Markets and Government • There is no annuity and/or life insurance, only risk free assets ◦ upon death, the risk-free assets convert to bequest ◦ bequest is transfered equality to all individuals alive • Government ◦ Collects taxes on labor earnings, consumption and corporate profit ◦ Makes transfers to individuals in pre- and post- retirement ages ◦ Makes exogenously given purchases • Budget constraint of the government G + ( r − n ) D + All Transfers = All Taxes steady state government debt – exogenous Roozbeh Hosseini(UGA) 7 of 34
Individual Optimization Problem • Individual of type θ solves T ∑ β t P t ( θ ) [ u ( c t ) − v ( l t )] U ( θ ) = max t = 0 subject to ( 1 + τ c ) c t + a t + 1 = ϕ t ( θ ) l t − T y ( ϕ t ( θ ) l t ) + Tr t + S t ( E t ) ( 1 + r ) a t − T a (( 1 + r ) a t ) Roozbeh Hosseini(UGA) 8 of 34
Individual Optimization Problem • Individual of type θ solves T ∑ β t P t ( θ ) [ u ( c t ) − v ( l t )] U ( θ ) = max t = 0 subject to ( 1 + τ c ) c t + a t + 1 = ϕ t ( θ ) l t − T y ( ϕ t ( θ ) l t ) + Tr t + S t ( E t ) ( 1 + r ) a t − T a (( 1 + r ) a t ) a t + 1 : asset holding Roozbeh Hosseini(UGA) 8 of 34
Individual Optimization Problem • Individual of type θ solves T ∑ β t P t ( θ ) [ u ( c t ) − v ( l t )] U ( θ ) = max t = 0 subject to ( 1 + τ c ) c t + a t + 1 = ϕ t ( θ ) l t − T y ( ϕ t ( θ ) l t ) + Tr t + S t ( E t ) ( 1 + r ) a t − T a (( 1 + r ) a t ) ϕ t ( θ ) l t : labor earning Roozbeh Hosseini(UGA) 8 of 34
Individual Optimization Problem • Individual of type θ solves T ∑ β t P t ( θ ) [ u ( c t ) − v ( l t )] U ( θ ) = max t = 0 subject to ( 1 + τ c ) c t + a t + 1 = ϕ t ( θ ) l t − T y ( ϕ t ( θ ) l t ) + Tr t + S t ( E t ) ( 1 + r ) a t − T a (( 1 + r ) a t ) Tr t : transfer to workers pre-retirement Roozbeh Hosseini(UGA) 8 of 34
Individual Optimization Problem • Individual of type θ solves T ∑ β t P t ( θ ) [ u ( c t ) − v ( l t )] U ( θ ) = max t = 0 subject to ( 1 + τ c ) c t + a t + 1 = ϕ t ( θ ) l t − T y ( ϕ t ( θ ) l t ) + Tr t + S t ( E t ) ( 1 + r ) a t − T a (( 1 + r ) a t ) E t : the average labor earning history Roozbeh Hosseini(UGA) 8 of 34
Individual Optimization Problem • Individual of type θ solves T ∑ β t P t ( θ ) [ u ( c t ) − v ( l t )] U ( θ ) = max t = 0 subject to ( 1 + τ c ) c t + a t + 1 = ϕ t ( θ ) l t − T y ( ϕ t ( θ ) l t ) + Tr t + S t ( E t ) ( 1 + r ) a t − T a (( 1 + r ) a t ) S t : social security benefit – paid only after retirement Roozbeh Hosseini(UGA) 8 of 34
Individual Optimization Problem • Individual of type θ solves T ∑ β t P t ( θ ) [ u ( c t ) − v ( l t )] U ( θ ) = max t = 0 subject to ( 1 + τ c ) c t + a t + 1 = ϕ t ( θ ) l t − T y ( ϕ t ( θ ) l t ) + Tr t + S t ( E t ) ( 1 + r ) a t − T a (( 1 + r ) a t ) r : after tax return on asset Roozbeh Hosseini(UGA) 8 of 34
Individual Optimization Problem • Individual of type θ solves T ∑ β t P t ( θ ) [ u ( c t ) − v ( l t )] U ( θ ) = max t = 0 subject to ( 1 + τ c ) c t + a t + 1 = ϕ t ( θ ) l t − T y ( ϕ t ( θ ) l t ) + Tr t + S t ( E t ) ( 1 + r ) a t − T a (( 1 + r ) a t ) • There is a corporate tax profit τ K r = ( 1 − τ K ) ˜ r Roozbeh Hosseini(UGA) 8 of 34
Equilibrium • Equilibrium is set of allocations, factor prices and policies such that ◦ Individuals optimize – taking policies as given ◦ factors are paid marginal product ◦ government budget holds ◦ markets clear and allocations are feasible • Once we know equilibrium allocations we can find status-quo welfare T β t P t ( θ ) [ u ( c t ) − v ( l t )] ∑ W s ( θ ) ≡ t = 0 Roozbeh Hosseini(UGA) 9 of 34
