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</td><td>
</td><td>\begin{document}             % End of preamble and beginning of text
</td><td>
</td><td>
</td><td>\begin{titlepage}
</td><td>  \title{Does Precommitment Raise Growth?\\
</td><td>  The Dynamics of Growth and Fiscal Policy\thanks{The authors are
</td><td>grateful to two anonymous referees of this journal for helpful
</td><td>comments on an earlier draft.}}
</td><td>
</td><td>\maketitle\thispagestyle{empty}
</td><td>
</td><td>\vfill
</td><td>
</td><td>\begin{center}
</td><td>
</td><td>\begin{tabular}{ll}
</td><td>  Thomas Krichel & Paul Levine\\
</td><td>Department of Economics\verb++&
</td><td>  Department of Economics\\
</td><td>University of Surrey & University of Surrey \\
</td><td>  Guildford GU2 5XH & Guildford GU2 7XH \\
</td><td>United Kingdom & United Kingdom
</td><td>  \\ \verb+T.Krichel@surrey.ac.uk+&
</td><td>\verb+P.Levine@surrey.ac.uk+
</td><td>\end{tabular}
</td><td>\end{center}
</td><td>
</td><td>\vfill
</td><td>
</td><td>\begin{abstract}
</td><td>\noindent We develop an endogenous growth model driven by
</td><td>externalities from both private and public capital. The government levies
</td><td>distortionary taxation to finance a publicly provided consumption good and
</td><td>public infrastructure. Firms face adjustment costs.
</td><td>We  compare the optimal and time-consistent policies in a
</td><td>linear-quadratic approximation of the model. Although the time-consistent
</td><td>equilibrium is sub-optimal in terms of ex ante intertemporal welfare, it yields
</td><td>higher long-run growth and welfare,
</td><td>through an accumulation of assets by the state and
</td><td>a cut in government consumption.
</td><td>
</td><td>\noindent{\sl Keywords}: Endogenous growth; Dynamics; Fiscal policy;
</td><td>Commitment
</td><td>
</td><td>
</td><td>\noindent{\sl JEL classification}: E62; H53
</td><td>\end{abstract}
</td><td>
</td><td>\vfill
</td><td>\end{titlepage}
</td><td>
</td><td>
</td><td>
</td><td>\section{Introduction}
</td><td>
</td><td>This paper studies a model of fiscal policy with endogenous growth
</td><td>which is dynamic in three ways. First, it models both private and
</td><td>public capital as stocks rather than flows. Second, it is
</td><td>non-Ricardian with distortionary taxes, finite lives and population
</td><td>growth. The choice between debt or taxation financing of a given path
</td><td>of government spending affects GDP growth and we do not impose a
</td><td>balanced budget.  These two features imply that the model has
</td><td>transitory dynamics, a characteristic that is absent from many papers
</td><td>in the fiscal policy and endogenous growth literature. Third, the
</td><td>model allows for dynamic fiscal policy; in particular we allow
</td><td>governments to choose a different tax rate and spending level in each
</td><td>period.
</td><td>
</td><td>The model incorporates a private sector and a government. The
</td><td>government may spend on a publicly-provided consumption good and
</td><td>on augmenting an infrastructure stock that provides for an
</td><td>externality in the production of firms. In the steady state, the
</td><td>government maintains this input in production as a constant
</td><td>fraction of GDP.  Therefore the marginal productivity of capital
</td><td>is bounded away from zero and perpetual growth is possible.  This
</td><td>type of model is pioneered by \citeN{robbar90po}.  He models a
</td><td>single-good world where production is a function of labour in
</td><td>inelastic supply, a capital stock that does not depreciate, and
</td><td>the flow of public services.  The main result of that paper is
</td><td>that welfare is maximized when growth maximized. Some of the
</td><td>subsequent literature has been concerned with welfare maximization
</td><td>in the context of more elaborate models where this result may no
</td><td>longer hold.
</td><td>
</td><td>\citeN{gerglo94dy} consider the case where both private and public
</td><td>capital stock depreciate fully during the period and preferences
</td><td>are logarithmic. Rather then assuming constancy of the tax rate,
</td><td>they derive the result that the optimal rate is constant.
</td><td>\citeN{saulau95le} extends \citeN{gerglo94dy} to include
</td><td>government consumption.  It turns out that the government
</td><td>consumption is lower, and that government investment is higher,
</td><td>under welfare maximization than under growth maximization. This
</td><td>point is also emphasised by \citeN{koifut93sc} who introduce
</td><td>infrastructure as a stock rather than a flow. This is more
</td><td>realistic but makes an intertemporal welfare analysis analytically
</td><td>intractable. Despite these technical problems, \citeN{gerglo99dy}
</td><td>show that an equilibrium exists under very general conditions,
</td><td>provided that the government can precommit to a sequence of
</td><td>government expenditures.  An example for such an equilibrium is
</td><td>developed in \citeN{stetur97md}. He models public capital as a
</td><td>stock, the services of which are subject to congestion.  There are
</td><td>no adjustment costs, public and private capital can be costlessly
</td><td>transformed one into the other. In that way the model ends up with
</td><td>a single state variable which is the ratio of the two stocks.
</td><td>There is a fairly elaborate array of financing options, including
</td><td>consumption taxes, income taxes and lump-sum taxes or transfers.
</td><td>He first computes the first-best where a benevolent planner will
</td><td>directly allocate consumption and investment in the two stocks.
</td><td>The instruments of fiscal policy are sufficient to replicate the
</td><td>first best. During the transition to the optimum steady state, the
</td><td>private capital stock growth rate overshoots and the growth rate
</td><td>of the infrastructure stock undershoots.  Thus both reach the
</td><td>common long-run growth rates from opposite sides.
</td><td>
</td><td>There is a whole strand of the literature that does not attempt a
</td><td>full welfare analysis and therefore can reach analytical results
</td><td>for give policy changes.  \citeN{stetur95dc}, \citeN{micdev95mc},
</td><td>\citeN{stetur97md}, and \citeN{walfis98ej} all use a model with
</td><td>government capital to examine the effects of a given fiscal
</td><td>policy.  For example, \citeN{walfis98ej} model public capital as a
</td><td>stock to consider the case where it is subject to congestion. In
</td><td>the absence of congestion, an increase in the infrastructure
</td><td>stock, financed through a lump-sum tax, increases the long-run
</td><td>private capital stock if both factors are complements in
</td><td>production. If the degree of congestion is large, the increase in
</td><td>public infrastructure will lead to a fall in the private capital
</td><td>stock, provided that the substitutability of both factors is low.
</td><td>In a similar vein, \citeN{stecas98dc} examine a variant of the
</td><td>basic model with endogenous labour supply and show that this model
</td><td>can account for much of the recent US growth experience. Finally
</td><td>there is a separate strand of literature that examines fiscal
</td><td>policy in real business cycle models, see for example
</td><td>\citeN{marbax93am}.
</td><td>
</td><td>%\citeN{stetur95dc} use the same basic production framework is the same as
</td><td>%\citeANP{gerglo94dy}'s, but no specific functional form is assumed for the
</td><td>%production function.  An additional level of generality is added by assuming
</td><td>%that the labour supply is elastic; however taxation is lump-sum.  The paper
</td><td>%comes to the agnostic conclusion that an increase in government investment may
</td><td>%lower growth and an increase in government consumption might raise the rate of
</td><td>%growth, two results that are counter-intuitive.  Using a different model,
</td><td>%\citeN{micdev95mc} argue that these results are a consequence of the lump-sum
</td><td>%tax assumption and the ambiguity disappears as soon as
</td><td>%distortionary taxation is introduced.
</td><td>
</td><td>Our model departs differs in significant ways from all the papers mentioned
</td><td>so far.  First we use the non-Ricardian
</td><td>\citeN{menyar65st}-\citeN{olibla85po}-\citeN{phiwei89pu} demand framework
</td><td>for a realistic assessment of fiscal policy.  Second, we allow for
</td><td>time-varying taxation and government spending {\sl and\/} explicit policy
</td><td>optimisation by the government in a situation where the first-best can not
</td><td>be a achieved in equilibrium.  Finally we characterize the optimum and
</td><td>time-consistent policy trajectories, even though the exact values are
</td><td>dependent on the parameters of the model.  There is no free lunch of
</td><td>course---all these extensions require numerical simulation and therefore
</td><td>our results to not have the general power of analytical results.
</td><td>
</td><td>%In an earlier attempt (\citeN{thokri95pl}), we did not include adjustment
</td><td>%costs in the stock of capital, nor did we
</td><td>%endogenize the determination of public expenditure. In this
</td><td>%paper we jointly consider the determination of expenditure and the way that
</td><td>%expenditure is financed.
</td><td>
</td><td>The rest of the paper is organized as follows.  Section \ref{sec:model}
</td><td>sets out our model.  Section \ref{sec:intertemporalaspects} uses
</td><td>simulations on a calibrated model to compare the optimal precommitment
</td><td>fiscal policy with the time-consistent policy.  Section
</td><td>\ref{sec:conclusions} concludes the paper.
