essays on optimal fiscal policy

Asimakopoulos, Stylianos (2014) Essays on optimal fiscal policy PhD thesis http://theses.gla.ac.uk/5282/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Glasgow Theses Service http://theses.gla.ac.uk/ theses@gla.ac.uk Essays on optimal …scal policy Stylianos Asimakopoulos Submitted in ful…llment of the requirements for the Degree of Doctor of Philosophy Adam Smith Business School College of Social Sciences University of Glasgow June 2014 Abstract This thesis examines the properties of optimal …scal policy in the long-run and over the business cycle in general equilibrium models with agents that diÔer with respect to their skills and with production processes embodying capital-skill complementarity To this end, the thesis is composed of four chapters which asses diÔerent aspects of optimal scal policy under various specications incorporating labour skill and production diÔerences as well as diÔerent assumptions regarding the policy instruments available to the government The …rst two chapters focus on the long-run, while the last two concentrate on business cycle dynamics The …rst and third chapters examine setups that allow households to diÔer with respect to their income and whose position in the labour market with respect to their skill is exogenously determined In contrast, the second and fourth chapters consider setups where the labour force belongs to a single household, which guarantees consumption irrespective of skill level, and unskilled labour can endogenously acquire skills to become skilled Chapter presents a detailed numerical analysis of the eÔects of optimal scal policy in an economy where the households are heterogeneous with respect to their labour and capital income The production structure is characterised by a CES function allowing for capital, skilled and unskilled labour In this setup, optimal …scal policy in the long-run implies a nonzero and positive tax rate on capital income together with highly progressive labour income taxes Moreover, the level of the optimal tax rate on capital income and the progressivity of labour income taxes are sensitive to the weight placed on the skilled agents in the objective function of the government Chapter analyses optimal factor income taxation when there are diÔerent returns to skilled and unskilled workers, who belong to the same household, and to capital in structures and equipment, under capital-skill complementarity and endogenous skill acquisition We …nd that when all factor inputs are taxed at separate rates, both capital income taxes are zero in the long-run, there is a subsidy to education and labour income taxes are progressive The progressivity in labour income taxes is reduced if investment in ii education cannot be subsidised, whereas if the government can only impose a single labour income tax, the tax on income from capital equipment will be non-zero These results remain valid even if the government is restricted to satisfy a given level of debt to output ratio, although with welfare losses Finally, we show that the transitional dynamics of the …scal instruments from the exogenous to optimal taxation are not aÔected by the restrictions to the scal policy menu Chapter examines how income taxes are optimally distributed over the business cycle in a model with high, middle and low income households when the government is restricted to balance its budget in each period The …ndings of an empirically relevant model indicate that under optimal …scal policy the income tax rate of the high income households has the lowest volatility and the income tax rate of the low income households exhibits the lowest counter-cyclicality If the …scal policy menu also includes a consumption tax, the progressivity of the income tax rates is even higher and the results regarding the volatilities of the income taxes are overturned We further …nd that the progressivity of the income tax rates is optimally increased after an output-enhancing shock Chapter undertakes a normative investigation of the quantitative properties of optimal tax smoothing in a business cycle model with state contingent debt, capital-skill complementarity, endogenous skill formation and stochastic shocks to public consumption, as well as total factor and capital equipment productivity We also examine the properties of optimal taxation under a restriction on the debt to output ratio Our main …nding is that, an empirically relevant