Problem Set 1 "Working with the Solow model"

Transcription

1 John Hassler, Sring 993 Problem Se "Working wih he Solow model" Le's define he following exogenous variables: s = savings rae δ = dereciaion rae of hysical caial n = oulaion growh rae g = rae of growh of labor augmening echnology. L = effecive labor suly = e (g+n) (Normalizing sandard labor suly o uniy) and he following endogenous: ~ k = caial sock er effecive labor uni (K/L). ~y = ouu er effecive labor uni (Y/L). k = caial sock er caia y = ouu er caia. a) f ~ k grows a rae g (i.e., growh rae of L. ~ k ~ = g ), wha is he growh rae of k and K. Wha is he k b) Provide some reasonable values for s, δ, n and g (consul Maddison and Romer, 989) c) The gross resource consrain for a closed economy is: FKL (, ) = C+ δ K+ K (.) Using Cobb-Douglas echnology we easily find ha: α α K = sk L δk (.) Make sure you undersand ha his is reasonable. Derive, se by se, he resource consrain in erms of k ~. (Hin: Divide by L, hen calculae he relaion beween k ~ and K K noing ha k ~ L =, subsiue and your done). d) Using your arameer values from c and a reasonable value for α, find he seady sae level of ~ k. n seady sae wha is he gross caial ouu raio K/Y?. The differenial equaion for k ~ is: ~ ~ () () α ~ k = sk ( g + n+ δ ) k (.3)

2 Problem se, John Hassler a) s his a sandard differenial equaion, ha you have seen in Mafu? Barro and Sala-i-Marin make a log-linear aroximaion o he growh rae of ~ k around i's seady sae (equaion.). They wrie: ~ k ~ = log( k ~ ) log( k ~ *) k λ3 8 (.4) b) Give an exlanaion in words for λ ha my moher (who is no an economis) would undersand. c) Calculae his derivaive from equaion (.3) a k ~ = k ~ *. Wha is is numerical value using your arameer values. d) Assume ha from an iniial seady sae, a devasaing war leads o a reducion in k ~ o one eighh of is seady sae value. Calculae he ouu ga (Y()/Y*()) immediaely afer his shock. Wha is now k ~?. Calculae he change in ineres rae (marginal roduciviy of caial) before and afer he war. How big is he ouu ga afer one year and afer 0 years? Do you hink his recovery is fas or slow?. Can you calculae he ime i akes unil half he ouu ga is gone? When is i oally gone? e) Comare a counry ha sar ou wih a caial sock k equal o he seady sae you calculaed above bu ha has half he savings rae you assumed. Calculae he seady sae level of ~ k for his counry and he iniial growh rae of ~ k and Y. The excellen rogram Mahemaica heled me find a soluion o he differenial equaion ~ ~ () () α ~ k = sk ( g + n+ δ ) k : ~ s () ( α)( g+ n+ δ) k = + ce g + n+ δ /( α) (.6) a) Calculae lim k ~ () b) Seing o 0 gives you k0 ~ () as a funcion of he inegraion consan c. Use his o subsiue for c in he exression for k ~ (). 4. Volunary bonus quesion. a) Comue k ~ from equaion (.6). Comare he value of ~ () k ~ wih he linear k () ~ ~() k * = aroximaion of Barro and Sala-i-Marin. s heir aroximaion reasonably accurae? k

3 John Hassler, Sring 993 Problem Se "The Ramsey Model for a Small Oen Economy" This roblem exends he sandard Ramsey model you saw in class. Before you do he roblem, comare his model o he Solow model in erms of model assumions. Do hey address he same quesions? When is which model more useful? Assumions We use he same noaion as in Problem Se. For simliciy we se g and n o 0. This means ha we don' have o worry abou he differences beween er caia, er effecive labor uni and aggregae unis. We are now looking a an oen economy. n order no o ge "bang-bang" adjusmens we inroduce an invesmen cos funcion so ha i becomes relaively more cosly o do heavy invesmens for a shor eriod han smaller invesmen for a long eriod. Firms Technology is given by he roducion funcion f ( k )= αk βk, (.) bu caial can no be insalled freely. nsead an invesmen cos i gi ( ) = γ, (.) has o be aid. All invesmen is financed ou of earnings and he reresenaive firm maximizes he resen value of dividends: r max e 4f ( k) i + g( i) 7 w 8d, < ia 0 0 (.3) % k K = i δk s.. & Kk = k ' 0 Since he counry is small and oen he ineres rae, r, is given and consan. Households Household own he firms and suly one uni of labor inelasically. The reresenaive household solves: Afer Gofries, Persson and Lundvik 989

