for public
expenditure and for public debt interest payment. This is the result of Woodford (1990) in a simpler environment.
21
To denote steady-state variables, we simply drop the subscript t.
23
To the best of our knowledge, the second item in Proposition 3 is new in the literature.
We prove that the capital tax is always non-negative and that its value is determined by the
severity of credit constraints. If credit constraints do not bind for any agent –for instance if they
are chosen to be below the natural borrowing limit, as defined by Aiyagari (1994)– then the
equilibrium capital tax will be 0.22 Conversely, the steady-state capital tax will be positive if
and only if some agents are credit-constrained. Indeed, when credit constraints are binding for
some agents, credit-constrained agents cannot borrow as much as they would like to, while nonconstrained agents save too much to self-insure. Both effects contribute to create an oversupply
of liquidity. To correct this oversupply, the government raises the capital tax, which decreases
the post-tax interest rate and thus the incentives to save.23 Aiyagari (1995) proves that capital
tax is positive in a similar environment, but without aggregate shocks. Our contribution is to
connect the capital tax rate to the severity of credit constraints.
The three relationships in equations (46)–(48) determine the marginal product of capital as
well as labor and capital taxes as a function of the heterogeneous valuation of liquidity across
islands. The level of public debt is then determined as a residual of the government budget
constraint. The public debt can be either positive or negative, and one of the main factors
affecting it is the Frisch elasticity of the labor supply. As can be seen from equation (48) and
confirmed in numerical simulations, when the labor supply is very elastic, the labor tax is very
small. Then, agents will hold a large quantity of savings for precautionary motives and the credit
constraint will rarely be binding in equilibrium. From equality (47) in Proposition 3, we know
that capital taxes will also be low. In consequence, the government can hardly rely on taxation
to finance public spending, and it must therefore accumulate resources to eventually finance
public spending out of interest payments. Public debt is therefore very likely to be negative in
that case. In contrast, if labor supply is inelastic, labor taxes will be high and the government
will be able to finance interest payment using taxation. The public debt will then be positive.
In the special case when the credit constraint does not bind, we can derive an additional
result about the indeterminacy of public debt.
22
The denominator in equation (48) is indeed bounded away from zero because the economy is finite, i.e., not
all marginal utilities can be simultaneously zero.
23
This logic is already present in Woodford (1990), and discussed in Davila et al. (2012). The last paper differs
from ours, because the authors analyze the optimality of the capital stock in a situation where the planner can
change agents’ saving decisions without distortion, while we specifically focus on distorting fiscal instruments.
24
Corollary 1 (steady-state without binding credit limit) If the credit constraint does not
bind in the steady-state equilibrium, any lump-sum transfer and public debt {T, B} such that
3
4
1
≠ 1 B = ≠G ≠ T
—
(49)
is a solution of the Ramsey problem of the government in steady-state.
Corollary 1 is a direct consequence of Proposition 3. Indeed, as no agent is credit constrained,
the capital tax is null, · K = 0, and the post-tax gross interest rate amounts to 1/—. Agents
are then able to accumulate a sufficient quantity of assets to perfectly self-insure themselves
in the steady-state.24 The liquidity valuation gap is zero and there is no role for labor taxes.
Public debt and lump-sum transfer need then to verify equality (49), which guarantees that the
government budget is balanced. However, public debt is not determined.
Corollary 1 is already found in Bhandari, Evans, Golosov, and Sargent (2016b), who show
that the public debt is indeterminate when the borrowing constraint is below the natural limit –in
which case the borrowing constraint does not bind.25 In our economy with capital accumulation,
we additionally derive that in that case, steady-state taxes are zero, · K = · L = 0, and that the
government accumulates assets to finance public expenditures G and lump-sum transfer T with
interest payments –i.e., public debt B is negative.
6.3
Understanding the dynamics in a simple case
To further understand fiscal policy dynamics, we simplify the model as follows. First, N = 1,
implying only two types of agents, employed (denoted by e) and unemployed (denoted by u).
The probability to stay employed is fie , while the probability to remain unemployed is fiu ,
with fie + fiu ≠ 1 > 0. Second, productivity levels are ◊e = 1 and ◊u = 0. Third, the credit
limit is ā = 0. Fourth, the production function is Cobb-Douglas, with a constant depreciation
rate, F (K, L) = ›K – L1≠– ≠ ”K, where › is the constant productivity level. Fourth, and more
importantly, the utility function u(x) is assumed to be linear when x is high enough: there exists
a threshold xú such that u(x) = log(x) for x < xú and u(x) = x ≠ xú + log(xú ) for x Ø xú . We
24
Solving Euler equations implies that the marginal utilities of consumption are the same for all agents. The
quantities (⁄eN ) and (ÂeN ) are thus the same for all agents.