restriction which does not allow the relative supply of skilled labour to adjust in response to aggregate shocks, signi…cantly changes the cyclical properties of optimal labour taxes This result remains valid even in the presence of a budget rule that restricts the debt to output ratio We show that the key to understanding this result is that the government …nds it optimal to adjust labour income tax rates to alter the average net returns to skilled and unskilled labour hours iii Contents Preface Chapter 1: Optimal …scal policy under skill heterogeneity and capital-skill complementarity 1.1 Introduction 1.2 The model 1.2.1 Firms 1.2.2 Households 1.2.3 The government budget constraint 1.2.4 Aggregate resource constraint and market clearing 1.3 Decentralized competitive equilibrium 1.4 Calibration 1.5 Exogenous …scal policy 1.6 Optimal …scal policy 1.7 Exogenous vs optimal …scal policy 1.7.1 Steady-state 1.8 Comparison of welfare and income inequality 1.9 Analysis of the results 1.10 The model without capital-skill complementarity 1.11 The model with capital-skill complementarity 1.12 Optimally chosen government consumption 1.13 Assessing income inequality 1.14 Partisan policy 1.15 Concluding remarks 8 13 13 15 17 17 18 19 22 22 25 25 28 30 32 35 37 39 41 46 Appendices 48 Appendix A Chapter A.1 First order conditions A.1.1 Households A.1.2 The …rm A.2 Decentralized competitive equilibrium 48 48 48 54 56 iv A.2.1 The …rst-order conditions of skilled households A.2.2 Budget constraint of skilled agents A.2.3 The …rst-order conditions of unskilled households A.2.4 Firms’…rst-order conditions A.2.5 The behaviour of the exogenous processes A.2.6 Government’ budget constraint s A.2.7 The aggregate resource constraint A.2.8 The production function A.2.9 The steady-state A.3 Optimal …scal policy A.4 Welfare gains between policy regimes Chapter 2: Optimal factor income taxation with endogenous skill supply 2.1 Introduction 2.2 The model 2.2.1 The representative …rm 2.2.2 The representative household 2.2.3 The government budget constraint 2.2.4 Aggregate resource constraint and market clearing conditions 2.2.5 Competitive equilibrium with exogenous …scal policy 2.3 Calibration 2.4 Exogenous steady-state 2.5 Optimal …scal policy 2.5.1 Implementability constraint 2.5.2 The primal approach 2.5.3 Flexible debt: steady-state and lifetime welfare 2.5.4 Transition path 2.5.5 Budget rule 2.6 Concluding remarks Appendices 56 57 57 57 58 58 59 59 59 62 65 66 66 70 71 73 77 77 78 78 81 81 82 83 87 92 93 97 98 v Appendix B Chapter 98 B.1 DCE system of equations 98 B.2 Compensating consumption supplement 99 Chapter 3: Optimal income taxation over the business cycle101 3.1 Introduction 101 3.2 Model 107 3.2.1 Households 109 3.2.2 Production and …rms 114 3.2.3 The government budget constraint 117 3.2.4 Resource constraint and market clearing conditions 118 3.3 Exogenous policy 119 3.3.1 Decentralized competitive equilibrium 119 3.3.2 Data analysis and targets 120 3.4 Calibration 123 3.4.1 Population shares 123 3.4.2 Tax-spending policy 123 3.4.3 Production and capital and labour markets 124 3.4.4 Utility function 126 3.4.5 Technology 127 3.5 Solution and results 127 3.6 Optimal taxation over the business cycle 130 3.6.1 The problem of the government 130 3.6.2 Properties of optimal taxes over the business cycle 132 3.6.3 Changing the set of …scal instruments 135 3.7 The optimal distribution of the tax burden over the business cycle 137 3.7.1 The case with three income taxes 137 3.7.2 The case with diÔerent scal policy menu 140 3.8 Robustness of results 142 3.9 Conclusions 144 Appendices 146 vi Appendix C Chapter 146 C.1 The skill premium 146 Chapter 4: Tax smoothing in a business cycle model capital-skill complementarity 4.1 Introduction 4.2 Model 4.2.1 Notation 4.2.2 Households 4.2.3 First order conditions for households 4.2.4 Firms 4.2.5 Government budget and market clearing 4.3 The Ramsey problem 4.3.1 Present value of budget constraint 4.3.2 Implementability constraint 4.3.3 Pseudo value function 4.4 Quantitative implementation 4.4.1 Functional forms 4.4.2 Exogenous policy and calibration 4.4.3 Calibration 4.5 Deterministic Ramsey 4.6 Stochastic processes 4.7 Stochastic Ramsey 4.8 Cyclical properties 4.8.1 Second moments 4.8.2 Impulse responses 4.9 Imposing a budget rule 4.9.1 Cyclical properties under the budget rule 4.10 Conclusions Appendices with 148 148 153 153 154 155 157 157 158 158 160 160 162 162 164 164 