Public Economics Ben Heijdra Chapter 3: Taxation and Intertemporal Choice

Public Economics: Chapter 3 1 Public Economics Ben Heijdra Chapter 3: Taxation and Intertemporal Choice

Aims of this topic Public Economics: Chapter 3 2 To introduce and study the basic Fisherian model of saving To show some equivalence results for different taxes To apply the model for tax policy To extend the Fisherian model to account for: endogenous labour supply human capital accumulation bequests To present some empirical evidence on the intratemporal and intertemporal elasticities characterizing the model

Public Economics: Chapter 3 3 A Basic Intertemporal Model of Consumption and Saving Not very difficult to introduce an intertemporal dimension to the consumption-saving choice [pioneered by Irving Fischer (1930), The Theory of Interest] Split historical time into two segments: Period 1 is the present Period 2 is the remaining future (obviously, by construction, there is no period 3). Perfect foresight about wages, prices, interest rates, and taxes No bequests Exogenous labour supply Perfect capital markets (no constraints on borrowing or lending)

Public Economics: Chapter 3 4 The representative household s lifetime utility function is given by: Λ = U(C 1,C 2 ) (LU) C i is consumption in period i we assume positive but diminishing marginal utility of consumption in both periods, i.e. U i U/ C i > 0 and U ii 2 U/ C 2 i < 0. For now we place no restriction on the sign of U 12.

Budget identities [in nominal terms]: Public Economics: Chapter 3 5 A 1 = (1 + R 0 )A 0 + W 1 L P1 C 1 A 2 = (1 + R 1 )A 1 + W 2 L P2 C 2 = 0 A i represents financial assets at the end of period i [A 0 was accumulated in the past ] R i is the nominal interest rate in period i L i = L is labour supply in period i ( L is the exogenous time endowment) P i is the price of goods in period i W i is the wage rate in period i

Public Economics: Chapter 3 6 Choose the consumption good, C i, as the numeraire so that the real budget constraints can be written as: a 1 = (1 + r 0 )a 0 + w 1 L C1 a 2 = (1 + r 1 )a 1 + w 2 L C2 = 0 (1 + r i ) (1 + R i ) P i P i+1 is the real interest rate in period i a i A i /P i is real financial assets in period i w i W i /P i is the real wage rate in period i The household can freely borrow or lend at the going interest rate r 1, a 1 can have either sign and the budget identities can be consolidated into a single lifetime budget constraint [With binding borrowing constraint, a 1 0, we are back in static case]:

step 1: solve for a 1 : step 2: re-write slightly: Public Economics: Chapter 3 7 a 1 = C 2 w 2 L 1 + r 1 = (1 + r 0 )a 0 + w 1 L C1 C 1 + C 2 1 + r 1 = (1 + r 0 )a 0 + h 0 Ω (LBC) where h 0 is full human wealth representing the after-tax value of the time endowment: Some pertinent remarks: h 0 w 1 L + w 2 L 1 + r 1 The life-time budget constraint (LBC) says that for the solvent household the present value of spending on goods [left-hand side] equals initial total wealth [Ω on the right-hand side]

Public Economics: Chapter 3 8 1 1+r 1. We can thus use duality theory just as for the static model to derive Hicksian and Marshallian expressions The prices of C 1 and C 2 are, respectively, 1 and for C 1 and C 2. Obviously we expect to find income and substitution effects to play a crucial role The household chooses C 1 and C 2 in order to maximize lifetime utility (LU) subject to the lifetime budget constraint (LBC). The Lagrangian expression is: L U(C 1,C 2 ) + µ [ Ω C 1 C 2 1 + r 1 where µ is the Lagrange multiplier for the constraint (marginal lifetime utility of wealth) ]

Public Economics: Chapter 3 9 The first-order conditions are given by (LBC) and the Euler equation: U 1 (C 1,C 2 ) U 2 (C 1,C 2 ) = 1 + r 1 (EE) MRS between C 1 and C 2 is equated to the relative price of C 1 U i in general depends on both C 1 and C 2 (because U 12 0 is not excluded a priori) Expressions (LBC) and (EE) define implicit functions C i = C i (Ω,r 1 ) for i = 1, 2.

Public Economics: Chapter 3 10 By total differentiation of (LBC) and (EE) we obtain: dc 1 dc 2 = 1 0 dω + C 2 (1+r 1 ) 2 U 2 dr 1 (A) where is: 1 1 1+r 1 U 11 (1 + r 1 )U 12 U 12 (1 + r 1 )U 22 [Young s theorem says U 12 = U 21 ] The second-order conditions for utility maximization ensure that > 0.

Public Economics: Chapter 3 11 Let us first consider the effects of a marginal increase in wealth. We obtain from (A): C 1 Ω = U 12 (1 + r 1 )U 22 C 2 Ω = (1 + r 1)U 12 U 11 0 0 The effect of wealth changes on consumption in both periods is ambiguous in general [U ii < 0 but U 12 0]. If U 12 0 then C i / Ω > 0 for i = 1, 2, and present and future consumption are both normal goods. If U 12 < 0 then either present consumption or future consumption may be an inferior good ( C i / Ω < 0). It follows from (LBC) that at most one good can be inferior, i.e.: C 1 Ω + ( 1 1 + r 1 ) C2 Ω = 1

Public Economics: Chapter 3 12 Next we consider the effects of a marginal increase in the interest rate r 1. Two effects: relative price of future consumption decreases the value of human wealth (and thus total wealth) falls: Ω/ r 1 = w 2 L/(1 + r1 ) 2 < 0 [future wage income is discounted more heavily] By taking both effects into account we obtain from (A): C 1 r 1 = C 2 r 1 = ( ) ( ) ( ) ( ) U12 (1 + r 1 )U 22 a1 1 U2 1 + r 1 1 + r 1 ( ) ( ) ( ) (1 + r1 )U 12 U 11 a1 1 + U 2 0 1 + r 1 0 where we have used the second period budget identity (1 + r 1 )a 1 = C 2 w 2 L to simplify these expressions.

