Exercises: Monetary theory and policy

Transcription

1 Exercises: Moneary heory and policy Carl E. Walsh December 6-10, 2010 These excerises are designed o offer you an opporuniy o pracice using some of he models discussed during he lecures and o develop some exensions of hose models. Money demand and opimal inflaion 1. Consider he following money-in-he-uiliy model: Preferences are E i=0 and he nominal budge consrain is β i u(c +i, m +i, 1 N +i ) P e z N a + M 1 + (1 + i 1 ) B 1 + T = P C + M + B where T is a lump-sum ransfer, real money balances are m = M /P and N is he fracion of ime spen working. (a) Derive he firs order condiions for consumpion, labor supply, money holdings, and bond holdings. In each case, provide an inuiive explanaion of he firs order condiion. Explain inuiively wha each means. (Hin: One approach is o se up he decision problem explicily. Anoher is o hing inuiively abou he rade offs he household faces. For example, if he household gives up a lile leisure, wha is he marginal cos and he marginal benefi ha need o be equalized? If he household saves a bi more oday and spends he earnings nex period, wha is he marginal cos? Wha is he marginal benefi? If he household increases is money holdings marginally, wha are he coss and benefis?) (b) Define superneuraliy. (c) Suppose he uiliy funcion is given by ( C 1 σ ) ( ) m 1 φ u(c, m, 1 N ) = 1 σ 1 φ c Carl E. Walsh, ( ) N 1+η Ψ. 1 + η 1

2 Does his economy have he propery of superneuraliy? If money is no superneural, explain in words how an increase in he seadysae growh rae of he money supply would affec equilibrium employmen. (d) Suppose he uiliy funcion is given by ( C 1 σ ) ( ) m 1 φ u(c, m, 1 N ) = + Φ 1 σ 1 φ ( ) N 1+η Ψ. 1 + η Does his economy have he propery of superneuraliy? If your answer differs from ha of par (c), explain why. If money is no superneural, explain in words how an increase in he seady-sae growh rae of he money supply would affec equilibrium employmen. 2. Suppose he represenaive household maximizes { } E β i [u(c +i ) v(n +i )] i=0 subjec o a nominal budge consrain of he form W N + (1 + i 1 ) B 1 + (1 + i m )M 1 + T P C B M 0. and a cash-in-advance consrain of he form P C M 1 + T. The consumpion good is produced by a consan reurns o scale echnology. W is he nominal wage he household earns on labor N i supplies. The household eners period wih bond and money holdings B 1 and M 1. hese bonds earn a nominal ineres rae of i 1 while money earns a nominal ineres rae of i m. The ineres paid on money is assumed o be a consan. (a) Wrie Bellman s equaion for he household s problem. (Hin: The value funcion depends on b 1 = B 1 /P 1 and m 1 = M 1 /P 1, i.e., he value funcion is V (b 1, m 1 ).) (b) Derive he firs order condiions for he opimal choices of C, N, B, and M. Explain in words wha each condiion means. (c) Wha disorions are caused by a non-zero nominal rae of ineres? Wha is he opimal rae of inflaion in his economy as a funcion of i m? Explain. (d) Can i m be picked o eliminae he disorion creaed by i 1 > 0 if inflaion is zero? Explain. 2

3 3. Suppose ha money is required o purchase boh consumpion and invesmen goods. The CIA consrain hen becomes c +x m 1 /(1+π )+τ, where x is invesmen. Assume ha he aggregae producion funcion akes he form y = e z k 1n α 1 α. Show ha he seady-sae capiallabor raio is affeced by he rae of inflaion. Does a rise in inflaion raise or lower he seady-sae capial-labor raio? Explain. Sicky prices, new Keynesian models, and opimal policy 1. Consider he following sylized facs abou prices a he micro level: (1) Prices of many goods remain unchanged for significan periods of ime; (2) Price changes due o sales are imporan; (3) Small price changes as well as large ones are observed in he daa; (4) The probabiliy a price changes falls in he monh immediaely afer he price has changed. Discuss he consisence (or lack hereof) beween each of hese facs and (a) he Calvo model of price adjusmen; (b) a menu cos, sae-dependen pricing model. 2. Suppose he represenaive firm i ses is price o minimize a quadraic loss funcion ha depends on he difference beween he firm s acual log price in period, p i, and is opimal log price, p. If he firm can adjus a ime, i will se is price o minimize 1 2 E β j ( p i+j p 2 +j), subjec o he assumed process for deermining when he firm will nex be able o adjus. (a) If he probabiliy of reseing prices each period is 1 ω as in he Calvo model, and ˆp denoes he opimal price chosen by all firms ha can adjus a ime, show ha ˆp minimizes ω j β j ( E pi p 2 +j). (b) Derive he firs order condiion for he opimal choice of ˆp. (c) Using your resul from (b), show ha ˆp = (1 ωβ) ω i β i E p +i. Explain inuiively why he weighs on fuure opimal prices p +j depends on ω. (d) Show ha ˆp = (1 ωβ) p + ωβe ˆp +1. 3

