Economics 1410 Fall Section 7 Notes 1. Define the tax in a flexible way using T (z), where z is the income reported by the agent.

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1 Economcs 1410 Fall 2017 Harvard Unversty Yaan Al-Karableh Secton 7 Notes 1 I. The ncome taxaton problem Defne the tax n a flexble way usng T (), where s the ncome reported by the agent. Retenton functon: R() = T () or c = T (). Measures how much the agent can retan out of total ncome. Denote transfers to ncome wth T () so that T (0) s the transfer to a non-workng ndvdual. Margnal tax rate: T (): ndvdual keeps 1 T () for an addtonal $1 of earnngs. It measures how much an agent gets taxed out of one addtonal dollar of ncome. T () T (0) Partcpaton tax rate: p =. It s the fracton of ncome that an agent pays n taxes when they move from 0 ncome to. II. Taxaton wth no behavoral responses Ths gnores labor supply responses to taxaton. The agent has utlty u(c) such that u (c) > 0 and u (c) 0. Labor does not enter the utlty functon and s suppled nelastcally. The agent consumes everythng that s left so that c = T () holds. Problem setup: Dscrete case: N ndvduals wth fxed ncomes 1 <... < N. The government then maxmes a utltaran objectve functon: SW F = N u( T ( )) =1 subject to : N T ( ) = 0 =1 The budget constrant mples that taxes should fund transfers, and the socal welfare functon means the government s maxmng the sum of after-tax utltes (hence utltaran: every agent s weghted equally). 1 These notes are partally based on those of Matteo Parads. 1

2 Contnuous case: the economy s populated by several agents and ther ncome s dstrbuted accordng to h() wth support 0,. The problem of the government s: SW F = subject to : 0 0 u( T ())h()d T ()h()d E Note that E n the budget constrant refers to a revenue requrement. We solved the problem n the dscrete case n the lecture sldes. We can solve the problem n the contnuous case. We have the followng Lagrangan: L = u( T ())λt ()h() Where λ s constant aross ndvduals and measures the value of government revenues n equlbrum. The optmal choce of T () comes from the followng FOC: Then we are left wth the condton: L T () = u ( T ()) + λh() = 0 u ( T ()) = λ λ s a constant and all agents have the same utlty functon (preferences). Thus, the equlbrum condton mples that consumpton s equaled across ndvduals. Ths s the consequence of the utltaran socal welfare functon and the concavty of utlty. Because there are no behavoral responses, there are no effcency costs to equalng consumpton among ndvduals by redstrbutng from rch to poor. Suppose that s the average ncome. What wll the margnal tax rate above be? III. Optmal lnear tax rate wth behavoral responses Back to the dscrete case, the government chooses to maxme utltaran socal welfare. Note that Z s the total taxable ncome and t s a functon of (1 ), whle = w l s the ndvdual taxable ncome. (Sde queston: why does Z(1 ) enter the utlty functon?). Then the government maxmes: SW F = u ((1 )w l + Z(1 ), l ) The FOC s then: SW F = w l + Z = d(1 ) + Z d(1 ) 2

3 What happened to l? Envelope Theorem! The agent s already optmng her own utlty wth the choce of l. Intuton of Envelope Theorem: because the agent s takng taxes and prces as gven and optmng labor supply the entre tme, then the ndrect effects on utlty sum up to ero. When you have optmed, changng one parameter has only one effect and that s on the objectve, and wll not have an ndrect mpact through other parameters. We then re-arrange the problem after settng SW F = 0: 1 = ḡ + 0 = 0 = Z + Z + Z Z = ḡ + Zd(1 ) d(1 ) d(1 ) Z (1 ) = ḡ + Zd(1 )(1 ) 1 e Z,(1 ) = 1 ḡ 1 ḡ + e Z,(1 ) where ḡ = Z. Ths s the weghted margnal utlty of person, weghted by the ncome of the person, and averaged across ndvduals. (ḡ s 1 when the covarance between ncome and margnal utlty s ero: (1) same ncome, (2) ncome dstrbuted randomly). The optmal tax s decreasng n the elastcty and ḡ. When ncome s very elastc to taxes, the government wants to avod negatve effects on revenues and transfers comng from dstortons to the labor supply - ths s the effcency component of the formula. To ncorporate equty n the formula, ḡ s a measure of nequalty n the economy and s low when the ncome dstrbuton s very spread out. The government wll therefore want to ncrease taxes at the optmum when nequalty s hgh. What happens when the government s Rawlsan rather than utltaran? A Rawlsan government maxmes the welfare of the poorest ndvdual. Ths means the government wants to maxme the R component n the lnear taxaton utlty functon - we are back at the revenue maxmaton case where ḡ = 0. IV. Optmal top tax rate In ths case, we wll not specfy a model. We wll nstead consder how a tax change affects taxaton, wth the constrant that the sum of all effects s ero n equlbrum. Consder the case where s the ncome cutoff for the hghest margnal tax rate bracket. The government then sets for all ncome above. The average ncome above s and depends on the 3

4 tax rate. When the tax s rased, ndvduals wth ncomes below wll not be affected. We have three dfferent effects of the tax on ndvduals above : 1- Mechancal effect: the effect of the tax assumng that labor supply doesn t change: dm = d( ) 2- Behavoral effect: descrbes the reacton of top taxpayers to the hgher tax by reducng labor supply, whch results n a reducton n revenue: d db = d = d(1 ) d = 1 1 = 1 ɛ,1 d d d(1 ) d 3- Welfare effect: let ḡ be the constant socal margnal welfare weght for earners above. The tax changes causes a loss of welfare for top earners equvalent to: dw = dḡ( ) The optmal tax: addng all three effects and settng equal to ero: We can rearrange: db + dm + dw = d (1 ḡ)( ) ɛ,1 1 = 0 where a = 0 = (1 ḡ) ɛ,1 ɛ = (1 ḡ) 1 aɛ + (1 ḡ) = (1 ḡ) = 1 ḡ 1 ḡ + aɛ,1 1, whch measures the thckness of the upper tal of the ncome dstrbuton. What does the equaton say? 1- The optmal tax s decreasng n socal margnal welfare weght of top earners (ḡ): the more the government cares about hgh ncome ndvduals the less they wll be taxed. In the lmt case where the government doesn t place any weght on the margnal consumpton of 4

5 the top earner, we are at the revenue maxmng top rate 1 1+aɛ. 2- The optmal tax s decreasng n the elastcty of labor supply: hgher elastcty mples hgher effcency costs. Ths s the uncompensated elastcty. 3- The optmal tax s decreasng n the thnness of the tal: the government mposes a lower tax when ncome s more concentrated at rather than the rest of the upper tal. For extra readng: Pketty, Thomas, and Emmanuel Sae. Income. Handbook of Publc Economcs Optmal Taxaton of Labor 5