Public Economics: Chapter 3 13 Remarks: Without further restrictions on U 12 and a 1 the effects are ambiguous. By differentiating the lifetime budget equation (LBC) we find: C 1 r 1 + ( 1 1 + r 1 ) C2 r 1 = a 1 1 + r 1 i.e. for an agent who chooses to save (a 1 > 0) either present or future consumption (or both) rise if the interest rate rises. If a 1 > 0 and U 12 0 then C 1 / r 0 and C 2 / r > 0. If the agent s utility maximum happens to coincide with its endowment point (so that a 1 = 0) then it neither saves nor dissaves and it follows that C 1 / r < 0 and C 2 / r > 0.

Public Economics: Chapter 3 14 Homothetic case: U(C 1,C 2 ) = G [ Ū(C 1,C 2 ) ], where G[ ] is a strictly increasing function and Ū(C 1,C 2 ) is homogeneous of degree one in C 1 and C 2. Key properties: (P1): Ū1C 1 + Ū2C 2 = Ū (P2): Ū1 and Ū2 are homogeneous of degree zero in C 1 and C 2 (P3): Ū12 = (C 1 /C 2 )Ū11 = (C 2 /C 1 )Ū22 and thus Ū 11 = (C 2 /C 1 ) 2 Ū 22 (P4): σ d ln(c 1 /C 2 )/d ln(ū1/ū2) = Ū1Ū2/(ŪŪ12) 0 [substitution elasticity] Since Ūii < 0 it follows from (P3) that Ū12 > 0 (and thus U 12 > 0). Hence, present and future consumption are both normal goods!

Public Economics: Chapter 3 15 Back to the effect of a change in the interest rate r 1 : the first-order condition (EE) becomes Ū1/Ū2 = 1 + r 1. Since the Ūi are homogeneous of degree zero, this Euler equation pins down a unique C 1 /C 2 ratio as a function of 1 + r 1. by loglinearizing the Euler equation and the budget restriction (LBC) (holding constant (1 + r 0 )a 0, w 1 L, and w2 L) we obtain the following expression: ω 1 1 ω 1 = (a ) 1/Ω) ( dr1 1 1 σ 1 + r 1 dc 1 C 1 dc 2 C 2 where ω 1 C 1 /Ω and 1 ω 1 C 2 /((1 + r 1 )Ω) are the budget shares of, respectively, first- and second-period consumption.

Public Economics: Chapter 3 16 we obtain the comparative static effects: C 1 r 1 = C 2 r 1 = C 1 1 + r 1 C 2 1 + r 1 [ (1 ω 1 ) w 2 L ] (1 + r 1 )Ω (1 ω 1)σ ] [ (1 ω 1 ) w 2 L (1 + r 1 )Ω + ω 1σ where we have also used (1 + r 1 )a 1 = C 2 w 2 L. The three terms appearing in square brackets on the right-hand sides represent, respectively, the income effect, the human wealth effect, and the substitution effect. We illustrate these effects in Figure 3.1.,,

Public Economics: Chapter 3 17 ultimate effect of an increase in the interest rate r 1 is given by the move from E 0 to E 1. Hicksian decomposition: the move from E 0 to E is the substitution effect (SE) the move from E to E is the income effect (IE) if the household were to have no non-interest income in the second period (w 2 L = 0) the human wealth effect would be absent. if w 2 L > 0, the increase in the interest rate reduces the value of human capital and shifts the budget restriction inward. Hence, the human wealth effect (HWE) is represented by the move from E to E 1.

Public Economics: Chapter 3 18 C 2 AN! EO! WEP 1 A! WEP 0! EN E 1!! E 0 HWE 7 0 SE IE!! BN! B C 1 Figure 3.1: Income, Substitution, and Human Wealth Effects [homothetic case]

Public Economics: Chapter 3 19 Adding Taxes to the Model: Some Equivalence Results A proportional tax on wages (t L ) plus initial assets (e.g. inheritance) augments the budget constraints as follows: a 1 = (1 t L ) [ (1 + r 0 )a 0 + w 1 L] C1 a 2 = (1 + r 1 )a 1 + w 2 (1 t L ) L C 2 = 0 so that the consolidated budget constraint becomes: C 1 + C 2 1 + r 1 = (1 t L ) [ (1 + r 0 )a 0 + w 1 L + w 2 L 1 + r 1 ] Ω 1 (WT) A proportional tax on consumption (t C ) augments the budget constraints as follows: a 1 = (1 + r 0 )a 0 + w 1 L (1 + tc )C 1 a 2 = (1 + r 1 )a 1 + w 2 L (1 + tc )C 2 = 0

