Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 15 ppt

In the simplest version, the production factors are raw laborand physical capital on which the government levies distorting flat-rate taxes.The problem is to determine the optimal sequenc

The model is a competitive equilibrium version of the basic neoclassicalgrowth model with a government that finances an exogenous stream of govern-ment purchases In the simplest version, the production factors are raw laborand physical capital on which the government levies distorting flat-rate taxes.The problem is to determine the optimal sequences for the two tax rates In

a nonstochastic economy, Chamley (1986) and Judd (1985b) show in relatedsettings that if an equilibrium has an asymptotic steady state, then the optimalpolicy is eventually to set the tax rate on capital to zero This remarkable resultasserts that capital income taxation serves neither efficiency nor redistributivepurposes in the long run This conclusion is robust to whether the governmentcan issue debt or must run a balanced budget in each period However, if thetax system is incomplete, the limiting value of optimal capital tax can be differ-ent from zero To illustrate this possibility, we follow Correia (1996), and study

a case with an additional fixed production factor that cannot be taxed by thegovernment

In a stochastic version of the model with complete markets, we find determinacy of state-contingent debt and capital taxes Infinitely many plansimplement the same competitive equilibrium allocation For example, two al-ternative extreme cases are (1) that the government issues risk-free bonds andlets the capital tax rate depend on the current state, or (2) that it fixes thecapital tax rate one period ahead and lets debt be state-contingent While the

in-– 473 in-–

state-by-state capital tax rates cannot be pinned down, an optimal plan doesdetermine the current market value of next period’s tax payments across states

of nature Dividing by the current market value of capital income gives a

mea-sure that we call the ex ante capital tax rate If there exists a stationary Ramsey

allocation, Zhu (1992) shows that there are two possible outcomes For somespecial utility functions, the Ramsey plan prescribes a zero ex ante capital taxrate that can be implemented by setting a zero tax on capital income Butexcept for special classes of preferences, Zhu concludes that the ex ante capitaltax rate should vary around zero, in the sense that there is a positive measure

of states with positive tax rates and a positive measure of states with negativetax rates Chari, Christiano, and Kehoe (1994) perform numerical simulationsand conclude that there is a quantitative presumption that the ex ante capitaltax rate is approximately zero

To gain further insight into optimal taxation and debt policies, we turn toLucas and Stokey (1983) who analyze a model without physical capital Ex-amples of deterministic and stochastic government expenditure streams bringout the important role of government debt in smoothing tax distortions overboth time and states State-contingent government debt is used as a form of

“insurance policy” that allows the government to smooth taxes over states Inthis complete markets model, the current value of the government’s debt re-flects the current and likely future path of government expenditures rather thananything about its past This feature of an optimal debt policy is especiallyapparent when government expenditures follow a Markov process because thenthe beginning-of-period state-contingent government debt is a function only ofthe current state and hence there are no lingering effects of past governmentexpenditures Aiyagari, Marcet, Sargent, and Sepp¨al¨a (2002) alter that feature

of optimal policy in Lucas and Stokey’s model by assuming that the ment can only issue risk-free debt Not having access to state-contingent debtconstrains the government’s ability to smooth taxes over states and allows pastvalues of government expenditures to have persistent effects on both future taxrates and debt levels Based on an analogy from the savings problem of chapter

govern-16 to an optimal taxation problem, Barro (1979) had thought that tax revenueswould be a martingale cointegrated with government debt, an outcome possess-ing a dramatic version of such persistent effects, none of which are present inthe Ramsey plan for Lucas and Stokey’s model Aiyagari et al.’s suspension of

aug-a zero limiting taug-ax aug-applies aug-also to laug-abor income; thaug-at is, the return to humaug-ancapital should not be taxed in the limit Instead, the government should resort

to a consumption tax But even this consumption tax, and therefore all taxes,should be zero in the limit for a particular class of preferences where it is op-timal for the government under a transition period to amass so many claims

on the private economy that the interest earnings suffice to finance governmentexpenditures While results that taxes rates for non-capital taxes require ever

more stringent assumptions, the basic prescription for a zero capital tax in a

nonstochastic steady state is an immediate implication of a standard returns-to-scale production technology, competitive markets, and a complete set

The single good is produced with labor n t and capital k t Output can

be consumed by households, used by the government, or used to augment the

capital stock The technology is

c t + g t + k t+1 = F (k t , n t) + (1− δ) k t , (15.2.3) where δ ∈ (0, 1) is the rate at which capital depreciates and {g t } ∞

t=0 is anexogenous sequence of government purchases We assume a standard concave

production function F (k, n) that exhibits constant returns to scale By Euler’s theorem, linear homogeneity of F implies

F (k, n) = F k k + F n n (15.2.4)

