Answers to Final Exam questions 1. An agent wishes to

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1 Econ. 5b Spring 999 George Hall and Chris Sims Answers o Final Exam quesions. An agen wishes o subjec o: max f(x) x p x = Y where x is a vecor of goods, and p is a vecor of prices comformable o x. Le Y denoe he agen s oal income. Compuing a firs-order necessar condiion for an opimum ields: f(x) x λp = 0 Le x denoe he vecor of opimal choices of x. Consider a deviaion x + = x + x such ha In oher words p x + = Y p x = 0 So wha is he cos in uili from deviaing for he opimum? Take a Talor approximaion. f(x + ) f(x ) f(x) x + 2 f(x) x 2 x x x 2 λp x + 2 f(x) 2 x x x 2 2 f(x) 2 x x x 2 Thus a firs-order deviaion in he decision rule causes a second-order cos in uili. Second order does no necessaril mean small. I depends on he curvaure of he uili funcion and he size of x. If he uili funcion is seepl curved around he opimum, so ha uili falls quickl as one moves awa from he opimum, hen hese second-order coss could be large. Also if he deviaions awa from he opimum are large, hese coss could be large. Bu in he models we have sudied in class wih convenional funcional forms (e.g. quadraic uili wih linear laws of moions for he sae variables), even relaivel large deviaions from he opimum resul in relaivel rivial uili coss. One las noe: Of course, a subopimal decision ha coss a large corporaion one-enh of one percen of is profis is a small misake from he firm s perspecive,

2 bu quie large from he perspecive of a manager if s/he can improve on he decision and capure some of he increased profi for him/herself. 2. A no-ponzi condiion is a consrain on agens ha prevens hem from leing heir borrowing grow indefiniel a he ineres rae. Wihou such a condiion, he agen can alwas finance arbiraril large expendiures wihou ever repaing, simpl borrowing more o repa old loans. A Ponzi scheme is a form of fraud in which his pe of behavior is aemped. A ransversali condiion is a necessar and/or sufficien condiion for opimali ha relaes o erminal condiions in a dnamic model. In a pical finie-horizon model of wealh accumulaion i will require ha all wealh be used up b he end of he planning horizon. A similar requiremen is ordinaril needed in infinie-horizon models, bu is harder o make precise. In an economic growh model i usuall akes he form of requiring ha wealh no grow unboundedl large or no grow as fas as he discoun rae. Boh no-ponzi condiions and TVC s limi he rae of growh of wealh, bu he are quie disinc. The no-ponzi condiion prevens wealh from growing more and more negaive a an exponenial rae, while in a growh model he TVC prevens wealh from growing more and more posiive a an exponenial rae. Also, a no- Ponzi consrain is perceived b he agen as imposed from ouside, as a legal requiremen or as a rule imposed b crediors. A TVC is imposed b an agen on his own behavior in order o maximize his objecive funcion. 3. (a) (i) A compeiive raional expecaions equilibrium is a riple (U,, x) such ha he Phillips curve is saisfied and x =. (ii) The Ramse problem is: max.5[(ū θ( x))2 + γ 2 ] subjec o: = x. (iii) The Ramse oucome is he inflaion rae ha solves he Ramse problem. (iv) A bes response funcion for he governmen is = arg max.5[(ū θ( x))2 + γ 2 ]. Following he noes and Sargen s Conques manuscrip, we denoe he governmen s bes response b = B(x). (v) A Nash equilibrium is a pair (x, ) such ha x = and = B(x). (b) Firs we will do Ramse. The governmen knows ha he household will observe is choice of and hus he household will se x =. Taking his ino accoun he governmen wishes o: subjec o: max.5[(ū θ( x))2 + γ 2 ] = x. 2

3 If we subsiue he consrain ino he governmen s objecive funcion, he governmen s problem becomes: max.5[ū 2 + γ 2 ] The soluion o his problem is o se = 0. Therefore a he Ramse oucome x = = 0, and he unemplomen rae is equal o he naural rae. Now le s solve for he Nash equilibrium. The household chooses x. The governmen akes x as given (fixed) and wishes o: max.5[(ū θ( x))2 + γ 2 ] Seing he firs derivaive equal o zero ields: We do a lile algebra....5[2(ū θ( x))( θ) + 2γ] = 0 γ = θ(ū θ( x)) γ = θū θ2 + θ 2 x (γ + θ 2 ) = θū + θ2 x And we ge he governmen s bes response is: = θ γ + θ Ū + θ2 2 γ + θ x 2 In a compeiive equilibrium we know so = x = θ γ + θ Ū + θ2 2 γ + θ. 2 Doing a lile more algebra leads o ) ( θ2 = γ + θ 2 θ + θ 2 Ū So in he Nash equilibrium, x = = θ γ Ū and U = Ū. In he Ramse oucome, he governmen s disase for inflaion, γ, has no effec on he inflaion rae. The governmen alwas ses he inflaion rae o 0. In he Nash equilibrium, he sronger he governmen s disase for inflaion (he larger is γ), he lower is he inflaion rae. As γ ges approaches infini, he Nash equilibrium approachs he Ramse oucome. 3