Public Economics: Chapter 3 20 so that the consolidated budget constraint becomes: (1 + t C ) [ C 1 + C 2 1 + r 1 ] = (1 + r 0 )a 0 + w 1 L + w 2 L 1 + r 1 Ω 2 (CT) Comparing (WT) and (CT) reveals that effect on choice set is the same if: Remarks: 1 t L = 1 1 + t C no effect on optimal consumption plans: relative price of future consumption unaffected private saving plans are affected: time path tax revenues affected warning: equivalence result hinges on time-constancy of t C! If t C1 t C2 then Euler equation is affected (like interest tax)

Public Economics: Chapter 3 21 A proportional tax on interest income (t R ) affects the budget constraints as follows: a 1 = [1 + r 0 (1 t R )]a 0 + w 1 L C1 a 2 = [1 + r 1 (1 t R )]a 1 + w 2 L C2 = 0 so that the consolidated budget constraint becomes: C 1 + C 2 1 + r 1 (1 t R ) = [1 + r w 2 L 0 (1 t R )]a 0 + w 1 L + 1 + r 1 (1 t R ) Ω 3 (IT) A proportional wealth tax (t Wi ) affects the budget constraints as follows: a 1 = (1 + r 0 t W1 )a 0 + w 1 L C1 a 2 = (1 + r 1 t W2 )a 1 + w 2 L C2 = 0

Public Economics: Chapter 3 22 so that the consolidated budget constraint becomes: C 1 + C 2 w 2 L = (1 + r 0 t W1 )a 0 + w 1 L + Ω 4 (AT) 1 + r 1 t W2 1 + r 1 t W2 Comparing (IT) and (AT) reveals that effect on choice set is the same if: 1 t W1 = 1 r 0 t R and 1 t W2 = 1 r 1 t R Remarks: holds only in single-asset situation [see Topic 4] wealth tax must take into account that interest rate may be time-variant if a i < 0 (borrowing) then wealth tax actually leads to receipts from the government (as t Wi a i 1 > 0 in that case)

Public Economics: Chapter 3 23 Application: The Effects of Consumption Taxes Study the effects of (potentially time-varying) consumption taxes t C1 and t C2 Often-used case in the literature assumes intertemporal additive separability in preferences. Life-time utility function becomes: Λ ( ) = U(C 1 ) + ( 1 1 + ρ ) U(C 2 ) U( ) is the instantaneous utility function (often called felicity function) ρ > 0 is the constant pure rate of time preference, representing the effects of impatience. The higher ρ, the heavier future utility is discounted, and the more impatient is the household

Public Economics: Chapter 3 24 Assume that the felicity function features a constant intertemporal substitution elasticity: U (C t ) 1 1 1/σ C 1 1/σ t The consolidated lifetime budget constraint is: for σ 1 ln C t for σ = 1 (1 + t C1 )C 1 + (1 + t C2)C 2 1 + r 1 = (1 + r 0 )a 0 + h 0 Ω (LBC) where h 0 is human wealth: h 0 w 1 L + w 2 L 1 + r 1

Public Economics: Chapter 3 25 Household chooses C 1 and C 2 to maximize Λ ( ) subject to the life-time budget constraint. The Lagrangian is: L C1 1/σ 1 1 1 1/σ + +µ ( ) 1 1/σ 1 C 2 1 1 + ρ 1 1/σ [ Ω (1 + t C1 )C 1 (1 + t C2) C 2 1 + r 1 where µ is the Lagrange multiplier. The first-order conditions consist of (LBC) and: C 1/σ 1 = µ (1 + t C1 ) ( ) 1 C 1/σ 2 = µ (1 + t C2) 1 + ρ 1 + r 1 ]

Public Economics: Chapter 3 26 or: Remarks: C 2 C 1 = [( ) ( 1 + tc1 1 + r1 1 + t C2 1 + ρ )] σ (EE) if r 1 > ρ then ceteris paribus C 2 /C 1 high: postpone consumption by saving now (more so the higher is σ) if t C1 > t C2 then ceteris paribus C 2 /C 1 high: postpone consumption by saving now (more so the higher is σ) intertemporal price affected if t C1 t C2 if t C is time-invariant then it drops out of (EE) [see also above]

Public Economics: Chapter 3 27 Quantitative analysis with linearized expressions and marginal tax changes: linearization of (LBC) [holding constant r i, w i, and thus Ω] yields: [ ] [ ] ω 1 C1 + t C1 + (1 ω 1 ) C2 + t C2 = 0 (A) with: linearization of (EE) yields: C 1 dc 1 C2 C 1 dc 2 C 2 t C1 dt C1 1+t C1 t C1 dt C1 1+t C1 ω 1 C 1(1+t C1 ) 1 ω Ω 1 C 2(1+t C2 ) (1+r 1 )Ω C 2 C 1 = σ [ t C1 t C2 ] (B)

Public Economics: Chapter 3 28 combining (A)-(B) yields: ω 1 1 ω 1 1 1 C 1 C 2 = ω 1 σ t C1 1 ω 1 σ t C2 (C) Remark: three cases can be studied: (a) present tax shock t C1 > 0 and t C2 = 0 (b) future tax shock t C2 > 0 and t C1 = 0 ( t (c) equal tax shock C1 = t C2 = ) t C > 0