</td><td>
</td><td>\section{The Model}\label{sec:model}
</td><td>
</td><td>Our model treats both households and firms as intertemporal maximizers in a
</td><td>fairly standard fashion. We describe the behaviour of households, firms
</td><td>and the government in a closed economy. 
</td><td>
</td><td>\subsection{ Households}
</td><td>
</td><td>We consider a population of identical households who face a constant
</td><td>probability of death $\mort$ per period. It grows at the exogenous rate
</td><td>$\popu$. All individuals enjoy logarithmic felicity
</td><td>$\Feli(\timt)=\ln\Cons(\timt)+\eta\, \ln\Govc(\timt)$ from consuming a
</td><td>private consumption commodity $\Cons(\timt)$ and a publicly provided
</td><td>consumption commodity $\Govc(\timt)$. They discount at a rate $\pref$ and
</td><td>face a constant probability $\mort$ of dying in each period.
</td><td>By virtue of this exponential lifetime assumption, their expected
</td><td>lifetime utility is independent of age and given by
</td><td>\begin{equation}
</td><td>  \label{yabl2}
</td><td>  \Util(\timt)=
</td><td>  \sum_{\titP=\timt}^\infty
</td><td>  \left({1-\mort\over1+\pref}\right)^{\titP-\timt}
</td><td>  \Feli(\titP)
</td><td>\end{equation}
</td><td>where we use the uppercase letters to express magnitudes on a per-person
</td><td>level.
</td><td>
</td><td>Every household is endowed with a unit of labour that she supplies
</td><td>inelasticly to the market in exchange for a post-tax wage $W(\timt)$.  At
</td><td>the end of any period $\titm$, we can define her {\sl human\/} wealth
</td><td>$H(\titm)$ as the present value of the current and all future expected
</td><td>wages, discounted at the post-tax interest rate
</td><td>$\irpo(\timt)=\irat(\timt)\,(1-\tort(\timt))$, where $\tort(\timt)$ is the
</td><td>tax rate on all income (labour and capital) of households:
</td><td>\begin{equation}
</td><td>  \label{yabl4}
</td><td>  H(\titm)=\sum_{\titP=\timt}^\infty{(1-\mort)^{\titP-\titp}\,W(\titP)
</td><td>    \over1+\irpo(\tiPm)}
</td><td>\end{equation}
</td><td>Human wealth is the same for all living individuals, irrespective
</td><td>of their age, because they all face the same death rate and
</td><td>because the wage is not dependent on age. That does not mean,
</td><td>however, that households of all ages will have the same
</td><td>consumption, because recently born households have no non-human
</td><td>wealth, which they only start accumulating after birth. Non-human
</td><td>wealth $X(\timt)$ takes the form of physical capital or government
</td><td>bonds, and because of arbitrage between both types of assets, they
</td><td>must earn the same return.  At time $\timt$ the household born in
</td><td>$\titP<\timt$ has some non-human wealth $X(\titP,\titm)$ at her
</td><td>disposal that she accumulated up to the end of period $\titm$.
</td><td>When the household dies she leaves an {\sl
</td><td>  unintentional\/} bequest, her non-human wealth, at the beginning of the
</td><td>period where death occurs. To model this set-up the following construction
</td><td>is introduced. There is an insurance company that takes the financial
</td><td>post-tax wealth of each dead household. It then distributes these assets as
</td><td>a premium $\inpr$ paid on the holdings of assets.  If the insurance company
</td><td>has no operating cost, the premium satisfies the zero-profit condition:
</td><td>$1+\inpr-1/(1-\mort)=0$.  Hence we have the dynamics of non-human wealth as:
</td><td>\begin{equation}
</td><td>  \label{yabl10}
</td><td>  X(\titP,\timt)={(1+\irpo(\titm))\,X(\titP,\titm)\over1-\mort}
</td><td>  +W(\timt)-\Cons(\timt)
</td><td>\end{equation}
</td><td>If we solve the period-to-period budget constraint (\ref{yabl10}) forwards
</td><td>in time and
</td><td>make the conventional transversality assumption that the present value of
</td><td>future wealth will tend to zero, the lifetime budget constraint of a
</td><td>household born at time $\titP$ is:
</td><td>\begin{equation}\label{yabl7}
</td><td>  X(\timt,\timt)=\sum_{\titP=\titp}^\infty
</td><td>  {(1-\mort)^{\titP-\timt}
</td><td>    \left[C(\titP)-W(\titP)\right]
</td><td>    \over1+{\irpo(\titm,\titP-1)}}=0
</td><td>\end{equation}
</td><td>since there are to bequests and where we have defined the interest
</td><td>rate between period $\timt$ and $\titP-1$ as:
</td><td>\begin{equation}\label{yabl00}
</td><td>  1+\irpo(\timt,\titP-1)
</td><td>  =\prod_{\tiPP=\timt}^{\titP}(1+\irpo(\tiPP-1))
</td><td>\end{equation}
</td><td>and $\irpo(\timt,\timt)=\irpo(\timt)$.
</td><td>The household's problem is to maximize (\ref{yabl2}) under (\ref{yabl7}).
</td><td>The familiar first order condition is:
</td><td>\begin{equation}
</td><td>  \label{yabl11}
</td><td>  {C(\titP)\over C(\timt)}
</td><td>  ={1+\irpo(\titm)\over1+\pref}
</td><td>\end{equation}
</td><td>
</td><td>This completes the study of the individual household. All households of the
</td><td>same age are identical, but households of different ages have different
</td><td>non-human wealth.  We therefore need to aggregate over different age
</td><td>levels. This leads to a ``Yaari-Blanchard'' demand function\footnote{
</td><td>  Details of the aggregation procedure are given in the working paper
</td><td>  version \citeN{thokri96does}. See also \citeN{olibla85po} for a
</td><td>  continuous-time version of (\ref{eq:tabyaariblanchard}). Note that in the
</td><td>  Ricardian case $\mort=\popu=0$ and
</td><td> \eqref{eq:tabyaariblanchard} gives us the familiar
</td><td>  Keynes-Ramsey rule.},  which may be written as:
</td><td>\begin{align}
</td><td>  \begin{split}
</td><td>    0&=\left({\mort+\popu\over1+\popu}-{1+\pref\over1-\mort}\right)
</td><td>    \cons(\timt)
</td><td>    +{1+\irat(\titm)\,(1-\tort(\timt))\over1+\grow(\timt)}\,\cons(\titm)\\
</td><td>    &\quad-{(\mort+\pref)\,(\mort+\popu)\over(1-\mort)
</td><td>      (1+\popu)}\,\weal(\timt)
</td><td>  \end{split}\label{eq:tabyaariblanchard}
</td><td>\end{align}
</td><td>Here the lower-case variable $\cons(\timt)$
</td><td>refers to aggregate level consumption in per-GDP form\footnote{
</td><td>All lower-case variables---apart from the interest, tax and 
</td><td>growth rates---are in per GDP form. All stocks refer to end-of-period.}
</td><td>$\grow(\timt)=[\Inco(\timt)-\Inco(\titm)]/\Inco(\titm)$ is the rate of
</td><td>growth of GDP.
</td><td>
</td><td>Wealth is composed of physical capital $\kapi(\timt)$ and government debt
</td><td>$\debt(\timt)$:
</td><td>\begin{equation}
</td><td>  \weal(\timt)=\debt(\timt)+\kapi(\timt)\label{eq:tabwealth}
</td><td>\end{equation}
</td><td>The next two subsections deal with the accumulation of
</td><td>these assets.
</td><td>
</td><td>\subsection{ Firms}
</td><td>
</td><td>%All income to households is taxed at rate $\tort(\timt)$, therefore
</td><td>%consumption depends on the post-tax interest rate
</td><td>%$\irat(\titm)\,\tort(\timt)$. Interest is earned on wealth $\weal(\timt)$
</td><td>%which, according to (\ref{eq:tabwealth}) is composed of government debt
</td><td>%$\debt(\timt)$ and capital $\kapi(\timt)$.
</td><td>
</td><td> Capital evolves under the
</td><td>impact of depreciation and investment as in \eqref{eq:tabkapievolution}.
</td><td>Investment is given by the solution of a profit maximization problem as
</td><td>follows.  Imagine a large number of identical firms. We use uppercase
</td><td>notation for per firm levels.  For every date $\titP>\timt$, the problem of
</td><td>each firm is to choose investments $\Inve(\titP)$, and employment
</td><td>$\Labo(\titP)$ that maximize the discounted sum of future profits:
</td><td>\begin{equation}\label{eq:firmtarget}
</td><td>  \sum_{\titP=\timt}^\infty
</td><td>  {Y(\titP)-\wage(\titP) \,\effl(\titP)\,\Labo(\titP)-
</td><td>  \Inve(\titP)
</td><td>  \left[1+\adco\!\left(({\Inve(\titP)-(\grow(\titP)+\depr)\,\Kapi(\tiPm)
</td><td>        )/\Kapi(\titP-1)}\right)\right]\over1+\irat(\titm,\titP-1)}
</td><td>\end{equation}
</td><td>where $\irat(\titm,\titP-1)$ is defined in the same way as
</td><td>\eqref{yabl00}.