25
They also find that the optimal level of public debt is determinate when the credit limit is above the natural
borrowing limit and thus binds for some agents, as in our model.
25
provide parameter restrictions in Appendix F, such that the marginal utility of consumption of
unemployed agents is 1/cu,t whereas the one of employed agents is 1, what considerably simplifies
the algebra.26 The details of model properties can be found in Appendix.
Assume now that the economy in steady state is hit in period t by a small positive shock which
reduces the government’s valuation of liquidity from the steady-state value µSS to µSS ≠ ”µSS
,
t
then the change in the post-tax real interest rate is ”rt ƒ
Se
Su ”µt .
When the government is richer
(and its valuation of liquidity decreases), it implements a fiscal policy to substantially increase
the post-tax real interest rate. To provide an order of magnitude, if the unemployment rate is
5%, then Se /Su = 19. This greater capital return, translating into lower capital taxes, enables
the government to increase consumption of the unemployed agents with a positive wealth. The
1
effect on labor tax is ”·tL ƒ ≠ Ï1 1 ≠ (1 + Ï) · SS
22
”µt where · SS is the steady-state labor tax.
The government decreases labor taxes to reduce distortions in the economy. As realistic values
for the elasticity Ï of labor supply are between 0.3 and 1, the labor tax variation is typically
small. Finally, the effect on public debt appears as a residual and depends on parameter values.
If the decrease in tax rates is high enough, tax income falls and public debt increases.
7
Quantitative investigation of the optimal tax system
We now simulate the model using standard parameter values to study quantitative properties
of the optimal fiscal policies.
7.1
Parametrization
The period is a year. The discount factor is set equal to — = 0.96 as in Heathcote (2005) to
obtain an annual discount rate close to 4%. The curvature of the utility function is set to ‡ = 2
(Hall, 2010), while the Frisch elasticity of labor supply is set to Ï = 0.5, which is consistent
with empirical estimates (Chetty, Guren, Manoli, and Weber 2011). We nevertheless provide
sensitivity analysis for these last two parameters, as many different values can be found in the
literature. The scaling parameters of the labor supply is ‰ = 1 as a benchmark, and we use this
as an adjustment factor to match the same steady-state allocations in comparative statics.
26
Compared to quasilinear preferences studied in Bhandari, Evans, Golosov, and Sargent (2016a), keeping
some concavity for low utility levels allows finding a steady-state where credit constraint binds and public debt is
determined. Challe and Ragot (2016) introduce this utility function in incomplete insurance market economies.
26
The aggregate state is assumed to follow an AR(1) process: st = fls st≠1 + Ást , where the
innovation (Ást )tØ0 is a white-noise process with a normal distribution, N (0, ‡s2 ). The production
function has a Cobb-Douglas functional form with a constant capital depreciation ”:
F (K, L, s) = ›(s)K – L1≠– ≠ ”K,
where ›(s) = exp(s) is the technology level. The quantity – is the capital share, which is set to
0.36, while ” is the annual depreciation rate, set to 10% (Heathcote, 2005). Public consumption
matches the average ratio of public spending over GDP in the US for the period 2000-2016,
which is 19%. This implies G = 0.30. The autocorrelation of the technology shock fls is 0.81,
which corresponds to a standard quarterly correlation of 0.95 for the TFP shock, while the
standard deviation is ‡s = 0.01. Table 1 summarizes the calibration of model parameters.
N
—
„
‡
‰
”
–
G
fls
‡s
4
0.96
0.5
2
1
0.1
0.36
0.30
0.815
0.01
Table 1: Parameter calibration
For the labor process, we use the specification of Domeij and Heathcote (2004), which is a
simple three-state process reproducing an AR(1) wage process with an annual autocorrelation
of 0.90 and a standard deviation for the innovation equal to 0.224. These two values correspond
to their US empirical counterparts, estimated on PSID data. The three levels for the hourly
productivity are respectively e1 = 0.213, e2 = 0.848, and e3 = 3.940, and the corresponding
transition matrix, which is constant, is:27
S
T
0.099 0.001 X
W 0.9
W
X
X
M =W
W 0.006 0.988 0.006 X
U
Choice of N .
0.001 0.099 0.001
V
Quantitative investigation of the model suggests that an economy featuring
N = 4, and thus 34 = 81 different agents, is a good benchmark. Indeed, we have also solved the
model for N = 6 (729 agents) and checked that the steady-states were quantitatively similar.
