Problem Se "Working wih he Solow model" Le's define he following exogenous variables: s δ n savings rae depreciaion rae of physical capial populaion growh rae L labor supply e n (Normalizing labor supply per capia and ime 0 populaion o uniy ) x rae of growh of labor augmening echnology. ~ L effecive labor supply e (n+x) and he following endogenous: ~ k capial sock per effecive labor uni (K/ ~ L ). ~y oupu per effecive labor uni (Y/ ~ L ). k y capial sock per capia oupu per capia a)wha is he growh rae of L ~. If k ~ grows a rae γ (i.e., k ~ / k ~ γ ), wha is he growh rae of k, K and k ~ α. Wha is a proper uni of γ? b) The gross resource consrain for a closed economy can be wrien. sf( K, L ~ )Gross Invesmen (.) How should we define gross invesmen? Subsiue your definiion ino (.) and hen α ~ α subsiue a Cobb-Douglas producion funcion (Y K L ) for F. Use his o derive, sep by sep, an expression for he growh rae of k ~. (Hin: Divide by L ~, hen calculae k ~ by noing ha ~ K K k ~ L, and you ge and expression conaining ~. Then subsiue and L your done). c) Provide some reasonable values for s, δ, n and x. d) Find he seady sae level of k ~, k ~ *. Use your parameer values from c) o find he seady sae capial oupu raio K/Y? If his raio is consan wha does i imply for he growh raes of K and Y. Is he labor oupu raio also consan? e) In seady sae wha is he growh rae of GDP and of wages? Wha abou he ineres rae, i.e., he marginal produc of capial?
Barro and Sala-i-Marin make a log-linear approximaion o he growh rae of ~ k around i's seady sae (equaion.3). They wrie:. ~ ~ k k ~ β log ~ k k * (.) where β α6x + n + δ 6 (.3) a) Saring from seady sae, if you decrease he capial sock by %, wha is a good approximaion (no exac answers please) for he growh rae of ~ k. How much is i given he parameer values in your answer o he previous quesion? b) Assume ha from an iniial seady sae, a devasaing war leads o a reducion in ~ k o one half of is seady sae value. Calculae he oupu gap (œy/y*-) immediaely afer his shock. Wha is now he growh rae of ~ k and of K, given he approximaion in (.)? Calculae he change in ineres rae. Is i reasonable? c) How big is he oupu gap afer one year and afer 0 years? Can you calculae he ime i akes unil half he oupu gap is gone? Is his slow or fas, do you hink? When is he gap oally gone. (Hin, define he variable d log k ~ ~ 3 / k * 8. Wha is he inerpreaion of d? Compue d from is definiion. Subsiue ino (.) and you ge a very simple differenial equaion ha you can solve explicily.) d) If you increase α wha happens o he rae of convergence and o he seady sae capial sock. Wha do you hink happens as α approaches. Volunary exra quesion Use he exac soluion o he differenial equaion for ~ k given in foonoe 5 in Barro and Sala-i-Marin o see how far off he log-linear approximaion is for some reasonable parameer values.
Problem Se "The Ramsey Model for an Open Economy wih Imperfec Markes" This problem exends he sandard Ramsey model you saw in class. Before you do he problem, compare his model o he Solow model in erms of model assumpions. Do hey address he same quesions? When is which model more useful? Assumpions We use he same noaion as in Problem Se. For simpliciy we se x and n o 0. This means ha we don' have o worry abou he differences beween per capia, per effecive labor uni and aggregae unis. We are now looking a an open economy. The firm sells is good on he world marke bu i faces a downward sloping demand curve. The price P of he firm's oupu is a negaive funcion of how much i produces. The good ha i produces canno be sored so everyhing he firm produces has o be sold immediaely. The firm buys capial from abroad a a price P k. There is, however, an imperfecion in he inernaional ransporaion secor. The more capial he firm buys per uni of ime, he more expensive ransporaion becomes. Households buy consumpion goods a a perfec world marke a price. Firms Technology and demand are given by y P αk P( y ). (.) To inves a he flow i he firm has o pay ip k (+T(i)) per uni of ime. T(i) is he ransporaion cos and T'(i)>0 for i>0. The firm also has a fixed number of employees and pays w in wages o hem. The firm pays is profis o he household in he form of dividends, d. I maximizes he presen value of dividends and hus solves I r max e 4P( y) y ipk + T( i) 7 w8d, < ia 0 0 % y αk K s.. & k i δk Kk k ' 0 (.)