</td><td>
</td><td>The function $\adco(\cdot)$ in the expression \eqref{eq:firmtarget} gives
</td><td>adjustment costs that the firm pays when investing. We make the usual
</td><td>assumptions that $\adco'(\cdot)>0$, $\adco''(\cdot)>0$, and $\adco(0)=0$.
</td><td>Since $\Inve(\timt)=(\grow+\depr)\,\Kapi(\titm)$ on a balanced
</td><td>growth path, the last assumption implies that there are no adjustment
</td><td>costs in this set-up.
</td><td>
</td><td>Output is given by the Cobb-Douglas production function:
</td><td>\begin{equation}\label{eq:cobbdouglasproduction1}
</td><td> Y(\timt)=\Kapi(\titm)^\alpha\left[\effl(\timt)\,\Labo(\timt)\right]^{1-
</td><td>   \alpha}
</td><td>\end{equation}
</td><td>Here $\effl(\timt)$ is the efficiency of the labour force $\Labo(\timt)$.
</td><td>We adopt the approach to endogenous growth pioneered by \citeN{kenarr62st}
</td><td>and \citeN{paurom86po} and allow the productivity of each worker to depend
</td><td>not only on the capital internal to the firm but also on externalities from
</td><td>the average capital  available to the other firms $\Kapi(\titm)$, and from
</td><td>the infrastructure put in place by the government $\Kago(\titm)$, i.e.
</td><td>\begin{equation}\label{eq:extern}
</td><td>  \effl(\timt)=\bar\effl^{1/(1-\alpha)}\,
</td><td>  {\Kago(\titm)^{\gamma_1}\,\Kapi(\titm)^{1-\gamma_1}
</td><td>\over\Labo(\timt)}.
</td><td>\end{equation}
</td><td>This feature of the model drives long-run endogenous growth. The
</td><td>aggregate production function now becomes:
</td><td>\begin{equation}\label{eq:cobbdouglasproduction}
</td><td> Y(\timt)=\bar\effl(\timt)\,\Kago(\titm)^{1-\gamt}\,\Kago(\titm)^{\gamt}
</td><td>\end{equation}
</td><td>where $\gamt=\alpha+(1-\alpha)\,(1-\gamma_1)$.
</td><td>In per-GDP form, this is the equation is written as
</td><td>\begin{equation}
</td><td>  \grow(\titp)=\bar\effl\,\kago(\timt)^{1-\gamma_2}\,
</td><td>  \kapi(\timt)^{\gamma_2}-1.
</td><td>\label{eq:tabproduction}
</td><td>\end{equation}
</td><td>Performing the profit maximization, and aggregating over
</td><td>all firms, we get:
</td><td>\begin{equation}
</td><td>    \begin{split}
</td><td>    0&={\alpha\,(1+\grow(\titp))\over\kapi(\timt)}
</td><td>    +\psi\,{(1+\grow(\titp))^3\,\inve(\titp)^3\over\kapi(\timt)^3}\\
</td><td>    &\quad-\psi\,{(\grow(\titp)+\depr)
</td><td>        \,(1+\grow(\titp))^2\,\inve(\titp)^2
</td><td>        \over\kapi(\timt)^2}\\
</td><td>  &\quad+(1-\depr)\,\tobq(\titp)-(1+\irat(\timt))\,\tobq(\timt)
</td><td>    \end{split}\label{eq:tabinve}
</td><td>\end{equation}
</td><td>where $\tobq$ is Tobin's $\tobq$. It is defined as
</td><td>\begin{equation}
</td><td>    \begin{split}
</td><td>      \tobq(\timt)&=1+{\psi\over2}\left({(1+\grow(\timt))\,\inve(\timt)
</td><td>          -(\depr+\grow(\timt))\,\kapi(\titm)
</td><td>          \over\kapi(\titm)}\right)^2\\
</td><td>      &\quad+{(1+\grow(\timt))\,\inve(\timt)\over\kapi(\titm)}
</td><td>      \,\psi\,{(1+\grow(\timt))\,\inve(\timt)
</td><td>        -(\depr+\grow(\timt))\,\kapi(\titm)\over
</td><td>        \kapi(\titm)}
</td><td>  \end{split}\label{eq:tabtobq}.
</td><td>\end{equation}
</td><td> Note that
</td><td>this implies that investment, like consumption, is forward-looking.
</td><td>The evolution of the capital stock is given by:
</td><td>\begin{equation}
</td><td>  \kapi(\timt)={1-\depr\over1+\grow(\timt)}\,\kapi(\titm)+\inve(\timt)
</td><td>  \label{eq:tabkapievolution}
</td><td>\end{equation}
</td><td>
</td><td>%\eqref{eq:tabequilibrium} states the market clearing condition.
</td><td>
</td><td>
</td><td>\subsection{ Government}
</td><td>
</td><td>Government debt---the second component of wealth in
</td><td>\eqref{eq:tabwealth}---is issued by the government to satisfy its budget
</td><td>identity:
</td><td>\begin{equation}
</td><td>    \debt(\timt)={1+\irat(\titm)\over1+\grow(\timt)}\,\debt(\titm)
</td><td>    +\gosp(\timt)-\taxa(\timt)\label{eq:tabgovernmentbudgetconstraint}
</td><td>\end{equation}
</td><td>Tax revenue $\taxa(\timt)$ is defined as
</td><td>\begin{equation}
</td><td>    \taxa(\timt)=\tort(\timt)\left[1-{\depr\,\kapi(\titm)\over
</td><td>        1+\grow(\timt)}\right]
</td><td>    \label{eq:tabtaxrevenue}
</td><td>\end{equation}
</td><td>where $\tort(\timt)$ is the tax rate. The term in square brackets assures
</td><td>that capital stock depreciation is tax-deductible.  Government spending
</td><td>$\gosp(\timt)$ is split into consumption spending
</td><td>$\govc(\timt)$ and government investment $\govi(\timt)$. Government %
</td><td>investment generates the same kind of adjustment costs as private
</td><td>investment.  Total government spending is therefore given by:
</td><td>\begin{equation}
</td><td>    \gosp(\timt)=\govc(\timt)+\govi(\timt)
</td><td>    \left[1+{\psi\over2}\!\left({(1+\grow(\timt))\,
</td><td>          \govi(\timt)-
</td><td>          (\depr+\grow(\timt))\,\kago(\titm)\over\kago(\titm)}\right)^2\right]
</td><td>    \label{eq:tabgovernmentspending}
</td><td>\end{equation}
</td><td>Infrastructure is assumed to depreciate at the same rate as
</td><td>private capital. Therefore the evolution of infrastructure is
</td><td>given by: 
</td><td>\begin{equation}
</td><td>    \kago(\timt)={1-\depr\over1+\grow(\timt)}\,\kago(\titm)+\govi(\timt)
</td><td>    \label{eq:tabkagoevolution}
</td><td>\end{equation}
</td><td>
</td><td>We assume that the government is perfectly benevolent; however there
</td><td>is no representative household in our overlapping generations model,
</td><td>but rather a spectrum of young and old households and those yet to be
</td><td>born.  Following a suggestion by \citeN{guical88me}\footnote{They
</td><td>showed that a general optimization problem that takes account of
</td><td>generational diversity could be broken down into a problem of
</td><td>maximizing a function of aggregate consumption and a second problem of
</td><td>distributing aggregate consumption between generations. } we use
</td><td>aggregate consumption to represent households of different
</td><td>generations, to arrive at a social welfare function $\tilde{\util}$
</td><td>for the government with the form:
</td><td>\begin{equation}\label{eq:govtarget}
</td><td>  \begin{aligned}
</td><td>  \tilde\util(\timt)&=\sum_{\titP=\timt}^\infty
</td><td>   \left({1-\mort\over1+\pref}\right)^{\titP}
</td><td>  \,\tilde{\feli}(\titP)\quad\text{where}\\
</td><td>  \tilde{\feli}(\titP)&=\ln\cons(\titP)+
</td><td>    \eta\,\ln\govc(\titP)
</td><td>  +(1+\eta)\,
</td><td>  \sum_{\tiPP=\timt+1}^{\titP}
</td><td>\ln(1+\grow(\tiPP))
</td><td>  \end{aligned}
</td><td>\end{equation}
</td><td>To derive the solvency constraint---as opposed to the identity
</td><td>\eqref{eq:tabgovernmentbudgetconstraint}---for the government,
</td><td>first consider the ``growth-adjusted'' real interest rate over
</td><td>$[\titm,\timt]$ as
</td><td>$\rho(\timt)=(1+\irat(\titm))/(1+\grow(\timt))-1$. Then solving
</td><td>\eqref{eq:tabgovernmentbudgetconstraint} forward in time, we
</td><td>transform the budget identity into a solvency constraint at time
</td><td>$\timt$, analogous to \eqref{yabl7}
</td><td>\begin{equation}
</td><td>  \label{eq:budgetconstraint}
</td><td>  \debt(\titm)=\sum_{\titP=0}^{\infty}
</td><td>   {\taxa(\tptP)-\gosp(\tptP)\over(1+\rho(\timt))\,
</td><td>    (1+\rho(\titp))\dots(1+\rho(\tptP))}
</td><td>\end{equation}
</td><td>provided that the transversality or ``no-Ponzi'' condition
</td><td>\begin{equation}
</td><td>  \label{eq:noponzi}
</td><td>   \lim_{\titP\to\infty}\,{\debt(\tptP)\over(1+\rho(\timt))\,
</td><td>   (1+\rho(\titp))\dots(1+\rho(\tptP))}=0
</td><td>\end{equation}
</td><td>holds. In (\ref{eq:budgetconstraint}) and (\ref{eq:noponzi}) we assume that
</td><td>eventually $\rho(\timt)>0$.  This is a feature of the Yaari-Blanchard
</td><td>consumption/savings model and rules out dynamic inefficiency. According to
</td><td>(\ref{eq:budgetconstraint}) a government in debt with $\debt(\timt)>0$ must,
</td><td>sometime in the future, run primary surpluses to be solvent.  The
</td><td>transversality condition (\ref{eq:noponzi}) does not require a {\sl
</td><td>  stable\/} debt/GDP ratio but merely that, in the long run, it does not
</td><td>increase faster than the growth-adjusted real interest rate $\rho(\timt)$.