Public Economics: Chapter 3 29 The effect on net saving can be deduced from the first-period (or second-period) budget identity: S 1 a 1 a 0 Loglinearized (see also below) we get: = r 0 a 0 + w 1 L (1 + tc1 )C 1 = (1 + t C2)C 2 w 2 L 1 + r 1 a 0 ds [ ] 1 Ω = ω 1 C1 + t C1 [ ] = (1 ω 1 ) C2 + t C2

Public Economics: Chapter 3 30 Case (a): raising the current consumption tax From (C) we find: C 1 C 2 = = 1 (1 ω 1) ω 1 t C1 1 ω 1 σ [ω 1 + (1 ω 1 ) σ] t C1 (σ 1)ω 1 present consumption falls unambiguously (postponement effect) effect on future consumption ambiguous due to offsetting IE and SE [in Cobb-Douglas case σ = 1 and C 2 = 0] saving rises (falls) if future consumption rises (falls)

Public Economics: Chapter 3 31 Figure 3.2 illustrates the effects for the case of low substitutability [0 < σ < 1] Total effect: from E 0 to E 1 SE: from E 0 to E IE: from E to E 1 Note: for σ = 0 (no substitution at all) new equilibrium would be at e 1 ; for σ = 1 (Cobb-Douglas) it would be at e 2 [EC 2 (not drawn) would pass through e 2 ]

Public Economics: Chapter 3 32 C 2 WEP 1 WEP 0 A! e 2!! E 1! EN!! E 0 e 1 7 0! 0! C! B C 1 Figure 3.2: Raising the Current Consumption Tax [low σ]

Public Economics: Chapter 3 33 Case (b): raising the future consumption tax From (C) we find: C 1 C 2 = = 1 (1 ω 1) 1 ω 1 (σ 1) (1 ω 1) [1 ω 1 + ω 1 σ] 1 ω 1 σ t C2 t C2 future consumption falls unambiguously (reverse postponement effect) effect on present consumption ambiguous due to offsetting IE and SE [in Cobb-Douglas case σ = 1 and C 1 = 0] saving falls (rises) if present consumption rises (falls)

Public Economics: Chapter 3 34 Figure 3.3 illustrates the effects for the Cobb-Douglas case (σ = 1) [indifference curve omitted] Total effect: from E 0 to E 1 SE: from E 0 to E IE: from E to E 1 Note: for σ = 0 (no substitution at all) new equilibrium would be at e 1

Public Economics: Chapter 3 35 C 2 WEP 0 A! WEP 1 C! e 1! E 0!! EN! E 1! 0! B C 1 Figure 3.3: Raising the Future Consumption Tax [Cobb-Douglas (σ = 1) case]

Public Economics: Chapter 3 36 Case (c): intertemporally-neutral increase in the consumption tax ( ) We assume that t C = t C1 = t C2 > 0 so that the Euler equation (EE) is unaffected From (C) we find: C 1 C 2 = = 1 (1 ω 1) 1 ω 1 1 t C1 1 1 0 t C intertemporal substitution elasticity does not matter! present and future consumption both fall unambiguously (due to IE)

Public Economics: Chapter 3 37 no effect on saving as can be deduced from: ds [ ] 1 Ω = ω 1 C1 + t C1 [ ] = (1 ω 1 ) C2 + t C2 Figure 3.4 illustrates the effects for the general case (σ may be anything) [indifference curve omitted] Total effect: from E 0 to E 1 SE: absent IE: from E 0 to E 1

Public Economics: Chapter 3 38 C 2 WEP 0 A! C! E 0! E 1!! 0! D! B C 1 Figure 3.4: Intertemporally Neutral Increase in the Consumption Tax

Public Economics: Chapter 3 39 A Basic Intertemporal Model of Labour Supply Not very difficult to introduce an intertemporal dimension to the consumption leisure choice [pioneered by Lucas & Rapping in the late 1960s] Continue to assume: Period 1 is the present, period 2 is the remaining future Perfect foresight about wages, prices, interest rates, and taxes No bequests Perfect capital markets

Public Economics: Chapter 3 40 Lifetime utility function [intertemporally additively separable]: Λ ( ) = U(C 1, L L 1 ) + U( ) is the instantaneous utility function ( 1 1 + ρ ) U(C 2, L L 2 ) ρ > 0 is the pure rate of time preference, representing the effects of impatience The higher ρ, the heavier future utility is discounted, and the more impatient is the household. C i is consumption and L i is labour supply in period i (i (1, 2)), L is the exogenous time endowment, and L L i is leisure

Budget identities [in nominal terms]: Public Economics: Chapter 3 41 A 1 = (1 + R 0 )A 0 + (1 t L1 )W 1 L 1 P 1 C 1 A 2 = (1 + R 1 )A 1 + (1 t L2 )W 2 L 2 P 2 C 2 = 0 A i represents financial assets at the end of period i [A 0 was accumulated in the past ] R i is the interest rate in period i t Li is the labour income tax in period i P i is the price of goods in period i W i is the wage rate in period i

Public Economics: Chapter 3 42 Choose the consumption good, C i, as the numeraire so that the real budget constraints can be written as: a 1 = (1 + r 0 )a 0 + w 1L 1 C 1 a 2 = (1 + r 1 )a 1 + w 2L 2 C 2 = 0 (1 + r i ) (1 + R i ) P i P i+1 is the real interest rate in period i a i A i /P i is real financial assets in period i w i W i /P i is the real wage rate in period i w i (1 t Li )w i is the after-tax real wage rate in period i

