Economics 8105 Macroeconomic Theory Recitation 6

Transcription

1 Economics 8105 Macroeconomic Theory Reciaion 6 Conor Ryan Ocober 11h, 2016 Ouline: Opimal Taxaion wih Governmen Invesmen 1 Governmen Expendiure in Producion In hese noes we will examine a model in which governmen expendiure is used in producion. In a ypical ax-disored environmen, governmen expendiure is hrown ino he ocean. The governmen akes away from he oal resources and does nohing o he economy oher han levy possibly disorionary axes. In his model, he governmen good is a ype of invesmen ha can be used in governmen producion. 1.1 Opimal Governmen Spending Imagine a social planner is free o choose governmen purchases wihou needing o worry abou funding hose purchases. We can hink of governmen purchases as governmen owned capial or infrasrucure. The social planner simply wans o maximize uiliy subjec o he resource consrain. Les assume ha uiliy is given by consan relaive risk-aversion (CRRA) preferences and he producion funcion is cobb-douglas in capial and governmen 1

2 invesmen. Assume ha boh ypes of invesmen have full depreciaion. max c,k +1,g +1 such ha β c1 σ 1 σ c + k +1 + g +1 Ak α g c, k +1, g +1 0 k 0, g 0 > 0 given Le µ be he lagrange muliplier on he resource consrain. The firs order condiions are given by Noice ha, β c σ = µ (c ) µ = µ +1 (1 α)ak+1g α +1 α (g +1 ) µ = µ +1 αak+1 α 1 g+1 (k +1 ) µ = (1 α)ak µ +1g α +1 α +1 µ = αak+1 α 1 g+1 µ +1 (1 α)ak+1g α +1 α = αak+1 α 1 g+1 g = 1 α α k If he social planner doesn have any consrains on axaion, he opimal level of governmen expendiure will be g = α. 1.2 Funding he Expendiures hrough a TDCE Quesion. Can we implemen his opimal soluion hrough a TDCE? If he governmen is able o raise money hrough lump sum axes, he soluion o he social planner s problem defined in secion 1.1 can be obained hrough a compeiive environmen. You can demonsrae his o yourself by showing ha he problems are equivalen. A more ineresing case is when he governmen is only able o raise money hrough disorionary axes. Le s examine a case in which he governmen mus fund is expendiures hrough axes on capial. We will assume ha he firm s ake he governmen invesmen as given when making heir own decisions. Since governmen purchases are like invesmen goods, he governmen will ake g 0 as given, similar o he household aking k 0 as given. 2

3 A Tax Disored Compeiive Equilibrium in his environmen is an allocaion for he HH: z H = {(c, k +1, x )} an allocaion for he firm: z F = {(y f, k f )} a sysem of prices: p = {(p, r )} a governmen policy: g = {(g +1, τ )}, such ha (HH) Given p and g, z H solves max c,k,x s.. p c + p x β c1 σ 1 σ r (1 τ )k + π k +1 x, c, k +1 0, k 0 > 0, given (Firm) Given p and g, z F solves max {(y f,kf )} s.. y f k f, y f [ ] p y f r k f Ak fα 0, g, (Mk) For all, (Goods Marke) c + x + g +1 = y f Ak fα g (Capial Marke) k = k f (Gov) p g +1 = r τ k 3

4 (Profi) π = y f r k f Noice ha in his problem, he firm makes a profi in every period. The producion funcion sill has consan reurns o scale in is inpus, bu he firm receives one of is inpus for free. We can solve he firms problem o ge a characerizaion of profis. r = p αak α 1 g π = p Ak α g r k π = p Ak α g π = p Ak α g r p k p p αak α g π = p (1 α)ak α g (1) This is a well known resul of cobb-douglas producion funcions. The firm will divide is revenue among is inpus according o he elasiciy erms α and 1 α. The equilibrium of his can be characerized by β c σ = λp (2) p = r +1 (1 τ +1 ) (3) r = p αak+1 α 1 g+1 (4) c + k +1 + g +1 = Ak α g (5) in addiion o he governmen budge, he ransversaliy condiion, and he iniial condiions for g 0 and k 0. Noe ha we could use he HH budge insead of he governmen budge, since one implies he oher given everyhing else we know. Combining hese equaions, we can characerize he Euler Equaion. ( c c +1 ) σ = β(1 τ +1 )αak α 1 +1 g +1 (6) If we assume ha his economy will converge o seady sae, hen τ τ, c c, k k, g g. In he limi, 1 = β(1 τ )αak α 1 g (7) 1.3 Building The Ramsey Problem Now we wan o hink he bes sequence of axes on capial in order o fund he sream of g given by secion 1.1. The firs se up seing up he Ramsey problem is deriving he implemenabiliy consrain, which consrains he se of allocaions o hose ha can be 4