</td><td>However in a world with even very small departures from perfectly
</td><td>functioning capital markets, the notion of unbounded government debt/GDP
</td><td>ratios does not appeal.  A stronger concept of solvency is that debt/GDP
</td><td>ratios stabilize. We enforce this condition through a small penalty
</td><td>attached to debt in the government's loss function which reflects the costs
</td><td>of issuing debt or acquiring assets if $\debt$ is negative. We also
</td><td>include the cost of collecting taxes, therefore replacing
</td><td>$\tilde{\feli}$ in the
</td><td>social welfare function \eqref{eq:govtarget} by $\tilde{\tilde{\feli}}$:
</td><td>\begin{equation}
</td><td>  \label{eq:singleperiodwelfare}
</td><td>  \tilde{\tilde{\feli}}(\timt)
</td><td>  =\tilde{\feli}(\timt)-\eta_{\debt}\,(\debt(\timt))^2
</td><td>  -\eta_{\tort}\,(\tort(\timt))^2
</td><td>  -\eta_{\Delta\tort}\,(\Delta\tort(\timt))^2
</td><td>\end{equation}
</td><td>The third term in (\ref{eq:singleperiodwelfare}) with a small value for
</td><td>$\eta_\debt$ is sufficient to ensure a stable debt/GDP ratio, i.e.~strong
</td><td>solvency.  The final two terms penalize both large changes and large levels
</td><td>in the tax rate. We think of the inclusion of these extra terms as imposing
</td><td>a constraint on the liabilities or assets the government can acquire and on
</td><td>the extent of taxation it can impose in any one period. All these terms
</td><td>cover features not modelled explicitly.
</td><td>
</td><td>The government's optimization problem at time $\timt$ is the maximization
</td><td>of \eqref{eq:govtarget}, with $\tilde v$ replaced by $\tilde{\tilde{v}}$
</td><td>given by \eqref{eq:singleperiodwelfare}.  Maximisation takes place with
</td><td>respect to the government's choice variables
</td><td>$\govc(\titP),\govi(\titP),\tort(\titP)$, $\forall$ $\titP\ge\timt$,
</td><td>subject to the model of the private sector and the condition that
</td><td>commodity markets clear:
</td><td>\begin{equation}
</td><td>  \cons(\timt)+\inve(\timt)\left[1+{\psi\over2}
</td><td>    \left({\inve(\timt)\,(1+\grow(\timt))-(\depr+\grow(\timt))\,
</td><td>        \kapi(\titm)
</td><td>        \over\kapi(\titm)}\right)^2\right]+\gosp(\timt)=1
</td><td>  \label{eq:tabequilibrium}
</td><td>\end{equation}
</td><td>This equation completes the model. 
</td><td>
</td><td>\subsection{ Equilibria}
</td><td>
</td><td>There are two equilibrium concepts depending on whether the government can
</td><td>precommit to a given trajectory for fiscal instruments over the future. If the
</td><td>government can precommit it can exercise the greatest leverage over the
</td><td>private sector. An announced path of instrument settings is credible
</td><td>and affects private sector behaviour immediately in the desired way. For
</td><td>instance the announcement of low taxes in the future will immediately raise
</td><td>savings, lower the real interest rate and increase private investment.
</td><td>
</td><td>The solution to the optimal policy with precommitment is found
</td><td>by standard optimal control techniques using Lagrangian multipliers.
</td><td>Although the private sector is atomistic and therefore can not
</td><td>act strategically, the equilibrium concept corresponds to an open-loop
</td><td>Stackelberg equilibrium for dynamic games between strategic
</td><td>players described in chapter 7 of \citeN{tambar95dynamic}.
</td><td>%Commitment may be achieved through some externally imposed reputational
</td><td>%mechanism but we do not consider these in this paper.
</td><td>
</td><td>
</td><td>When a government cannot commit itself to a future policy, it must act
</td><td>each period to maximize its welfare function, given that a similar
</td><td>optimization problem will be carried out in the next period.  Formally, the
</td><td>policymaker maximizes at time $\timt$ a welfare function
</td><td>$\tilde\util(\timt)$ such that:
</td><td>\begin{equation}
</td><td>  \label{timeconsistent}
</td><td>  \tilde\util(\timt)=\tilde{\tilde\feli}(\timt)+\difa\,\tilde\util(\titp)
</td><td>\end{equation}
</td><td>where $\tilde{\tilde\feli}(\timt)$ is the single-period felicity given in
</td><td>\eqref{eq:singleperiodwelfare} and $\tilde\util(\timt)$ is evaluated on the
</td><td>assumption that an identical optimization exercise is carried out from time
</td><td>$\titp$ onwards. The solution to this problem is found by dynamic
</td><td>programming and, unlike the precommitment policy, leads to a
</td><td>time-consistent trajectory or rule for instruments\footnote{See Appendix C
</td><td>  of \citeN{thokri96does} for details.}.  This equilibrium concept
</td><td>corresponds to a feedback Nash equilibrium for dynamic games, see
</td><td>\citeN{tambar95dynamic}, chapter 6. It has the property of being
</td><td>Markov-perfect---that is the government instruments and the private-sector
</td><td>forward-looking variables depend only on current values of the state
</td><td>variables. Following this solution procedure, a rational
</td><td>(utility-optimizing) government will never wish to deviate from the
</td><td>policies designed at the beginning of the planning period.
</td><td>
</td><td>
</td><td>\section{Calibration and Results}\label{sec:intertemporalaspects}
</td><td>
</td><td>\subsection{ Calibration}
</td><td>
</td><td>%%\label{sec:calibration}
</td><td>The model is calibrated around a steady-growth state fitted to the economy
</td><td>of the United States in 1990.  The growth rate is the central variable of
</td><td>the model whose deduction as an endogenous variable would be subject to
</td><td>multiple numerical solutions.  Therefore we first choose $\grow=2.5\%$. We
</td><td>also fix $\irat=5\%$. We chose the mortality $\mort=2\%$, and overall
</td><td>population growth $\popu=1\%$, to take account of immigration.
</td><td>
</td><td>We collect basic national accounting data from the US Department of
</td><td>Commerce's Economic Bulletin Board.  For the capital stock, we have data
</td><td>available from \citeN{oecdst97flows} about the net capital stock $\Kapi=
</td><td>12706.7$ billion \$US. This is the figure we choose for the private capital
</td><td>stock, i.e.~$\kapi\approx2.4$.  Taking $\inve$ to stand for fixed
</td><td>investment and using the
</td><td>observed figure for the capital stock $\kapi$ we calibrate
</td><td>the depreciation rate.  We use our assumption that the rate of depreciation
</td><td>of private and public capital are equal to deduce the public infrastructure
</td><td>stock from the government expenditure on infrastructure.  To estimate that
</td><td>expenditure, we collect data from the IMF Government Finance Statistics
</td><td>yearbook.  We assume that the categories 4, 5 and 12 are the expenditure
</td><td>contributing to the capital stock of the government and aggregate federal
</td><td>state and local government expenditure.  We can then compute $\infr$, the
</td><td>proportion of investment expenditure, as $\infr\approx36\%$.
</td><td>Adding the federal, state and local
</td><td>debt and dividing by GDP gives $\debt=53.3\%$.
