14.471: Fall 2012:PS3 Solutions

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1 4.47: Fa 202:PS3 Soutions Deember 4, 202. Commodity Taxation Consider an eonomy onsisting of identia individuas with preferenes given by U (x,x 2,)= α og (x )+ α 2 og (x 2 )+( α α 2 ) og ( ). Individuas suppy abor,, at a maret wage of unity, and purhase goods and 2, both of whih have fixed produer pries equa to. The government s revenue requirement is R, measured in the fixed produer pries. (a) Assume that the government is restrited to using inear ommodity taxes on goods and 2 to raise revenue. Set up the Ramsey optima tax probem for this eonomy, and find the optima tax rates on goods and 2 as a funtion of R and the individua preferene parameters. Let us use the prima approah. The impementabiity onstraint (ombining the budget onstraint and the FOC) is: U U U x + x 2 + = 0 x x 2 α α 2 α α 2 x + x 2 = 0 x x 2 α + α 2 =. Therefore the prima version of the Ramsey probem for this eonomy is max α og (x )+ α 2 og (x 2 )+( α α 2 ) og ( ) s.t. = α + α 2 x + x 2 + R =, where the first onstraint is the impementabiity onstraint and the seond onstraint is the resoure onstraint. Using the impementabiity onstraint to eiminate, the probem an be redued to The first order onditions for x and x 2 are max α og (x )+ α 2 og (x 2 )+( α α 2 ) og ( α α 2 ) s.t. R = α + α 2 x + x 2. α /x = λ α 2 /x 2 = λ. The resoure onstraint then impies α + α 2 λ =. α + α 2 R

2 Sine onsumer pries are proportiona to margina utiities and the margina utiities are equa the onsumer pries must be equa. Sine the produer pries are aso equa, there is a ommon tax rate for both goods. The government revenue requirement then impies α + α 2 R = t (x + x 2 )=t λ or R t =. α + α 2 R (b) Obtain expressions for the margina utiity of money of the individua, α, and the margina ost of government revenue, λ. How do hanges in the government revenue requirement affet these osts and the differene between the two? α an be determined from the nowedge that it is equa to the ratio of the margina utiity of any good to the prie of that good, or U α α + α 2 R λ α = = = λ = λ = =. x R q x +t + α + α 2 λ α +α 2 R The margina ost of government revenue is λ, whih was omputed in part (a) under either soution method: α + α 2 λ =. α + α 2 R The margina utiity of inome is unaffeted by the government revenue requirement whie the margina ost of government revenue is inreasing in R (sine = α + α 2 is the tota output, R must be ess than α + α 2.) Thus the margina effiieny ost of the tax system (as measured by the margina ost of government revenue ess the margina utiity of inome for the individua) is inreasing in R. This resut maes intuitive sense as the government must raise revenue using distortionary taxes and the effiieny ost of distortionary taxes is onvex in the eve of the tax. If the government oud raise funds with ump sum taxes, the margina ost of government revenue woud be λ = α and the margina effiieny ost of the tax system woud be α α = Capita Taxation This question ass you to numeriay expore the wefare impiations of apita taxation in the Ramsey setup disussed in ass. There is a representative agent with preferenes u (, ) = σ θ ( ) θ σ over onsumption and abor. The disount fator is β. Tehnoogy is Cobb-Dougas with F (, ) =A α α, and apita depreiates at a rate δ. Perform your auations for σ = 2, σ = 3, and σ = 4. (a) Consider a steady state ompetitive equiibrium of this eonomy where the tax on apita is κ = 0and there is aso no tax on abor inome. What is the steady state gross interest rate R? Using the steady state resoure onstraint and the first order onditions from the onsumer s utiity maximization probem and the firm s profit maximization probem, find 4 equations that impiity determine the steady state wage w, onsumption, abor suppy, and apita sto as a funtion of the parameters. Caibrate the eonomy by hoosing parameter vaues for θ, A, α, δ, and β that impy empiriay reasonabe vaues for the endogenous variabes in the steady state (refer to Cooey (995), hapter ). Sove your equations for,,, and w. 2