5 achieved hrough he ax srucure ha we have specified. We sar wih he presen value household budge consrain. p c + p k +1 p c p c r 0 (1 τ 0 )k 0 + r (1 τ )k + π r (1 τ )k p k +1 + π r +1 (1 τ +1 )k +1 p k +1 + π Using he no arbirage condiion in (3), we can see ha he sum erm on righ hand side is a elescoping sum, as r +1 (1 τ +1 ) p = 0. Thus, we have p c r 0 (1 τ 0 )k 0 + lim T p T k T +1 + p c r 0 (1 τ 0 )k 0 + π π (8) where he second sep comes from he ransversaliy condiion. Now we need o wrie (8) in erms of allocaions so ha we can include i in a planner s problem. Addiionally, noe ha τ 0 is a lump sum ax on he iniial endowmen of he household. Any opimal ax scheme will ax his iniial endowmen as much as possible as i does no disor any fuure decisions, so we can se τ 0 = 1 for simpliciy. Using he firs order condiion for consumpion, (2), in period and he iniial period wih p 0 = 1, we can ge an expression for price: β c σ c σ 0 = p (9) Subsiuing (9) and (1) ino (8), we ge he implemenabiliy consrain in erms of only allocaions: β c σ c σ 0 c β c σ c σ 0 (1 α)ak α g β c σ (c (1 α)ak α g ) 0 Now we can se up he Ramsey problem. In general, a Ramsey problem akes a sream of governmen expendiures as given as maximizes uiliy over he possible ways o fund ha 5

6 expendiure. In his case, i would be feasible o ask he Ramsey problem o solve for he opimal level of governmen expendiure as well. max c,k +1,g +1 such ha β c1 σ 1 σ β c σ (c (1 α)ak α g ) 0 (10) c + k +1 + g +1 Ak α g (11) c, k +1, g +1 0 (12) k 0, g 0 > 0 given Now we wan o characerize he soluion. Le λ be he muliplier on (10), µ be he muliplier on (11), and noe ha he non-negaiviy consrains will no be binding. The lagrangian of his funcion is L = (β [ c 1 σ 1 σ + λc σ For noaional ease, I will define ((1 α)ak α g c ) ] ) + µ (Ak α g c k +1 g +1 ) W (c, k, g ) = c 1 σ ( 1 1 σ λ) + λc σ (1 α)ak α g F (k, g ) = Ak α g Then he firs order condiions are β W c () = µ (13) β W k () + µ F k () = µ 1 (14) β W g () + µ F g () = µ 1 (15) Wih hese firs order condiions, he resource consrain, he ransversaliy condiion, and he iniial condiions we can characerize he soluion o he Ramsey Problem. 1.4 Deriving Opimal Taxaion Resuls A he hear of Ramsey s mehod is deriving resuls abou which axaion schemes migh be opimal. Once we have our allocaions from secion 1.3, we can go back o he TDCE and ask which axes implemen hose allocaions. One consrucive example is o ask how o implemen he seady sae soluion of he Ramsey problem in he limi. Thus, we wan 6

7 o characerize he soluion o he problem when c c rp k +1 k rp g +1 g rp In order o derive hese soluions, i is useful o characerize he parial derivaives of W in equaions (13) hrough (15). W c () = c σ (1 λ(1 σ)) λσc σ 1 (1 α)ak α g W k () = λc σ W g () = λc σ (1 α)αak α 1 g (1 α)(1 α)ak α g α Firs, we can use (15) and (14) o characerize he opimal g relaive o he privae capial sock. αak α 1 g [ β λc σ β λc σ β λc σ β W k () + µ F k () = β W g () + µ F g () (1 α)αak α 1 g (1 α)(1 α)ak α g α (1 α) + µ ] = (1 α)ak α g α αak α 1 + µ αak α 1 g g = + µ (1 α)ak α g α [ ] β λc σ (1 α) + µ = (1 α)ak α g α g = 1 α α k So we ge he same proporion of privae and governmen capial, which is encouraging. This means ha g rp = 1 α α k rp Now les focus on he ineremporal subsiuion. By dividing (13) across ime, we can ge an Euler Equaion. W c () W c ( + 1) = β µ µ +1 From he firs order condiion for capial, we can ge an expression for µ µ +1. Thus, we ge β +1 W k ( + 1) + µ +1 F k ( + 1) = µ β +1 W k ( + 1) + F k ( + 1) = µ µ +1 µ +1 β +1 W k ( + 1) β +1 W c ( + 1) + F k( + 1) = µ µ +1 W k ( + 1) W c ( + 1) + F k( + 1) = µ µ +1 W c () W c ( + 1) = β(f k( + 1) + W k( + 1) W c ( + 1) ) 7

8 In he seady sae, he lef hand side of he equaion is 1 and we can plug in he funcional forms o ge [ 1 = β αakrp α 1 g [ rp + c σ 1 = β λc σ rp (1 α)αakrp α 1 grp ] rp (1 λ(1 σ)) λσc σ 1 rp (1 α)akrpg α rp λ(1 α)αak αakrp α 1 grp rp α 1 g ] rp + [ 1 = βαakrp α 1 grp λ(1 σ) λσ () c rp λ(1 α) Ak α rpg rp 1 λ(1 σ) λσ(1 α) Akα rp g rp c rp Now we can compare his equaion o he seady sae of he TDCE environmen, from (6), 1 = β(1 τ )αak α 1 g Thus, we see ha we mus have ha he axaion in he limi converges o ] τ = λ(1 α) 1 λ(1 σ) λσ(1 α) Akα rpgrp c rp This erm is a lile difficul o inerpre, bu i is probably no zero. So he ypical resul from Chamley and Judd ha axaion in he resul should be equal o zero does no hold in his case. As a saniy check, examine he differen parameers and hink abou he value of ax for λ = 0 or α = 1. Wha do hese values imply? Does i make sense ha capial axaion in he limi migh be negaive? 8