</td><td>
</td><td>The exogenous parameters are summarized in the top two lines of
</td><td>Table \ref{tab:calibration}. These lines also state two additional
</td><td>pieces of heuristics that we use to calibrate the model.  First,
</td><td>we calibrate the government felicity parameter $\eta$ on the ratio
</td><td>of public versus private consumption.  This would be the result of
</td><td>a static optimization exercise: maximize $\ln C +\eta \ln G$
</td><td>subject to $G+C$ being constant is $G=\eta C$; i.e., in per-GDP
</td><td>form $g=\eta\,c$. Notice that we are not assuming that dynamically
</td><td>optimal policies P or T are being pursued in the central
</td><td>calibration about which we linearise the model. The reason for
</td><td>this is that we need a linearised model in order to compute these
</td><td>regimes. However we do carry out sensitivity analysis on $\eta$
</td><td>and other parameters. Second we calibrate the relative
</td><td>productivity of the infrastructure to the ratio of infrastructure
</td><td>to the total (i.e.~public plus private) capital
</td><td>stock.\footnote{There has been much
</td><td>  debate about this parameter.  The empirical results of \citeN{davash89mo}
</td><td>  and \citeN{alimun90ne} suggest that $1-\gamt$ is 39\% and 34\%,
</td><td>  respectively.  However we feel that these estimates should be upper
</td><td>  bounds for $1-\gamt$ because other studies have found much lower value.
</td><td>  In the extreme case, \citeN{johtat90sl} suggest that $1-\gamt$ is not %
</td><td>  statistically significant from zero. We are however confident that our
</td><td>  result carry over to a wide variety of scenarios because we have
</td><td>  conducted extensive sensitivity analysis (see Appendix A).}  The other
</td><td>lines in Table \ref{tab:calibration} illustrate how we derive the remaining
</td><td>values from the steady-state relationships.
</td><td>
</td><td>\begin{table}[]
</td><td>\tabcolsep=2.25pt
</td><td>\newlength\ec
</td><td>\setlength{\ec}{15pt}
</td><td>\begin{tabular}{@{}rl@{\hspace{\ec}}rl@{\hspace{\ec}}rl@{\hspace{\ec}}rl@{\hspace{\ec}}rl@{\hspace{\ec}}rlrl@{}}
</td><td> $\grow$&$=2.5\%$&$\irat$&$= 5\%$&$\mort$&$=2\%$&$\popu$&$=1\%$&$\inve$&$=15\%$
</td><td>&$\eta$&$={\govc/\cons}$&$\approx18\%$\\
</td><td>$\kapi$&$=241\%$&$\gosp$&$=18\% $&$\infr$&$=36\%$&$\debt$&$=53.3\%$&$\bar\epsilon$&$=73\%$&$\gamt$&$={\kapi/(\kago+\kapi)}$&$\approx75\%$\\
</td><td>$\depr$&
</td><td>  \multicolumn{11}{l}{$=(1+\grow)\,\inve/\kapi-\grow$}&
</td><td>  $\approx\phantom{0}6\%$\\
</td><td>  $\kago$&\multicolumn{11}{l}{$=\infr\,\gosp\,(1+\grow)/(\grow+\depr)$}&$\approx 79\%$\\
</td><td>  $\bar\epsi$&\multicolumn{11}{l}{$=(1+\grow)\,{\kago}^{-\gamt}\,\kapi^{\gamt-1}$}&$\approx73\%$\\
</td><td>  $\alpha  $&\multicolumn{11}{l}{$={(\irat+\depr)\,\kapi/(1+\grow)}$}&$\approx26\%$\\
</td><td>  $\cons$&\multicolumn{11}{l}{$=1-\inve-\gosp$}&$\approx67\%$\\
</td><td>  $\govc$&\multicolumn{11}{l}{$=\gosp\,(1-\infr)$}&$\approx12\%$\\
</td><td>  $\tort$&\multicolumn{11}{l}{$=\debt\,(\irat-\grow)+(1+\grow)\,\gosp\,(1+\grow-\depr\,\kapi)$}&$\approx22\%$\\
</td><td>  $\pref$
</td><td>  &\multicolumn{11}{l}{$=(1+\grow)\,(\debt+\kapi(\,\mort\,(1+\popu)
</td><td>  +\cons\bigl[\irat\,(1-\tort)\,(1+\popu)\,(1-\mort)+(1-\mort)$}\\
</td><td>  &\multicolumn{11}{l}{$\quad\,(\popu-\mort)
</td><td>  +(\popu+\mort)\,\mort\,(1-\grow)\bigr]
</td><td>  /\left[\cons\,(1+\popu)+(\mort+\popu)\,(\kapi+\debt)\right]/(1
</td><td>  +\grow)\,$}&$\approx\,1.8$\%
</td><td>\end{tabular}
</td><td>\caption{Calibration}\label{tab:calibration}
</td><td>\end{table}
</td><td>
</td><td>
</td><td>Finally we choose the parameters $\eta_{\tort}$, $\eta_{\Delta\tort}$, and
</td><td>$\eta_{\debt}$ in (\ref{eq:singleperiodwelfare}). If we put
</td><td>$\eta_{\tort}=\eta_{\Delta\tort}=0$ which implies no constraint on the size
</td><td>of the tax rate in any one period, but enforce strong solvency by setting
</td><td>$\eta_{\debt}$ equal to a small value (in fact we find that 0.1 is
</td><td>sufficient for this purpose) we obtain optimal trajectories under
</td><td>precommitment for which the tax rate in the first period is over 100\%,
</td><td>although the tax rate falls sharply thereafter. This oddity reflects a
</td><td>number of deficiencies in our model including the absence of other tax
</td><td>distortions, the absence of explicit modelling of collection costs,
</td><td>political constraints on high tax rates etc, as well as the shortcomings of
</td><td>a linear-quadratic approximation.  Fortunately quite small values of
</td><td>$\eta_{\tort}$ and $\eta_{\Delta\tort}$ remedy this feature of the
</td><td>simulation.  We choose $\eta_{\tort}=\eta_{\Delta\tort}=1$.  These
</td><td>are small values because in our quadratic approximation the marginal rate
</td><td>of substitution between the consumption/GDP ratio $\cons$ and $\tort$ along
</td><td>the modified utility curve is
</td><td>$-\eta_{\tort}\,\tort^*\,\cons^2/\cons^*=.12\,\eta_{\tort}$ for our
</td><td>calibration.
</td><td>
</td><td>
</td><td>\subsection{ Results}
</td><td>
</td><td>\begin{table}[h]
</td><td>\begin{center}
</td><td>\begin{tabular}{crr||crr}
</td><td>   & precommitment &time-cons.&& precommitment &time-consistent\\\hline
</td><td>   $\debt_\infty$&$-71.451$&$-75.492$&$\tort_\infty$&$ -4.107$&$ -9.269$  \\
</td><td>   $\grow_\infty$&$  0.342$&$  0.876$&$\irat_\infty$&$  0.005$&$  0.301$  \\
</td><td>   $\cons_\infty$&$  1.062$&$  4.855$&$\govc_\infty$&$ -2.129$&$ -7.589$  \\
</td><td>   $\inve_\infty$&$  0.493$&$  0.741$&$\govi_\infty$&$  0.574$&$  1.993$  \\
</td><td>   $\kapi_\infty$&$ -2.942$&$-13.842$&$\kago_\infty$&$  4.009$&$ 16.576$  \\
</td><td>  $\tilde\util(\infty)$&$7.257$&$14.202$&$\tilde\util_0$&$ -217.6$&$-218.2$\\
</td><td>\end{tabular}
</td><td>\caption[results]{Precommitment and Time-Consistent
</td><td>  Policies.  Variables are in per cent and measured as deviations about the
</td><td>  original steady-state.  For example, $\grow_\timt=\grow(\timt)-\grow$ where
</td><td>  $\grow(\timt)$ is actual and $\grow$ is steady-state growth.
</td><td>  $\tilde\util(\infty)$ is the steady-state welfare
</td><td>  evaluated as in \eqref{welsteady}.
</td><td>  $\tilde\util_0$ is the quadratic approximation of
</td><td>  welfare at time 0, as calculated by the simulation software. This
</td><td>  number is not given as a per cent.
</td><td>}
</td><td>\label{tab:results}
</td><td>\end{center}
</td><td>\end{table}
</td><td>
</td><td>Table \ref{tab:results} reports the long-run steady-state values
</td><td>of key variables for the precommitment (P) and time-consistent (T)
</td><td>regimes as deviations about the original steady state.  In both
</td><td>regimes, debt becomes negative i.e., the government accumulates
</td><td>assets. Taxes fall in the long run, but they fall by more than
</td><td>government spending and the income that the assets accumulated
</td><td>generate make up for the difference. The most important difference
</td><td>between the regimes is the size of the long-run growth rate.  Both regimes
</td><td>improve over the base line in terms of growth, but the growth rate
</td><td>in the T regime is over .5\% higher. The immediate reason for this
</td><td>can be seen by examining changes in the per-GDP government and
</td><td>private capital stocks ($k^g$ and $k$) respectively. In
</td><td>linear-deviation form \eqref{eq:tabproduction} becomes
</td><td>\begin{equation}
</td><td>\label{lineargrowth}
</td><td> n_{t+1}=\frac{n}{k^g+k}\,[k_t^g+k_t]
</td><td>\end{equation}
</td><td>using $\gamt=\kago/(\kapi+\kago)$ from the calibration 
</td><td>in Table \ref{tab:calibration}. 