Since he counry is small and open he ineres rae, r, is given and consan. There is also access o a perfec capial marke for boh firms and households. Households Households own he firms and supply one uni of labor inelasically. The represenaive household solves: I / σ c ρ max e d, c / σ < A 0 0 s.. % a w+ d + ra c K & a a 0 K r lim e a 0 ' K (.3) Quesions. a) Is he oal cos of invesing, ip k (+T(i)), convex or concave in i for posiive i? Wha does ha imply for he relaive cos of volaile and smooh invesmen plans? Solve he maximizaion problem of he firm in he following seps. b) Se up he curren value Hamilonian. Represen he soluion as wo differenial equaions, one for k and one for q, he curren value shadow price of capial and an invesmen funcion ii(q,p k ). c) Make a linearisaion of he sysem around he seady sae. For wha could you use his linearisaion? When can i be dangerous? d) From now on assume ha he price funcions have he following simple srucure βy Py ( ) γ i Ti () (.4) Solve for he seady saes of k and q, (k* and q*) and draw a phase diagram.
Solve he problem of he household. a) Sar by seing up he Hamilonian, now in presen values for a change. Take ime derivaives of H c 0 and subsiue in from he equaion for H a. This gives you a nice differenial equaion for c. b) Wha happens if ρ r? Is ha reasonable? Wha is he inerpreaion of σ? How does hings change wih changes in σ. c) Solve for he level of c using he ransversaliy equaion in (.3) (Hin; presen value of consumpion equals presen value of income plus saring wealh). 3. Suppose rρ and ha he economy is in seady sae. Use he phase diagram o see wha happens dynamically if we ge a permanen negaive shock o he demand for expor goods ( β increases). Wha happens o consumpion and he curren accoun over ime. (Hin; use he following version of he iner emporal budge consrain: p p p 0 p p p p k c w + d + ra c P( y ) y i P + T( i) 6+ ra 0 (.4) where superscrip p denoes permanen values, defined by: I I p p r x r 4x 9e d x 7e d (.5) r 0 0 The presen discouned value is he same as for he acual pah of x. Now wrie he equaion for he curren accoun, where b equals he sock of deb o he res of he world. b P( y ) y i P + T( i) 6 c rb (.6) k Subsiue (.4) ino he equaion for he curren accoun and noe ha a-b, (he household canno sell is ownership of he firm and he value of he firm is no included in a).using his for 0 we can hen wrie he useful equaion: p p p p p b P( y ) y P( y ) y i P + T( i) i P + T( i ) ( c c ) (.7) 0 4 9 4 6 k k 4 99 Equaion (.7) can be used o inuiively derive wha happens dynamically afer he shock.
4. Wha happens if he shock is i ) emporary (happens a 0 and ends a s) and; ii ) anicipaed permanen (he informaion abou he shock arrives a 0 and he acual shock happens a s). (Hin. Draw he phase diagrams and noe ha k can newer jump (why?) and ha q can jump only when new informaion (like informaion abou a fuure produciviy shock) arrives. In he case of he anicipaed shock noe ha a he ime of he shock he phase diagram shifs and ha a exacly he ime of he shif we have o be a he sable saddle pah for he sysem no o explode. So when he informaion arrives a 0, he "old" phase diagram coninues o hold. Now q jumps so ha {q,k} his he new sable pah a exacly s).)
