TAXATION AND SAVINGS. Robin Boadway Queen s University Kingston, Canada. Day Six

Transcription

1 TAXATION AND SAVINGS by Robin Boadway Queen s University Kingston, Canada April, 2004 Day Six

2 Overview TAXATION AND SAVINGS We will first summarize the two-period life cycle savings model and derive some expressions for the effect of changes in the interest rate on savings We then consider the effect of exogenous tax changes on the representative consumer Three broad-based proportional taxes will be considered: wage, consumption and capital income taxes This will be done assuming perfect certainty and foresight as well as perfect capital markets The main focus will be on the two-period case The extension to a multi-period setting (including continuous time) is straightforward Next, the implications of a government budget constraint will be introduced This necessarily involves an overlapping generations (OLG) setting Two alternative formulations of the government budget constraint will be considered First, the government will be assumed to collect the same present value of tax revenue from each household Tax substitutions will be considered for their effect on private and total savings This involves both the use of government debt and the ability of the government to use cohort-specific taxation This analysis is done primarily for heuristic reasons since the assumptions required are not too realistic Then, an aggregate government budget constraint will be assumed The simplest case is equal tax revenues in each period Tax substitutions will be analyzed in this context both from a normative and a positive point of view A special case is that of social security (public pensions) Finally, some extensions of the analysis will be briefly considered, including capital market constraints, uncertain lifetimes, and government debt Though most of the analysis will use the overlapping-generations (OLG) framework, the issue of operative bequests (Ricardian equivalence) will be considered THESE EXTENSIONS ARE TO BE DONE Taxes and Household Saving Behavior: The Two-Period Case Assumptions fixed labor supplies (to concentrate on the saving decision) perfect capital markets and perfect foresight no uncertainty no bequests a life cycle consisting of two periods (1, 2) of equal length Definitions and Notation w 1,w 2 = earnings in periods 1 and 2 ( 0) 1

3 a = financial asset wealth m = lifetime wealth (assets plus human wealth) r = market interest rate c 1,c 2 = consumption in first and second periods u(c 1,c 2 ) = household lifetime utility (increasing and strictly quasi-concave, with both goods normal) s = savings (in period 1), which can be > < 0 t r,t c,t w = proportional tax rates on interest, consumption, and earnings The tax rates are exogenous and constant over time, and the government can fully commit to them Household Budget Constraint The per period budget constraints are as follows: c 1 = [(1 t w )w 1 + a s](1 t c ), c 2 =[s(1 + (1 t r )r)+(1 t w )w 2 ](1 t c ) Note that the tax rate on consumption is based on tax-inclusive consumption, so it is like a direct tax on consumption rather than a sales tax This is done for simplicity: the consumption tax could be defined as a proportion of before-tax consumption (or equivalently, as a proportion of producer prices) Combining these two constraints by eliminating s yields the household s lifetime budget constraint: c 1 1 t c + c 2 (1 + (1 t r )r)(1 t c ) =(1 t w)w 1 + a + (1 t w)w 2 1+(1 t r )r This says that the present value of consumption expenditures inclusive of tax equals the present value of the stream of after-tax earnings and assets It is useful to rewrite this budget constraint as follows: c 1 + pc 2 = [(1 t w )w 1 + a + p(1 t w )w 2 ](1 t c ) m(w 1,pw 2,a,t w,t c ) where p =1/((1 t r )r) is the relative price of c 2 and m( ) is after-tax lifetime wealth (asset wealth plus human wealth) Note the following things about this budget constraint: consumption taxes and earnings taxes affect the budget constraint in a similar way: both reduce wealth without affecting relative prices The only difference between them is that the reduction in the after-tax income stream differs over the two periods 2

4 earnings here should include all non-capital additions to lifetime wealth They include net inheritances and gifts, for example (Note that gifts and bequests given may be included as consumption) the tax on capital income affects the relative price of c 2 It also affects wealth through the human wealth term If earnings are all in the first period, the second term disappears an income tax can be characterized as t r = t w and t c =0 The Household s Problem The household s lifetime maximization problem can be written: max {c 1,c 2 } u(c 1,c 2 ) subject to c 1 + pc 2 = m(w 1,pw 2,a,t w,t c ) (α) where α is the Lagrangian multiplier This yields the uncompensated demand functions c 1 (p, m( )),c 2 (p, m( )) and the marginal utility of income α(p, m( )) Saving is given by s =(1 t w )w 1 + a c 1( ) 1 t c The indirect utility function is v(p, m( )) u(c 1 (p, m( )),c 2 (p, m( ))) SHOW ON A DIA- GRAM The dual to this problem is the household s expenditure minimization problem; min {c 1,c 2 } c 1 + pc 2 subject to u(c 1,c 2 )=u (γ) where γ is the Lagrangian multiplier and u is a pre-determined level of utility The solution to this problem gives the compensated demand functions c 1 (p, u),c 2 (p, u) and the inverse of the marginal utility of income γ(p, u) The expenditure function is e(p, u) c 1 (p, u)+ pc 2 (p, u) By the envelope theorem, we obtain Hotelling s Lemma: e(p, u) p = c 2 (p, u) If u chosen in this dual problem corresponds to the level of utility obtained from the above utility-maximization, or primal, problem, e(p, u) = m( ) 3