Let u c be the derivative of u(c t ,  t ) with respect to consumption; u
is

the derivative with respect to  We use u c (t) and F k (t) and so on to denote the time- t values of the indicated objects, evaluated at an allocation to be

understood from the context

15.2.1 Government

The government finances its stream of purchases {g t } ∞

t=0 by levying flat-rate,

time-varying taxes on earnings from capital at rate τ k

t and from labor at rate

τ n

t The government can also trade one-period bonds, sequential trading ofwhich suffices to accomplish any intertemporal trade in a world without uncer-

tainty Let b t be government indebtedness to the private sector, denominated in

time t -goods, maturing at the beginning of period t The government’s budget

constraint is

g t = τ t k r t k t + τ t n w t n t+b t+1

where r t and w t are the market-determined rental rate of capital and the wage

rate for labor, respectively, denominated in units of time t goods, and R t is the

gross rate of return on one-period bonds held from t to t + 1 Interest earnings

on bonds are assumed to be tax exempt; this assumption is innocuous for bondexchanges between the government and the private sector

A nonstochastic economy 477

15.2.2 Households

The representative household maximizes expression ( 15.2.1 ) subject to the

fol-lowing sequence of budget constraints:

first-order conditions are

poral trades in a world without uncertainty, condition ( 15.2.12 ) constitutes a

no-arbitrage condition for trades in capital and bonds that ensures that thesetwo assets have the same rate of return This no-arbitrage condition can be ob-

tained by consolidating two consecutive budget constraints; constraint ( 15.2.6 ) and its counterpart for time t + 1 can be merged by eliminating the common quantity b t+1 to get

where the left side is the use of funds, and the right side measures the resources at

the household’s disposal If the term multiplying k t+1 is not zero, the household

can make its budget set unbounded by either buying an arbitrarily large k t+1

when (1− τ k

t+1 )r t+1+ 1− δ > R t, or, in the opposite case, selling capital

short with an arbitrarily large negative k t+1 In such arbitrage transactions,the household would finance purchases of capital or invest the proceeds from

short sales in the bond market between periods t and t + 1 Thus, to ensure

the existence of a competitive equilibrium with bounded budget sets, condition

( 15.2.12 ) must hold.

If we continue the process of recursively using successive budget constraints

to eliminate successive b t+j terms, begun in equation ( 15.2.13 ), we arrive at

the household’s present-value budget constraint,

As discussed in chapter 13, the household would not like to violate these

transver-sality conditions by choosing k t+1 or b t+1 to be larger, because alternative ble allocations with higher consumption in finite time would yield higher lifetimeutility A consumption/savings plan that made either expression negative wouldnot be possible because the household would not find anybody willing to be onthe lending side of the implied transactions

An alternative way of establishing the equilibrium conditions for the rental

price of capital and the wage rate for labor is to substitute equation ( 15.2.4 ) into equation ( 15.2.17 ) to get

Π = [F k (t) − r t ] k t + [F n (t) − w t ] n t

If the firm’s profits are to be nonnegative and finite, the terms multiplying k t

and n t must be zero; that is, condition ( 15.2.18 ) must hold These conditions

imply that in any equilibrium, Π = 0

15.3 The Ramsey problem

15.3.1 Definitions

We shall use symbols without subscripts to denote the one-sided infinite sequence

for the corresponding variable, e.g., c ≡ {c t } ∞

Definition: A government policy is a 4-tuple of sequences (g, τ k , τ n , b)

Definition: A competitive equilibrium is a feasible allocation, a price system,

and a government policy such that (a) given the price system and the ment policy, the allocation solves both the firm’s problem and the household’sproblem; and (b) given the allocation and the price system, the government

govern-policy satisfies the sequence of government budget constraints ( 15.2.5 ).

There are many competitive equilibria, indexed by different governmentpolicies This multiplicity motivates the Ramsey problem

Definition: Given k0and b0, the Ramsey problem is to choose a competitive equilibrium that maximizes expression ( 15.2.1 ).

To make the Ramsey problem interesting, we always impose a restriction

on τ k

0, for example, by taking it as given at a small number, say, 0 Thisapproach rules out taxing the initial capital stock via a so-called capital levy

that would constitute a lump-sum tax, since k0 is in fixed supply One often

imposes other restrictions on τ t k , t ≥ 1, namely, that they be bounded above

by some arbitrarily given numbers These bounds play an important role inshaping the near-term temporal properties of the optimal tax plan, as discussed

by Chamley (1986) and explored in computational work by Jones, Manuelli, andRossi (1993) In the analysis that follows, we shall impose the bound on τ k

t

only for t = 0 1

1 According to our assumption on the technology in equation (15.2.3), capital

is reversible and can be transformed back into the consumption good Thus, the

Zero capital tax 481

15.4 Zero capital tax

Following Chamley (1986), we formulate the Ramsey problem as if the ment chooses the after-tax rental rate of capital ˜r t, and the after-tax wage rate