Problem Se 3 Endogenous Growh Problem Learning by Doing in a Two Secor Model Consider a closed economy wih a fixed number of people - normalized o one. The people ge uiliy from wo sources; by being aken care of by people employed in a service secor and from a good produced in a manufacuring secor. The insananeuos uiliy funcion is U log( s ) + log( c ) (3.) Where s is he number of hours of service per uni of ime and c is he consumpion of manufacured goods per uni of ime. The individuals maximize fuure uiliy under a budge consrain; I Ue 0 ρ d s.. Pc + s w, (3.) where P is he price of manufacured goods, w is he wage and we fix individual labor supply o uniy and normalize he price of services o uniy. People can choose o work eiher in he service secor or in he manufacuring secor. Labor markes and consumpion markes are compeiive. Producion in he manufacuring secor equals consumpion and is given by c A l (3.3) c where A is he level of produciviy and l c is he share of people working in he manufacuring secor. Each firm akes he level of produciviy as given. There is learning by doing in he manufacuring secor, however, so A evolves over ime according o: A υ c (3.4) Quesions a) Before doing any number crunching hink abou he following. In his economy, wha will happen o he fuure consumpion possibiliy se if all people decide o consume more manufacured goods oday? If only one person consumes more manufacured goods? Wha do you hink of he specificaion of learning by doing, does i make sense?
b) When we normalize he price of services o uniy wha is he wage raes in he wo secors? c) Wih perfec compeiion in he manufacuring secor wha is he price of manufacured goods? d) Solve he maximizaion problem of he consumer. (Hin: All individuals are equal. This implies ha if here is a capial marke he ineres rae will have o adjus so ha no one will borrow. We can hus as well assume ha here is no capial marke. The maximizaion problem of he consumer is he saic.) Wha is he growh rae in his secor. Wha happens o he growh rae if more people work in he manufacuring secor? f) Now le's find he opimal amoun of people in he manufacuring secor. Firs, do you hink i should be higher or lower han he marke soluion? To find he soluion, follow hese seps:. Le here be a benevolen cenral planner who maximizes he individuals' uiliy bu akes ino accoun he learning by doing effec. Assume here is no possibiliy o sore he manufacured good.. Wrie he Hamilonian and he opimaliy condiions. Noe how he Hamilonian differs from he saic problem of he individual. Guess ha in an opimal seady sae, he labor shares are consan and hen differeniae he firs opimaliy condiion (H l ) w.r.. ime. Wha is hen he relaionship beween he growh rae of he shadow value of he consrain and he growh rae of A? Eliminae he growh rae of he shadow value and he shadow value from he second opimaliy condiion (H A ) and you ge an expression for he growh rae of A involving υ, l c and ρ 4. There is anoher expression for he growh rae of A coming from he learning by doing consrain. Use his o eliminae he growh rae of A. (3.4)Problem. The "School is Never Ou" model There is a coninuum of workers-consumers on he inerval [0,], i.e. an infinie number of agens indexed by he real numbers i beween 0 and. Each individual i has one uni of labor o spend per day (raher per insan). Since all individual are alike we can drop he i superscrip where we don' need i. Her decision is how much of he day o spend in school (-a i ) and how much o work (a i ). She can, however, never qui shool. When she goes o school she accumulaes human capial (h i ) a a rae proporional o he ime spen in school. h β( a) h (3.)
The ineresing hing wih he model is ha he produciviy of individual i depends on he human capial of everybody else. By working wih smar people your produciviy increases. The producion of a single individual is hus given by: α α 7 7 y ah AH (3.3) Where A and H denoes aggregae levels of a and h. H A I i hdi I0 0 (3.4) The producion canno be sored, so he individual consumer-producer consumes wha she produces each momen in ime so she solves i a di I max log( y) e a s.. h β( a) h ρ d y ah 7 AH 7 (3.5) h 0 α α 0 h Quesions a) Is here decreasing or consan reurns o human capial in his model? How much of he exernaliy is aken ino accoun by his small selfish agens? This can be found be evaluaing H. h i b) Assume a common value of a for all agens, (is his reasonable?), hen compue he growh rae of he economy as a funcion of a. c) Derive he equilibrium growh rae of he economy by leing he agens choose a opimally. (Hin; Se up he maximizaion problem of he consumer-producer and subsiue for c in he uiliy funcion. Se up he Hamilonian wih is opimaliy condiion and he solve for he growh rae of human capial assuming a seady sae where a is consan. d) Derive he opimal growh rae of he economy. Can you explain your resuls?