5 Comparative statics The effect of interest rate changes on household savings can be obtained by a comparative static analysis of the above problem The problem is not a conventional one since the relative price p affects income m (through a human wealth effect) The derivation of the comparative statics using the Slutsky equation must take this into account Consider first the derivation of the Slutsky equation At the consumer s optimum, we have: c i (p, u) =c i (p, m( )), and c i (p, u) =c i (p, e(p, u)) i =1, 2 Differentiating the second of these and using Hotelling s Lemma, we have c i p = c i c i u p + c 2 m m i =1, 2 This is the conventional Slutsky equation, but it does not tell the whole story since m itself depends on p Differentiating the uncompensated demand function c i (p, m( )) with respect to p and using the above Slutsky relationship, we have: c i p = c i p + c i m m m p = c i c i p c 2 u m +(1 t c i c)(1 t w )w 2 i =1, 2 m The three terms on the right-hand side of this equation are the substitution effect, the income effect and the human wealth effect, respectively SHOW ON A DIAGRAM Applying this to second-period consumption, we have: c 2 p = c 2 p (c 2 (1 t c )(1 t w )w 2 ) c 2 u m < 0ifs > 0 The substitution effect is negative If household savings are positive (s >0), c 2 > (1 t c )(1 t w )w 2 so the second term is negative as well (since we are assuming that both c 1 and c 2 are normal goods) In this case, an increase in p (reduction in the interest rate) will cause c 2 to fall unambiguously The effect on saving comes from considering the effect on c 1 : c 1 p = c 1 p (c 2 (1 t c )(1 t w )w 2 ) c 1 u m In this case, the substitution effect is positive, while the second term is still negative for a net saver Note that sign s/ r =sign c 1 / p Therefore, the interest elasticity of saving can be positive or negative depending on the relative magnitude of the two terms The 4

6 larger is the human wealth effect (ie, the higher are second period earnings), the larger is the interest elasticity of savings (or the smaller negative) Note that saving could fall with rise in the interest rate, even though second-period consumption necessarily rises That is because a given amount of saving can finance more future consumption when the interest rate has increased First-period earnings only Consider the standard case in which w 2 = 0 In this case, the usual Slutsky equation applies: c 1 p = c 1 c 1 p c 2 u m so the effect of saving depends on offsetting income and substitution effects It is straightforward to convert this into the following rate of change (elasticity) form: c 1 p p c 1 = pc 2 c 1 + pc 2 (σ η) where σ is the elasticity of substitution of c 2 for c 1 in the utility function, and η is the income elasticity of demand for c 1 Thus, σ picks up the substitution effect and η picks up the income effect The term in front is the share of lifetime wealth devoted to second period consumption In the case of a Cobb-Douglas utility function, σ = η = 1, so c 1 / p = 0 and therefore, the interest elasticity of savings is zero In the CES case, η = 1, so the interest elasticity of savings > < 0asσ > < 1 A common assumption about intertemporal utility functions is that they take the additively separable form: u(c 1,c 2 )=u(c 1 )+ u(c 2) 1+δ where δ is the utility discount rate If u(c i ) is of the constant elasticity form, this additively separable utility function is homothetic, so η = 1 The Continuous Time Case Consider the simple case of an additive lifetime utility function with per period utility functions identical and of a constant elasticity form: U = T 0 e δt u(c t )dt = T 0 δt cσ 1 t e σ 1 dt where T is the length of life and δ is the utility discount rate (All discounting is instantaneous) Time 0 could correspond to real time so this problem could be for a person with T left to live Time of death is perfectly certain for now 5