27
Compared to Domeij and Heathcote (2004), we introduce a small probability to switch from states 1 to 3,
and from states 3 to 1. This does not change the properties of the AR(1) process, but speeds up the algorithm,
as all islands are visited.
27
It turns out that in the economy with N = 6, the capital tax amounts to 10%, compared to
11% when N = 4, while the labor tax is 37%, compared to 36% for N = 4. These differences in
steady-state tax values between economies with N = 4 and N = 6 are one order of magnitude
lower than the tax changes implied by reasonable variations in preference parameters, on which
there is no strong consensus. We are thus confident that an economy with N = 4 captures the
key properties of the dynamics of the fiscal system in our model.
Solution method.
Our truncated equilibrium enables us to solve the model using a standard
perturbation method. First, we start with determining the steady-state allocation without
aggregate shocks, using an iteration over the post-tax interest rate, as described in Appendix
G. Second, we linearize all equations around the steady-state (including first-order conditions
of the Ramsey problem), to obtain the dynamics of the fiscal system and constrained-optimal
allocation after a small technology shock. The technology shock is small enough such that, for
all simulations, the set of credit-constrained islands remain unchanged along the business cycle.
For N = 4, the dynamic system is composed of 521 variables (including all Lagrange coefficients), which is simulated with Dynare. The gains of the perturbation method are twofold.
First, we can choose a continuous state-space for the technology shock. Second, we can use the
same mature methodology as the one used in the DSGE literature to compute IRFs and second
moments. For future work, this means that we can rely on tools (e.g., Dynare) that have already
been developed and make our results directly comparable with a large branch of the literature.
7.2
Steady-state results and sensitivity analysis
Steady-state output.
We start with describing the steady-state allocation and fiscal system.
As shown theoretically in Proposition 3, the pre-tax interest rate r̃ is pinned down by the
discount factor —. Since r̃ = FK (K, L), this implies that the capital-output ratio is K/Y = 2.5,
and that the equilibrium pre-tax wage w̃ = FL (K, L) = 1.1. The labor supply and total output
are then easy to derive, as there is no wealth effect due to the GHH utility function.
The optimal capital tax amounts to · K = 11%, while the labor tax is · L = 36%. The
debt-to-GDP ratio equals 148%. Although different from actual taxes in the US (reported to be
·UKS = 39.7% and ·ULS = 26.9%, by Domeij and Heathcote, 2004), these values are comparable
to their empirical counterparts. In addition, though high, the public debt level does not reach
28
unrealistically high or negative values.
How do our results compare with the literature studying the optimal steady-state debt level?
Aiyagari and McGrattan (1998) find an optimal debt over GDP of 60%, maximizing steady-state
welfare. Açikgöz (2015) shows that the results are quantitatively different if one solves for the
Ramsey program instead of maximizing steady-state welfare. He finds an optimal quantity of
debt to be above 300% of GDP. Dyrda and Pedroni (2016) find a negative optimal debt-toGDP ratio, below ≠300% of GDP. As shown in the sensitivity analysis below, these substantial
quantitative differences can partly be explained by parameter choices, such as the elasticity of
labor supply or the income process, for which there is no strong consensus.
Concerning the capital tax, Proposition 3 shows that it is directly related to the severity
of credit constraints. In steady-state, only 4% of agents are credit-constrained and all of them
have the smaller productivity level e1 . This is sufficient to induce a positive capital tax, though
quite small. Indeed, more than 90% of the government steady-state revenues come from labor
taxes. Table 2 summarizes steady-state quantities.
r̃(%)
w̃
· K (%)
· L (%)
4.2
1.1
11
36
B/Y (%) K/Y (%)
148
2.5
L
Y
0.9
1.6
Table 2: Steady-state outcome
Sensitivity analysis. We perform a sensitivity analysis for some key parameters. For all
experiments, we recalibrate the labor-supply parameter ‰ to obtain the same public spendingto-GDP ratio. All results are gathered in Table 3. The first line of the table recalls the model
outcome for the benchmark calibration. The second line increases the Frisch elasticity of the
labor supply to 0.75. As expected, the labor tax decreases, while the capital tax increases
to balance the budget of the government. The public debt decreases, due to the increased
distortions implied by raising the capital tax. Note that if we further increase the Frisch elasticity
up to 2, we obtain a negative optimal public debt (not reported in Table 3), as we explained
in Section 6.2. The third line of Table 3 presents the result when the concavity of the utility
function decreases from 2 to 1. In this case, agents save more and the “severity” of the credit
constraint decreases, as the concavity of the utility function is lower. The capital tax decreases
to 5% and the labor tax increases to 39%. The public debt slightly increases as total savings
29
increase (while the capital stock remains barely unaffected).