Appendix o P-Se 3 Regarding coninuous populaions A coninuous populaion is surely an absracion. If we index people wih all raional numbers beween 0 and each have o be infiniely small, e.g., have zero human capial in order for aggregae human capial o be finie. Neverheless his may be a useful absracion. To ry o give some inuiion, look a a populaion consising of N (where N is very big) individuals. To calculaed heir aggregae human capial we can divide he populaion ino I groups of equal size, indexed by i {,...I}. Then we figure ou he average level of human capial per capia in each group, h I i. For higher I we ge a finer division of he populaion I and for each I we choose we ge a sequence of h i for i {,...I}. I have drawn wo figures I I of h i for I 5 and 5 below. Noe ha if we increase I, h i will change for each i bu i will keep is approximae order of magniude consan. 0.7 0.6 0.5 0.4 0.3 0. 0. 0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0 In he sense ha he average over all i remains consan when I increase bu he variance increases.
The amoun of human capial in each group is hen: So o ge oal human capial we sum over he groups: N I h I i, (3.3) H I i N I h I i (3.4) This is equal o he are of all he bars in he figures, noing ha he base of each bar is N/I. We can also compue how much an increase in he average human capial level for one group i affecs aggregae human capial. I is going o be: dh dh I i N (3.5) I Thus we see ha: dh dh I i N (3.6) I decreases when I increases (he pariion becomes finer). Suppose we increase I very much, sooner or laer hen NI. Afer ha we canno go on in realiy since hen we have o sar spliing people (we could in fac divide people ino pars and le he pars share equally he human capial of he individual, in his case i doesn', however, make much sense o hink of increasing he human capial of a par of an individual). On he oher hand, if N is large we can increase I very much before we run ino his problem and we may be forgiven if we forge ha consrain. If we he le I go o infiniy we have: lim I N dh lim I I dh i I I I i i 0 N lim 0 I I I i I h N h di (3.7) In some cases, like he one in P-se 3, i is much easier o do calculaions wih I going o infiniy. Then we may formally disregard he exernaliy when opimizing for he individual. For I large bu finie his is only on approximaion. Noa bene he connecion o small firms and perfec markes.
Problem Se 4 The Overlapping Generaions Model Quesion. Consider an overlapping generaions model in discree ime, where people work and earn a wage w in he firs period and are reired consuming heir savings s plus ineres in he second. A generaion born a ime consumes c when young and d + when old. They hus solve he problem: max 4Uc ( ) + ( + θ) Ud ( ) + 9 c, d+ s.. c + s w d s ( + r ) + + (4.) All variables are per capia which is aken o mean ha we divide wih he size of he relevan generaion. There is producion wih consan reurns o scale. Perfec markes ensures ha: r f '( k) w f ( k ) r k (4.) The savings in period consiues he capial for he nex generaion o work wih in period +. The populaion grows a rae n so k n s + ( + ) (4.3) a) Show ha indirec uiliy increases in w and r +.. b) Solve he problem of he consumer for a general uiliy funcion. Express your resul as a savings funcion only depending on k, k + and parameers. Give me an equaion ha implicily defines a seady sae. c) Can we say anyhing abou he sabiliy of he seady sae? Now assume logarihmic uiliy. Find a savings funcion. Show ha i does no depend on k + and explain why..
3. Now also assume Cobb-Douglas producion so f ( k) k α. (4.8) Use he savings funcion o ge a difference equaion which you solve for a seady sae and find a condiion for sabiliy. 4. Now le's use his model o sudy he effec of a budge defici used o finance ransfers o he curren ax payers, i.e., he curren siuaion in Sweden. According o he Ricardian Equivalence Theorem by Barro, such a policy should have no effec. Bu wha is he predicions of his model? Le's formalize he quesion as follows. The governmen borrows b a ime from he currenly young. The receips are immediaely paid back o he currenly young as ransfers. The governmen is no allowed o go bankrup so in + he old ge back wha he governmen borrowed plus ineres. a) Firs assume ha nex generaion resores he governmen's financial posiion by paying back b (+r + ) o he old. Wrie down he budge consrains for he young in and +. Find he sign of he effecs on capial socks, wages and uiliy for he generaions born in o + for a small b and saring from a seady sae. b) Now assume ha he generaion born in + and onwards pays ineres bu rolls over he principal by leing he governmen borrow o finance paying back he principal o he old. Wrie down he budge consrains for he young in and +. Find he sign of he effecs on capial socks, wages and uiliy for he generaions born in o + for a small b and saring from a seady sae. c) Assume ha he generaions born from + and onwards roll over all deb including ineres raes. How will b s evolve for s>? Wha happens if r s <n? Explain.