Public Economics: Chapter 3 43 The household can freely borrow or lend at the going interest rate r 1, a 1 can have either sign and the budget identities can be consolidated into a single lifetime budget constraint. [With binding borrowing constraint, a 1 0, we are back in static case] step 1: solve for a 1 : a 1 = C 2 w 2L 2 1 + r 1 = (1 + r 0 )a 0 + w 1L 1 C 1 step 2: re-express in terms of C i and leisure L L i : ) ) C 1 + w1 ( L C 2 + w L1 + 2 ( L L2 = (1 + r 0 )a 0 + h 0 (LBC) 1 + r 1 where h 0 is net human wealth representing the after-tax value of the time endowment: h 0 w 1 L + w 2 L 1 + r 1

Some pertinent remarks: Public Economics: Chapter 3 44 The life-time budget constraint (LBC) says that for the solvent household the present value of spending on goods and leisure [left-hand side] equals initial total wealth [right-hand side] The prices of C 1, ( L L1 ), C2, and ( L L2 ) are, respectively, 1, w 1, 1 1+r 1, and w 2 1+r 1. We can thus use duality theory just as for the static model to derive Hicksian and Marshallian expressions for C i and ( L Li ) Obviously we expect to find income and substitution effects to play a crucial role [this is left as an exercise for the ambitious student] Usually possible to solve the model with two-stage budgeting

Public Economics: Chapter 3 45 Digression on Two-Stage Budgeting: TOOL Useful technique both for theoretical and empirical work Basic idea: split up dynamic problem into a static part (easy) and a dynamic part (almost as easy) static part: divide full consumption over components dynamic part: choose optimal time path for full income The TSB procedure is valid if preferences are intertemporally separable and the felicity function is homothetic Example: constant elasticity of substitution (both intra-temporally and inter-temporally)

Public Economics: Chapter 3 46 Felicity function is of the iso-elastic form: U (F t ) 1 1 1/σ F 1 1/σ t for σ 1 ln F t for σ = 1 (A) where σ represents the intertemporal substitution elasticity (σ 0) and where F t is sub-felicity, depending on consumption and leisure. Sub-felicity is also iso-elastic: F t [ ε (C t ) (η 1)/η + (1 ε) ( L Lt ) (η 1)/η ] η/(η 1) for η 1 (C t ) ε ( L Lt ) 1 ε for η = 1 (B) where η is the intratemporal substitution elasticity between consumption and leisure.

Public Economics: Chapter 3 47 We define full consumption as total spending in a period: X t C t + (1 t Lt )w t ( L Lt ) (C) The real budget constraints can then be written as: Stage 1: Optimal Static Choice a 1 = (1 + r 0 ) a 0 + w 1 L X 1 (D1) a 2 = (1 + r 1 ) a 1 + w 2 L X 2 = 0 (D2) in the static stage the household chooses consumption, C t, and leisure, L L t, in order to maximize sub-felicity, F ( C t, L L t ) (given in (B)) given the constraint (C) holding constant full consumption, X t. Formally (for the easy case with η = 1) the problem is: max F t (C t ) ε ( L ) 1 ε Lt {C t, L L t} subject to (C)

Public Economics: Chapter 3 48 Lagrangian expression: L 1 (C t ) ε ( L Lt ) 1 ε + µt [ Xt C t w t ( L Lt )] where µ t is the Lagrange multiplier. first-order necessary conditions (FONCs): L 1 = εf t µ C t C t = 0 t L 1 ( L ) = (1 ε) F t µ Lt L L t wt = 0 t eliminate µ t by combining the two FONCs: (1 ε) / ( L Lt ) ε/c t = w t (SC) Hence: the MRS between leisure and consumption equated to the relative price of leisure (the after-tax real wage rate). This is the static condition (SC hereafter).

Public Economics: Chapter 3 49 By combining (SC) and the budget constraint (C) we find the level solutions for consumption and leisure conditional on full consumption: w t ( L Lt ) = (1 ε)xt (E2) C t = εx t (E1) Note: Cobb-Douglas subfelicity features constant spending shares! We can now deduce the true price index linking X t and F t. Steps: postulate link: P Ft F t = X t (F) where P Ft is the (yet unknown) cost of living index.

Public Economics: Chapter 3 50 substituting (E1)-(E2) into the subfelicity function (for the CD case): ( ) ε ( 1 w t ε 1 ε F t comparing (F) and (G) shows that P Ft is: = (εx t ) ε ( (1 ε)xt w t ) 1 ε ( ) 1 ε 1 ε = ε ε X t w t ) 1 ε F t = X t (G) P Ft = ( 1 ε ) ε ( w t 1 ε ) 1 ε (TPI) where 1 and w t represent the price of, respectively, consumption and leisure.

Public Economics: Chapter 3 51 Note (1): the adventurous student can deduce the solutions for C t, L L t, and P Ft for the general case with η 1. [See solution below] Note (2): expression (F) is really an expenditure function, i.e. we can write E (1,w t,f t ) = P Ft F t and recover the Hicksian demand for consumption and leisure in the usual fashion. Similarly, the indirect utility function can be recovered by writing V (1,w t,x t ) = X t /P Ft from which we can recover the Marshallian demands.