Parameters
Output
„
‡
‰
N
· K (%)
· L (%)
B/Y (%)
Benchmark
calibration
0.50
2
1.00
4
11
36
148
Change in „
0.75
2
0.90
4
18
32
103
Change in ‡
0.50
1
1.03
4
5
39
155
Table 3: Sensitivity analysis
7.3
Dynamic properties
We now investigate the dynamic properties of the optimal fiscal system after a technology
shock. Figure 1 plots impulse-response functions after an unexpected increase of 1% in TFP.
All variables are provided as proportional deviations from steady-state, except the tax rates ·tL ,
and ·tK , which are given in level deviations from steady-state.
The first panel in Figure 1 plots the TFP shock. GDP increases by 1.2%. Aggregate
consumption (Ctot) increases on impact by almost 1% and then decreases smoothly toward
its steady-state value. The capital stock (K) increases progressively, with a maximal increase
below 1%, whereas the aggregate savings of private agents (A) increases rapidly, by 1.25%
at the maximum. The public debt (B) increases (up to 2%) to absorb excess savings. The
private sector saves too much after a technology shock. Indeed, wage inequality increases after
such a shock (as the real wage increases), which generates additional incentives to save for highproductivity agents in order to smooth out consumption in case of switching to low productivity.
The government absorbs these excess savings by an increase in public debt and cut capital taxes
(tauk) for the reason discussed in Section 6.3.
The third line plots the labor and capital tax rates. The labor tax (taul) decreases a little
bit after the technology shock, while the capital tax (tauk) decreases a lot for two periods before
converging rapidly back to its steady-state value. Cutting capital taxes enables the government
to transfer resources to low-income agents, who have a positive wealth. The positive technology
shock relaxes the budget constraint of the government, because the economy is wealthier. The
30
Figure 1: Aggregate IRFs after a 1% increase in TFP
Lagrange multiplier of the government budget (mu) decreases. The last line plots after tax real
wage (w), real interest rate (r) and total labor (L).
Table 4 provides unconditional second-order moments generated by the simulated model. As
consistent with our analysis of IRFs, the volatility of the labor tax is low and the volatilities
of the capital tax and of the public debt are high compared to output. Public debt is volatile
because it reverts slowly to its mean. The persistence of capital tax is very low compared to
labor taxes and public debt, as observed discussing IRFs.
The investigation of the quantitative properties of the model shows that some results are
robust to reasonable variations in parameter values. This is the case of the high volatility of
capital tax, the slow mean-reversion dynamics of public debt, and the range of capital and labor
taxes. Other results, such as the steady-state level of public debt, are not precisely pinned down
in this class of model at this stage, as already discussed by Aiyagari and McGrattan (1998).
31
Standard deviations
sd(Y )
sd(· K )
sd(· L )
sd(B)
0.06
0.23
0.003
0.17
Autocorrelations
corr(Y, Y≠1 )
0.93
K ) corr(· L , · L )
corr(· K , ·≠1
≠1
-0.04
0.99
corr(B, B≠1 )
0.94
Table 4: Second-order moments
8
Concluding remarks
We have proved that the competitive equilibrium in an incomplete insurance-market economy
with aggregate shocks can be represented as the allocation of a family-head program. The gain
of this representation is that it generates a finite-dimensional state-space equilibrium, in which
the Ramsey outcome can be studied with aggregate shocks and with various fiscal tools. We
apply this framework to study optimal fiscal policy, when positive transfers, distorting taxes on
capital and labor, and public debt are available.
The methodology presented in this paper could be applied in different settings. First, additional heterogeneity, such as the structure of qualification, could be introduced to make more
extensive use of empirical estimates of key parameters. Second, we could also include other
distortions, such as limited participation, search-and-matching frictions on the labor market or
nominal frictions, which are all extensively used in macroeconomic models. The tools developed
in the present paper, based on the dynamic structure of Lagrange coefficients, could allow us to
derive new theoretical and quantitative results for optimal monetary policy.
32
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Woodford, M. (1990): “Public Debt as Private Liquidity,” American Economic Review, 80(2),
382–388.
Appendix
A
Proof of Proposition 1
Consider an agent endowed with the N + 1-period history eN +1 = (êN , e) œ E N +1 . The history
eN +1 can also be written as eN +1 = (eN , eN ). In the former notation, eN +1 is seen as the history
ẽN œ E N with the successor state e œ E, while in the latter notation, eN +1 is seen as the state
eN œ E followed by history eN œ E N . The solutions to the maximization program (19)–(21)
are the policy rules denoted c = gcN +1 (a, eN +1 ), aÕ = gaNÕ +1 (a, eN +1 ), l = glN +1 (a, eN +1 ) and the
multiplier ‹ = g‹N +1 (a, eN +1 ) satisfying the following first-order conditions:
S
Uc (c, l) + ‹ = —E U
ÿ
eÕ œE
T
! Õ Õ"
Me,eÕ Uc c , l (1 + r)V ,
l = (‰w◊e )Ï ,
‹(aÕ + ā) = 0 and ‹ Ø 0.