Appendix o P-Se 4 A Simple Example wih Deah Probabiliy Here is an example ha shows ha i is no he chance of deah bu raher he birh of new generaions ha breaks Ricardian Equivalence in an overlapping generaions model. Le's assume ha here is a probabiliy o die which is -p, ha rφ0, and ha here is a perfec capial marke so if one saves and don' die one ges a reurn of /p on he savings. In addiion n individuals are born in period. The individual in period hen maximizes max U c 4 7 7 c, c 8 s.. c w c p + pu c τ τ p The budge consrain for he governmen is hen G τ + ( p+ n) τ. In he second period here are p people around, so he ax receips are pτ. Then a balanced ax shif saisfies τ + ( p+ n) τ 0 τ ( p+ n) τ Subsiue his ino he budge consrain of he individual and we ge c w c p w c p w c p τ + τ ( τ + τ ) p τ + ( ( p+ n) τ ) ( τ + τ ) p τ n τ + τ p p So he chance o ge away wih some ax paymens is exacly balanced by ha higher axes have o be paid if one survives if n0. I.e., expeced ax paymens are hen independen of he deah probabiliy. If, on he oher hand, n>0 he consumpion possibiliy se is increased if axes are shifed forward in ime.
Problem Se 5 Consumpion under uncerainy Quesion. Consider a person who lives for wo periods and has he same uiliy funcion in boh periods bu discouns uiliy in he second wih +θ. She faces a consan ineres rae r and her income in period one is y. In period wo her income ~ y is random wih mean y. She maximizes expeced uiliy seen from period.. Define he maximizaion problem of he consumer and show ha he firs order condiion is r U '( c ) + EU '( c ) 7 (5.) +θ where E defines he expecaions operaor. Will equaion (5.) be saisfies also in a muli-period model? (If you can, define a value funcion and use i o prove he answer. Oherwise make an informed guess and provide some heurisic argumens.) Now assume quadraic uiliy. U αc βc. (5.6) Find he FOCs. Solve for he consumpion c of he consumer. How does he consumpion depend on he degree of income risk? Explain your resul and discuss he usefulness of his uiliy specificaion. 3. Using he quadraic uiliy funcion, le's do a Hall-ype consumpion ime series regression. c µ + φc + ε (5.8) Wha is he rue regression coefficien I? Esimae he parameers of he regression from Swedish consumpion daa. Discuss poenial problems and remedies in OLS esimaion.
Include more RHS variables ha are known in period -, e.g., c -? Do your resul accord wih he heoreical predicions? Now le's look a some oher uiliy funcions. Firs le he individual have he following uiliy funcion. 4. αc e Uc () α (5.) Find he degree of absolue risk aversion of his individual. Now, se for simpliciy r and T o zero. Le he second period risk be addiive and normally disribued so ha ~ d y y + ε, ε N40, σ 9 (5.) Solve for c. Hin; Remember he following relaion: d 4 x9 x N x, σ βx βx+ β σ x/ Ee e (5.3) Define precauionary savings, s(v), as he difference beween consumpion wih no risk and wih risk. s( σ) 4c σ 09 c. (5.4) Is precauionary savings increasing in he variance and D? Is i increasing in income? Now le 5. α c Uc () α (5.7) Find he degree absolue risk aversion of his individual, how does i depend on consumpion?