Optimal Policy Reform • So far we have imposed no restriction on policies • We can choose them to match he US system • Or, we can choose them to be optimal • Optimal means they deliver status-quo welfare at the lowest cost • We characterize optimal policies next Roozbeh Hosseini(UGA) 10 of 34
A Cost Minimization Problem min PDV of Net Transfers to a Generation { T y ( · ) , T a ( · ) ,... } s.t. � � 1- given policies T y ( · ) , T a ( · ) , ... , individual optimize 2- resulting allocation delivers no less welfare than status-quo Roozbeh Hosseini(UGA) 11 of 34
A Cost Minimization Problem min PDV of Net Transfers to a Generation { T y ( · ) , T a ( · ) ,... } s.t. � � 1- given policies T y ( · ) , T a ( · ) , ... , individual optimize 2- resulting allocation delivers no less welfare than status-quo • This is a very complicated problem choice variables are functions constraint set is function of those functions! Roozbeh Hosseini(UGA) 11 of 34
A Cost Minimization Problem min PDV of Net Transfers to a Generation { T y ( · ) , T a ( · ) ,... } s.t. � � 1- given policies T y ( · ) , T a ( · ) , ... , individual optimize 2- resulting allocation delivers no less welfare than status-quo • This is a very complicated problem choice variables are functions constraint set is function of those functions! • Instead, we use primal approach write the problem only in terms of allocations Show details Roozbeh Hosseini(UGA) 11 of 34
A Cost Minimization Problem Planning Problem T P t ( θ ) � ∑ ( 1 + r ) t [ c t ( θ ) − ϕ t ( θ ) l t ( θ )] dF ( θ ) min t = 0 s.t. Roozbeh Hosseini(UGA) 12 of 34
A Cost Minimization Problem Planning Problem T P t ( θ ) � ∑ ( 1 + r ) t [ c t ( θ ) − ϕ t ( θ ) l t ( θ )] dF ( θ ) min t = 0 s.t. T β t P t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ∑ U ( θ ) = t = 0 Roozbeh Hosseini(UGA) 12 of 34
A Cost Minimization Problem Planning Problem T P t ( θ ) � ∑ ( 1 + r ) t [ c t ( θ ) − ϕ t ( θ ) l t ( θ )] dF ( θ ) min t = 0 s.t. T β t P t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ∑ U ( θ ) = t = 0 T β t P t ( θ ) ϕ ′ T t ( θ ) l t ( θ ) U ′ ( θ ) = v ′ ( l t ( θ )) + β t P ′ ∑ ∑ t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ϕ t ( θ ) t = 0 t = 0 Roozbeh Hosseini(UGA) 12 of 34
A Cost Minimization Problem Planning Problem T P t ( θ ) � ∑ ( 1 + r ) t [ c t ( θ ) − ϕ t ( θ ) l t ( θ )] dF ( θ ) min t = 0 s.t. T β t P t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ∑ U ( θ ) = t = 0 T β t P t ( θ ) ϕ ′ T t ( θ ) l t ( θ ) U ′ ( θ ) = v ′ ( l t ( θ )) + β t P ′ ∑ ∑ t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ϕ t ( θ ) t = 0 t = 0 Roozbeh Hosseini(UGA) 12 of 34
A Cost Minimization Problem Planning Problem T P t ( θ ) � ∑ ( 1 + r ) t [ c t ( θ ) − ϕ t ( θ ) l t ( θ )] dF ( θ ) min t = 0 s.t. T β t P t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ∑ U ( θ ) = t = 0 T β t P t ( θ ) ϕ ′ T t ( θ ) l t ( θ ) U ′ ( θ ) = v ′ ( l t ( θ )) + β t P ′ ∑ ∑ t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ϕ t ( θ ) t = 0 t = 0 U ( θ ) ≥ W s ( θ ) Roozbeh Hosseini(UGA) 12 of 34