(50)
(51)
(52)
We now use a guess-and-verify strategy. The transfer is constructed such that all agents with
the same N -period history will have the same after-transfer wealth. The measure of agents with
37
history eN follows the same law of motion as (7) in the island economy and this measure is also
equal to SeN . If all agents with the same history (êN , e), e œ E have the same beginning-of-period
wealth aêN , the after-transfer wealth, denoted âeN , of agents with history eN ≤ êN will be
âÕeN =
ÿ
ẽN œE N
SẽN
SeN
(53)
Õ
ẽN ,eN aẽN ,
for agents with the same history to hold the same wealth. By construction, âeN follows a
dynamics similar to equation (8) of the after-pooling wealth ãt,eN in the island economy. The
transfer denoted
that enables all agents with the same history to have the same wealth is:
ú
N +1
ú
N +1
)
N +1 (e
= (1 + r) (âeN ≠ aêN ) ,
where we use eN +1 = (êN , e) = (eN , eN ). The transfer
ú
N +1
(54)
defined in (54) swaps the beginning-
of-period wealth (1 + r)aêN by the average wealth (1 + r)âeN . By construction, all agents with
current history eN will have an identical after-transfer wealth, independently of the previousperiod history êN . Since there is a continuum with mass SẽN of agents with history ẽN , in which
each individual agent is atomistic, all agents take the transfer
ú
N +1
as given.
Finally, it is easy to check that the transfer scheme is balanced in each period. Using the defiN N
N
nition (53) of âeN , we obtain for eN = (eN
N ≠1 , . . . , e1 , e0 ) œ E , SeN âeN =
q
êœE
S(ê,eN
N
N ≠1 ,...,e1 )
ÿ
M eN ,eN a(ê,eN
1
0
N
N ≠1 ,...,e1 )
S(ẽ,eN ) úN +1 (ẽ, eN )
êN œE N
SêN
êN ,eN aêN
=
. Therefore, for a given eN œ E N , we obtain:
= (1 + r)
ẽœE
q
C
ÿ
ẽœE
1
S(ẽ,eN ) âeN ≠ a(ẽ,eN
N
N ≠1 ,...,e1 )
2
D
=0
where the last equality comes from the definition of âeN in equation (53).
B
Proof of Proposition
The proof runs in three steps. In the remainder, we use the following notation. For N > k > 0,
ek = (ek≠1 , . . . , e0 ) œ E k , eN,k = (eN , . . . , ek ) œ E N +1≠k , and (eN,k , ek ) = (eN , . . . , ek , ek≠1 , . . . , e0 ).
B.1
A contraction lemma
We denote by Conv(A) the convex hull of the set A µ R, and µ the Lebesgue measure on R.
38
Lemma 2 (contraction lemma) Assume that A µ [≠ā, amax ] and that conditions of PropoÓ
Ô
sition 2 are fulfilled. Let, for any eN̄ œ E N̄ , B = gaNÕ +1 (a, (êN,N̄ , eN̄ ))|êN,N̄ œ E N +1≠N̄ , a œ A .
We have then µ (Conv(B)) Æ Ÿ ◊ µ (Conv(A)).
Proof. Since B µ R, we have by definition of the convex hull, Conv(A) = [min(A), max(A)] and
Conv(B) = [min(B), max(B)]. Let aÕ = max(A) and a = min(A), then µ (Conv(A)) = aÕ ≠ a
and B µ [gaNÕ +1 (a, (êN,N̄ , eN̄ )), g(aÕ , (ẽN,N̄ , eN̄ ))] for some êN,N̄ , ẽN,N̄ œ E N +1≠N̄ . Therefore, we
obtain µ (Conv(B)) Æ gaNÕ +1 (aÕ , (ẽN +1≠N̄ , eN̄ )) ≠ gaNÕ +1 (a, (êN +1≠N̄ , eN̄ )). Applying the Lipschitz
property (27) directly yields µ (Conv(B)) Æ Ÿ ◊ µ (Conv(A)).
