O PTIMAL T AXATION WITH B EHAVIORAL A GENTS ( WITH OTHER MODELS AT THE END ) Emmanuel Farhi and Xavier Gabaix Harvard Lecture 3: September 2018
I NTRODUCTION ◮ Behavioral version of three pillars of optimal taxation theory: ◮ Ramsey (linear taxation to raise revenues and redistribute) ◮ Pigou (linear taxation to correct for externalities) ◮ Mirrlees (nonlinear taxation to raise revenues and redistribute) ◮ General behavioral biases, including: ◮ Misperceptions of taxes ◮ “Internalities” ◮ Mental accounts
R ELATED L ITERATURE ◮ Behavioral Public Finance: Gruber and Koszegi 01, Liebman and Zeckhauser 04, O’Donoghue and Rabin ’06, Chetty, Kroft Looney 09, Finkelstein 09, Bernheim Rangel 09, Mullainathan Schwartzstein Cogdon 12, Allcott Mullainathan Taubinsky 14, Allcott Taubinsky 15, Baicker, Mullainathan, Schwartzstein 15, Lockwood and Taubinsky 15, Taubinsky and Rees-Jones 17, Lockwood 17, Yu 17 Moser and Olea 17 ◮ Inattention / salience: Sims 03, Gabaix and Laibson 02, 06, Mankiw Reis 02, Reis 06, Abel, Eberly and Panageas 09, Ma´ ckowiak and Wiederholt 10, Masatlioglu and Ok 10, Veldkamp 11, Matejka and McKay 14, Spiegler and Piccione 12, Woodford 12, Bordalo, Gennaioli and Shleifer 12,13, Caplin and Dean 14, Gabaix 14, Schwarzstein 15 ◮ Law and Economics: Thaler and Sunstein ’09, Gamage & Shanske ’11, Galle ’14, Goldin ’15. ◮ Behavioral economics: k − level models, Crawford et al., J´ ehiel et al., Hong and Stein, Koszegi and Szeidl, Schwartzstein, Furster-Laibson-Mendel, Eyster-Rabin,... ◮ Bounded Rationality: Sargent 93, Rubinstein 98, Tirole 11, Aguiar Serrano 14, Gul Pesendorfer Strzalecki 15 ...
E MPIRICAL MOTIVATION ◮ Income tax: do you know your marginal tax rate? People are confused about their true marginal tax rate, and indeed use instead their average tax rate: De Bartolome ’95, Liebman and Zeckhauser ’04 ◮ Evidence for inattention: Chetty, Kroft Looney 09, Taubinsky and Rees-Jones 15: People are partially inattentive to taxes ◮ Government react to that inattention by increasing taxes: Finkelstein 09 on EZ-pay ◮ Lots of evidence of “neglect” of various kinds: e.g. people underweigh the present value of cost of gas when they buy a car: Allcott and Wozny ’14 vs Busse Knittel Zettelmeneyer 13 . ◮ Evidence on mental accounts: Thaler ’85, Hastings and Shapiro ’13 ◮ Evidence on hyperbolic discounting / temptation
D ECISION UTILITY VS E XPERIENCED UTILITY MODEL u s ( c ) s.t. q · c ≤ w c ( q , w ) = arg max c ◮ u s = " decision " utility ( s for s ubjective) ◮ u = " experienced " utility ◮ Ex. internalities from temptation, hyperbolic discounting,...