Substituting this expression into equation ( 15.2.5 ) consolidates the firm’s

first-order conditions with the government’s budget constraint The government’s

policy choice is also constrained by the aggregate resource constraint ( 15.2.3 ) and the household’s first-order conditions ( 15.2.11 ) The Ramsey problem in

Lagrangian form becomes

where R t= ˜r t+1+ 1− δ , as given by equation (15.2.12) Note that the

house-hold’s budget constraint is not explicitly included because it is redundant whenthe government satisfies its budget constraint and the resource constraint holds

capital stock is a fixed factor for only one period at a time, so τ k

0 is the onlytax that we need to restrict to ensure an interesting Ramsey problem

The first-order condition with respect to k t+1 is

θ t = β {Ψ t+1 [F k (t + 1) − ˜r t+1 ] + θ t+1 [F k (t + 1) + 1 − δ]} (15.4.2)

The equation has a straightforward interpretation A marginal increment of

capital investment in period t increases the quantity of available goods at time

t + 1 by the amount [F k (t + 1) + 1 − δ], which has a social marginal value θ t+1

In addition, there is an increase in tax revenues equal to [F k (t+1) −˜r t+1] , whichenables the government to reduce its debt or other taxes by the same amount.The reduction of the “excess burden” equals Ψt+1 [F k (t + 1) − ˜r t+1] The sum

of these two effects in period t + 1 is discounted by the discount factor β and set equal to the social marginal value of the initial investment good in period t , which is given by θ t

Suppose that government expenditures stay constant after some period T ,

and assume that the solution to the Ramsey problem converges to a steady state;

that is, all endogenous variables remain constant Using equation ( 15.2.18a ), the steady-state version of equation ( 15.4.2 ) is

θ = β [Ψ (r − ˜r) + θ (r + 1 − δ)] (15.4.3)

Now with a constant consumption stream, the steady-state version of the

house-hold’s optimality condition for the choice of capital in equation ( 15.2.11b ) is

A substitution of equation ( 15.4.4 ) into equation ( 15.4.3 ) yields

Since the marginal social value of goods θ is strictly positive and the marginal

social value of reducing government debt or taxes Ψ is nonnegative, it follows

that r must be equal to ˜ r , so that τ k = 0 This analysis establishes thefollowing celebrated result, versions of which were attained by Chamley (1986)and Judd (1985b)

Proposition 1: If there exists a steady-state Ramsey allocation, the ciated limiting tax rate on capital is zero

asso-Its ability to borrow and lend a risk-free one period asset makes it feasible

to for the government to amass a stock of claims on the private economy that

Limits to redistribution 483

is so large that eventually the interest earnings suffice to finance the stream ofgovernment expenditures.2 Then it can set all tax rates to zero It should be emphasized that this is not the force that underlies the above result that τ k

should be zero asymptotically The zero-capital-tax outcome would prevail even

if we were to prohibit the government from borrowing or lending by requiring

it to run a balanced budget in each period To see this, notice that is we had

set b t and b t+1 equal to zero in equation ( 15.4.1 ), nothing would change in our derivation of the conclusion that τ k = 0 Thus, even when the government must

perpetually raise positive revenues from some source each period, it remains optimal eventually to set τ k to zero

15.5 Limits to redistribution

The optimality of a limiting zero capital tax extends to an economy with erogeneous agents, as mentioned by Chamley (1986) and explored in depth by

het-Judd (1985b) Assume a finite number of different classes of agents, N , and for

simplicity, let each class be the same size The consumption, labor supply, and

capital stock of the representative agent in class i are denoted c i

The government can make positive class-specific lump-sum transfers S t i ≥

0 , but there are no lump-sum taxes As before, the government must rely on rate taxes on earnings from capital and labor We assume that the governmenthas a social welfare function that is a positively weighted average of individual

flat-utilities with weight α i ≥ 0 on class i Without affecting the limiting result for

the capital tax, we assume that the government runs a balanced budget TheLagrangian of the government’s optimization problem becomes

+ θ t [F (k t , n t) + (1− δ) k t − c t − g t − k t+1]+

i=1 x i t , for x = c, n, k, S Here we have to include the budget

constraints and the first-order conditions for each class of agents

The social marginal value of an increment in the capital stock dependsnow on whose capital stock is augmented The Ramsey problem’s first-order

condition with respect to k i

asserting that the limiting capital tax must be zero in any convergent efficient tax program

Pareto-Judd (1985b) discusses one extreme version of heterogeneity with two classes

of agents Agents of class 1 are workers who do not save, so their budget straint is

Primal approach to the Ramsey problem 485

that if the government only values the welfare of workers (α1> α2= 0) , therewill not be any recurring redistribution in the limit Government expenditureswill be financed solely by levying wage taxes on workers

It is important to keep in mind that the results pertain only to the limitingsteady state Our analysis is silent about how much redistribution is accom-plished in the transition period