</td><td>Hence from \eqref{lineargrowth} growth increases if $k_t^g+k_t$ increases,
</td><td>i.e., if government capital stock increases by more than private capital
</td><td>stock decreases. This happens under both regimes, but more so under T,
</td><td>which is why growth also increases by more. However it should be noted that
</td><td>the low value for $\kapi$---which is the private capital stock per GDP---is
</td><td>also a result of GDP expanding faster in the T regime and this is confirmed
</td><td>by the higher investment in this regime.
</td><td>
</td><td>To complete the story we need to understand why capitals stocks
</td><td>change in this way. However first
</td><td>we consider the long-run implications of policy
</td><td>for welfare. The welfare of an individual who is born and lives in
</td><td>the steady state is equal to
</td><td>\begin{equation}\label{welsteady}
</td><td>  \tilde u(\infty)= \frac{1+\pref}{\mort+\pref}\,
</td><td>  [\ln\cons+\eta\,\ln\govc
</td><td>  +{(1+\eta)\,(1-\mort)\over\pref+\mort}\,\ln(1+\grow)]
</td><td>\end{equation}
</td><td>
</td><td>From \eqref{welsteady} the steady-state intertemporal welfare
</td><td>depends on utility from current consumption, $c+\eta\,g^c$, and
</td><td>growth $n$. From Table 3.2 with
</td><td>$\eta=0.18$ both these components of intertemporal welfare are
</td><td>higher in T relative to P in the long run for reasons we discuss
</td><td>below; hence long-run steady-state welfare is also higher.
</td><td>
</td><td>\renewcommand{\arraystretch}{.2}
</td><td>
</td><td>
</td><td>Now we can evaluate the {\sl growth-rate equivalent\/} of a
</td><td>steady-state regime change. Suppose that it would be possible to
</td><td>jump from the initial steady state to the steady state of the P
</td><td>and/or the T regime. By how much would the growth rate in the
</td><td>initial equilibrium have to raise in order to reflect that change?
</td><td>If we use the results from Table \ref{tab:results}, we find that
</td><td>the long run of the optimal regime corresponds to an increase in
</td><td>growth by .35\%. That is important, but not spectacular. The
</td><td>change from the initial steady state to the T regime steady state
</td><td>has a growth rate equivalent of .87\%.  The change from the
</td><td>steady-state of the P regime that of the T regime therefore
</td><td>corresponds to an increase in growth by .52\%.
</td><td>
</td><td>We have tested the properties that the long run of the T regime
</td><td>involves higher growth and welfare than the P regime over a wide
</td><td>range of parameter settings. Results are reported in Appendix A.
</td><td>Our sensitivity analysis suggests that our main findings are
</td><td>remarkably robust with respect to the calibration of the model.
</td><td>
</td><td>
</td><td>Now let us return to the question of why T and P regimes differ in the
</td><td>accumulation of private and public capital. Figures 3.1 to 3.9 show the
</td><td>trajectories, for the key variables, all reported in deviation form about
</td><td>their baseline values. The overall profile of taxation and expenditure
</td><td>under the optimal (time-inconsistent) policy is as follows. A large burst
</td><td>of taxation in the first periods is followed by a decline in the tax rate.
</td><td>However we also see a later increase in taxation, such that the limiting
</td><td>steady-state tax rate is still positive.  The explanation for this profile
</td><td>is quite familiar.  The installed capital stock is predetermined at the
</td><td>beginning of the control period, i.e.~the start of control is not expected
</td><td>by the private sector.  Therefore a tax on that stock mimics a lump-sum
</td><td>tax. Using a heavy tax in the beginning therefore minimizes the welfare
</td><td>cost of taxation.\footnote{Note that this result is not dependent on the
</td><td>  finite-life aspects of the model.  All that matters is that taxation is
</td><td>  distortionary.} Thus for both P and T regimes government investment is
</td><td>financed by a combination of a increase in the tax rate, $\tort$ and a
</td><td>reduction in government consumption, $\govc$. Both these changes are
</td><td>concentrated at the beginning of the planning period. By implementing the
</td><td>tax increase in this way (in effect an initial tax surprise) its
</td><td>distortionary impact on private investment is contained. Private
</td><td>consumption ($c$) falls in the short run but increases in the long run. The
</td><td>fall in $\cons$ and $\govc$ crowd-in the increase in the public capital-GDP
</td><td>ratio $\kago$ which more than compensates for the reduction in $\kapi$,
</td><td>brought about by an interest rate rise, so that growth increases.
</td><td>
</td><td>All these changes take place in both P and T regimes when
</td><td>implemented starting at the baseline; however the time-consistency
</td><td>constraint means that the changes are more pronounced under T; $\cons$
</td><td>and $\govc$ fall
</td><td>and taxation rises by more in the short-run under T and rise by more
</td><td>in the long-run, crowding-in a bigger increase in public
</td><td>capital stock and allowing for a much lower tax rate. Policy is of
</td><td>the general form of a sacrifice in the short run and reaping the
</td><td>benefit of higher growth and output level in the long run.
</td><td>Imposing time-consistency by optimizing first at the end of the
</td><td>planning period and working towards the beginning improves the
</td><td>welfare at the end in the T regime relative to P. Thus we end up
</td><td>with more initial sacrifice, but more long-run benefit in T.
</td><td>
</td><td>Another way of viewing the time-consistency constraint is that it
</td><td>provides an incentive to continue to raise taxes until no more
</td><td>taxes are needed to finance expenditure. \citeN{mauobs91dynamic}
</td><td>is an early contribution that established this result in a far
</td><td>simpler model without endogenous growth.  Our study shows that the
</td><td>essence of \citeANP{mauobs91dynamic}'s results carries over to a
</td><td>much more developed model incorporating endogenous growth. If we
</td><td>believe that the accumulation of debt is an important feature of
</td><td>observed economic policy, considering time-consistent policies
</td><td>does not bring the predictions of the model closer to the
</td><td>empirical facts; in fact it drives them  away since asset
</td><td>accumulation of the government is larger.
</td><td>
</td><td>
</td><td>The new element that we add to the picture is the decision between
</td><td>government consumption and investment expenditure. A na\"\i{}ve
</td><td>view would be to blame time consistency for insufficient
</td><td>investment. Our numerical experiments suggest that this is not
</td><td>correct and that in fact the time-consistent policy
</td><td>overaccumulates public capital. Loosely speaking we are adding a
</td><td>second layer of overinvestment into the dynamic behaviour.  It is
</td><td>already known that for any given path of government expenditure,
</td><td>the time-consistent policy overaccumulates financial assets (with
</td><td>respect to the optimal policy).  When we introduce the additional
</td><td>degree of freedom to allow the government either to consume or
</td><td>invest, we find overinvestment in physical assets.
</td><td>
</td><td>
</td><td>%\footnote{ We left $\tort_1$ off the figures. The
</td><td>%actual
</td><td>%  figures are $\tort_1=51.25\%$ for the optimal and $\tort_1=49.96\%$ for
</td><td>%  the time-consistent regime. Had we included these values the trajectories
</td><td>%  would have been much more compressed.}
</td><td>If we believe that ``out there in the real world'' governments in
</td><td>fact underinvest, we can not take comfort from the time-consistent
</td><td>solution when searching for a theoretical underpinning for this
</td><td>view, unless we allow the government to discount {\sl much\/} more
</td><td>heavily than the private sector. To fix ideas, let the government
</td><td>discount at a factor that is $\xi\le1$ times the discount factor
</td><td>of the private sector. Simulations show that as we decrease $\xi$,
</td><td>i.e., we make the government more and more impatient, the long-run
</td><td>asset accumulation in both regimes declines, but the asset
</td><td>position the time-consistent regime is more sensitive to the
</td><td>decline in $\xi$. Thus $\xi=.6$---a quite severe
</td><td>distortion---implies a long-run debt/GDP ratio of is $-3\%$ for
</td><td>the time-consistent regime, but $-2\%$ for the optimal regime.
</td><td>Note that the difference between growth rates is also affected.
</td><td>The optimal long run growth rate is $.9\%$ over the baseline, but
</td><td>the long-run time-consistent growth rate is $.7\%$ over the
</td><td>baseline. This suggests that the better long-run welfare
</td><td>improvement is valid when the government does not discount much
</td><td>more heavily than the private sector, but the stylized fact of
</td><td>government accumulation of debt can be captured by allowing the
</td><td>government to discount more heavily than the private sector.
</td><td>
</td><td>
</td><td>\renewcommand{\arraystretch}{1}
</td><td>
</td><td>\section{Conclusion}\label{sec:conclusions}
</td><td>Our main result is that precommitment can actually lead to {\sl
</td><td>lower\/} long-run growth and welfare and the time-consistent
</td><td>solution is associated with an overaccumulation of assets by the
</td><td>government, unless the latter discounts more heavily than the
</td><td>private sector. Ex ante precommitment must yield higher
</td><td>intertemporal welfare and problems regarding implementation of
</td><td>optimal but time-inconsistent policies have focused on the
</td><td>establishment of some commitment mechanism that would make them
</td><td>credible. Our results suggest that the failure to find such a
</td><td>mechanism will actually be beneficial to future generations and
</td><td>can obviate the problems of short-termism associated with
</td><td>democratic decision-making.