A Cost Minimization Problem Planning Problem T P t ( θ ) � ∑ ( 1 + r ) t [ c t ( θ ) − ϕ t ( θ ) l t ( θ )] dF ( θ ) min t = 0 s.t. T β t P t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ∑ U ( θ ) = t = 0 T β t P t ( θ ) ϕ ′ T t ( θ ) l t ( θ ) U ′ ( θ ) = v ′ ( l t ( θ )) + β t P ′ ∑ ∑ t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ϕ t ( θ ) t = 0 t = 0 U ( θ ) ≥ W s ( θ ) status-quo welfare for each θ Roozbeh Hosseini(UGA) 12 of 34
Properties of the Efficient Allocations • Next, we investigate some properties of efficient allocations • What margins should be distorted and why? • Note that distortions � = taxes necessarily • But are informative statistics about efficient allocations Roozbeh Hosseini(UGA) 13 of 34
Distortions • Intra-temporal distortion: distorting labor supply margin v ′ ( l t ( θ )) 1 − τ labor = ϕ t ( θ ) u ′ ( c t ( θ )) • Inter-temporal distortion: distorting “annuity margin” u ′ ( c t ( θ )) 1 − τ annuity = β ( 1 + r ) u ′ ( c t + 1 ( θ )) Roozbeh Hosseini(UGA) 14 of 34
Distortions • Intra-temporal distortion: distorting labor supply margin v ′ ( l t ( θ )) 1 − τ labor = ϕ t ( θ ) u ′ ( c t ( θ )) • Inter-temporal distortion: distorting MRS b/w c t and c t + 1 u ′ ( c t ( θ )) 1 − τ annuity = β ( 1 + r ) u ′ ( c t + 1 ( θ )) Roozbeh Hosseini(UGA) 14 of 34
Intra-temporal Distortions • Mirrlees-Diamond-Saez formula (Standard) � 1 − F ( θ ) � 1 τ labor = ǫ ( θ ) + 1 g ( θ ) 1 − τ labor θ f ( θ ) Roozbeh Hosseini(UGA) 15 of 34
Intra-temporal Distortions • Mirrlees-Diamond-Saez formula (Standard) � 1 − F ( θ ) � 1 τ labor = ǫ ( θ ) + 1 g ( θ ) 1 − τ labor θ f ( θ ) Behavioral response: captured by elasticity of labor supply Roozbeh Hosseini(UGA) 15 of 34
Intra-temporal Distortions • Mirrlees-Diamond-Saez formula (Standard) � 1 − F ( θ ) � 1 τ labor = ǫ ( θ ) + 1 g ( θ ) 1 − τ labor θ f ( θ ) Tail trade-off: taxing type θ : reduces output in proportion to θ f ( θ ) , but relaxes incentive constraints for all types above Roozbeh Hosseini(UGA) 15 of 34
Intra-temporal Distortions • Mirrlees-Diamond-Saez formula (Standard) � 1 − F ( θ ) � 1 τ labor = ǫ ( θ ) + 1 g ( θ ) 1 − τ labor θ f ( θ ) Social value of resource extraction from type θ and above � dF ( θ ′ ) � ¯ u ′ ( c ( θ )) 1 − u ′ ( c 0 ( θ ′ )) θ � g t ( θ ) = u ′ ( c 0 ( θ ′ )) 1 − F ( θ ) λ θ Roozbeh Hosseini(UGA) 15 of 34
Inter-temporal Distortions • Annuity margin (New) β ( 1 + r ) u ′ ( c t + 1 ( θ )) = 1 − p ′ u ′ ( c t ( θ )) t + 1 ( θ ) 1 − F ( θ ) 1 − τ annuity ( θ ) = g ( θ ) p t + 1 ( θ ) f ( θ ) Roozbeh Hosseini(UGA) 16 of 34
Inter-temporal Distortions • Annuity margin (New) β ( 1 + r ) u ′ ( c t + 1 ( θ )) = 1 − p ′ u ′ ( c t ( θ )) t + 1 ( θ ) 1 − F ( θ ) 1 − τ annuity ( θ ) = g ( θ ) p t + 1 ( θ ) f ( θ ) p ′ t + 1 ( θ ) > 0 ⇒ annuity is “taxed” Roozbeh Hosseini(UGA) 16 of 34
Inter-temporal Distortions • Annuity margin (New) β ( 1 + r ) u ′ ( c t + 1 ( θ )) = 1 − p ′ u ′ ( c t ( θ )) t + 1 ( θ ) 1 − F ( θ ) 1 − τ annuity ( θ ) = g ( θ ) p t + 1 ( θ ) f ( θ ) p ′ t + 1 ( θ ) > 0 ⇒ under-insurance is optimal Roozbeh Hosseini(UGA) 16 of 34
Inter-temporal Distortions • Annuity margin (New) β ( 1 + r ) u ′ ( c t + 1 ( θ )) = 1 − p ′ u ′ ( c t ( θ )) t + 1 ( θ ) 1 − F ( θ ) 1 − τ annuity ( θ ) = g ( θ ) p t + 1 ( θ ) f ( θ ) p ′ t + 1 ( θ ) > 0 ⇒ under-insurance is optimal • Intuition: for higher ability future consumption has higher weight Roozbeh Hosseini(UGA) 16 of 34
Implementation: Finding Optimal Taxes • So far we only talked about distortions ◦ these are properties of allocations ◦ they are not tax functions Roozbeh Hosseini(UGA) 17 of 34