15.6 Primal approach to the Ramsey problem

In the formulation of the Ramsey problem in expression ( 15.4.1 ), Chamley reduced a pair of taxes (τ k

t , τ n

t ) and a pair of prices (r t , w t) to just one pair ofnumbers (˜r t , ˜ w t) by utilizing the firm’s first-order conditions and equilibriumoutcomes in factor markets In a similar spirit, we will now eliminate all pricesand taxes so that the government can be thought of as directly choosing a feasibleallocation, subject to constraints that ensure the existence of prices and taxessuch that the chosen allocation is consistent with the optimization behavior ofhouseholds and firms This primal approach to the Ramsey problem, as opposed

to the dual approach in which tax rates are viewed as governmental decisionvariables, is used in Lucas and Stokey’s (1983) analysis of an economy withoutcapital Here we will follow the setup of Jones, Manuelli, and Rossi (1997)

To facilitate comparison to the formulation in equation ( 15.4.1 ), we will

now only consider the case when the government is free to trade in the bondmarket The constraints with Lagrange multipliers Ψt can therefore be replacedwith a single present-value budget constraint for either the government or therepresentative household (One of them is redundant, since we are also imposingthe aggregate resource constraint.) The problem simplifies nicely if we choose

the present-value budget constraint of the household ( 15.2.14 ), in which future

capital stocks have been eliminated with the use of no-arbitrage conditions Forconvenience, we repeat the household’s present-value budget constraint here:

t )w t in equation ( 15.6.1 ) with the household’s

marginal rates of substitution

A stepwise summary of the primal approach is as follows:

1 Obtain the first-order conditions of the household’s and the firm’s lems, as well as any arbitrage pricing conditions Solve these conditions for

prob-{q0

t , r t , w t , τ k

t , τ n

t } ∞ t=0 as functions of the allocation {c t , n t , k t+1 } ∞

t=0

2 Substitute these expressions for taxes and prices in terms of the allocationinto the household’s present-value budget constraint This is an intertem-poral constraint involving only the allocation

3 Solve for the Ramsey allocation by maximizing expression ( 15.2.1 ) subject

to equation ( 15.2.3 ) and the “implementability condition” derived in step

2

4 After the Ramsey allocation is solved, use the formulas from step 1 to findtaxes and prices

15.6.1 Constructing the Ramsey plan

We now carry out the steps outlined in the preceding list of instructions

Step 1 Let λ be a Lagrange multiplier on the household’s budget constraint

( 15.6.1 ) The first-order conditions for the household’s problem are

Primal approach to the Ramsey problem 487

As before, we can derive the arbitrage condition ( 15.2.12 ), which now reads

Profit maximization and factor market equilibrium imply equations ( 15.2.18 ).

Step 2 Substitute equations ( 15.6.3 ) and r0= F k (0) into equation ( 15.6.1 ), so

that we can write the household’s budget constraint as

∞

t=0

β t [u c (t) c t − u
(t) n t]− A = 0, (15.6.5) where A is given by

Step 3 The Ramsey problem is to maximize expression ( 15.2.1 ) subject to

equation ( 15.6.5 ) and the feasibility constraint ( 15.2.3 ) As before, we proceed

by assuming that government expenditures are small enough that the problemhas a convex constraint set and that we can approach it using Lagrangian meth-

ods In particular, let Φ be a Lagrange multiplier on equation ( 15.6.5 ) and

for this problem are3

t=0, and a multiplier Φ that satisfies the

system of difference equations formed by equations ( 15.6.9 )–( 15.6.10 ).4

Step 4: After an allocation has been found, obtain q0

t from equation ( 15.6.3a ),

r t from equation ( 15.2.18a ), w t from equation ( 15.2.18b ), τ n

t from equation

( 15.6.3b ), and finally τ k

t from equation ( 15.6.4 ).

3 Comparing the first-order condition for k t+1 to the earlier one in

equa-tion ( 15.4.2 ), obtained under Chamley’s alternative formulaequa-tion of the Ramsey problem, note that the Lagrange multiplier θ t is different across formulations

Specifically, the present specification of the objective function V subsumes parts

of the household’s present-value budget constraint To bring out this difference,

a more informative notation would be to write V j (t, Φ) for j = c, n rather than just V j (t)

4 This system of nonlinear equations can be solved iteratively First, fix Φ,

and solve equations ( 15.6.9 ) and ( 15.6.10a ) for an allocation Then check the implementability condition ( 15.6.10b ), and increase or decrease Φ depending

on whether the budget is in deficit or surplus

Taxation of initial capital 489

15.6.2 Revisiting a zero capital tax

Consider the special case in which there is a T ≥ 0 for which g t = g for all

t ≥ T Assume that there exists a solution to the Ramsey problem and that it

converges to a time-invariant allocation, so that c, n , and k are constant after some time Then because V c (t) converges to a constant, the stationary version

of equation ( 15.6.9a ) implies

Equalities ( 15.6.11 ) and ( 15.6.12 ) imply that τ k= 0

15.7 Taxation of initial capital

Thus far, we have set τ0k at zero (or some other small fixed number) Now

suppose that the government is free to choose τ0k The derivative of J in equation ( 15.6.8 ) with respect to τ0k is