4 (e) Assume he price arge p = p +γy +ε where ε is a random disurbance. The log aggregae price level is p = (1 ω)ˆp + ωp 1. Using he resul in par (d), obain an expression for aggregae inflaion as a funcion of expeced fuure inflaion and oupu gap y, and ε. (f) Is he impac of oupu on inflaion increasing or decreasing in ω, he measure of he degree of nominal rigidiy? Explain. 3. In sae coningen models of price adjusmen, wha is mean by he selecion effec? Explain why sae coningen pricing can reduce he real effecs of a change in he nominal money supply. 4. Suppose he household s uiliy depends on a composie consumpion good ha consiss of differeniae producs. There are a coninuum of such producs c j wih price p j. The composie consumpion good ha eners he household s uiliy funcion is defined as [ 1 C = 0 ] θ c θ 1 θ 1 θ j dj θ > 1. (1) (a) Wrie down he household s decision problem of purchasing he individual c j goods o minimize he cos of achieving a given level of C. (b) Derive he opimal demand for good j as a funcion of is relaive price and oal consumpion C. Be sure o carefully define he appropriae aggregae consumpion price index. 5. Suppose he cenral bank cares abou inflaion variabiliy, oupu gap, variabiliy and ineres rae variabiliy. The objecive of he cenral bank is o minimize E 1 2 [ β i π 2 +i + λ x x 2 +i + λ i (i +i i ) 2]. i=0 The srucure of he economy is given by π = βe π +1 + κx + e (2) ( ) 1 x = E x +1 (i E π +1 r ), σ (3) where e and r are exogenous, serially uncorrelaed sochasic shocks. Le ψ denoe he Lagrangian muliplier on consrain (2) and le θ be he muliplier on (3). (a) Derive he firs order condiions for he opimal policy of he cenral bank under discreion. Show ha θ is non-zero if λ i > 0. Explain he economic inuiion behind his resul. 4

5 (b) Explain how he oupu gap and inflaion respond under he opimal policy o a posiive realizaion of e. How do hey respond o a posiive realizaion of r? (c) Define wha is mean by an opimal commimen policy from a imeless perspecive. How does his differ from a fully opimal commimen policy? (d) Now assume θ = 0. Derive he firs order condiions for he opimal commimen policy from a imeless perspecive. Eliminae any Lagrangian mulipliers o obain an expression ha only involves inflaion and oupu gap erms. Explain why he lagged oupu gap affecs curren policy under he imeless perspecive. 6. Consider a basic new Keynesian model wih Calvo adjusmen of prices and flexible nominal wages. (a) In his model, inflaion volailiy reduces he welfare of he represenaive agen. Explain why. (b) In he absence of cos shocks, opimal policy would ensure inflaion and he oupu gap boh remain equal o zero. Wha does his imply for he behavior of oupu? Why can oupu flucuae effi cienly despie sicky prices? (c) Suppose boh prices and nominal wages are sicky (assume a Calvo model for wages). Will volailiy in he rae of wage inflaion be welfare reducing? Explain. (d) Is zero inflaion and a zero oupu gap sill feasible? Explain. 7. Consider a new Keynesian model wih sicky prices and wages. Assume boh prices and wages adjus according o he simple Calvo model, hough wih differen degrees of sickiness. The linearized equilibrium condiions of his model expressed as log-deviaions around he zero-inflaion seady sae can be wrien as ( ) 1 x = E x +1 (i E π +1 r n ) σ π = βe π +1 + κ p (ω z ) π w = βe π w +1 + κ w (mrs ω ) ω = ω 1 + π w π where x is he oupu gap, π is he inflaion rae, π w is he rae of nominal wage inflaion, i is he nominal ineres rae, ω is he real wage, z is he marginal produc of labor, mrs is he marginal rae of subsiuion beween leisure and consumpion, and r n is he equilibrium real ineres rae in he flexible wage and price equilibrium. To close he model, a specificaion of moneary policy mus be added. 5

6 (a) Carefully explain he meaning of each equilibrium condiion. Where does each come from (i.e., whose behavior does i presen?, wha assumpions lead o he equaion in quesion?) (b) Wha facors generae ineffi ciencies in his economy? Explain carefully. (c) Suppose he moneary auhoriy minimizes a loss funcion based on a second-order approximaion o he welfare of he represenaive household. Does he moneary auhoriy face a rade-off beween sabilizing π, π w, and x if produciviy shocks are he only source of exogenous flucuaions in his economy? If i does face a rade off, explain why, and carefully explain wha facors deermine he relaive weighs he moneary auhoriy should give o sabilizing inflaion, wage inflaion, and he oupu gap. 8. Consider a new Keynesian model wih sicky prices and wages. Assume boh prices and wages adjus according o a simple Calvo model, hough wih differen degrees of sickiness. (a) Suppose fiscal axes and subsidies are used o eliminae he disorions caused by imperfec compeiion in goods and labor markes and he moneary auhoriy minimizes a loss funcion based on a second-order approximaion o he welfare of he represenaive household. Carefully explain wha facors affec he relaive weighs ha should be placed on reducing inflaion volailiy, wage inflaion volailiy, and oupu gap volailiy in his economy. (b) Now assume prices are flexible bu wage adjusmen is modeled using he Calvo model. Wrie down he (linearized) equilibrium condiions for his economy. (c) Assume he cenral bank can implemen he opimal precommimen policy from a imeless perspecive and does so o minimize ( ) 1 2 E β j [ ( π w +j ) 2 + λx 2 +j ] where π w is wage inflaion and x is he oupu gap. Wha are he firs order condiions for he cenral bank s decision problem? Eliminae Lagrangian mulipliers and derive he opimal argeing rule. Does he cenral bank face rade-offs beween sabilizing wage inflaion and sabilizing he oupu gap in he face of produciviy disurbances? Explain. Financial fricions 6