Implementation: Finding Optimal Taxes • So far we only talked about distortions ◦ these are properties of allocations ◦ they are not tax functions • Tax function: a map between a tax base and tax obligations Roozbeh Hosseini(UGA) 17 of 34
Implementation: Finding Optimal Taxes • So far we only talked about distortions ◦ these are properties of allocations ◦ they are not tax functions • Tax function: a map between a tax base and tax obligations • We propose a set of taxes ◦ A nonlinear tax (subsidy) on assets: T a , t (( 1 + r ) a t ) ◦ A nonlinear tax on labor earnings: T y , t ( y t ) ◦ A type-independent retirement transfer: S t Roozbeh Hosseini(UGA) 17 of 34
Implementation: Finding Optimal Taxes • So far we only talked about distortions ◦ these are properties of allocations ◦ they are not tax functions • Tax function: a map between a tax base and tax obligations • We propose a set of taxes ◦ A nonlinear tax (subsidy) on assets: T a , t (( 1 + r ) a t ) ◦ A nonlinear tax on labor earnings: T y , t ( y t ) ◦ A type-independent retirement transfer: S t • We can solve these tax functions numerically Show details Roozbeh Hosseini(UGA) 17 of 34
Calibration 1. Parametrize and estimate earning ability ϕ t ( θ ) 2. Parametrize and calibrate model of mortality P t ( θ ) 3. Parametrize and calibrate government policy – to US status-quo 4. Parametrize and calibrate preference and technology Roozbeh Hosseini(UGA) 18 of 34
Calibration 1. Parametrize and estimate earning ability ϕ t ( θ ) 2. Parametrize and calibrate model of mortality P t ( θ ) 3. Parametrize and calibrate government policy – to US status-quo 4. Parametrize and calibrate preference and technology • Do 1, 2 and 3 independent of the model • Use the model to do 4 Roozbeh Hosseini(UGA) 18 of 34
Earning Ability Profiles • Use labor income per hour as proxy for working ability (PSID) • Assume log ϕ t ( θ ) = log θ + log ˜ ϕ t with ϕ t = β 0 + β 1 t + β 2 t 2 + β 3 t 3 log ˜ • θ has Pareto-Lognormal distribution w/ parameters ( µ θ , σ θ , a θ ) a θ = 3 is tail parameter → standard σ θ = 0.6 is variance parameter → variance of log wage in CPS µ θ = − 1/ a θ is location parameter Show Profiles Roozbeh Hosseini(UGA) 19 of 34
Survival Profiles • Assume Gompertz force of mortality hazard λ t ( θ ) = m 0 θ m 1 ( exp ( m 2 t ) / m 2 − 1 ) and P t ( θ ) = exp ( − λ t ( θ )) m 1 which determines ability gradient m 2 determines overall age pattern of mortality m 0 is location parameter • Use SSA’s male mortality for 1940 birth cohort • Use Waldron (2013) death rates (for ages 67-71) Roozbeh Hosseini(UGA) 20 of 34
Death Rates by Lifetime Earning Deciles Roozbeh Hosseini(UGA) 21 of 34
Status-quo Government Policies • Government collects three types of taxes ◦ non-linear progressive tax on taxable income – we use T ( y ) = y − φ y 1 − τ , the HSV tax function ( τ = 0.151, φ = 4.74) ◦ FICA payroll tax – we use SSA’s tax rates ◦ linear consumption tax – McDaniel (2007) • there is also a social security and Medicare benefit ◦ we use SSA’s benefit formula ◦ 3% of GDP, paid equally to all retirees Roozbeh Hosseini(UGA) 22 of 34
Preferences • Utility over consumption and hours u ( c ) − v ( l ) = log ( c ) − ψ l 1 + 1 ǫ 1 + 1 ǫ • We choose ǫ = 0.5 • ψ and β are chosen to match aggregate moments. Roozbeh Hosseini(UGA) 23 of 34