∂J

∂τ k

0

= Φu c (0) F k (0) k0, (15.7.1)

which is strictly positive for all τ k

0 as long as Φ > 0 The nonnegative Lagrange

multiplier Φ measures the utility costs of raising government revenues throughdistorting taxes Without distortionary taxation, a competitive equilibriumwould attain the first-best outcome for the representative household, and Φwould be equal to zero, so that the household’s (or equivalently, by Walras’Law, the government’s) present-value budget constraint would not exert anyadditional constraining effect on welfare maximization beyond what is present

in the economy’s technology In contrast, when the government has to use some

of the tax rates {τ n

t , τ k t+1 } ∞ t=0, the multiplier Φ is strictly positive and reflectsthe welfare cost of the distorted margins, implicit in the present-value budget

constraint ( 15.6.10b ), which govern the household’s optimization behavior.

By raising τ k

0 and thereby increasing the revenues from lump-sum taxation

of k0, the government reduces its need to rely on future distortionary taxation,and hence the value of Φ falls In fact, the ultimate implication of condition

( 15.7.1 ) is that the government should set τ k

0 high enough to drive Φ down

to zero In other words, the government should raise all revenues through a

time- 0 capital levy, then lend the proceeds to the private sector and financegovernment expenditures by using the interest from the loan; this would enable

the government to set τ t n = 0 for all t ≥ 0 and τ k

t = 0 for all t ≥ 1.5

15.8 Nonzero capital tax due to incomplete taxation

The result that the limiting capital tax should be zero hinges on a completeset of flat-rate taxes The consequences of incomplete taxation are illustrated

by Correia (1996), who introduces an additional production factor z t in fixed

supply z t = Z that cannot be taxed, τ z

t = 0

The new production function F (k t , n t , z t) exhibits constant returns to scale

in all of its inputs Profit maximization implies that the rental price of the newfactor equals its marginal product

p z t = F z (t)

The only change to the household’s present-value budget constraint ( 15.6.1 ) is

that a stream of revenues is added to the right side,

∞

t=0

q t0p z t Z.

5 The scheme may involve τ k

0 > 1 for high values of {g t } ∞

t=0 and b0 However,such a scheme cannot be implemented if the household could avoid the taxliability by not renting out its capital stock at time 0 The government would

then be constrained to choose τ k

V (c t , n t , k t , Φ) = u (c t , 1 − n t)

+ Φ{u c (t) [c t − F z (t) Z] − u
(t) n t } (15.8.2)

In contrast to equation ( 15.6.7 ), k t enters now as an argument in V because

of the presence of the marginal product of the factor Z (but we have chosen to suppress the quantity Z itself, since it is in fixed supply).

Except for these changes of the functions F and V , the Lagrangian of the Ramsey problem is the same as equation ( 15.6.8 ) The first-order condition with respect to k t+1 is

Condition ( 15.8.4 ) and the no-arbitrage condition for capital ( 15.6.12 ) imply

an optimal value for τ k,

As discussed earlier, Φ > 0 in a second-best solution with distortionary taxation,

so the limiting tax rate on capital is zero only if F zk= 0 Moreover, the sign of

τ k depends on the direction of the effect of capital on the marginal product of

the untaxed factor Z If k and Z are complements, the limiting capital tax is

positive, and it is negative in the case where the two factors are substitutes.Other examples of a nonzero limiting capital tax are presented by Stiglitz(1987) and Jones, Manuelli, and Rossi (1997), who assume that two types oflabor must be taxed at the same tax rate Once again, the incompleteness ofthe tax system makes the optimal capital tax depend on how capital affects themarginal products of the other factors

exoge-chases g t (s t ) We use the history of events s t to define history-contingent

com-modities: c t (s t ) ,  t (s t ) , and n t (s t) are the household’s consumption, leisure,

and labor at time t given history s t ; and k t+1 (s t) denotes the capital stock

carried over to next period t + 1 Following our earlier convention, u c (s t) and

F k (s t ) and so on denote the values of the indicated objects at time t for history

s t, evaluated at an allocation to be understood from the context

The household’s preferences are ordered by

(15.9.2)

15.9.1 Government

Given history s t at time t , the government finances its exogenous purchase

g t (s t) and any debt obligation by levying flat-rate taxes on earnings from capital

at rate τ k

t (s t ) and from labor at rate τ n

t (s t) , and by issuing state-contingent

debt Let b t+1 (s t+1 |s t

) be government indebtedness to the private sector at the

beginning of period t + 1 if event s t+1 is realized This state-contingent asset

is traded in period t at the price p t (s t+1 |s t ) , in terms of time- t goods The

government’s budget constraint becomes

A stochastic economy 493

where r t (s t ) and w t (s t) are the market-determined rental rate of capital andthe wage rate for labor, respectively