4 4. (a) I is a menu cos model because i implies ha changes in prices absorb oupu. The erm ξ ( (P P )/ P ) 2 represens he cos of changing prices, and is herefore he one ha makes i a menu cos model. The fac ha i is a monopolisic compeiion model is he fac ha in i prices are decision variables for firms, who see a possibili (no realized in equilibrium) ha he could make he price of heir own produc deviae from ha of he average firm. Each firm has a demand curve, represened b equaion (7) on he exam, which relaes producion relaive o he average level of producion (L α / L α ) o price relaive o he average price (P / P ). As θ + he demand curve flaens and approaches he compeiive case of perfecl elasic demand. (b) The parameer is ξ. You should have given five minues of explanaion of his answer, hough, and his migh have been easies afer ou had derived he FOC s. Wih ξ 0, more variable ime pahs of prices shrink he feasible se of consumpion pahs, so i is clear ha wih ξ 0 neurali does no hold. Tha i does hold wih ξ = 0 requires a more complicaed argumen: Wih ξ = 0, he FOC s and consrains can be broken ino one se ha involves onl real variables (wih real wage, w and real ineres rae included in he real variables), no including τ, and anoher se consising of he governmen budge consrain, fiscal polic, and monear polic, ha involves also B, P and τ. The former block can deermine he real equilibrium b iself, while he laer block, unless i implies infeasibili or violaion of privaesecor ransversali, can modif he ime pahs of P, τ and B arbiraril wihou changing he real allocaion. Noe in paricular ha he monopolisic compeiion componen of he model is reained, wihou producing an nonneurali. (c) In his model all liabiliies of he governmen bear ineres, bu he price level is sill he rae a which he rade for goods. Since governmen liabiliies do no provide ransacions services, he are held onl for he reurn he provide, which in urn is made possible b he governmen s power o ax. If he governmen has he power and will o increase axes devoed o deb service as i increases ousanding deb, i can do so wihou raising he price level. If insead i increases he volume of is ousanding nominal deb wihou increasing axes, he value of he deb in erms of real commodiies (he inverse of he price level) will decline, i.e. here will be inflaion. (d) The consumer Euler equaions are C: L: B: = λ () C L = λ w (2) [ ] λ λ+ = βr E + µ. (3) P P + 4

5 The firm Euler equaions are π: Φ = ζ (4) ( ) P L: ζ αl P α L α w = αν (5) L α P : [ L α P P ζ 2ξζ + 2βξE P P 2 P + P ζ + P 2 ] = ν θ P θ P θ 5. (6) The new variables λ, µ, ζ, and ν are Lagrange mulipliers, on he consumer budge consrain, he consumer B > 0 consrain, he firm budge consrain, and he firm demand curve, respecivel. To use hese equaions o answer (4b), appl he equilibrium condiions ha barred variables mus mach heir individual-firm counerpars, mulipl (5) b L and (6) b Φ P, and solve o eliminae Lagrange mulipliers, producing 2ξ P P P P P + 2βξE C = w (7) L [ ] = βr E (8) C P P + C + [ ] Φ+ P + P = (θ ) L α θ Φ P α w L. (9) Wih ξ = 0, (7) and (9) boh involve onl real variables and involve no lags or expecaions. The are in erms of C, L, and w alone. The can be combined wih he social resource consrain, which wih ξ = 0 is jus C = L α, o produce a saic, 3-equaion ssem in hree variables ha deermines consan values for hem. The res of he ssem hen has no effec on he values of hese real variables. (e) Wih he log uili funcion ha has been specified, equaion (8) on he exam specifies ha he discoun facor firms use o value fuure profis (Φ + /Φ in (9) on he exam) is he same as ha used b consumers (in (8) on he exam) o value o value fuure reurns on heir bond invesmens. If his condiion did no hold, hen an enrepreneur could creae a securi wih a paern of nex-period paoffs ha would be valued differenl b firms and consumers. This would creae an arbirage opporuni. The absence of an possibili of creaing a securi wih a paern of paous ha is valued differenl b differen agens defines a complee marke, and i is equivalen o saing ha all agens use he same sochasic discoun facor. Noe ha here, because of he monopolisic compeiion, here is no presumpion ha marke equilibria are opimal, and hus no presumpion ha i is sociall desirable ha here be complee markes. 5. This quesion asks ou o check exisence and uniqueness b he usual roocouning crieria. Sadl, I realized as I prepared his answer ha he roos given