To ge analyical soluion o his problem we need muliplicaive risk ha is log-normally disribued, i.e., we may hink of risk as a risky reurn and ha he percenage reurn has a normal disribuion. So c y + y c 7 (5.8) d log ε N 0, σ 4 9 ε where y is known in he firs period. Solve for c. Hin; log(c ) is normal so you can use he hin o he previous quesion. Now define relaive precauionary savings as c sr ( σ) c σ 0 (5.9) Is relaive precauionary savings increasing in he variance and D? Is i increasing in income? Which of he wo uiliy funcions you have worked wih do you hink capures aiudes owards risk bes?
Problem Se 6 Is here a Swedish Equiy Premium Puzzle? * In his problem se we are going o use a sandard model of asse pricing. We are also going o assume a specific sochasic process for how consumpion evolves over ime. We will calibrae he parameers of he process is calibraed so as o replicae he mean, variance and auocorrelaion of Swedish real consumpion. Then we can compue he expeced risky and risk free reurns and compare hem wih acual reurns in Sweden. Assumpions The represenaive consumer maximizes E α c β (6.) α s 0 The firm produces a good ha has o be consumed immediaely. The counry is closed so by assuming a process for producion we also know how consumpion evolves. To simplify we assume ha he growh rae of consumpion follows a firs order Markov process, i.e., i only depends on las periods growh. We simplify furher by assuming ha growh only can ake wo values; high or low. We assume ha consumpion growh is ruled by c c % λ wih probabiliy φ, if λ λ K Kλ wih probabiliy - φ, if λ λ λ & Kλ wih probabiliy - φ, if λ λ K ' λ wih probabiliy φ, if λ λ (6.) So he probabiliy of a repeiion of las periods growh is I and of a change -I. Noe ha O is saionary, i.e., is uncondiional momens are consan over ime. Quesions a) Show ha he uncondiional (ergodic) probabiliies of O λ and λ are 0.5 and 0.5. Hin; he probabiliies can be found by solving he following sysem of equaions. π π! " φ φ # φ φ $! π " π # (6.3) " # $! $ * The approach follows Mehra and Presco, JME 985.
where he S's are he waned probabiliies. Try o undersand why hese equaions give you he uncondiional probabiliies. b) Parameerize by seing λ + µ δ λ + µ + δ (6.4) Show ha he uncondiional mean, variance and auocorrelaion of O are > + µδ,, φ C. Hin; Use he relaion Em Em λ λ π + Em λ λ π (6.5) where m is he momen you are looking for. c) Esimae he sample mean, variance and auocorrelaion of O from he Swedish daa on consumpion. Use he resuls calibrae he model, ha is o assign numbers o ; µδφ,, @. Now we can also compue he process for he price of he firm in he model. The price will generally depend on he curren level of consumpion and on expecaions abou fuure consumpion growh. Wih he simple Markov process for consumpion, expecaions only depend on las periods growh. Furhermore he CRRA uiliy funcion implies ha he price is homogeneous of degree in he curren level of consumpion. This means ha he price of he firm only depends on curren consumpion and on las periods growh, and ha i is linear in curren consumpion. Le P(c,O) denoe he price of he firm. We can hen wrie sandard pricing funcion for asses as Pc pc E U c 7 + i 4, λ 9 β 4pc+ + c λ λ + 9 U c 7! Pc p c E U c + 7 i 4, λ 9 β 4pc+ + c λ λ + 9 U c! 7 " # # $ " # # $ (6.9) p i can now be inerpreed as he price of he firm relaive o he curren level of consumpion, given ha he growh from las o curren period was O i. d) Divide boh sides of he pricing funcion wih c, subsiue o ge rid of c and c +. Derive a wo equaion sysem ha (implicily) defines p i in erms of {E, I,O,O,D}. e) If O O i and O + happens o be O j, he reurn on equiy from o + can be wrien
~r ij j i p c+ + c p c + (6.3) i pc Ge rid of consumpion in he las equaion and derive an expression for he uncondiional expeced reurn on equiy. Plug in he esimaed values of ; µδφ,, @ and se E 0.95 and D,5 and 0. Compue he uncondiional expeced reurn on equiy. f) Show ha he uncondiional risk less rae of reurn equals Er + β φ λ + φ λ β φ λ + φ λ (6.6) α α α α 4 9 6 4 9 4 9 6 4 9 g) Compue he models prediced equiy premium for he hree differen risk aversion coefficiens. h) Esimae average real reurn on equiy and on bonds from he Swedish financial daa. Wha is he average equiy premium in he sample? i) Compare he sample equiy premium wih he predicions of he model. Wha do you conclude?