</td><td>
</td><td>We have provided intuition for our results in the context of a
</td><td>specific model. However if we step back from the discussion of the
</td><td>model, we can find some compelling reasons for why this result may
</td><td>be quite general. Imagine first an ex-ante optimal,
</td><td>time-inconsistent policy that involves ``indulgence'' initially,
</td><td>and ``sacrifice'' in the future.  What would does the
</td><td>time-consistent policy look like? It is useful to consider a
</td><td>hypothetical ``cheating'' policy in which the time-inconsistent
</td><td>future policy trajectories are announced and believed by private
</td><td>sector, but the government then engages in re-optimization given
</td><td>these expectations. Then the tendency to indulge would continue
</td><td>during all earlier periods. But of course since past indulgence
</td><td>has eroded the possibility to indulge in the current period;  as
</td><td>we move to the future, we indulge less in every period, because
</td><td>current indulgence reduces the possibility for future indulgence.
</td><td>In the long run we run down our opportunities to indulge to zero.
</td><td>Clearly the long-run welfare in such a cheating policy will be
</td><td>poor. In the time-consistent equilibrium the private sector
</td><td>anticipates the possibility of re-optimization so no cheating can
</td><td>occur. To achieve time-consistency and eliminate the incentive to
</td><td>cheat, the government must then offer more indulgence in the early
</td><td>period and more sacrifice later. Thus the reversal from indulgence
</td><td>to sacrifice will be more for the time-consistent policy than in
</td><td>the time-inconsistent case. The intertemporal welfare for the
</td><td>latter will be (by construction) better at the beginning of the
</td><td>planning period and it will also be superior to the
</td><td>time-consistent solution in the long-run.
</td><td>
</td><td>But now assume that the opposite is true, that the optimal policy
</td><td>consists in making a sacrifice in the early periods and allow for
</td><td>indulgence in the later periods. Again, time-inconsistency in the
</td><td>form of an incentive to cheat and make more sacrifice in the early
</td><td>stage exists along
</td><td>this policy path. In the time-consistent
</td><td>solution there will be more sacrifice in the early periods, but in
</td><td>later periods, the sacrifice will bring fruit and allow for higher
</td><td>consumption possibilities. In contrast to be previous case, now
</td><td>the time-consistent policy brings {\sl higher\/} welfare than the
</td><td>time-inconsistent policy in the long run.
</td><td>
</td><td>Are most economic optimisation solution leading to trajectories of
</td><td>the ``indulge then sacrifice'' type or the ``sacrifice then
</td><td>indulge'' type?  We not aware of any broad study of this question,
</td><td>but it seems to us that the latter type is much more prominent
</td><td>than the former. The latter situation arises for instance in
</td><td>models such as that is this paper, where there is capital
</td><td>accumulation problem and initial capital falls short of an
</td><td>overaccumulation level.  It is also typically true in many models
</td><td>where the government can issue debt or accumulate assets and where
</td><td>initial government assets are smaller than the present value of
</td><td>government expenditure.  Our conclusion should therefore hold in
</td><td>wide variety of models.
</td><td>
</td><td>
</td><td>\bibliography{bib}
</td><td>
</td><td>\appendix
</td><td>
</td><td>\newpage
</td><td>
</td><td>\section{Sensitivity analysis}
</td><td>
</td><td>In this appendix, we present a sensitivity analysis for the model.  We
</td><td>change values for the fundamental parameters of the model and recompute the
</td><td>steady state that results from the optimal and time consistent policies.
</td><td>Note that we present the value that the variables take rather than the
</td><td>deviation from the steady state. This value will be different
</td><td>from the value in the original exercise because there is a different steady
</td><td>state and because there is a different policy that is associated with that
</td><td>steady state. There just reporting differences would have been misleading.
</td><td>
</td><td>In the first column, we present the value of the fundamental variable that
</td><td>has changed. From the 3rd to the 13th column we give the value of the
</td><td>steady state of the variable indicated in the top row. For any shift in a
</td><td>fundamental parameter we report the steady state of the variable in the
</td><td>optimal regime `P' and in the time-consistent regime `T'.
</td><td>
</td><td>All our results carry through these simulations.
</td><td>
</td><td>\begin{longtable}{@{}r|rrrrr@{\hspace{5.5pt}}rrrrr@{\hspace{5.5pt}}r@{\hspace{5.5pt}}r@{}}
</td><td>&&$\irat$&$\grow$&$\kapi$&$\kago$&$\debt$&$\tort$&$\cons$&$\inve$&$\govi$&$\govc$&$\util$\\
</td><td>\hline
</td><td>$\gamt=$&P&$5.17$&$2.97$&$233$&$83.6$&$-17.9$&$18.8$&$67.7$&$15.4$&$7.23$&$9.66$&$11.16$\\
</td><td>$.60240$&T&$5.84$&$3.81$&$212$&$95.9$&$-22$&$13.7$&$71.6$&$15.4$&$8.84$&$4.18$&$27.06$\\
</td><td>\hline
</td><td>$\gamt=$&P&$4.89$&$2.76$&$241$&$80.2$&$-17.8$&$18.4$&$67.9$&$15.6$&$6.8$&$9.69$&$5.14$\\
</td><td>$.90360$&T&$4.92$&$3.09$&$238$&$93.2$&$-21.7$&$13.1$&$71.8$&$16.0$&$8.10$&$4.12$&$5.47$\\
</td><td>\hline
</td><td>$\inve=$&P&$4.95$&$2.83$&$237$&$83.6$&$-20.0$&$18.1$&$71.3$&$12.4$&$7.13$&$9.22$&$11.05$\\
</td><td>$.11984$&T&$5.07$&$3.04$&$232$&$90$&$-16.5$&$16.1$&$73.0$&$12.4$&$7.8$&$6.81$&$14.50$\\
</td><td>\hline
</td><td>$\inve=$&P&$5.05$&$2.87$&$238$&$82.6$&$-16.9$&$18.8$&$65.7$&$17.5$&$7.06$&$9.78$&$5.03$\\
</td><td>$.16852$&T&$5.61$&$3.76$&$222$&$101$&$-27$&$11.1$&$71.1$&$18.1$&$9.27$&$1.58$&$7.99$\\
</td><td>\hline
</td><td>$\irat=$&P&$4.07$&$2.78$&$238$&$82.7$&$-16.5$&$19.3$&$67.7$&$15.4$&$7.01$&$9.98$&$22.23$\\
</td><td>$.04000$&T&$4.41$&$3.38$&$227$&$95.7$&$-21.2$&$12.2$&$73$&$15.7$&$8.52$&$2.78$&$29.13$\\
</td><td>\hline
</td><td>$\irat=$&P&$5.93$&$2.91$&$238$&$83.3$&$-19.2$&$18.2$&$67.7$&$15.6$&$7.15$&$9.51$&$-3.56$\\
</td><td>$.06000$&T&$6.18$&$3.38$&$227$&$95.2$&$-22.2$&$14.3$&$70.6$&$15.8$&$8.48$&$5.2$&$2.20$\\
</td><td>\hline
</td><td>$\depr=$&P&$5.04$&$2.82$&$239$&$93.1$&$-15.8$&$19$&$67.2$&$15.5$&$7.02$&$10.3$&$6.68$\\