6 in he problem saemen do no correspond o he problem 4 model as claimed. This does no affec he applicaion of he roo-couning crieria as described below and should no have affeced anone s abili o answer he quesion, unless ou aemped o derive he linearized model and find is roos, a ask ou were no asked o underake. See he noe below for a discussion of wha he problem would have looked like wih he correc roos. (a) The model has wo endogenous error erms. Therefore, so long as he model does no have a special srucure ha creaes singulariies, we expec o find ha we need one unsable roo o Γ 0 Γ for each endogenous error erm, i.e. wo. You were old o consider an roo larger han one in absolue value as unsable. So he firs parameer seing, which was claimed o produce roos of.0500,.0000, , , implies exisence of a unique equilibrium and he second one, wih roos of.0500,.0000, i, i, implies non-exisence because i has hree roos exceeding one in absolue value. The absolue value of he wo imaginar roos is.036. (b) Now he firs parameer seing is claimed o produce onl one unsable roo, while he second produces wo. This implies non-uniqueness for he firs seing and exisence wih uniqueness for he second seing. (c) The smaller ssem, ignoring B and he governmen budge consrain, is giving he wrong answers. This ges o he main poin of he FTPL lieraure: ignoring he GBC leads o misaken conclusions. For he firs parameer seing, he apparen non-uniqueness disappears when we realize ha all bu one of he poenial equilibria impl explosive behavior for he omied B/P variable and would hus violae ransversali. For he second parameer seing, he equilibrium ha apparenl exiss does no in fac, because i would impl explosive behavior of governmen deb ha violaes ransversali. Noe on wha he problem would have looked like wih correc roos: M own suspicions ha here was a misake were aroused b he fac ha here were large negaive or imaginar-wih-small-real-par roos. This means ha he dnamics are fas relaive o he ime uni, and usuall we se up discree ime models so ha he dnamics are slow relaive o he ime uni. I is a good general rule, which I should have paid more aenion o here, ha when our model gives roos impling rapid oscillaions, eiher ou have made a misake or here is somehing odd in our model s formulaion. In he model of problem 4, wih he saed ineres rae and ax policies (keeping hem consan) exisence and uniqueness hold for an posiive value of ξ and here are no negaive or imaginar roos. If he roos had been calculaed correcl he would have been 0.349,.0000, , and.0500 for he ξ =. case and ,.0000, ,.0500 for he ξ =.2 case. As before, he roos of.05 disappear when he governmen budge consrain and B are dropped from he model. Thus wih he correc roos here is exisence and uniqueness for boh values of ξ, and a false 6

7 resul of indeerminac when he GBC and B are dropped from he model, again for boh values of ξ. 6. This model is esseniall he one sudied b Mark Hugge (993). I is discussed in some deail in chaper 5 of LS. Noe ha since here is no aggregae uncerain in his econom, R = R. (a) The household s Bellman equaion is: { } subjec o v(w, s) = max c,w u(c) + β s P(s, s )v(w, s ) W = RW + (s) c W F (b) For a given F, a saionar equilibrium is a fixed ineres rae R, wo polic funcions, W = f(w, s) and c = g(w, s), and a ime-invarian disribuion λ(w, s) such ha: (i) The polic funcions solve he household s opimum problem. (ii) The disribuion λ(w, s) is saionar: λ(w, s ) = P(s, s )λ(w, s) s W Ω(W,s) where Ω(W, s) = {W : W = f(w, s)}. (iii) The loan marke clears: λ(w, s)f(w, s) = 0. W,s (iv) The goods marke clears: λ(w, s)g(w, s) = λ(w, s)(s). W,s W,s (c) One algorihm o solve for his saionar equilibrium is (i) Guess an ineres rae R. (ii) Solve he household s problem for he wo polic funcions W = f(w, s) and c = g(w, s). (iii) Compue he associaed saionar disribuion λ(w, s). (iv) Check o see if he loan marke clears: K = W,s λ(w, s)f(w, s). 7 (v) If K > 0, go back o sep and r a smaller R. If K < 0, go back o sep and r a larger R. If K = 0, done.