B.2
Proof of the convergence of
ú
N +1
Let N > 0. Proposition 1 shows that when the transfer is
ú
N +1 ,
there are E N possible asset
holdings denoted (aÕêN )êN œE N . We denote AN the set of all possible asset holdings. We define
(N )
AN (eN ) = {aÕêN œ AN |eN ≤ êN }, for eN œ E N ,
(55)
which is the set of all possible beginning-of-period asset holdings of agents with current history
eN (they made their choices in previous period, while they had the history êN ). Since the aftertransfer wealth level âeN of (53) is by construction an average of before-transfer wealth levels
1
(N )
2
aÕêN , we have âeN œ Conv AN (eN ) .
To simplify notations, we define for B µ E N , fi(B) as the set of possible predecessors of
histories included in the set B. When B = {eN } is a singleton, fi(eN ) = {êN |eN ≤ êN }. We
(N )
(N )
rewrite (55) as: AN (eN ) = {aÕêN œ AN |êN œ fi(eN )}. For any aÕêN œ AN (eN ), there exists
ẽN œ E N such that êN ≤ ẽN and aÕêN = gaNÕ +1 (aÕẽN , (ẽN , e1 )): in other words, aÕêN is the optimal
choice of an agent who had in the previous period the N -history ẽN , which is thus a possible
past of eN . Using the notation fi 2 = fi ¶ fi, ẽN œ fi 2 (eN ), we can define :
(N )
(N )
AN ≠1 (eN ) = {aÕẽN œ AN |ẽN œ fi 2 (eN ) and gaNÕ +1 (aÕẽN , (ẽN , e1 )) œ AN (eN )},
which is the set of all possible asset holdings two periods ago for agents with current history eN .
39
Similarly, we define for any 0 < k < N ,
(N )
(N )
AN ≠k (eN ) = {aÕẽN œ AN |ẽN œ fi k+1 (eN ) and gaNÕ +1 (aÕẽN , (ẽN , ek )) œ AN ≠k+1 (eN )},
(N )
which allows us to construct a sequence of sets (AN ≠k )k=0,...,N . In the previous notation fi k+1
(N )
denotes fi ¶ . . . ¶ fi (k + 1 times). Note that we could equivalently define AN ≠k+1 (eN ) as:
(N )
(N )
AN ≠k+1 (eN ) = {gaNÕ +1 (aÕẽN , (ẽN , ek ))|ẽN œ fi k+1 (eN ) and aÕẽN œ AN ≠k (eN )}.
1
1
22
(N )
If 1 Æ k Æ N ≠N̄ , we deduce, applying Lemma (2), µ Conv AN ≠k+1 (eN )
and iterating from k = 1 to k = N ≠ N̄ ≠ 1 to k = N ≠ 1, one finds:
1
2
(N )
1
(N )
1
(N )
2
µ Conv(AN (eN )) Æ ŸN ≠N̄ µ Conv(AN̄ (eN )) .
(56)
Since amax (≠ā) is the largest (lowest) wealth levels by definition, AN µ [≠ā, amax ] and for
1
(N )
2
(N )
all k, AN ≠k (eN ) µ AN µ [≠ā, amax ]. This implies that we have µ Conv(AN̄ (eN )) Æ amax +ā.
1
2
(N )
Second we have showed that âeN , aêN œ Conv AN (eN ) for any êN œ fi(eN ). This implies that
1
1
(N )
22
|âeN ≠ aêN | Æ µ Conv AN (eN )
. In consequence, we have from equation (56):|âeN ≠ aêN | Æ
ŸN ≠N̄ (amax +ā), which can be made arbitrarily small (0 < Ÿ < 1), when N increases. We deduce
from the definition (54) that we have lim N æŒ supeN +1 œE N +1 |
B.3
ú
N +1 )|
N +1 (e
= 0.
Convergence of the value function
Let Á > 0. There exists N̄ such that for all N Ø N̄ : |
V (+Á) (a, e) =
max
aÕ Ø≠ā,cØ0,lØ0
ú
N +1 |
S
U (c, l) + —E U
ÿ
< Á. Define:
ẽN ≤eN
aÕ + c = w◊e l + (1 + r)a + T + Á.
T
Me,eÕ VN +1 (aÕ , eÕ )V ,
(57)
(58)
Similarly, V (≠Á) (a, e) is the value function where the budget constraint is diminished by ≠Á. (Á
has to be low enough for all agents’ resources to be positive, which is possible as ◊e > 0.)