167 170 172 173 173 177 182 184 187 189 Appendix D Chapter 189 D.1 Household’ …rst-order conditions 189 s vii D.2 D.3 D.4 D.5 Deterministic Ramsey system The eÔects of kt and t on the Ex ante capital tax Uncontingent debt D.5.1 Ex-post capital tax D.5.2 Private assets tax viii skill premium 189 191 193 194 194 195 List of Tables 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Calibration Exogenous steady-state results Great ratios and welfare Great ratios and skill premium Steady-state results of exogenous and optimal …scal policy Great ratios and welfare under exogenous and optimal …scal policy Summary of quantitative results Summary of income inequality results Calibration Steady state of exogenous …scal policy Comparison of steady state optimal tax results Comparison of steady state optimal taxation results with …xed debt to output ratio Business cycle statistics of main endogenous variables Data averages and business cycle statistics of policy variables Model parameters Steady state of the exogenous policy model Business cycle statistics of the exogenous policy model Optimal tax policy Optimal tax policy with consumption tax Model parameters Steady state of exogenous policy Steady state of optimal policy Parameters for stochastic processes Stochastic results Autocorrelations Steady state of optimal policy with …xed debt to output ratio Stochastic results under …xed debt to output ratio Autocorrelations under …xed debt to output ratio ix 21 22 23 23 27 28 31 40 80 81 89 96 120 121 125 128 129 133 137 166 167 169 171 174 175 183 186 187 The eÔects of kt and D.3 t on the skill premium DiÔerentiating the skill premium, given by (181) in the main text,with respect to kt we have (note that we not use the st notation to keep the presentation simpler): s wt u wt @ @kt Ak kt t = fAk t u t ) ht ) ((1 ( t hs ) t (1 ) (1 (1 ) ) ( t hs ) + t a Ak kt t a a g This is positive if a > ; a; < 1; < ; < DiÔerentiating (181) with respect to t gives: s wt u wt @ @ t hu ( t hs ) (1 t t = ) ( t hs ) + t Ak kt t + t A k kt t (1 hs t t ((1 ) ( t hs ) + t a( t hs ) Ak kt t t s + a ( t ht ) + k a t At kt ) (1 u t ) ht ) Ak kt t ) (1 Focusing on the term in the squared brackets we need to test if that term is positive If this is true then the above derivative is negative due to the negative sign at the beginning of the right hand side (1 ) ( t hs ) + Ak kt a( t hs ) t t t k s At kt + a ( t ht ) + + t Ak kt a t A k kt t t having in the …rst row the terms with labour and in the rest the terms with capital: (1 ) ( t hs ) + a ( t hs ) a( t hs ) t t t Ak kt + Ak kt + t t k k + t At kt a t At kt 191 a] + ]+ a ] a)] + )] + a)] ( t hs ) [1 +a t k + At kt [ + t A k kt [ t ) (1 ( t hs ) [(1 t k + At kt [(1 + t Ak kt [( t " ( t hs ) [(1 t k + At kt [(1 ) (1 )+ a)] + a)] t( # The …rst row is positive because (1 ) > and (1 a) > Therefore, if (1 )+ t( a) > the term in the squared brackets is positive (1 )+ (1 )> (1 ( (1 ( ) ) > 1, since > s wt u wt @ …nally have that @ To summarize: s wt u wt @ @ t t " t ((1 t t ( a) ( ) > ) )1 ) (1 ( t hs ) [(1 t k + At kt [(1 hs t a) > ) t Since < > t < we < hu ( t hs ) (1 t t = ( )> (1 Note that: t ) (1 )+ ) (1 ) ( t hs ) + t # Ak kt t a a)] + a)] t( u t ) ht ) is negative if [(1 )+ t( > 1, since > ) a)] > or > t , which is true because > and < t < 192 D.4 Ex ante capital tax Assume that the government uses a capital tax that is not state-contingent, so that its value for period t + is decided using the history st De…ne this uncontingent tax as k (st ) and note that it needs to satisfy the Eulert+1 equation from (149) in the main text, so that the Ramsey allocations are preserved: n h e uc (st ) = Et uc (st+1 ) Fk (st+1 ) k t+1 ) t+1 (s e where we have used Fk (st+1 ) = rt+1 (st+1 ) Hence, n h e uc (st ) = Et uc (st+1 ) Fk (st+1 ) By comparing (D24) with (D23), we see that n Et uc (s t+1 h e ) Fk (st+1 ) io io +1 implying that: k t t+1 (s ) Et uc (st+1 ) = h i k t+1 e )Fk (st+1 ) t+1 (s e Et uc (st+1 )Fk (st+1 ) (D24) needs to satisfy: +1 k t+1 ) t+1 (s (D23) needs to satisfy: +1 k t t+1 (s ) k t t+1 (s ) n h e = Et uc (st+1 ) Fk (st+1 ) k t t+1 (s ) k t t+1 (s ) io +1 = (D25) io : (D26) This gives k (st ) the ex ante capital tax interpretation, since, by multiplyt+1 ing both numerator and denominator in (D26) by kt+1 (st ), this expression provides the expected tax revenue from capital income as share of the expected capital income, where the expectation is calculated using information