15.9.2 Households

The representative household maximizes expression ( 15.9.1 ) subject to the

fol-lowing sequence of budget constraints:

able at time t , i.e., history s t,

r t+1 s t+1

+ 1− δ& (15.9.6)

And once again, this no-arbitrage condition can be obtained by consolidating

the budget constraints of two consecutive periods Multiply the time– t + 1 sion of equation ( 15.9.4 ) by p t (s t+1 |s t ) and sum over all realizations s t+1 The

ver-resulting expression can be substituted into equation ( 15.9.4 ) by eliminating



s t+1 p t (s t+1 |s t )b t+1 (s t+1 |s t) Then, to rule out arbitrage transactions in

cap-ital and state-contingent assets, the term multiplying k t+1 (s t) must be zero;

this approach amounts to imposing condition ( 15.9.6 ) Similar no-arbitrage

arguments were made in chapters 8 and 13

As before, by repeated substitution of one-period budget constraints, wecan obtain the household’s present-value budget constraint:

0



r0+ 1− δk0+ b0, (15.9.7)

where we denote time- 0 variables by the time subscript 0 The price system

q t0(s t) conforms to the following formula, versions of which were displayed inchapter 8:

Indeterminacy of state-contingent debt and capital taxes 495

where the limits are taken over sequences of histories s tcontained in the infinite

history s ∞

15.9.3 Firms

The static maximization problem of the representative firm remains the same

as before Thus, in a competitive equilibrium, production factors are paid theirmarginal products,

s t+1 } t ≥0 with an associated competitive allocation{c t (s t ), n t (s t ), k t+1 (s t);∀s t } t ≥0.

Note that the labor tax is uniquely determined by equations ( 15.9.5a ) and ( 15.9.11b ) However, there are infinitely many plans for state-contingent debt

and capital taxes that can implement a particular competitive allocation

Intuition for the indeterminacy of state-contingent debt and capital taxes

can be gleaned from the household’s first-order condition ( 15.9.5c ), which states

that capital tax rates affect the household’s intertemporal allocation by changingthe current market value of after-tax returns on capital If a different set ofcapital taxes induces the same current market value of after-tax returns oncapital, then they will also be consistent with the same competitive allocation

It remains only to verify that the change of capital tax receipts in different statescan be offset by restructuring the government’s issue of state-contingent debt.Zhu (1992) shows how such feasible alternative policies can be constructed.Let { t (s t);∀s t } t ≥0 be a random process such that

for t ≥ 0 Compared to the original fiscal policy, we can verify that this

alter-native policy does not change the following:

1 The household’s intertemporal consumption choice, governed by

first-order condition ( 15.9.5c ).

2 The current market value of all government debt issued at time t , when discounted with the equilibrium expression for p t (s t+1 |s t) in equation

( 15.9.5b ).

3 The government’s revenue from capital taxation net of maturing

gov-ernment debt in any state s t+1

Thus, the alternative policy is feasible and leaves the competitive allocationunchanged

Since there are infinitely many ways of constructing sequences of randomvariables { t (s t)} that satisfy equation (15.10.1), it follows that the competitive

allocation can be implemented by many different plans for capital taxes andstate-contingent debt It is instructive to consider two special cases where there

is no uncertainty one period ahead about one of the two policy instruments We

first take the case of risk-free one-period bonds In period t , the government

issues bonds that promise to pay ¯b t+1 (s t ) at time t + 1 with certainty Let the

amount of bonds be such that their present market value is the same as that forthe original fiscal plan,

implied by equation ( 15.10.2c ):

 t+1 s t+1

= b t+1 (s t+1 |s t)− ¯b t+1 (s t)

r t+1 (s t+1 ) k t+1 (s t) . (15.10.4)

The Ramsey plan under uncertainty 497

We can check that equations ( 15.10.3 ) and ( 15.10.4 ) describe a permissible policy by substituting these expressions into equation ( 15.10.1 ) and verifying

that the restriction is indeed satisfied

Next, we examine a policy where the capital tax is not contingent on therealization of the current state but is already set in the previous period Let

¯t+1 (s t ) be the capital tax rate in period t + 1 , conditional on information

at time t We choose ¯ τ t+1 (s t) so that the household’s first-order condition

Thus, the alternative policy in equations ( 15.10.2 ) with capital taxes known one

period in advance is accomplished by setting

 t+1 s t+1

= ¯τ t+1 k s t

− τ k t+1 s t+1

.