3 The onsumer probem formuated using the present vaue budget onstraint and without abor taxes, is max β t u ( t, t ) s.t. q t ( t w t t T t ) R 0 0 t=0 t=0 where q t = t. i= R i This eads to first order onditions for t, t+, and t β t u ( t, t )=λq t β t+ u ( t+, t+ )=λq t+ β t u ( t, t )= λq t w t. Combining these and taing advantage of the definition of q t we get the usua intratempora and intertempora optimaity onditions u ( t, t )=βr t+ u ( t+, t+ ) w t u ( t, t )= u ( t, t ). We assume the existene of a steady state. In the steady state, onsumption and abor are onstant, thus impying R = /β. The gross interest rate is R = +( κ)(r δ), whih impies R r = + δ. κ We assume the presene of profit maximizing firms with the speified tehnoogy, whih impies r = F (, ) =αa α α w = F (, ) =( α) A α α. Soving for the apita abor ratio using the renta rate, r α =. αa The resoure onstraint in steady state is + = AF (, )+( δ), whih then impies α = AF, δ = A δ. Differentiating the given utiity funtion, the margina utiities with respet to onsumption and abor are u (, ) = θ ( ) θ σ θ θ ( ) θ u (, ) = θ ( ) θ σ θ ( θ)( ) θ. Thus the intratempora optimaity ondition is w θ ( ) θ σ θ θ ( ) θ = θ ( ) θ σ θ ( θ)( ) θ wθ θ ( ) θ = θ ( θ)( ) θ wθ ( ) = ( θ) wθ ( ) = ( θ) θw =. ( θ) + θw 3

4 Vaues for,, and y an then be omputed from =, =, and y = F (, ). We therefore have a trianguar system of equations that aows for the omputation of a steady state variabes of interest as funtions of the parameters: R = β R r = + δ κ r α = αa α w = A ( α) α = A δ θw = ( θ) + θw = = y = A α α. We aibrate the mode foowing Cooey. Capita evoution in the steady state impies = I + ( δ) or δ = I/. In 2007 (using data from before the reession in the hope that it better approximates a steady state), investment was $3.8 triion and apita was $47.9 triion. Thus impying δ = Next, β = R =(+( κ)(r δ)) = +( κ) α δ. y We use the approximation of α =0.33. Output in 2007 was $4. triion. We approximate the apita tax rate as Thus β = Finay, the intratempora optimization ondition (inuding abor taxes), requires θ ( α)( τ ) y =. θ Using Cooey s hoie of = 0.3, approximating the abor tax rate as τ = 0.28, and observing that onsumption in 2007 was $9.8 triion eads to θ = Finay, we set y A = =8.6. α α With vaues for θ, A, α, δ, and β we an now ompute vaues for the endogenous variabes in this eonomy. (b) Using the parameters found in (a), ompute the steady state for apita taxes κ [ 0.5, 0.5], assuming that the tax revenue is returned in a ump sum fashion to onsumers. Pot steady state output, apita, onsumption, and abor y (κ), (κ), (κ), and (κ) as a funtion of the apita tax κ. Compute the vaue of the steady state as Ṽ ( (κ)) = (κ) θ ( (κ)) θ σ ( β)( σ) 4