7 The lifetime budget constraint is: T 0 e (1 t r)rt c t dt =(1 t c )a 0 + T 0 e (1 t r)rt (1 t c )(1 t w )w t dt = m 0 (α) where m 0 is lifetime wealth at time zero and α is the Lagrangian multiplier The household maximizes U subject to the budget constraint The first-order conditions are: e δt u (c t ) αe (1 tr)rt =0 t where u (c t ) = c σ t Differentiating this with respect to t (noting that α is timeindependent), we obtain: u (c t )ċ t u =(1 t r )r δ (c t ) Since σ = u (c t )c t /u (c t ), this becomes: ċ t = (1 t r)r δ c t σ This differential equation has a simple solution: c t = e ((1 t r)r δ)t/σ c 0 Substituting this back into the budget constraint gives c 0 = km 0, where k, the marginal propensity to consume out of wealth, is: k = The following properties of c 0 apply: Ambiguity of interest rate effect: k/ r > < 0 Human wealth effect: m 0 / r < 0 Effect of T : k/ T < 0 ((1 t r )r δ)/σ (1 t r )r exp[(((1 t r )r δ)/σ (1 t r )r)t ] 1 c 0 /(1 t c ) is independent of t c (since km 0 /(1 t c ) is independent of t c ) Effect of Tax Changes on Savings Return now to the two-period case Consider the general case in which earnings can occur in both periods In this simple case, saving is given by: s =(1 t w )w 1 + a c 1(p, m( )) 1 t c 6

8 where p = ( 1 1+(1 t r )r, m = (1 t w )w 1 + a + (1 t ) w)w 2 (1 t c ) 1+(1 t r )r Note that p t r = rp 2, m = m, t c 1 t c m t w = (w 1 + pw 2 )(1 t c ) Change in a consumption tax s = c 1 t c (1 t c ) 2 1 c 1 m 1 t c m t c c 1 = (1 t c ) 2 (η 1) < > 0 as η < > 1 Thus, in the homothetic case, saving is unaffected by the change in t c Effectively, consumption expenditures c 1 /(1 t c ) are unchanged as t c changes: a rise in t c is just offset by a fall in c 1 Change in a wage tax s = w 1 1 c 1 t w 1 t c m ( 1 c 1 m = m t w ) w 1 + pw 2 c 1 m > < 0 This shows the importance of the timing of tax liabilities If w 2 =0, s/ t w < 0 The larger is w 2 relative to w 1 (ie, the later in life do earnings occur), the larger positive (or smaller negative) will be the effect of t w on s Change in an interest tax s = 1 c 1 p = rp2 c 1 t r 1 t c p t r 1 t c p < > 0 The larger is σ relative to η and the larger is the human wealth effect (w 2 ), the larger negative (smaller positive) is s/ t r To summarize: There are two important components to the effect of taxes on life-cycle saving: a relative price effect and a life-cycle timing effect Equal Revenue Comparisons: The Individual Case 7

9 Consider now revenue-neutral tax comparisons We begin with the case in which the present value of lifetime revenue is held constant at the individual level The main point of this case can be illustrated for the simple case in which second-period earnings are zero, although the analysis extends in a straightforward way to the case where there are earnings in both periods We consider in turn the substitution of a wage tax for an income tax, and then a wage tax for a consumption tax Both cases can be illustrated geometrically Wage Tax versus Income Tax The comparison is between a wage tax at the rate t w and an income tax t = t w = t r that yields the same present value of revenue to the government The underlying assumption is that the government must finance a given stream of expenditures over time Note the important point that in order to keep the present value of tax revenue constant for all households while maintaining a constant stream of expenditures, the government will have to engage in debt transactions as it reforms its tax system Thus, a switch from an income tax to a wage tax will result in the government having more tax revenue in the first period and less in the second leading it to do some public saving Figure 1 is used to illustrate the present-value-revenue-neutral substitution of a wage tax for a pre-existing income tax The line labelled AB is the pre-tax budget line It has a slope of (1 + r) = 1/p and an endowment point A corresponding with lifetime wealth of OA With an income tax imposed, the budget line is CE with a slope of (1 + (1 t)r) The household chooses the point I, with present and future consumption of OK and OL Tax revenue is AC in the present and LN in the future This has a present value to the government of PK (evaluated at r) Savings under the income tax is KC The line QR is the locus of points yielding the same present value tax revenues as the income tax, since all points on it are below the pre-tax budget line by the same present value Thus, the equal yield wage tax also has the budget line QR, with all tax revenue QA obtained in the first period The household chooses point II with first-period consumption OS Household saving is SQ, which could be greater or less that saving under the income tax, KC Consider now the effect on national saving Recall that government expenditures are being held constant Since the stream of tax revenues generated is different under the two taxes, the government must be varying its debt in order to keep the present value of tax revenues constant In particular, under the wage tax, first-period tax revenues are higher by QA CA=QC, so government savings must increase by this amount The change in 8