Stage 2: Optimal Dynamic Choice Public Economics: Chapter 3 52 in the dynamic stage the household chooses full consumption in the two periods (X 1 and X 2 ) in order to maximize lifetime utility Λ ( ) subject to the budget constraints. Formally (for the easy case with σ = 1) the problem is: ( ) ( ) ( ) max ln X1 1 X2 + ln {X 1,X 2,a 1 } 1 + ρ subject to: P F1 P F2 a 1 = (1 + r 0 )a 0 + w 1 L X 1 0 = (1 + r 1 )a 1 + w 2 L X 2

Lagrangian expression: Public Economics: Chapter 3 53 ( ) ( ) ( ) X1 1 X2 L 2 ln + ln P F1 1 + ρ P F2 [ +λ 1 a1 (1 + r 0 )a 0 w1 L ] + X 1 +λ 2 [ 0 (1 + r1 )a 1 w 2 L + X 2 ] where λ 1 and λ 2 are the Lagrange multipliers for the two budget constraints. first-order necessary conditions (FONCs): L 2 = 1 λ 1 = 0 X 1 X ( 1 ) L 2 1 1 = λ 2 = 0 X 2 1 + ρ X 2 L 2 = λ 1 λ 2 (1 + r 1 ) = 0 a 1

eliminate λ 1 and λ 2 to get: Public Economics: Chapter 3 54 1/X 1 1/ [(1 + ρ)x 2 ] = 1 + r 1 (DC) Hence: the MRS between present and future full consumption is equated to the relative price of present full consumption (the gross interest rate). This is the key dynamic condition (DC) characterizing the optimum. Note, by rewriting we obtain the Euler equation for full consumption: X 2 = 1 + r 1 X 1 1 + ρ (EE)

the consolidated budget constraint is: where h 0 is net human wealth: Public Economics: Chapter 3 55 X 1 + X 2 1 + r 1 = (1 + r 0 ) a 0 + h 0 (LBC) h 0 w 1 L + w 2 L 1 + r 1

Public Economics: Chapter 3 56 By combining (DC) [or equivalently (EE)] with (LBC) we find the full consumption levels: ( ) 1 + ρ X 1 = [(1 + r 0 ) a 0 + h 0 ] 2 + ρ ( ) 1 + r1 X 2 = [(1 + r 0 )a 0 + h 0 ] 2 + ρ Since we know X 1 and X 2 we also know C 1, C 2, L L 1, and L L 2. We are done! Note (1): again we find constant spending shares (because σ = 1) Note (2): the adventurous student can deduce the solutions for X 1 and X 2 for the general case with σ 1 and / or η 1.

Public Economics: Chapter 3 57 Back to the Basic Intertemporal Labour Supply Model Work with the general utility specification (A)-(B): U (F t ) F t F 1 1/σ t 1 1 1/σ, (A) [ ε (C t ) (η 1)/η + (1 ε) ( L ) ] (η 1)/η η/(η 1) Lt (B) The diligent student will be able to verify the optimal solutions static: w t ( L Lt ) = (1 ωct ) X t ω Ct ε η ε η + (1 ε) η [w t ] 1 η C t = ω Ct X t

Public Economics: Chapter 3 58 true price index: dynamic: P Ft [ ε η + (1 ε) η [w t ] 1 η] 1/(1 η) ( 1 ε for η 1 ) ε ( w t 1 ε) 1 ε for η = 1 F 2 F 1 = ( PF2 P F1 X 2 X 1 P F2F 2 P F1 F 1 = ) σ ( 1 + r1 ( PF2 ) σ 1 + ρ ) 1 σ ( 1 + r1 P F1 1 + ρ ) σ

closed-form solution for X 1 : Public Economics: Chapter 3 59 X 1 = ξ 1 [(1 + r 0 )a 0 + h 0 ] [ ( ) σ ( ) ] 1 σ 1 1 1 + PF1 (1+r 1 ) 1+ρ P F2 ξ 1 for σ 1 1+ρ 2+ρ for σ = 1 The key thing to note how many different types of wage (or tax) elasticities can be distinguished. The labour income tax affects: the static division of X t over C t and ( L Lt ). The intratemporal substitution elasticity η matters here the true cost-of-living index and thus...

Public Economics: Chapter 3 60...the dynamic divisions F 2 /F 1 and X 2 /X 1. Both η and the intertemporal substitution elasticity σ matter here The derivation of the effects of labour taxes is left as an exercise. Some final remarks: quantitative linearization technique can again be used in general we expect to find IE, SE, and HWE econometricians have estimated multi-period dynamic labour supply models in their various formats, i.e. with different conditioning variables [see Blundell & MaCurdy (1999) for details]

Public Economics: Chapter 3 61 Extensions to the Two-Period Model human capital accumulation and the effects of taxation borrowing constraints bequests

Human Capital Accumulation Public Economics: Chapter 3 62 transfer resources through time by investing in human capital (educational and training effort) lifetime utility: Λ = U ( C 1,C 2, L L 1 I 1 ) C t is consumption in period t L 1 is work in period 1 I 1 is training time in period 1 ( L L 1 I 1 is current leisure) L 2 = L (exogenous future labour supply)