5 for given κ. We want to evauate the wefare ost of this apita tax distortion by omparing this to the vaue that onsumers woud obtain if the apita tax were set to zero. A naive approah to this woud be to ompare Ṽ ( (κ)) with Ṽ ( (0)). Compute λ (κ) suh that (( + λ (κ)) (κ)) θ ( (κ)) θ σ = V ( (0)). ( β)( σ) Pot λ (κ). How does your resut depend on σ? Interpret. Matab ode to reate the pots. Note that the steady state vaues of output, apita, onsumption, and abor do not depend on the vaue of σ. This an be seen by omparing the graphs or referening the equations derived in part (a). This resut maes sense beause the parameter σ determines the urvature of the utiity funtion. In the steady state onsumption and abor suppy never hange and the apita-abor ratio is pinned down by the disount rate and the apita taxes, so that the urvature of the utiity is of no importane. As an be seen from the pot of λ (κ), the inrease in onsumption assoiated with a tax reform that sets κ = 0 is strity inreasing in κ. Furthermore, onsumption dereases if the initia apita tax is negative. Given our previous exposure to the Chamey-Judd resut, we now this finding is inorret. This probem arises beause the omparison ignores the transition from a given apita sto to a partiuar steady state apita sto. In steady state soutions in whih apita is subsidized, onsumption is higher than in the κ = 0 steady state but the agent woud prefer to eat some of the apita sto now and transition to a steady state with ess apita. () The onusion in (b) ignores the transition from the apita sto (κ) to (0) one the apita tax κ is redued to zero. The vaue funtion aounting for this soves the Beman equation V () = max u (, )+ βv ( ),, subjet to + = F (, )+( δ). Sove numeriay for V () and evauate it at (κ). Again, find λ (κ) suh that (( + λ (κ)) (κ)) θ ( (κ)) θ σ = V ( (κ)). ( β)( σ) Pot λ (κ). How does your resut depend on σ? How does it ompare to your resut in (b)? Interpret. Matab ode (thans to Forian and Greg for their wor) to reate the pots. The ode numeriay soves for the vaue funtion V () using vaue funtion iteration. The inrease in onsumption assoiated with a tax reform that sets κ = 0 is now strity greater than zero and equa to zero ony at an initia apita tax of zero, whih is the optima eve. Whie a apita subsidy oud inrease the steady state apita abor ratio, and therefore the wage and onsumption, the agent prefers to eat the apita now. Furthermore, the pot of λ (κ) now depends on the parameter σ. Higher vaues of σ orrespond to greater urvature in the period utiity funtion, and indiate that the agent paes greater vaue on a smooth onsumption trajetory. Therefore, for a given initia suboptima apita sto a higher vaue of σ maes the osts of the transition to the optima apita sto more important. This orresponds to the redued vaue of λ (κ) as σ inreases. 3. Pigouvian taxation, poution and jobs Pease read the paper Shimer (202) A Framewor for Vauing the Empoyment Consequenes of Environmenta Reguation. a) Map the mode(s) into an Arrow-Debreu Warasian eonomy and then appy the genera resuts on Pigouvian taxation that we disussed in ass Let us map the Shimer mode into our notation from ass: 5

6 The vetor of goods onsumed by individua i: x i =( i,d i,)=( i,d i, ) The externaity/tota/average onsumption of dirty goods x = d i di onsumer pries q = (p,p d ) p d produer pries p =( p, ) +τ +τ d the utiity funtion of individua i: u i = V (u( i,d i )),D) Assume that the externaity is arge enough that optimaity requires a redution in poution (reaoation). Now use the formua at the bottom of page 2 of Ivan s eture notes on orretive taxation to find the optima p/q = +τ d Pigouvian tax and use that we are in the equiibrium with reaoation (whih impies p d/q ): d +τ p /q +τ d = = p d/q d +τ V + V b) Compare the resuts from a) with the resuts in Shimer (202) and disuss. Let us define VD V di di = σ d,d as the (average) MRS between dirty goods and poution and then we get +τ d = σ d,d +τ this an be rewritten as equation (2) on p.5 of Shimer: D di di τ d τ = σ d,d ( + τ d ) Interestingy, the optima tax does not depend on the ost of reaoation. ) Compare the resuts from a) with the urrent EPA praties for vauing job osses of reguation and disuss. The EPA sores reguation drafts on the basis of expeted unempoyment osts. Aording to this mode, this is not optima sine ony the MRS between dirty goods and poution shoud matter. 4. Short questions Are the foowing statements true, fase or unertain? Pease expain your answer. The Pigouvian prinipe does not appy with heterogeneous agents. Fase. The Pigouvian prinipe sti appies (see Leture Notes on Corretive Taxation - Pigouvian Taxation with Many Agents) if we have ump-sum taxes varying aross individuas. The Pigouvian prinipe does appy but is modified as a funtion of the assumptions on how agents ontribute to the externaity and what their demand struture is. In ass, we overed 3 ases. First, if everyone ontributes in the same way to aggregate onsumption externaity, then the surharge shoud be based on the weighted sum aross agents of the margina utiity af a househods with respet to average onsumption. Seond, if preferenes are additivey separabe and if the agent s onsumption impats the additive externaity in a househod-speifi manner, then the surharge shoud equa a weighed average of externaities, weighted by onsumer demand derivatives with weights ying in the unit interva. Third, if utiity is not separabe then the surharge shoud equa a weighted average of externaities, weighted by onsumer demand derivatives with weights whih oud ie outside of the unit interva. 6