7 1. Adverse selecion: Suppose invesors have access o a risky projec whose expeced reurn is R. Assume R = { R x wih probabiliy 1 2 R + x wih probabiliy 1 2 An increase in x is an increase in he projec s riskiness. (Changes variance bu does no change expeced reurn.) Le L be he loan amoun, r is he ineres rae, and C is he borrower s collaeral. (a) Wha is he expeced profi o he lender? (Assume he lender ges all he collaeral if he borrower defauls.) Show ha he expeced profi o lender is decreasing in x (i.e., expeced reurn o lender is lower for riskier projecs). (b) Wha is he expeced profi o he borrower? Show ha he expeced profi o he borrower is increasing in x (i.e., expeced reurn o borrower is higher for riskier projecs). (c) Suppose borrowers differ in he riskiness of heir projecs bu lenders canno observe he borrower s x. Tha is, hey all have projecs wih expeced reurns of R bu heir x can differ. Define x as he value of x such ha he expeced reurn o he borrower wih an x of x is zero. Show ha he expeced profi o he borrower is posiive for any borrower whose x > x. (d) Show ha x is increasing in r. Explain why his means ha some borrowers wih less risky projecs (lower x) will find i unprofiable o borrow when r increases. (e) Explain why he expeced profi o he lender migh fall as r increases due o adverse selecion. 2. Suppose he borrower can inves eiher in projec A, which pays off R a in he good sae and 0 in he bad sae, or in projec B, which pays off R b > R a in he good sae and 0 in he bad sae. Le he probabiliy of success for projec A be p a and assume i is p b for projec B, wih p a > p b. Assume he expeced payoff from A is higher: p a R a > p b R b. The borrower mus pu up collaeral C. (a) Wha is he borrower s expeced reurn from invesing in projec A? (b) Wha is he borrower s expeced reurn from projec B? (c) Find he loan rae rl such ha he expeced reurns o he borrower from he wo projecs are equal. (d) Show ha as r l rises above rl, he borrower will shif from he less risky projec o he more risky projec. Explain why his occurs. 3. In he Diamond-Dybvig model, assume here is free enry o banking. Using he numerical example from lecure, wha is he value of r d 2 ha maximizes bank profis, given r d 1 = 1.28? Why isn his an equilibrium? 7

8 Moneary and fiscal policy 1. Consider a simple model of opimal ax and borrowing. The governmen wans o minimize he presen discouned value of ax disorions subjec o he need o raise suffi cien revenue o finance expendiures. Tha is, subjec o min E E β j ( τ 2 1,+j + λτ 2 ) 2,+j β j (τ 1,+j + τ 2,+j ) = R where R is he oal presen discouned value of revenue ha mus be raised. (a) Derive he firs order condiions for he governmen s opimal choice of τ 1, and τ 2,. (b) Show ha he expeced fuure value of each ax is equal o is curren level. (c) Why do axes following a random walk (acually, a maringale) process? 2. Fiscal policy a he ZLB: Consider he following simple NK model: ( ) 1 x = E x +1 (i E π +1 r n ) σ π = βe π +1 + κx r n = z φ (E g +1 g ) where g is governmen purchases of oupu. (a) Show ha if he zero lower bound on he nominal ineres rae is ignored, an equilibrium in which π = x = 0 is feasible if i = i = z φ (E g +1 g ). Explain why changes in g have no effec on he oupu gap or inflaion in his case. (b) Now suppose z akes on a large negaive value such ha i becomes negaive. In his case, i = 0 > i. Assume he negaive shock lass for only one period, so x +1 = π +1 = 0. Find he equilibrium values of x and π. Show ha an increase in g ha is believed o be emporary will raise he oupu gap and inflaion. Show ha an expeced reducion in fuure governmen spending will also raise curren oupu. Explain he inuiion behind each resul. 8

9 (c) Now suppose again ha he shock lass only one period, bu now suppose ha x +1 and π +1 migh no equal zero. However, assume x +2 = π +2 = 0. Assume i = 0, i +1 = i, and i +2 = i +2 = z +2 φ (E g +3 g +1 ). Find x and π as a funcion of fiscal policy and he cenral bank s choice of i. (Hin: work backwards bu firs finding x +1 and π +1 ). Show ha by seing i < i +1, he cenral bank can increase x and π. (d) Wha do your resuls from par (c) imply abou he impac of a emporary rise in g on x and π? 9