at period t To obtain the ex ante rate stated in equation (187) of the main text, we …rst expand the Euler-equation (D23): e uc (st ) = Et uc (st+1 )Fk (st+1 ) + Et uc (st+1 ) (1 Et uc (st+1 ) ) 193 k t+1 e )Fk (st+1 )+ t+1 (s (D27) e and note that Et uc (st+1 ) k (st+1 )Fk (st+1 ) in (D27) equals k (st )Et uc (st+1 ) t+1 t+1 ek (st+1 ), using (D26) Substituting this expression back into (D27) we F obtain: k t t+1 e )Fk (st+1 )+ t+1 (s )Et uc (s e uc (st ) = Et uc (st+1 )Fk (st+1 ) + Et uc (st+1 ) (1 ): (D28) Finally solving (D28) for k (st ) gives the ex ante capital tax rate reported t+1 in equation (187) of the main text D.5 Uncontingent debt D.5.1 Ex-post capital tax The treatment of state-uncontingent debt and presentation follows Chari et al (1994) and Ljungqvist and Sargent (2012, ch 16) Assume that the government issues uncontingent debt, bt+1 (st ) which has a risk-free return Rt (st ) The budget constraint of the government in period t is written as: gt (st ) = s a t + s (st )wt (st ) t st hs (st ) + t (st ) g [ t (st )] + u u (st )wt (st ) k t t t ) t (s )rt (s )kt (s + t st bt+1 (st ) hu (st )+ t bt (st ) Rt (st ) (D29) The budget constraint of the household in period t is given by: ct (st ) +k t+1 (st ) + s w t st + (1 t bt+1 (st ) _ Rt (st ) st hs st + t ) kt st + a t + [1 + k t u t st s t (st )] g [ t (st )] = (1 st u w t st rt st kt st 1 t +bt st st (st ) ) hu st + t (D30) which implies that the …rst-order condition with respect to holding bonds is given by: uc (st+1 ) = Et (D31) uc (st ) Rt (st ) 194 Note that the right-hand side of (D4) needs to be the same as the right-hand side of the …rst-order condition with respect to bonds in the case of statecontingent debt, so that the implied allocations from the two problems (i.e with and without state-contingent debt) are the same In turn, this implies that the risk-free (or uncontingent) return needs to satisfy: X = pt st+1 j st t) Rt (s st+1 pst (D32) To obtain an expression for br+1 (sr ) for a given period r, we work as follows We multiply the budget constraint of the household in (D30) for periods r and r + by r (sr ) and r+1 (sr+1 ) respectively, sum the resulting budget constraint in r + over all possible realisations sr+1 and add it to the budget constraint in period r We then use the …rst-order conditions of the household to simplify the expression and continue this forward process until time period T ! By imposing the appropriate transversality conditions we obtain an expression for br+1 (sr ) as a function of identi…ed equilibrium paths given in: _ br+1 (sr ) =Rr (sr ) _ P P t r t (st )[uc (st )ct (st )+uhs (st )hs (st )+uhu (st )hu (st )+ t (st )] t t r r r (s )uc (s ) t=r+1 st Rr (sr ) kr+1 (sr ) (D33) where, t (st ) is de…ned in the main text under equation (167) Hence we can use (D31) to obtain Rt (st ), (D33) to …nd bt+1 (st ) and …nally (D29) to calculate the ex-post capital tax reported in equation (188) of the main text D.5.2 Private assets tax Assume that the government issues uncontingent debt, bt+1 (st ), which has a risk-free return Rt (st ), satisfying (D32), but which is taxed using a statecontingent tax t+1 (st+1 ) The budget constraint of the government is now 195 written as: gt (st ) = s + a t + s (st )wt (st ) st g bt+1 (st ) Rt (st ) st hs (st ) + t t t (s t ) + t u u (st )wt (st ) t hu (st )+ t st k t t t )+ t (s )rt (s )kt (s st bt s t (D34) while the budget constraint of the household becomes: ct (st ) + kt+1 (st ) + bt+1 (st ) Rt (st ) + [1 + a t (st )] g [ t (st )] = t t u t u t t s t s s = [1 t (s )] wt (s ) t (s )] wt (s ) t (s ) ht (s ) + [1 t t t k t u ) kt (st ) + [1 t (s )] ht (s ) + (1 t (s ) rt (s ) t t kt (st ) + [1 ) t (s )] bt (s (D35) which implies that the …rst-order condition with respect to holding bonds becomes: X = Rt (st ) st+1 pst t+1 st+1 p st uc (st+1 ) uc (st ) t+1 st+1 (D36) The introduction of the new assets tax has to be such that the equilibrium allocations obtained without it are respected Hence the asset tax must be such that makes the right-hand side of (D4) and (D36) equal Hence, the asset tax must satisfy: Et uc (st+1 ) t+1 st+1 = (D37) which implies that at time period t, the expected value of the asset tax in period t + 1, valued in terms of utility, has to be equal to zero Therefore, (D37) implies that Rt (st ) in this case is given by (D31) as well Moreover, by working as above, we substitute household 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