Public Economics: Chapter 3 63 training technology: H 2 = G (I 1 )H 1 G ( ) is the human capital production function (G ( ) > 0 for I 1 0); G > 0 > G ; lim I1 0 G (I 1 ) =. H 1 is exogenously given initial stock of human capital if household possesses H t units of human capital and works L t raw hours, then the labour effort in efficiency units is H t L t and gross wage income is w t H t L t (for t = 1, 2).

periodic budget constraints: Public Economics: Chapter 3 64 a 1 = (1 t L1 )w 1 H 1 L 1 C 1 0 = [1 + r 1 (1 t R )]a 1 + (1 t L2 ) w 2 H 2 L C2 t R is a proportional tax on interest income t Lt is the labour income tax. lifetime budget constraint (no borrowing constraints): C 1 + C 2 1 + r 1 (1 t R ) = (1 t L1)w 1 H 1 L 1 + (1 t L2)w 2 H 2 L 1 + r 1 (1 t R )

Public Economics: Chapter 3 65 household chooses C 1, C 2, L 1, and I 1 in order to maximize lifetime utility subject to the training technology and the lifetime budget constraint. The first-order conditions associated with an interior solution to this maximization problem are: U/ C 1 U/ C 2 = 1 + r 1 (1 t R ), (A) U/ ( L L1 I 1 ) = (1 t L1 ) w 1 H 1, (B) U/ C 1 U/ ( L ) L1 I 1 = (1 t L2)w 2 G (I 1 ) H 1, (C) U/ C 1 1 + r 1 (1 t R ) (A) is the Euler equation (B) is the condition for optimal labour supply (C) is the condition for optimal training effort

Public Economics: Chapter 3 66 combining (B) and (C) we get an expression determining I 1 : H 1 does not feature G (I 1 ) = 1 t L1 1 t L2 w 1 w 2 [1 + r 1 (1 t R )]. form of utility function does not matter t L1 or t L2 both lead to G (I 1 ) or I 1 t R leads to G (I 1 ) or I 1 (reduces the cost of borrowing!) if t L1 = t L2 it drops out; general income tax increase leads to increase in I 1!

Borrowing Constraints Public Economics: Chapter 3 67 According to Sandmo (1985), the two-period model suffers from at least two potentially serious abstractions: lending and borrowing rates are the same there are no quantity constraints on borrowing In reality both assumptions are not true

Public Economics: Chapter 3 68 Let the borrowing rate in period 1 be r B1 and the lending rate r L1 and let r L1 < r B1. Then the second-period budget identity would be affected according to [ignore initial assets]: a i 1 = (1 t L1 ) w 1 L C i 1 (1 + r L1 )a i 1 + (1 t L2 ) w 2 L for a i 1 > 0 C2 i = (1 + r B1 ) a i 1 + (1 t L2 )w 2 L for a i 1 0 In terms of Figure 3.5 the life-time budget constraint (LBC) would feature a kink at the point where a i 1 changes sign [point B]. The choice set shrinks from ABF to ABE

Public Economics: Chapter 3 69 C 2 i! A (1 + r L1 )$ L = (1 + r B1 )$ B B! (1 + r B1 )$ C C!! 0! D! E F! C 1 i Figure 3.5: Capital Market Constraints and the Choice Set

Public Economics: Chapter 3 70 If there is, in addition, a quantity constraint on the maximum amount that can be borrowed, say a i 1 a MAX then the LBC features an additional kink to the right of point B [at point C]. The choice set shrinks further from ABE to ABCD likely that many household have optimum at the kink. See Figure 3.6 (explained in detail in the text) many substitution effects (of taxes) may not be relevant for such households econometric research must take such features into account

Public Economics: Chapter 3 71 f($ i )!!! i a 1 =!a i MAX i i!a MAX < a 1 < 0 i a 1 = 0 i a 1 > 0 0 $ C $ B $ L $ i Figure 3.6: Capital Market Constraints and Household Patience

Bequests Public Economics: Chapter 3 72 The basic model is easily extended for the existence of intergenerational linkage via altruistic bequests With this model one can study the impact of estate taxation Core model is due to Barro (who used it to investigate Ricardian equivalence) Our basic consumption model (with exogenous labour supply) would be changed as follows Budget identities parent : C Y 1 + a 1 + b 1 C O 2 = (1 + r 0 )b 0 + w 1 L = (1 + r 1 )a 1 + w 2 L

Budget identities child : Public Economics: Chapter 3 73 C Y 2 + a 2 + b 2 C O 3 = (1 + r 1 )b 1 + w 2 L = (1 + r 2 )a 2 + w 3 L etcetera (for grandchild, great-grandchild...). Remarks: b 1 is the bequest given to the offspring of the household [the offspring receives (1 + r 1 )b 1 at beginning of its first period of life (inclusive of interest)] (1 + r 0 )b 0 was the inheritance the household itself received Ct Y is consumption by young ( Y ) household in period t Ct O is consumption by old ( O ) household in period t

Public Economics: Chapter 3 74 In the absence of capital market imperfections (a t unrestricted) the consolidated constraints are relevant: etcetera. C1 Y + b 1 + CO 2 w 2 L = (1 + r 0 )b 0 + w 1 L + 1 + r 1 1 + r }{{ 1} h 1 C2 Y + b 2 + CO 3 w 3 L = (1 + r 1 )b 1 + w 2 L + 1 + r 2 1 + r }{{ 2} h 2 (LBC1) (LBC2)