10 national savings from replacing the income tax with an equal-yield wage tax is as follows: national savings = household saving + government saving =SQ KC + QC =SK> 0 = c Thus, national saving rises by the fall in household consumption Wage Tax versus Consumption Tax Now consider a switch from a wage tax to an equal-present-value consumption tax levied at the rate t c Figure 2 illustrates this case The consumption tax also shifts the budget line inwards, and the government collects revenues in both periods An equal revenue switch causes the budget line to shift in from the no-tax case by the same amount: there is no relative price effect so the budget line is the same under both taxes, QR In both cases, the household chooses point II Saving under the wage tax is SQ, and under the consumption tax WA In the wage tax case, tax revenues are AQ, all in the first period In the consumption tax case, revenue is found by projecting a ray from the origin through II to the pre-tax budget line Present and future tax revenues are the horizontal and vertical displacements, SW and VX The present value of revenue is the same, QA The change in household saving is: household savings = WA SQ > 0 =QA SW = first period tax revenue Government revenue rises to offset the fall in personal saving, and aggregate consumption and saving both remain the same This illustrates that a pure change in the timing of a tax with no relative price effect and every person paying the same present value of tax revenue has no aggregate effect Any aggregate effect comes from the effect of the relative price change on aggregate consumption A further special case of this is a funded public pension plan, to be discussed further below This analysis can be readily applied to the case where the household obtains earnings in both periods, or more generally, to a multi-period life-cycle model Equal Per Period Aggregate Revenue Comparisons Consider now the case in which the government obtains equal total tax revenues per period from all households With the path of government expenditures unchanged, this implies that there is no change in government debt It is necessary to take an explicit OLG 9

11 perspective in this case since revenues in any period are obtained simultaneously from all cohorts alive at the time To simplify matters, we continue to study the two-period lifecycle case, so only two cohorts are alive at one time Also, it suffices to ignore relative price effects since the essential difference that arises when moving to aggregate per period budget balance is the timing effect of tax reforms We study first the case of an equal revenue substitution of a consumption tax for a wage tax, and then look at a pure intergenerational transfer in the form of an unfunded public pension plan Wage Tax versus Consumption Tax Consider a two-period OLG model in which persons of each cohort have the same first period earnings, and population grows at a constant rate As well, the interest rate is fixed The results for this case can be readily inferred from the above analysis The major change from the previous case is that there is no change in government saving accompanying the tax reform: only household saving changes Imagine starting out in a steady state with a wage tax in place generating a given amount of government revenue per capita At some point in time, the wage tax is replaced with a consumption tax, and t c is adjusted so that revenues per capita are the same in every period Note that this tax reform is equivalent to instituting an ongoing intergenerational transfer from the old to the young We can distinguish between short-run and long run effects In the short run, when the tax reform is instituted, the older cohort is made worse off since they now incur a consumption tax that they otherwise would not have to pay There will be a transition period to a new steady state In the steady state, aggregate savings will increase because of the timing effect: all cohorts are paying taxes later in life Whether or not welfare will increase in the long run depends on whether the present value of tax revenues under the wage tax will exceed that under the consumption tax This is equivalent to asking whether an ongoing intergenerational transfer from the old to the young will make households better off in the steady state, given that this transfer increases the stock of capital or asset wealth To see what determines that, consider the following second example of a pure intergenerational transfer Unfunded Public Pension An unfunded public pension is an ongoing intergenerational transfer from the young to the old, the opposite of the above tax reform There will be a short-run gain for those who are old when the scheme is introduced, and aggregate savings will fall Suppose the tax paid by each person when young is τ, and the pension benefit received in the second period is b Since these are not related to c or w, it is easy to infer their effect Moreover, let population 10