15.11 The Ramsey plan under uncertainty

We now ask what competitive allocation should be chosen by a benevolent ernment; that is, we solve the Ramsey problem for the stochastic economy.The computational strategy is in principle the same given in our recipe for anonstochastic economy

gov-Step 1, in which we use private first-order conditions to solve for prices andtaxes in terms of the allocation, has already been accomplished with equations

( 15.9.5a ), ( 15.9.8 ), ( 15.9.9 ) and ( 15.9.11 ) In step 2, we use these expressions to

eliminate prices and taxes from the household’s present-value budget constraint

( 15.9.7 ), which leaves us with

where A is still given by equation ( 15.6.6 ) Proceeding to step 3, we define

− c t (s t)− g t (s t)− k t+1 (s t)

(

where {θ t (s t);∀s t } t ≥0 is a sequence of Lagrange multipliers For given k0 and

b0, we fix τ0k and maximize J with respect to {c t (s t ),

These expressions reveal an interesting property of the Ramsey allocation If the

stochastic process s is Markov, equations ( 15.11.4 ) suggest that the allocations from period 1 onward can be described by time-invariant allocation rules c(s, k) ,

Ex ante capital tax varies around zero 499

15.12 Ex ante capital tax varies around zero

In a nonstochastic economy, we proved that if the equilibrium converges to asteady state, then the optimal limiting capital tax is zero The counterpart to

a steady state in a stochastic economy is a stationary equilibrium Therefore,

we now assume that the process on s follows a Markov process with transition probabilities π(s  |s) ≡ Prob(s t+1 = s  |s t = s) As noted in the previous section,

this assumption implies that the allocation rules are time-invariant functions of

(s, k) If the economy converges to a stationary equilibrium, the stochastic

process {s t , k t } is a stationary, ergodic Markov process on the compact set

S×[0, ¯k] where S is a finite set of possible realizations for s t, and ¯k is an upper

bound on the capital stock.7

Because of the indeterminacy of state-contingent government debt and ital taxes, it is not possible uniquely to characterize a stationary distribution of

cap-realized capital tax rates but we can study the ex ante capital tax rate defined

invoking the equilibrium price of equation ( 15.9.5b ), we see that this expression

is identical to equation ( 15.10.5 ) Recall that equation ( 15.10.5 ) resolved the

indeterminacy of the Ramsey plan by pinning down a unique fixed capital tax

rate for period t + 1 conditional on information at time t It follows that

the alternative interpretation of ¯τ t+1 k (s t ) in equation ( 15.12.1 ) as the ex ante

capital tax rate offers a unique measure across the multiplicity of capital taxschedules under the Ramsey plan Moreover, it is quite intuitive that one wayfor the government to tax away, in present-value terms, a fraction ¯τ k

t+1 (s t) ofnext period’s capital income is to set a constant tax rate exactly equal to thatnumber

Let P ∞(·) be the probability measure over the outcomes in such a

station-ary equilibrium We now state the proposition of Zhu (1992) that the ex ante

7 An upper bound on the capital stock can be constructed as follows,

¯

k = max{¯k (s) : F¯k (s) , 1, s

= δ¯ k (s) ; s ∈ S}.

capital tax rate in a stationary equilibrium either equals zero or varies aroundzero.

Proposition 2: If there exists a stationary Ramsey allocation, the ex antecapital tax rate is such that

P ∞ [V c (c t , n t , Φ)/u c (c t ,  t) = Λ] = 1 for some constant Λ

A sketch of the proof is provided in the next subsection Let us just add herethat the two possibilities with respect to the ex ante capital tax rate are not

vacuous One class of utilities that imply P ∞(¯τ t k = 0) = 1 is

u (c t ,  t) = c

1−σ t

1− σ + v ( t ) , for which the ratio V c (c t , n t , Φ)/u c (c t ,  t) is equal to [1 + Φ(1− σ)], which plays

the role of the constant Λ required by Proposition 2 Chari, Christiano, andKehoe (1994) solve numerically for Ramsey plans when the preferences do notsatisfy this condition In their simulations, the ex ante tax on capital incomeremains approximately equal to zero

To revisit Chamley (1986) and Judd’s (1985b) result on the optimality of

a zero capital tax in a nonstochastic economy, it is trivially true that the ratio

V c (c t , n t , Φ)/u c (c t ,  t) is constant in a nonstochastic steady state In a stationaryequilibrium of a stochastic economy, Proposition 2 extends this result: for someutility functions, the Ramsey plan prescribes a zero ex ante capital tax ratethat can be implemented by setting a zero tax on capital income But exceptfor such special classes of preferences, Proposition 2 states that the ex ante

capital tax rate should fluctuate around zero, in the sense that P ∞(¯τ k

t > 0) > 0

and P ∞(¯τ k

t < 0) > 0

Ex ante capital tax varies around zero 501

15.12.1 Sketch of the proof of Proposition 2

Note from equation ( 15.12.1 ) that ¯ τ k

Since a stationary Ramsey equilibrium has time-invariant allocation rules

c(s, k) , n(s, k) , and k  (s, k) , it follows that ¯ τ k

Note that the operator Γ is a weighted average of H[s  , k  (s, k)] and that it has the property that ΓH ∗ = H ∗ for any constant H ∗.