</td><td>$.0500$&T&$5.59$&$3.74$&$222$&$116$&$-28.0$&$10.7$&$73$&$16.3$&$9.41$&$1.33$&$6.88$\\
</td><td>\hline
</td><td>$\depr=$&P&$4.97$&$2.88$&$237$&$75.1$&$-20.4$&$18.2$&$68.6$&$15.5$&$7.16$&$8.81$&$8.04$\\
</td><td>$.0700$&T&$5.15$&$3.18$&$230$&$82.3$&$-18.3$&$15.4$&$70.8$&$15.4$&$8.02$&$5.74$&$12.77$\\
</td><td>\hline
</td><td>$\mort=$&P&$5.02$&$2.86$&$238$&$83.4$&$-14.8$&$17.9$&$68.5$&$15.5$&$7.13$&$8.87$&$15.63$\\
</td><td>$.01600$&T&$5.29$&$3.39$&$227$&$95.4$&$-24.3$&$12.6$&$72.3$&$15.8$&$8.5$&$3.49$&$21.72$\\
</td><td>\hline
</td><td>$\mort=$&P&$4.98$&$2.83$&$239$&$82.3$&$-21.9$&$19.3$&$67.1$&$15.5$&$7.01$&$10.3$&$0.22$\\
</td><td>$.02400$&T&$5.30$&$3.38$&$227$&$95.5$&$-19.9$&$14.3$&$71$&$15.7$&$8.5$&$4.78$&$7.47$\\
</td><td>\hline
</td><td>$\sigm=$&P&$4.98$&$2.93$&$238$&$66.6$&$-18.2$&$16.6$&$69.3$&$15.7$&$5.74$&$9.32$&$11.52$\\
</td><td>$.28616$&T&$5.24$&$3.54$&$227$&$76.8$&$-22.5$&$9.78$&$74.3$&$16.1$&$6.93$&$2.69$&$14.99$\\
</td><td>\hline
</td><td>$\sigm=$&P&$5.03$&$2.76$&$238$&$99.1$&$-17.5$&$20.4$&$66.5$&$15.3$&$8.49$&$9.7$&$3.01$\\
</td><td>$.42924$&T&$5.35$&$3.23$&$227$&$114$&$-21.4$&$16.8$&$69.2$&$15.4$&$10.1$&$5.30$&$10.44$\\
</td><td>\hline
</td><td>$\debt=$&P&$5.02$&$2.81$&$238$&$82.9$&$-17.8$&$18.9$&$67.7$&$15.4$&$7.05$&$9.85$&$4.17$\\
</td><td>$.42800$&T&$5.32$&$3.34$&$227$&$95.5$&$-21.5$&$13.9$&$71.5$&$15.7$&$8.47$&$4.41$&$11.16$\\
</td><td>\hline
</td><td>$\debt=$&P&$4.99$&$2.88$&$238$&$83$&$-17.9$&$18.4$&$67.9$&$15.6$&$7.10$&$9.47$&$8.08$\\
</td><td>$.64200$&T&$5.28$&$3.41$&$227$&$95.5$&$-22.3$&$13.2$&$71.7$&$15.8$&$8.52$&$3.98$&$14.48$\\
</td><td>\hline
</td><td>$\grow=$&P&$5.02$&$2.3$&$238$&$87.4$&$-18.3$&$20.4$&$66.4$&$15.4$&$7.03$&$11.1$&$-15.76$\\
</td><td>$.02000$&T&$5.38$&$2.98$&$225$&$103$&$-22.0$&$13.5$&$71.4$&$15.9$&$8.79$&$3.96$&$-8.18$\\
</td><td>\hline
</td><td>$\grow=$&P&$4.99$&$3.43$&$238$&$79.1$&$-17.0$&$16.3$&$69.7$&$15.6$&$7.16$&$7.49$&$36.08$\\
</td><td>$.03000$&T&$5.26$&$3.82$&$229$&$88.9$&$-22.3$&$13.4$&$71.9$&$15.7$&$8.26$&$4.21$&$42.02$\\
</td><td>\hline
</td><td>$\popu=$&P&$5.01$&$2.85$&$238$&$83.2$&$-16.3$&$18.3$&$68.1$&$15.5$&$7.10$&$9.29$&$9.80$\\
</td><td>$.00800$&T&$5.3$&$3.38$&$227$&$95.5$&$-22.9$&$13.1$&$71.9$&$15.8$&$8.49$&$3.87$&$16.16$\\
</td><td>\hline
</td><td>$\popu=$&P&$5$&$2.83$&$238$&$82.6$&$-19.6$&$19$&$67.5$&$15.5$&$7.04$&$10.0$&$2.69$\\
</td><td>$.01200$&T&$5.30$&$3.38$&$227$&$95.5$&$-21.0$&$13.9$&$71.3$&$15.7$&$8.5$&$4.49$&$9.62$\\
</td><td>\hline
</td><td>$\kapi=$&P&$5.06$&$2.82$&$209$&$82.3$&$-15.8$&$19.4$&$67.1$&$15.4$&$7.01$&$10.5$&$4.72$\\
</td><td>$2.11$&T&$5.72$&$3.81$&$193$&$103$&$-27.6$&$11.4$&$72.8$&$16$&$9.46$&$1.73$&$10.24$\\
</td><td>\hline
</td><td>$\kapi=$&P&$4.95$&$2.87$&$267$&$83.4$&$-19.4$&$17.9$&$68.6$&$15.6$&$7.14$&$8.68$&$7.53$\\
</td><td>$2.71$&T&$5.12$&$3.19$&$259$&$91.7$&$-19.1$&$14.8$&$70.9$&$15.7$&$8.05$&$5.38$&$11.88$\\
</td><td>\hline
</td><td>$\eta=$&P&$5.03$&$2.79$&$238$&$82.9$&$-18.1$&$20.1$&$66.7$&$15.4$&$7.03$&$10.9$&$3.14$\\
</td><td>$.196$&T&$5.34$&$3.30$&$227$&$95.5$&$-22.2$&$15.4$&$70.1$&$15.6$&$8.44$&$5.86$&$11.83$\\
</td><td>\hline
</td><td>$\eta=$&P&$4.97$&$2.91$&$238$&$83$&$-17.5$&$17$&$69.0$&$15.7$&$7.13$&$8.19$&$11.73$\\
</td><td>$.156$&T&$5.25$&$3.47$&$227$&$95.5$&$-21.6$&$11.2$&$73.3$&$16$&$8.56$&$2.17$&$13.58$\\
</td><td>\end{longtable}
</td><td>
</td><td>
</td><td>%\end{spacing}
</td><td>
</td><td>
</td><td>
</td><td>\newpage
</td><td>
</td><td>\begin{tabular}{@{\hskip -.5in}c@{\hskip -1.75in}c}
</td><td>        \input grusa.govc.tex&
</td><td>        \input grusa.govi.tex\relax\\
</td><td>\hskip -1.5in  {\small Figure 1: Government Consumption} &
</td><td>\hskip -1.75in {\small Figure 2: Government Investment}\\
</td><td>\phantom{an additional blank line}& \\
</td><td>\end{tabular}
</td><td>
</td><td>
</td><td>
</td><td>\begin{tabular}{@{\hskip -.5in}c@{\hskip -1.75in}c}
</td><td>        \input grusa.debt.tex&
</td><td>        \input grusa.tort.tex\relax\\
</td><td>\hskip -1.5in  {\small Figure 3: Debt} &
</td><td>\hskip -1.75in {\small Figure 4: Tax rate}\\
</td><td>\phantom{an additional blank line}& \\
</td><td>\end{tabular}
</td><td>
</td><td>
</td><td>\begin{tabular}{@{\hskip -.5in}c@{\hskip -1.75in}c}
</td><td>        \input grusa.cons.tex&
</td><td>        \input grusa.inve.tex\relax\\
</td><td>\hskip -1.5in  {\small Figure 5: Consumption} &
</td><td>\hskip -1.75in {\small Figure 6: Investment}\\
</td><td>\phantom{an additional blank line}& \\
</td><td>\end{tabular}
</td><td>
</td><td>\begin{tabular}{@{\hskip -.5in}c@{\hskip -1.75in}c}
</td><td>        \input grusa.kapi.tex&
</td><td>        \input grusa.kago.tex\relax\\
</td><td>\hskip -1.5in  {\small Figure 7: Capital} &
</td><td>\hskip -1.75in {\small Figure 8: Infrastructure}\\
</td><td>\phantom{an additional blank line}& \\
</td><td>\end{tabular}
</td><td>
</td><td>\begin{tabular}{@{\hskip -.5in}c@{\hskip -1.75in}c}
</td><td>        \input grusa.grow.tex&
</td><td>        \input grusa.irat.tex
</td><td>\\
</td><td>\hskip -1.5in  {\small Figure 9: Growth rate} &
</td><td>\hskip -1.75in {\small Figure 10: Interest rate}
</td><td>\\
</td><td>\end{tabular}
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>\end{document}
</td><td>
</td><td>
</td><td>% LocalWords:  GDP robbar po Barro koifut sc gerglo dy saulau le stetur dc mc
</td><td>% LocalWords:  micdev larjon menyar olibla phiwei pu tabkapievolution kenarr pl
</td><td>% LocalWords:  paurom firmtarget tabinve tabtobq tabwealth tabkagoevolution tbp
</td><td>% LocalWords:  tabequilibrium tabproduction guical govtarget reoptimize gilsai
</td><td>% LocalWords:  tabgovernmentbudgetconstraint singleperiodwelfare LocalWords qu
</td><td>% LocalWords:  miqfai cb linearconsumption lineartobinsq Precommitment Krichel
</td><td>% LocalWords:  Guildford GU XH distortionary Ricardian JEL externality Yaari eq
</td><td>% LocalWords:  endogenize precommitment intertemporal Tobin's thokri Ponzi TC
</td><td>% LocalWords:  cobbdouglasproduction transversality precommit mauobs na ve IMF
</td><td>% LocalWords:  overaccumulates overinvestment underinvest overaccumulation yabl
</td><td>% LocalWords:  oecdst atomistic Stackelberg tambar reputational maximization
</td><td>% LocalWords:  md walfis ej stecas davash mo alimun ne johtat sl rrrrrrrrrrr pt
</td><td>% LocalWords:  tabyaariblanchard infinte rl rlrl welsteady lineargrowth termism
</td><td>% LocalWords:  rrrrr costlessly marbax inelasticly