M ISPERCEPTION MODEL ◮ True price q and perceived price q s ( q , w ) ◮ Agent behavior (Gabaix 2014): c ( q , w ) = arg c ∈ R n | q s ( q , w ) ( c ) s.t. q · c = w smax i.e. u ′ ( c ( q , w )) = λ q s ( q , w ) with λ such that q · c ( q , w ) = w ◮ The “trade-off” intuition works: u ′ = q s c 1 1 u ′ q s c 2 2 ◮ Budget constraint is satisfied: q · c ( q , w ) = w
B EHAVIORAL P RICE THEORY : G ENERAL M ODEL ◮ Two primitives: ◮ Marshallian demand function c ( q , w ) with q · c ( q , w ) = w ◮ " Experienced " utility function u ( c ) ◮ Indirect utility: v ( q , w ) = u ( c ( q , w )) . ◮ Misoptimization wedge τ b : = q − u c ( c ( q , w )) v w ( q , w ) ◮ τ b = 0 for traditional, rational agent. ◮ Slutsky matrix S C j ( q , w ) = c q j ( q , w ) + c w ( q , w ) c j ( q , w )
B EHAVIORAL P RICE THEORY : G ENERAL M ODEL ◮ Modified Roy’s identity v q j ( q , w ) v w ( q , w ) = − c j − τ b · S C j ◮ Example. Take c j = 1 pack per day, τ b j = $ 10 / pack (Gruber jj = − ψ c j and Koszegi 2004), S C q j = − 0.14 packs per dollar per day. Then, − τ b · S C j = − τ b j S C jj = 1.5 dollars per day. v qj ( q , w ) ◮ So, v w ( q , w ) = − 1 + 1.5 = 0.5 > 0 ◮ So, increasing the cigarette tax makes consumers better off.
M ORE BEHAVIORAL CONSUMER THEORY : C ONCRETE MODELS ◮ General model nests many concrete models ◮ Decision vs. Experienced utility model τ b = u s − u c c v s v w w ◮ τ b i > 0 for “tempting” goods: drugs, fats, etc. ◮ Slutsky: S ij = S s ij ◮ Misperception model τ b = q − q s ◮ τ b i > 0 for goods with non-salient taxes ∂ q s k ( q , w ) ◮ Slutsky, typically non-symmetric: S H ij = ∑ k S r ik ∂ q j
M ANY - PERSON R AMSEY (D IAMOND 1975) �� � � � � v h ( p + τ , w ) τ · c h ( p + τ , w ) − w + λ ∑ L ( τ ) = W h = 1... H h ◮ Optimal tax formula in " target form " in term of sufficient statistics: � λ − γ h � 0 = ∂ L ( τ ) = ∑ τ b , h ) · S C , h c h [ i + λ ( τ − � ] i ∂τ i h ◮ Sufficient statistics ◮ Social marginal welfare weight β h = W v h v h w ◮ Social marginal utility of income γ h = W v h v h w + λ τ · c h w ◮ Substitution elasticities S C , h i τ b , h = β h ◮ Weighted misoptimization wedge � λ τ b , h � λ − γ h � i , substitution λ τ · S C , h ◮ Mechanical c h (distortion i τ b , h · S C , h from fiscal externality), misoptimization � (distortion i from failure of envelope theorem) ◮ Extends to Pigou
N UDGES ◮ Nudge χ influences demand c ( q , w , χ ) , possibly utility u ( c , χ ) but not budget q · c = w . ◮ Concrete model: decision utility u s , perceived price q s , ∗ , nudgeability η ≥ 0 c ( q , w , χ ) = arg smax c | u s , B s u s ( c ) s.t. q · c ≤ w i.e. c is s.t. u s ′ ( c ) = Λ B s c ( q , c , χ ) with Λ s.t. q · c ( q , w , χ ) = w ◮ Nudge as a psychic tax: B s ( q , c , χ ) = q s , ∗ · c + χη c i , ◮ Nudge as an anchor: B s ( q , c , χ ) = q s , ∗ · c + η | c i − χ |
O PTIMAL N UDGES ◮ Optimal nudge formula � τ b , h � χ + β h u h 0 = ∂ L τ − τ ξ , h − � ∂χ = ∑ χ · c h [ λ ] v h w h ◮ Integrates nudges into canonical optimal taxation framework