Public Economics: Chapter 3 75 Preferences: common assumption is that of altruism towards one s offspring: Λ Y 1 ( ) = U ( C Y 1,C O 2 Λ Y 2 ( ) = U ( C Y 2,C O 3 ( ) ) 1 + Λ Y 2 ( ) 1 + δ ( ) ) 1 + Λ Y 3 ( ) 1 + δ (OF1) (OF2) etcetera. Λ Y 1 ( ) is life-time utility function of the household who is young in period 1 [the parent ] Λ Y 2 ( ) is life-time utility function of the household who is young in period 2 [the current child ] δ (> 0) parameterizes the parent s concern for the child s wellbeing [the higher is δ the less the parental concern is]

Public Economics: Chapter 3 76 We can see that the effective objective function for the current parents is the forward iteration of (OF1) taking into account (OF2) etcetera: Λ Y 1 ( ) = U ( ( ) [ ) 1 C1 Y,C2 O + U ( ( ) ] ) 1 C2 Y,C3 O + Λ Y 3 ( ) 1 + δ 1 + δ = U ( ( ) ) 1 C1 Y,C2 O + U ( ) C2 Y,C3 O 1 + δ ( ) 2 1 + U ( ) C3 Y,C4 O +... (DOF) 1 + δ consumption levels by all present and future generations matter δ parameterizes the degree of discounting applied to future generations

Public Economics: Chapter 3 77 In the absence of constraints on bequests (either b t unrestricted or restriction b t 0 never binding) consolidation over the dynastic family can be achieved (b 1, b 2 etcetera can be substituted): C Y 1 + etcetera, leading to: C Y 1 + CO 2 + C Y 2 1 + r 1 + CY 2 + b 2 + CO 3 1+r 2 h 2 1 + r 1 + CO 2 1 + r 1 = (1 + r 0 )b 0 + h 1 C O 3 + C Y 3 (1 + r 1 ) (1 + r 2 ) + = (1 + r 0)b 0 + h 1 + h 2 1 + r 1 + (DLBC) the left-hand side represents the present value of present and future consumption by all members of the household dynasty the right-hand side represents the present value of present and future resources of all members of the household dynasty

Public Economics: Chapter 3 78 Maximizing the dynasty objective function (DOF) subject to the dynasty life-time budget constraint (DLBC) is a multi-period generalization of our earlier problems. Cautionary and Other Remarks: a wealth transfer tax acts like an interest income tax if negative bequests are not taxed the multidimensional choice set of the dynasty is kinked: likely to be lot of households in the kink (with zero bequests) if b t 0 becomes binding Ricardian Equivalence fails and estate taxation will not have an effect at all

Public Economics: Chapter 3 79 Some Empirical Evidence on Key Elasticities Interest elasticity of saving uncompensated savings effect of change in interest rate ambiguous in 1960s and 1970s interest rate effect estimated by using ad hoc consumptionor savings function. Interest elasticity of current consumption typically positive but low from 1980s onward Euler equations: estimate intertemporal subst. elasticity, σ: lnc t+1 lnc t σ ( r t ρ t Ct+1 t Ct 1 + t Ct+1 by using data on C t, r t (and possibly taxes). Typical finding is that σ is close to zero (σ 1/3 seems reasonable) calibration approach: use estimate for σ to compute interest elasticities. Human wealth effect may be quite important )

Intertemporal substitution in labour supply pioneers Lucas & Rapping (1969). Public Economics: Chapter 3 80 econometricians estimate Frisch demands for consumption and leisure: U C ( Ct, L L t ) = λt ( U L L Ct, L ) L t = λt wt ( ) 1 + rt λ t = 1 + ρ where λ t is the marginal utility of initial wealth in period t estimable labour supply equation: λ t+1 ln L t = σ L lnβ t + σ L ln w t σ L (r t 1 ρ), (1) that is, labour supply changes if (1) tastes change, (2) the after-tax wage changes, or (3) there is a gap between r t 1 and ρ.

Public Economics: Chapter 3 81 empirical findings low intertemporal substitution elasticity (sometimes wrong sign) Table 3.1 shows some typical results for prime-age male workers Different estimates due to different IVs or equation formats Note: the jury is still out on this issue. Better data and better estimation methods may unearth more support for the intertemporal substitution hypothesis

Public Economics: Chapter 3 82 Research by: Intertemporal demographic group: Substitution Elasticity MaCurdy 0.14 to 0.35 0.10 to 0.45 Becker 0.45 white men 0.10 non-white men Becker 0.17 white men 0.06 non-white men Smith 0.32 white men 0.23 non-white men Browning et al. 0.05 0.03 MaCurdy 0.07 to 0.28 Blundell at al. 0.5 to 1 married women Table 1. Intertemporal Substitution Elasticity in Labour Supply

Punchlines Public Economics: Chapter 3 83 two-period consumption-saving model of Fisher concepts: solvency condition, human wealth, lifetime utility, felicity function, Euler equation, intertemporal substitution decomposition: income effect, human wealth effect, substitution effect some important tax equivalencies tax policy analysis: the effects of the consumption tax (linearized model)

Public Economics: Chapter 3 84 intertemporal labour supply model method: two-stage budgeting human capital accumulation and the effects of taxes capital market imperfections: differential interest rates, credit rationing intergenerational linkages and altruistic bequests empirical evidence regarding the intertemporal substitution hypothesis