Using standard dynamic programming arguments with bounded returns (see Stokey, Lucas,
and Prescott 1989 Section 9.2), one has for all a œ [≠ā, amax ], e œ E, then V (+Á) (a, e) Æ
40
2
Æ Ÿµ Conv(AN ≠k (eN )) ,
V Bewley (a, e) Æ V (≠Á) (a, e) and V (+Á) (a, e) Æ VN +1 (a, eN +1 ) Æ V (≠Á) (a, e). The last inequality
úN +1 .
follows from the bounds on the transfers
As a consequence, for all a œ [≠ā, amax ], e œ E:
- - Bewley
- (a, e) ≠ VN +1 (a, eN +1 )- Æ -V (+Á) (a, e) ≠ V (≠Á) (a, e)- .
-V
It is then easy to show that the right hand side can be made arbitrarily small as Á tends toward
0, as the set [amax , a] ◊ E is compact and V (±Á) is continuous in Á.
C
Proof of Lemma 1
We use the methodology of Marcet and Marimon (2011) to simplify the Ramsey program.
Denoting — t mt (st )St,eN ⁄t,eN the Lagrange multiplier for island eN at date t, the objective of the
Ramsey program (28)–(35) can be rewritten as:
J = E0
Œ
ÿ
t=0
Q
—t
ÿ
eN œE N
St,eN U (ct,eN , lt,eN ) ≠ Et
S
aUc (ct,eN , lt,eN ) + ‹t,eN ≠ —Et U
ÿ
Œ
ÿ
—t
t=0
ÿ
t+1,eN ,êN Uc (ct+1,êN , lt+1,êN )(1
êN œE N
Noticing that ⁄t,eN ‹t,eN = 0 and using
t,eN
(59)
St,eN ⁄t,eN
eN œE N
=
q
êN œE N
St≠1,êN ⁄t≠1,êN
St,eN
t,êN ,eN
T
+ rt+1 )V
with
0,eN
= 0 as
defined in (36), (59) yields after some manipulations the objective in (37).
D
Derivation of first-order conditions for the Ramsey program
We compute the first-order conditions of the simplified Ramsey program (37)–(38). Let — t mt (st )µt
be the Lagrange multiplier of the government budget constraint (29). The Lagrangian is:
L = E0
≠ E0
Œ
ÿ
t=0
Œ
ÿ
t=0
—t
ÿ
eN œE N
1
St,eN U (ct,eN , lt,eN ) + Uc (ct,eN , lt,eN )
1
t,eN (1
+ rt ) ≠ ⁄t,eN
22
(60)
µt — t (Gt + Bt≠1 + rt At≠1 + wt Lt + Tt ≠ Bt ≠ F (At≠1 ≠ Bt≠1 , Lt , st≠1 )) ,
where ct,eN = (wt ◊eN )lt,eN ≠ at,eN + (1 + rt )
(‰wt ◊eN )Ï (using (8), (30) and (32)).
q
êN œE N
41
St≠1,êN
St,eN
t,êN ,eN at≠1,êN
≠ Tt and lt,eN =
First-order conditions with respect to Bt , rt , and Tt are straightforward.
Derivative with respect to at,eN .
Ât,eN =—Et
ÿ
It yields for all eN œ E N \ Ct :
(1 + rt+1 )
t+1,eN ,ẽN Ât+1,eN
ẽN œE N
≠ —Et µt+1 (rt+1 ≠ r̃t+1 ).
(61)
Note that we have using (40) Et µt+1 (rt+1 ≠ r̃t+1 ) = Et [µt+1 (rt+1 + 1)] ≠ µt . Combining it with
(61) and using
q
ẽN œE N
t+1,eN ,ẽN
= 1, we obtain equation (42).
Derivative with respect to wt .
µt Lt
3
We obtain:
4
ÿ
Ï
1+
(wt ≠ FL (Kt≠1 , Lt , st≠1 )) =
St,eN ◊eN lt,eN Ât,eN .
wt
N
N
(62)
e œE
1
·L
·L
2
Using wt ≠FL (Kt≠1 , Lt , st≠1 ) = wt ≠ w̃t = ≠ 1≠·t L w̃t , equation (62) becomes µt Lt 1 ≠ Ï 1≠·t L =
St,eN ◊eN lt,eN Ât,eN . Since Lt =
E
Proof of Proposition 3
eN œE N
t
q
q
eN œE N
t
St,eN ◊eN lt,eN , we finally deduce equation (43).
First-order conditions (40) and (43) immediately imply (46) and (48) at the steady-state. Now,
we sum individual consumption Euler equations (31) for all eN œ E N \ C (i.e., when ‹eN = 0):
ÿ
eN œE N \C
SeN Uc (ceN , leN ) = —(1 + (1 ≠ · k )r̃) U
We now split the sum as
—· k r̃
ÿ
eN œE N
S
ÿ
ÿ
SeN
T
eN ,ẽN Uc (cẽN , lẽN )
V.