Problem Se 7 A Simple Real Business Cycle Model * In his problem se we are going o consruc a small real business cycle model and use Swedish daa o calibrae i. Since i is very simple i may no perform perfec bu i will illusrae he main ideas and virues of his kinds of models. RBC models have some ypical feaures; They build on maximizing agens. A mainained hypohesis is ha real shocks drive he business cycle. They ry o model a propagaion mechanism such ha a simple shock, ypically echnological, can cause persisen business cycles. Parameer values are as far as possible aken from oher "usually reliable sources" (which in his case will be my pure guesses). They choose he res of he parameer values so ha he model replicaes some ineresing momens of aggregae macro variables, ypically relaive sandard deviaions, correlaions and auocorrelaions. Ofen he models are analyically unsolvable so he model momens are generaed by simulaion. Consrucing he Model Le here be represenaive consumer wih log uiliy who owns a firm wih CD echnology. A ime he problem for he consumer is hen max E s 0 s.. C + K Y Y s β ln C s+ 7 + α ZK (7.) Where Z is a echnological shock ha we will specify laer. Necessary condiions for an opimal soluion is ha where / is a shadow value. Λ C Λ α αβeλ Z K + + + α + + 6 (7.) C K Z K * The approach follows McCallum, 989, in Modern Business Cycle Theory.
a) Wrie down a verbal inerpreaion of each of he necessary condiions. Now we will ry o find one of he possibly infinie soluions and hope ha one saisfies he ransversaliy condiion. If so, he soluion also saisfies sufficien opimaliy condiions. Firs we sar by guessing ha consumpion is proporional o producion, wih proporionaliy 3 so ha b) Find K + as a funcion of 3, Z and K. c) Show ha if our guess is correc C α ΠZ K (7.3) Π αβ 6 (7.4) Hin. Use firs he opimaliy condiion o eliminae / in he second. Then use he guess and your answer o b). From now on we use lower case leers o indicae logs and we assume ha he log of he echnology shock is whie noise. We can now esablish ha k+ π k + α6k + z c π c + α6c + z y π + α6y + z y (7.5) where π k log Π6 π c log Π π ( α)logπ y k (7.6) d) Is he soluion saisfying he ransversaliy condiion ha j lim E β Λ K 0? (7.7) j + j + j+ e) Using (7.5) i is possible o show ha his very simple model gives us perfecly correlaed series for k, c and y. Can you hink of a way o break his?
Calibraion Take he following parameer as given from "usually reliable sources" E 0.95. We now have wo parameers lef; he variance of z, defined as σ z and D. These are going o be chosen so as o mach he variance and auocorrelaion of consumpion. a) Derend consumpion daa by regressing he log of c on a consan and a ime rend. Define he residuals as he business cycle componen. Calculae he variance and auocorrelaion of he consumpion business cycle, σ c. b) Use a compuer o simulae he model. Generae a series of normally disribued shocks of suiable lengh. Generae simulaed series for a few values of σ z in he range from σ c /5 o σ c and for D in he range 0. o 0.7. Include a leas he four combinaions of endpoins. Se he firs observaions on simulaed c o π c / α. Then choose he se of parameers ha gives you he bes mach beween simulaed and sample momens. This is he end of he calibraion. Simulaion Now we have all he necessary parameers so we can see if he model is able o generae simulaed series wih saisical properies similar o daa. a) Simulae he model. Plo he resul including he shocks. Also plo he business cycle of c. See if you can find some similariies and differences. c) Compue he auocorrelaion for,, and 3 lags in he sample and in he simulaed daa. d) Discuss he model's performance and wha you hink could be done o improve i.