4 John Hassler,Sring 993 ρ max e log( c ), 7d < c A s % a = w + d + ra c K K & a = a 0 K r lim e a 0 = K ' (.4) Quesions. Think of he roducion funcion. s i reasonable (for all values of k)? Solve he maximizaion roblem of he firm. (Hin; se u he curren value Hamilonian. Se u he hree oimaliy equaions you have learn. Two of hem involves ime differenials). Reresen he soluion as an invesmen funcion i=i(q) and wo differenial equaions, one for k and one for q, he curren value shadow rice of caial. Exlain in words wha q reresens. Solve for he seady saes of k and q, (k* and q*) and draw a hase diagram.. Les sudy he rae of convergence o he seady sae. Firs we solve he differenial equaion for k. Remember ha he soluion for k can be wrien as : λ k = c e λ + c e + k *, (.9) where he λ's are he eigenvalues of he coefficiens marix in he differenial equaion sysem for k and q. Find exressions for hese eigenvalues and find here signs. Wha does he signs imly for he value of one of he inegraion coefficiens c i. (Hin hink of wha haens as goes o infiniy). Solve for he oher inegraion coefficien. λ i r = ± γ δ + r6 + 8β Solve he roblem of he household. Sar by seing u γ he Hamilonian, now in resen 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. Wha haens if ρ r? s ha reasonable? Then solve for he level of c using he ransversaliy equaion in (.4) (Hin; resen value of consumion equals resen value of income lus saring wealh). Provided ha a seady sae k* exiss.

5 John Hassler,Sring 993. Suose ρ=r and ha he economy is in seady sae. Use he hase diagram o see wha haens dynamically if we ge a ermanen increase in roduciviy (α increases). Wha haens o consumion and he curren accoun over ime. (Hin; use he following version of he iner emoral budge consrain: 0 c = w + d + ra c = f ( k) i( + g( i) 6 + ra 0 (.3) where suerscri denoes ermanen values, defined by: r x r 4x 9e d = x 7e d (.4) r 0 0 Subsiue (.3) ino he equaion for he curren accoun (.5) and noe ha a=-b, (he household canno sell is ownershi of he firm and he value of he firm is no included in a). b = f ( k ) c i ( + g( i )) rb (.5) Using his for =0 we can hen wrie he useful equaion: b = f ( k ) f ( k) i ( + g( i ) i( + g( i) ( c c ) (.6) nuiively he ermanen values are defined so ha if x is ke a x for all. The resen discouned value is he same as for he acual ah of x. 5. Wha haens if he roduciviy shock is i ) emorary (haens a =0 and ends a =s) and; ii ) aniciaed ermanen (he informaion abou he shock arrives a =0 and he acual shock haens a =s). (Hin. Draw he hase diagrams and noe ha k can newer jum (why?) and ha q can jum only when new informaion (like informaion abou a fuure roduciviy shock) arrives. n he case of he aniciaed shock noe ha a he ime of he shock he hase diagram shifs and ha a exacly he ime of he shif we have o be a he sable saddle ah for he sysem no o exlode. So when he informaion arrives a =0, he "old" hase diagram coninues o hold. Now q jums so ha {q,k] his he new sable ah a exacly =s).) 3