12 growth be n, so that the number of worker in period t is given by: L t =(1+n)L t 1 This implies that there are always 1 + n more young than old Given that τ and b are fixed for all periods and r is given, the economy moves immediately to a new steady state after one period of transition Of course, this is because of the two-period life-cycle assumption If households lived for many periods, it would take one full life cycle to achieve the new steady-state equilibrium in which all households face the same fiscal treatment over their lifetimes The lifetime budget constraint of the typical household is: c 1 + c 2 1+r = w 1 τ + b 1+r = w 1 + W where W b/(1 + r) τ is public pension wealth In the long run, households will be better or worse off according to whether W < > 0 The government budget constraint in period t is: bl t 1 = τl t = b =(1+n)τ Using this, public pension wealth becomes: W = ( ) 1+n 1+r 1 τ < > 0asn < > r Thus, an intergenerational transfer from the young to the old, which reduces the capital stock, makes the old better off if n>r, and vice versa This illustrates the notion of dynamic (in)efficiency Ifn>rforever, the economy is dynamically inefficient in the sense that all persons could be made better off forever by decreasing the capital stock 1 This can be engineered by any type of intergenerational transfer from the young to the old, including an unfunded public pension, the substitution of a wage tax for a consumption tax, an increase in debt, etc This is effectively a Ponzi game If n<r, this is no longer possible: an intergenerational transfer cannot be Pareto improving Any long run gain from reducing public pensions only comes about by making the older generations worse off in the transition This has a relationship with the Golden Rule If we compare steady state growth paths, per capita utility depends on the relationship between n and r, in other words on the level of savings This can be illustrated using the following simple example Suppose the 1 This also extends to the case where there is technical progress so the rate of growth g includes both population growth and productivity growth To study that case, one makes the tax proportional to wages so a steady state solution can be achieved 11

13 production function for the economy is F (K, L), which is linear homogeneous (constant returns to scale) By the latter property, it can be written in the following intensive form: F (K, L) L = F ( ) K L, 1 = f(k) = f (k) = F K(K, L) K L k = F K(K, L) =r + δ where k K/L, r is the interest rate and δ is the depreciation rate on capital Taking the time derivative of the definition of k, we have: k = dk L K L 2 L = K L = k L L + k = nk + k where n is the rate of growth of the labor force The resource constraint implies F (K, L) = C + K + δk, where C is aggregate consumption Dividing by L to convert this to per capita terms yields: c(k) =f(k) (n + δ)k k = c(k) =f(k) (n + δ)k in the steady state where c is per capita consumption steady state, we obtain: Therefore, differentiating with respect to k in the c k = f (k) n δ = r n, 2 c k 2 = f (k) < 0 This implies that for k = k such that r = n, per capita consumption is maximized in the steady state SHOW ON A diagram For k<k, r>nso an increase in the capital stock will increase per capita consumption in the long run However, since increasing k require some reduction in c during the transition, this will not be Pareto-improving Thus, for k k, the economy is dynamically efficient On the other hand, for k>k, a reduction in k will both increase c in the steady state as well as along the transition The economy is dynamically inefficient (Note this requires that k>k or r<nforever) In an OLG context, if r < n forever, an intergenerational transfer from the young to the old will be be Pareto improving However, if r > n, an intergenerational transfer from old to young will make all households better off in the long run, but will make some households worse off in the transition In the real world, r>nis more likely SUMMARIZE SUMMERS SIMULATIONS HERE Caveats The above argument about the effect of intergenerational transfers on aggregate savings and per capita welfare is based on a very simple model of household saving The following is a list of complicating factors that can change the way in which intergenerational transfers affects saving and/or welfare 12

14 The steady state assumption might not be reasonable Intergenerational transfers might be used as a way of providing intergenerational insurance to the extent that some cohorts are unlucky relative to others because of demographic change, productivity shocks, unemployment shocks, wars, disease, etc If there are externalities from investment due to endogenous productivity effects, an increase in saving can improve efficiency If retirement age is endogenous, intergenerational transfers from young to old can induce early retirement and cause an offsetting increase in saving If households undersave because of myopia, time inconsistent preferences, or strategic savings behavior, an intergenerational transfer from old to young may not increase saving much If households are subject to liquidity constraints that prevents them from dissaving when young, a tax on wages to finance transfers to the old may reduce consumption rather than saving If households are unable to insure against uncertainty in the length of life because of imperfect annuity markets, there will be precautionary saving leading to unintentional bequests; a public pension system is a form of annuity and will cause them to reduce their savings even if fully funded because the need for precautionary saving (self-insurance) is reduced If households leave bequests for altruistic reasons, transfers from the young to the old may be offset by an increase in bequests thereby offsetting the reduction in savings (Ricardian equivalence) THEN TURN TO MORE THAN ONE ASSET ADD SECTION ON HUMAN CAPITAL, AND MENTION THAT IT APPLIES TO DURABLES AND PERSONAL BUSINESSES THEN ADD RISKY ASSET THEN DO LAND 13

15 C 2 B D R E N L II I (1 + r) 0 S K P Q C A C 1 Figure 1 Wage versus Income Taxation 14

16 C 2 B R X V II (1 + r) 0 S W Q A C 1 Figure 2 Wage versus Consumption Taxation 15