Parameters Chosen Outside the Model Parameter Description Values/source Demographics T maximum age 75 (100 y/o) R retirement age 40 (65 y/o) n population growth rate 0.01 Preferences ǫ elasticity of labor supply 0.5 Productivity σ θ , a θ , µ θ PLN parameters 0.5,3,-0.33 Technology r return on capital/assets 0.04 Government policies τ ss , τ med , τ c tax rates 0.124,0.029,0.055 G government expenditure 0.09 × GDP D government debt 0.5 × GDP Roozbeh Hosseini(UGA) 24 of 34
Parameters Calibrated Using the Model Moments Data Model Wealth-income ratio 3 3 Average annual hours 2000 2000 Parameter Description Values/source β discount factor 0.981 ψ weight on leisure 0.74 Show Distribution of Earnings, Assets Roozbeh Hosseini(UGA) 25 of 34
Optimal Policy Reform • We can now use our calibrated model to ◦ Solve for status-quo allocations ◦ Solve for efficient allocations • Under both set of allocations we can calculate distortions • The difference between two sets of distortions motivates policy reform • We can also use the model to compute optimal tax functions Roozbeh Hosseini(UGA) 26 of 34
Inter-Temporal Distortions: Annuitization Margin u ′ ( c t ( θ )) 1 − τ annuity = β ( 1 + r ) u ′ ( c t + 1 ( θ )) Roozbeh Hosseini(UGA) 27 of 34
Intra-Temporal Distortions: Labor Supply Margin v ′ ( l t ( θ )) 1 − τ labor = ϕ t ( θ ) u ′ ( c t ( θ )) Roozbeh Hosseini(UGA) 28 of 34
Optimal Asset Taxes (Subsidies) Roozbeh Hosseini(UGA) 29 of 34
Optimal Labor Income Taxes Roozbeh Hosseini(UGA) 30 of 34
Aggregate Effects Shares of GDP Status-quo Reform (efficient) Consumption 0.70 0.65 Capital 3.00 3.67 Government Debt 0.50 0.07 Net worth 3.53 3.78 Tax Revenue (Total) 0.25 0.27 Labor income tax 0.15 0.16 Consumption tax 0.04 0.04 Capital tax 0.06 0.07 Government Transfers (Total) 0.14 0.10 To retirees 0.09 0.06 To workers 0.05 0.04 Asset subsidy 0 0.07 PDV of net transfers to each cohort falls by 9.3% Roozbeh Hosseini(UGA) 31 of 34
How Important Are Asset Subsidies? • What is the best that can be achieved without them? Roozbeh Hosseini(UGA) 32 of 34
How Important Are Asset Subsidies? • What is the best that can be achieved without them? • We can include the following restriction in our planning problem P t ( θ ) u ′ ( c t ) = β ( 1 + r ) P t + 1 ( θ ) u ′ ( c t + 1 ) Roozbeh Hosseini(UGA) 32 of 34
How Important Are Asset Subsidies? • What is the best that can be achieved without them? • We can include the following restriction in our planning problem P t ( θ ) u ′ ( c t ) = β ( 1 + r ) P t + 1 ( θ ) u ′ ( c t + 1 ) • The resulting allocations cost 0.5% more than status-quo Roozbeh Hosseini(UGA) 32 of 34
How Important Are Asset Subsidies? • What is the best that can be achieved without them? • We can include the following restriction in our planning problem P t ( θ ) u ′ ( c t ) = β ( 1 + r ) P t + 1 ( θ ) u ′ ( c t + 1 ) • The resulting allocations cost 0.5% more than status-quo • Implication: IF proper asset subsidies are not in place, phasing out old-age transfers is not a good idea! Roozbeh Hosseini(UGA) 32 of 34
How Important is Progressivity of Asset Subsidies? • Progressivity is a consequence of differential mortality Roozbeh Hosseini(UGA) 33 of 34
How Important is Progressivity of Asset Subsidies? • Progressivity is a consequence of differential mortality • How much of the cost saving can be achieved by linear subsidies? Roozbeh Hosseini(UGA) 33 of 34