7 The fundamenta probem of distortive taxation is that ump-sum taxes are unavaiabe. Fase beause (i) some ump-sum taxes exist in the rea word (e.g. negative ump-sum interept in the shedue due to inome-tax dedutions or transfers from wefare programs) and (ii) the unavaiabiity of a rih enough set of instruments of ump-sum taxes is driven by more fundamenta auses (e.g. tehnoogia onstraints, information probems) and is thus a syndrom rather than the deep, fundamenta ause. If ump sum taxes were feasibe in the Ramsey mode, then the first best aoation woud be attainabe. The seond-best probem then hooses the right mix of distortive taxes to maximize the representative agentís utiity subjet to the government s budget onstraint. However, the Ramsey framewor does not expiity apture the reasons for ruing out ump-sum taxation. Distributiona onerns are a natura reason to resort to distortionary taxation (Mirrees, 97). If worers are heterogeneous with respet to their abor produtivity, and if this trait is not observabe, or if for some reason taxes annot be onditioned upon them, then soiety annot attain amost any of the first-best aoations. By taxing observabe differenes suh as inome, redistribution is possibe, abeit at a oss in eff ieny. Suh a trade-off between redistribution and effiieny provides a mirofoundation for the roe of distortionary taxation. Then the "fundamenta probem" an be seen either as an information probem (abor produtivity is not observabe), or a a of instruments probem (ump sum taxation onditiona say on DNA is not aowed). Soure: Werning, Tax Smoothing with Redistribution (page 2) Suppose preferenes over onsumption and eisure are Cobb-Dougas. Then a proportiona tax on abor, used to finane government onsumption, does not affet abor suppy. Thus taxation in this ase is non-distortionary. Fase. To evauate distortions we need to oo at the substitution effet and not at the tota effet. The onsumption-eisure margin is distorted beause the reative prie hanges here even if tota effet on abor suppy is zero. The shortest way to see this is by notiing the wedge between the MRS and the MRT: MRS =+τ MRT An aternative way to see this is to ompute the impat on utiity under a ump-sum rebate of the tax (in GE). If the onsumer disposes of time endowment T, then fu inome equas I = PT where the prie of eisure is P = w( t) and where eisure is and abor L. Under Cobb-Dougas, utiity is given by: u = α α + βg. From the foowing demands, it is obvious that eisure does not hange and hene the unompensated abour suppy eastiity is zero = ( α)i αi α(pt) = = = αt P P But we an ompute the indiret utiity funtion by pugging in onsumption and eisure into the utiity L h P and show that the ompensated abbr suppy eastiity PL = L is positive. Hene if we (i) fix G, (ii) ompensate any variation in t by mathing a higher ump-sum distribution R, then (i) A higher t inreases dl R in whih ase dt < 0 beause of the inome effet of R/eisure is a norma good, (ii) wefare drops dv/dt < 0. Any inrease in the wage tax mathed by a ump-sum rebate woud derease utiity: the wage tax is distorting. To onude, there are 3 effets:. PE: a higher t has a positive inome effet on abor 2. PE: a ower t has a negative inome effet on abor 3. GE: a higher R has a negative inome effet on abor In PE, with Cobb-Dougas preferenes, effets and 2 ane eah other out. In GE, there is a net negative effet on abor and hene the wage tax is distorting when the tax revenues are redistributed. 7

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