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# Macroeconomic Theory I EconS 500

WSU

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This 6 page Class Notes was uploaded by Maurine Kuhic on Thursday September 17, 2015. The Class Notes belongs to EconS 500 at Washington State University taught by Mark Gibson in Fall. Since its upload, it has received 79 views. For similar materials see /class/205981/econs-500-washington-state-university in Economic Sciences at Washington State University.

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Date Created: 09/17/15

Mark J Gibson A Simple Optimal Policy Problem Consider a growth model with taxes on labor income and capital income Tax revenue is used to nance an exogenous amount of government expenditure Z A quot A A A A1 Ak A competitive equilibrium 1s ct pkg pt wt rt 1392 139 such that 822 I solve max 20 uctft st 201am kM 20 in 1 mm 11 00 6k2 625216 gt0 k0 k0 21 t Ja mmmn at I i1 16t g Flei22 zagzjl m e The government wants to choose the policy 139 1395 1f that maximizes social welfare subject to its budget constraint To do this the government must understand how its choice of policy affects equilibrium allocations and prices Let T be the set of policies for which a competitive equilibrium exists and for which 139 if Without the restriction that 139 if the optimal policy would be to tax only period0 capital income since initial capital is supplied inelastically For 139 e T let 021Zt139kt139pt139 wt139 rt139 be the associated competitive equilibrium The Ramsey problem is mu zg dmmami ARamsey equilibrium is 1 ct 1 21kt 1 pt 1 wt1 rt1 such that 1 solves maxTET 20 u 91 21 The Ramsey plan is f We are assuming that the government can commit to the plan Without commitment the government would reoptimize every period To simplify this problem we will reduce the number of equilibrium conditions We can take either the consumer s budget constraint or the govemment s budget constraint and eliminate prices and policies from it using the firstorder conditions from the consumer s problem and the firm s problem This is known as the primal approach The firstorder conditions from the consumer s problem are 5214654 1P tugcl 1P 1 75M int 1171 1 1 1 n1 5 We plug these conditions and the firm s profitmaximization conditions into the consumer s budget constraint 2 m k 2 p 1 rim 11 rm 6k 20 p2 cl 1 721 262 20 p2 11 TfXVz 6k2 k21 20 146106uzcp 20 uc 11r11kp 5kz Z0 t a 621Z211 1 7211Eek21 21 6 km ZZO 146106uzcp u60 011 ff1kolo5ko This equation is referred to as the implementability constraint Together with the resource constraint it ensures that an allocation is a competitive equilibrium This allows us to transform the Ramsey problem from a choice of policy to a choice of allocation The govemment s transformed Ramsey problem is to choose cl 2 k to solve maX 20 uctft st ct kt1 l 5g g Fk2Et 20 Maine uzq 4 ucozo1harmc010 6k0 c z k gt 0 k0 10 2 2 21 Let A be the Lagrange multiplier on the implementability constraint and let Wcz1 ucz1uczc uczz Then the Ramsey problem can be rewritten as max20WmM Moozo11 k 1kozo 6k0 stctkM l 5lggFktZt c z k gt0k00 2 2 21 The rstorder conditions for t l 2 are lm912 H tVVzczagwl luthUCng I M11Eekt19gtl 6 Proposition Chamley 1986 The optimal taX rate on capital income in a steady state is zero Proof The rstorder conditions for a Ramsey equilibrium require that for I l 2 Vmcpgul WcCW52lal111k2195215 Thus in a steady state 1 1FkkE 6 The rstorder conditions for a competitive equilibrium require that a Ce 2 CHI 2111 1211Flek219 21 6 Thus in a steady state 1 11 rkFkkZ 6 These two steadystate conditions only hold if H 0 I Mark J Gibson Stochastic Dynamic General Equilibrium Models The state of the economy in period t is a history of events s s0 s1 st The probability of a particular history is 7rs The initial event s0 is given All equilibrium objects are functions of the state of the economy Pure exchange economy Consider an economy with heterogeneous consumers in which there is uncertainty about endowments a21s39 Consumer i has the expected utility function 20 25 n s log c s39 AnArrow Debreu equilibrium is 510 f7s such that 510 solve max 2025 tn39stlogclst st 2025 f7s cls s 2025r f7stalst cls Z 0 2151s 21a1s With a sequential markets structure Qs is an Arrow security that delivers one unit of the consumption good to consumer i in state s The price of this security is qs39 A sequential markets equilibrium is 81s 51s qs such that 81s 81s solve max 2025 n s logcls st n2 sea mm sia s ws gtbs cls 2 0 bls39 Z B bls 0 215S39ZIWS39 Z s 0 Production economy Consider a growth model in which there is uncertainty about total factor productivity The representative consumer is endowed with one unit of labor in each period and k0 units of initial capital The consumer s expected utility is 20 25 n s ylog cs l ylogl s The resource constraint for this economy is Cs ks 1 l 5ks zs 19ks ZU 139 AnArrow Debreu equilibrium is 532 23 1332 rs Ms f7s such that 532 2s 1332 solve maxZZOZS n s ylogcs l ylogl Zs st 20 25 f7s cs ks 1 1 5ks 3 20 25 viS Zs fs ks cs 2 0 0 s is s 1 km 2 1 5ks ks 130 m ps azs 19 s a lim f W W 1 azs 6l s 2s 5s s 1 1 5 s zs 19 s 3912s 1quot1 Asequeritial markets equilibrium is 53 23 1332 rs WU such that 83 23 1332 solve maxZZOZS n s ylogcs l ylogl Zs st cs39ks 1 l 5ks S vixs s rs ks C32 Z 0 0 S 32 S l 16039 Z 1 5ks 1630 0 m azs 19 s 12s 1 Ms 1 otzs 19 s 12s 1 5s s 1 1 5 s zs 19 s 3912s 1quot1 If the probability of an event SM is only conditional on the previous event st rather than the entire history of events 3 then the stochastic process is Markov In this case there is another equilibrium concept recursive competitive equilibrium The concept of state is different here The state of the economy is not the entire history of events but rather the dynamic programming state The equilibrium objects are all functions of the state of the economy For the previous model the dynamic programming state is k 2 Let the evolution of 2 be governed by a Markov process with transition function qz z39 A recursive competitive equilibrium is 0k z 5kz I 39kz 2k z rkz viikz such that 5k z le39k z 2k z solve 3k z maX ylog c l y logl Z zz 0k39 z39qz 239 st 0 k39 l 5k S V11k z rk zk czo 0351 k3921 6k rk z az k l3912k zk Mk 2 1 azt9k 12kz 1 6k z 39k z 1 6k 26k 2k z1quot1

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