How Important is Progressivity of Asset Subsidies? • Progressivity is a consequence of differential mortality • How much of the cost saving can be achieved by linear subsidies? • We can include the following restriction in our planning problem P t ( θ ) u ′ ( c t ) = ( 1 − τ t + 1 ) β ( 1 + r ) P t + 1 ( θ ) u ′ ( c t + 1 ) and find optimal τ t ’s Roozbeh Hosseini(UGA) 33 of 34
How Important is Progressivity of Asset Subsidies? • Progressivity is a consequence of differential mortality • How much of the cost saving can be achieved by linear subsidies? • We can include the following restriction in our planning problem P t ( θ ) u ′ ( c t ) = ( 1 − τ t + 1 ) β ( 1 + r ) P t + 1 ( θ ) u ′ ( c t + 1 ) and find optimal τ t ’s • The resulting allocations cost 3% less than status-quo i.e., one third of the cost saving, relative to fully optimal Roozbeh Hosseini(UGA) 33 of 34
How Important is Progressivity of Asset Subsidies? • Progressivity is a consequence of differential mortality • How much of the cost saving can be achieved by linear subsidies? • We can include the following restriction in our planning problem P t ( θ ) u ′ ( c t ) = ( 1 − τ t + 1 ) β ( 1 + r ) P t + 1 ( θ ) u ′ ( c t + 1 ) and find optimal τ t ’s • The resulting allocations cost 3% less than status-quo i.e., one third of the cost saving, relative to fully optimal • Implication: differential mortality matters for optimal policy! Roozbeh Hosseini(UGA) 33 of 34
Conclusion • This paper has two main contributions: 1. It develops a methodology to study optimal policy reform that does not rely on an arbitrary social welfare function allows separation of efficiency gains from redistribution 2. It points to a novel reason for subsidizing assets To correct for in-efficiencies due to imperfect annuity markets Roozbeh Hosseini(UGA) 34 of 34
Conclusion • This paper has two main contributions: 1. It develops a methodology to study optimal policy reform that does not rely on an arbitrary social welfare function allows separation of efficiency gains from redistribution 2. It points to a novel reason for subsidizing assets To correct for in-efficiencies due to imperfect annuity markets • Contrast to asset subsidies in the current US system asset subsidies should not stop at retirement asset subsidies must be progressive Roozbeh Hosseini(UGA) 34 of 34
Distribution of Earnings Roozbeh Hosseini(UGA) 34 of 34
Distribution of Wealth Go to Back Roozbeh Hosseini(UGA) 34 of 34
A Cost Minimization Problem • We start by writing objective in terms of allocations only • From individual budget constraint PDV of Net Transfers is equal to T P t ( θ ) � ∑ min ( 1 + r ) t [ c t ( θ ) − ϕ t ( θ ) l t ( θ )] dF ( θ ) t = 0 for any set of tax and transfers Roozbeh Hosseini(UGA) 34 of 34
A Cost Minimization Problem • We start by writing objective in terms of allocations only • From individual budget constraint PDV of Net Transfers is equal to T P t ( θ ) � ∑ min ( 1 + r ) t [ c t ( θ ) − ϕ t ( θ ) l t ( θ )] dF ( θ ) t = 0 for any set of tax and transfers Intuition: Static Model c = ϕ ( θ ) l − T Roozbeh Hosseini(UGA) 34 of 34
A Cost Minimization Problem • We start by writing objective in terms of allocations only • From individual budget constraint PDV of Net Transfers is equal to T P t ( θ ) � ∑ min ( 1 + r ) t [ c t ( θ ) − ϕ t ( θ ) l t ( θ )] dF ( θ ) t = 0 for any set of tax and transfers Intuition: Static Model c − ϕ ( θ ) l = − T Roozbeh Hosseini(UGA) 34 of 34
A Cost Minimization Problem • For any set of policies, let { c t ( θ ) , l t ( θ ) } individual choices • Let U ( θ ) be utility associated with this allocation Roozbeh Hosseini(UGA) 34 of 34