Under some regularity conditions, H(s, k) attains a minimum H − and a

maximum H+ in the stationary equilibrium That is, there exist equilibrium

states (s − , k − ) and (s+, k+) such that

P ∞ [H (s, k) = H ∗ ] = 1 (15.12.8c)

First, take equation ( 15.12.8a ) and consider the state (s, k) = (s − , k −) that isassociated with a set of possible states in the next period, {s  , k  (s, k); ∀s  ∈ S}.

By equation ( 15.12.7a ), H(s  , k )≥ H − , and since H(s, k) = H −, condition

( 15.12.8a ) implies that H(s  , k  ) = H − We can repeat the same argument

for each (s  , k ) , and thereafter for the equilibrium states that they map into,and so on Thus, using the ergodicity of {s t , k t }, we obtain equation (15.12.8c)

with H ∗ = H − A similar reasoning can be applied to equation ( 15.12.8b ), but

we now use (s, k) = (s+, k+) and equation ( 15.12.7b ) to show that equation ( 15.12.8c ) is implied.

By the correspondence in expression ( 15.12.6 ) we have established part (a) of Proposition 2 Part (b) follows after recalling definition ( 15.12.3 ); the constant H ∗ in equation ( 15.12.8c ) is the sought-after Λ

Examples of labor tax smoothing 503

15.13 Examples of labor tax smoothing

To gain some insight into optimal tax policies, we consider several examples ofgovernment expenditures to be financed in a model without physical capital.The technology is now described by

c t s t

+ g t (s t ) = n t s t

Since one unit of labor yields one unit of output, the competitive equilibrium

wage will trivially be w(s t) = 1 The model is otherwise identical to the previousframework This very model is analyzed by Lucas and Stokey (1983), whoalso study the time consistency of the optimal fiscal policy by allowing thegovernment to choose taxes sequentially rather than once-and-for-all at time

0 8

The household’s present-value budget constraint is given by equation ( 15.9.7 )

except that we delete the part involving physical capital Prices and taxes are

expressed in terms of the allocation by conditions ( 15.9.5a ) and ( 15.9.8 )

Af-ter using these expressions to eliminate prices and taxes, the implementability

condition, equation ( 15.11.1 ), becomes

We then form the Lagrangian in the same way as before After writing out the

derivatives V c (s t ) and V n (s t) , the first-order conditions of this Ramsey problemare

8 The optimal tax policy is in general time inconsistent as studied in chapter

24 and as indicated by the preceding discussion about taxation of initial capital.However, Lucas and Stokey (1983) show that the optimal tax policy in the modelwithout physical capital can be made time consistent if the government can issuedebt at all maturities (and so is not restricted to issue only one-period debt as

in our formulation) There exists a period-by-period strategy for structuring

a term structure of history-contingent claims that preserves the initial Ramseyallocation {c t (s t ), n t (s t);∀s t } t ≥0 as the Ramsey allocation for the continuation

economy By induction, the argument extends to subsequent periods apply

the argument to the maturity structure of both real and nominal bonds in a

monetary economy Also see Persson, Persson, and Svensson (1988)

sequen-To uncover a key property of the optimal allocation for t ≥ 1, it is

instruc-tive to merge first-order conditions ( 15.13.3a ) and ( 15.13.3b ) by substituting out for the multiplier θ t (s t) :

(1 + Φ)u c (c, 1 − c − g) + Φcu cc (c, 1 − c − g)

− (c + g)u
c (c, 1 − c − g)

= (1 + Φ)u
(c, 1 − c − g) + Φcu c
(c, 1 − c − g)

− (c + g)u

(c, 1 − c − g), (15.13.4)

where we have invoked the resource constraints ( 15.13.1 ) and  t (s t )+n t (s t) = 1

We have also suppressed the time subscript and the index s t for the quantities

of consumption, leisure and government purchases in order to highlight a keyproperty of the optimal allocation In particular, if the quantities of govern-

ment purchases are the same after two histories s t and ˜s j for t, j ≥ 0, i.e.,

g t (s t ) = g j(˜s j ) = g , then it follows from equation ( 15.13.4 ) that the optimal choices of consumption and leisure, (c t (s t ),  t (s t )) and (c j(˜s j ),  j(˜s j)) , mustsatisfy the very same first-order condition Hence, the optimal allocation is a

function only of the current realized quantity of government purchases g and does not depend upon the specific history leading up that outcome This his-

tory independence can be compared to the analogous history independence ofthe competitive equilibrium allocation with complete markets in chapter 8.The following preliminary calculations will be useful in shedding furtherlight on optimal tax policies for some examples of government expenditurestreams First, substitute equations ( 15.9.5a ) and ( 15.13.1 ) into equation