T AKING S TOCK ◮ So far: ◮ General taxation motive (revenue raising, redistribution, correcting for externalities, internalities) ◮ Arbitrary behavioral biases ◮ Generalize canonical optimal tax formulas ◮ Sufficient statistics approach ◮ Now: ◮ More structure: specific behavioral model, taxation motive ◮ Concrete lessons for taxes
M ODIFIED R AMSEY I NVERSE E LASTICITY R ULE ◮ Representative agent with quasilinear utility u i ( c i ) u ( c ) = c 0 + ∑ i > 0 ◮ Misperception of taxes τ s i = m i τ i ◮ Limit of small taxes ( Λ = λ − 1 small) 1 i ) 2 ψ i y i + Λ ∑ L ( τ ) = − ∑ 2 ( τ s τ i y i i i where ψ i rational demand elasticity, y i expenditure with no tax
M ODIFIED R AMSEY I NVERSE E LASTICITY R ULE ◮ Behavioral elasticity m i ψ i ◮ Behavioral Ramsey formula Λ τ i = m 2 i ψ i − 1 2 m 2 i τ 2 i ψ i + Λ τ i gives − m 2 i τ i ψ i + Λ = 0. � Proof: max τ i ◮ Contrast with traditional Ramsey: i = Λ τ R ψ i
Q UANTITATIVE I LLUSTRATION Λ ◮ With heterogeneity, τ i = � 2 � m h E ψ i i ◮ Taubinsky and Rees-Jones (2017) find: � m h � = 0.25 E � m h � = 0.13, so that heterogeneity is very and var var ( m h ) E [ m h ] 2 = 0.13 large, 0.25 2 = 2.1. ◮ Take ψ = 1, Λ = 1.25%, so τ = 7.3%. ◮ If tax was fully salient, optimal tax would be divided by 6 ◮ If heterogeneity disappeared, optimal tax would be multiplied by 3.
P IGOU : “D OLLAR FOR D OLLAR ” P RINCIPLE ◮ Representative agent with quasilinear utility ◮ One taxed good with price p and externality − ξ c ◮ Inattention to tax τ s = m τ ◮ Welfare W social = U ( c ) − ( p + ξ ) c ◮ Consumer maximizes W private = U ( c ) − ( p + m τ ) c ◮ Opimtal tax is m τ = ξ , i.e. τ = ξ m ◮ Modifies " dollar for dollar " Pigouvian principle ◮ Constrast with Ramsey: Pigou 1 1 m vs Ramsey m 2
P IGOU : Q UANTITATIVE I LLUSTRATION ◮ Previous numbers by Taubinsky and Rees-Jones (2017) E [ m h ] ◮ τ ∗ = ξ E [ m h ] 2 + var ( m h ) . ◮ With heterogeneity, τ ∗ = 1.3 ξ . ◮ If the tax became fully salient (i.e. m h = 1 ), it would be divided by 1.3. ◮ If heterogeneity disappeared (i.e. m h = 0.25), the optimal tax would be multiplied by 3 .
P IGOU : T AXES VS . Q UANTITY R ESTRICTIONS ◮ Traditional presumption that Pigouvian taxes dominate quantity restrictions because allow agents to express intensity of preferences ◮ Heterogeneity ◮ misperception m h ◮ externality ξ h ◮ Quasilinear + quadratic utility ◮ bliss point c ∗ h ◮ “elasticity” Ψ = − 1 / U cc ( c ∗ h )
P IGOU : T AXES VS . Q UANTITY R ESTRICTIONS ◮ First Best achievable iff ξ h m h independent of h � � ◮ Optimal tax τ ∗ = E [ ξ h m h ] / E m 2 h ◮ Alternatively, optimal quantity restriction c ∗ = E [ c ∗ h ] ◮ Quantity restrictions better than taxation iff: � − ( E [ ξ h m h ]) 2 � � � ξ 2 m 2 1 h ) ≤ Ψ E E Ψ var ( c ∗ h h E [ m 2 h ] 1. enough heterogeneity in attention ( m h ) or externality ( ξ h ) 2. not too much heterogeneity in preferences ( c ∗ h ) 3. if high demand elasticity ( Ψ high) (cf. Weitzman).
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