A Cost Minimization Problem • For any set of policies, let { c t ( θ ) , l t ( θ ) } individual choices • Let U ( θ ) be utility associated with this allocation • Then T β t P t ( θ ) ϕ ′ T t ( θ ) l t ( θ ) U ′ ( θ ) = v ′ ( l t ( θ )) + β t P ′ ∑ ∑ t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ϕ t ( θ ) t = 0 t = 0 Roozbeh Hosseini(UGA) 34 of 34
A Cost Minimization Problem • For any set of policies, let { c t ( θ ) , l t ( θ ) } individual choices • Let U ( θ ) be utility associated with this allocation • Then T β t P t ( θ ) ϕ ′ T t ( θ ) l t ( θ ) U ′ ( θ ) = v ′ ( l t ( θ )) + β t P ′ ∑ ∑ t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ϕ t ( θ ) t = 0 t = 0 • This is called implementability constraint Go to Planning Problem Roozbeh Hosseini(UGA) 34 of 34
A Cost Minimization Problem • For any set of policies, let { c t ( θ ) , l t ( θ ) } individual choices • Let U ( θ ) be utility associated with this allocation • Then T β t P t ( θ ) ϕ ′ T t ( θ ) l t ( θ ) U ′ ( θ ) = v ′ ( l t ( θ )) + β t P ′ ∑ ∑ t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ϕ t ( θ ) t = 0 t = 0 • This is called implementability constraint Go to Planning Problem Intuition: Static Model U ( θ ) = max u ( c ) − v ( l ) s.t. c = ϕ ( θ ) l − T ( ϕ ( θ ) l ) Roozbeh Hosseini(UGA) 34 of 34
A Cost Minimization Problem • For any set of policies, let { c t ( θ ) , l t ( θ ) } individual choices • Let U ( θ ) be utility associated with this allocation • Then T β t P t ( θ ) ϕ ′ T t ( θ ) l t ( θ ) U ′ ( θ ) = v ′ ( l t ( θ )) + β t P ′ ∑ ∑ t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ϕ t ( θ ) t = 0 t = 0 • This is called implementability constraint Go to Planning Problem Intuition: Static Model � y � U ( θ ) = max u ( c ) − v s.t. c = y − T ( y ) ϕ ( θ ) Roozbeh Hosseini(UGA) 34 of 34
A Cost Minimization Problem • For any set of policies, let { c t ( θ ) , l t ( θ ) } individual choices • Let U ( θ ) be utility associated with this allocation • Then T β t P t ( θ ) ϕ ′ T t ( θ ) l t ( θ ) U ′ ( θ ) = v ′ ( l t ( θ )) + β t P ′ ∑ ∑ t ( θ ) [ u ( c t ( θ )) − v ( l t ( θ ))] ϕ t ( θ ) t = 0 t = 0 • This is called implementability constraint Go to Planning Problem Intuition: Static Model U ′ ( θ ) = ϕ ′ ( θ ) l ( θ ) v ′ ( l ( θ )) ϕ ( θ ) Roozbeh Hosseini(UGA) 34 of 34
Implementation: Finding Optimal Taxes • We have set of individual FOC’s P t ( θ ) u ′ ( c t ) β ( 1 + r ) P t + 1 ( θ )( 1 − T ′ a , t + 1 ) u ′ ( c t + 1 ) = ( 1 − T ′ y , t ) ϕ t ( θ ) u ′ ( c t ) v ′ ( l t ) = • We also have their budget constraints • Using these equations we can back-out tax and transfers such that efficient allocations are implemented Roozbeh Hosseini(UGA) 34 of 34
Implementation: Finding Optimal Taxes • We have set of individual FOC’s P t ( θ ) u ′ ( c t ) β ( 1 + r ) P t + 1 ( θ )( 1 − T ′ a , t + 1 ) u ′ ( c t + 1 ) = ( 1 − T ′ y , t ) ϕ t ( θ ) u ′ ( c t ) v ′ ( l t ) = • We also have their budget constraints • Using these equations we can back-out tax and transfers such that efficient allocations are implemented • Before, doing that we need to calibrate the model Go to Calibration Roozbeh Hosseini(UGA) 34 of 34
Unconditional Survival Probabilities Roozbeh Hosseini(UGA) 34 of 34
Earnings Ability Profiles Go Back Roozbeh Hosseini(UGA) 34 of 34
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