ẽN œE N eN œE N \C
q
eN œE N \C
SeN Uc (ceN , leN ) =
ÿ
eN œC
=
q
eN œE N
≠
S
q
eN œC .
After some manipulation, we obtain
S
SeN UUc (ceN , leN ) ≠ —(1 + (1 ≠ · k )r̃) U
ÿ
eN ,ẽN Uc (cẽN , lẽN )
ẽN œE N
where in the right hand side we recognize the “Euler equation” for constrained agents. Using
equation (31), we obtain equation (47) with —r̃ = 1 ≠ —.
42
TT
VV ,
F
The simple model
The number of employed and unemployed agents is Se = (1 ≠ fiu )/(2 ≠ fie ≠ fiu ) and Su =
(1 ≠ fie )/(2 ≠ fie ≠ fiu ). As the productivity of unemployed is 0, their labor supply is null lu = 0.
To focus on the interesting case, we first provide two parameter restrictions. First, we assume
3
1
–
( ≠ 1)Se (1 ≠ –)Ï
—
1/— ≠ 1 + ”
4 –Ï+1
1≠–
< G,
(63)
which implies that the first-best allocation cannot be implemented in steady state. It ensures
that ( —1 ≠ 1)K Ø G: interest payment on the whole capital stock cannot finance public spending
and positive taxes are necessary. This implies that credit constraint binds for unemployed agents
(otherwise Âe = Âu = 0 and the first best would be attained). Hence, ⁄u = 0 and · K > 0.
We show that T = 0 in steady state. Indeed, from (43), we have (all variables are in
steady-state below):
µt ≠ 1 = µÏ
·tL
1 ≠ ·tL
(64)
1≠—(1+r)fie
and from (42) µt ≠ Âu = (µt ≠ Âe ) —(1+r)(1≠fi
> 0. Then, if µt ≠ Âe < 0 then µt ≠ Âu < 0, and
e)
condition (45) cannot hold. As a consequence µt ≠ Âe Ø 0 and · L Ø 0, and from (45), T = 0.
We assume that ‰ = 1. After some tedious algebra (using the budget constraints 30), we
finds that ce ≠
(1 ≠ –)1+Ï
1
1+ 1
le Ï
1
1+ Ï
–
1/—≠1+”
> cu (i.e., xú exists, and the utility function is well defined) if we have
2 1+Ï
1≠–
Ï 1+Ï
e)
( 1+Ï
)
> (1 + Ï) (1≠fi—(1≠fi
. This condition also ensures that
u )(1≠—fie )
1/cu > 1, such that the utility function is globally concave. Finally, condition (44) implies that
µt = fie (a + ⁄t≠1 ) +
Su /Se
1 + rt
(65)
Differentiating (64) and (65) gives the relationships provided in the text.
G
Algorithm to solve the model
General method.
Our algorithm, which relies on a guess-and-verify strategy, is as follows.
1. We determine the steady-state of the Ramsey program (see below).
2. We write a code that writes the set of dynamic equations in Dynare. We use the Dynare
43
solver to double-check our steady-state computations and to simulate the model.
3. We finally verify that the set of credit-constrained islands does not change in the presence
of aggregate shocks.
Finding the steady-state.
We now describe in more detail the algorithm to find the
steady-state (step 2 of the general method).
1. We guess the set C of islands that are credit-constrained. We fix a post-tax interest rate
r and a transfer value T (typically T = 0) and a post-tax wage rate w. Then:
(a) We compute the labor supply on each island eN , using households’ first-order conditions (32). We then deduce the aggregate labor L. We compute the aggregate capital
K with FK (K, L) = — ≠1 ≠ 1.
(b) We determine individual consumption levels and asset holdings the using equations
(30), (31), and (8). We deduce a corresponding value for public spending, given by
G = F (K, L) ≠ rA ≠ wL ≠ T .
(c) We set a value of µ. We set values ÂeN , eN œ C (for credit-constrained islands). Using
(42), we then solve for ÂeN , eN œ E N \ C (unconstrained islands). We obtain then the
⁄eN , eN œ E N using (39) defining ÂeN . We finally iterate on ÂeN , eN œ C, until we
have ⁄eN = 0 for eN œ C (constrained islands). We iterate on µ until equation (44)
holds at the steady-state.
(d) We iterate on w until (43) holds at the steady-state.
2. We iterate on r until G/Y matches its target. Check that the value T = 0 is indeed the
equilibrium value, otherwise iterate. We finally verify that Euler inequalities are strict for
islands eN œ C to check that the set C of constrained islands is correct. Otherwise, we
iterate on C.
44