6 John Hassler, Sring 993 Problem Se 3 Endogenous Growh These wo roblems come from he 989 mid-erm exam in he advanced macro course a MT (Don' le ha discourage you, hey are easier han wha you have done so far.) Problem Learning by Doing Consider a closed economy wih a fixed number of eole - normalized o one - wih he following references. ρ U = log( c ) + log( u ) 7e d 0 0 (3.) Where c is consumion and u is leisure. The ouu canno be sored and he roducion funcion is given by: y = H l (3.) H is he level of human caial and as can be seen also equal o he roduciviy level and l is he labor inu. Markes are comeiive and each firm akes he level of roduciviy given. There is learning by doing, however, so H evolves over ime according o: H =υy (3.3) The consumer has one uni of ime o divide beween work and leisure each oin in ime. Quesions. Before doing any number crunching hink abou he following. n his economy, wha will haen o he fuure consumion ossibiliy se if eole decide o consume more oday? Assume ha he individual consumer is o small o consider his. Wha do you hink of he secificaion of learning by doing, does i make sense?

7 John Hassler, Sring 993. Se u he maximizaion roblem of he consumer and solve i. (Hin: Normalize he wage o one hen find he income of he consumer and he rice of he consumion good. The economy is closed and all households are equal. This imlies ha if here is a caial marke he ineres rae will have o adjus so ha no one will borrow. We can hus as well assume ha here is no caial marke. Then you can wrie a budge consrain for he consumer in each ime oin. Wih he assumion under., is he roblem saic or dynamic?). 3. Now solve for he equilibrium growh rae of he economy. 4. Now le's find he oimal growh rae of he economy. Firs, do you hink i should be higher or lower han he marke soluion? To find he soluion, follow hese ses:. Le here be a benevolen cenral lanner who maximizes U 0 bu akes ino accoun he learning by doing effec.. Wrie down he cenral lanning roblem, subsiuing for c so he only consrain is he learning by doing consrain and he choice variable is l. 3. Wrie he Hamilonian and he oimaliy condiions. Guess ha in an oimal seady sae, he labor suly is consan and hen differeniae he firs oimaliy condiion (H l ) w.r.. ime. Wha is hen he relaionshi beween he growh rae of he shadow value of he consrain and he growh rae of H? Eliminae he growh rae of he shadow value and he shadow value from he second oimaliy condiion (H A ) and you ge an exression for he growh rae of H involving υ, l and ρ 4. There is anoher exression for he growh rae of H coming from he learning by doing consrain. Use his o eliminae l from he growh rae of H from he oher exression. 5. Now you can solve for he oimal growh rae of H Lasly, noe he relaion beween he growh rae of c and he growh rae of H and you're done. (3.3)Problem. Educaion and growh 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 has one uni of labor and has a human caial level of h i. Each agen decides ha she allocaes a i share a o work and (-a i ) o sudy. Since all individual are alike we can

8 John Hassler, Sring 993 dro he i suerscri where we don' need i. This leads o he individual increasing her sock of human caial according o: h = β( a) h (3.0) The roducion of a single individual is hen given by: α α 7 7 y = ah AH (3.) Where A and H denoes aggregae levels of a and h. i H = hdi (3.) A = 0 0 i a di Noe he exernaliy here, he individual is more roducive if he oher individuals ogeher have more human caial. The roducion canno be sored, so he individual consumer-roducer consumes wha he roduces each momen in ime. ndividuals have logarihmic uiliy and a subjecive discoun rae of ρ. Quesions. s here decreasing or consan reurns o human caial 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. Assume a common value of a for all agens, (is his reasonable?), hen comue he growh rae of he economy as a funcion of a. 3. Derive he equilibrium growh rae of he economy by leing he agens choose a oimally. (Hin; Se u he maximizaion roblem of he consumer-roducer and subsiue for c in he uiliy funcion. Se u he Hamilonian wih is oimaliy condiion and he solve for he growh rae of human caial assuming a seady sae where a is consan. 3

9 John Hassler, Sring Derive he oimal growh rae of he economy. Can you exlain your resuls? 4

10 John Hassler, Sring 993 Aendix o P-Se 3 Regarding coninuous oulaions A coninuous oulaion is surely an absracion. f we index eole wih all raional numbers beween 0 and each have o be infiniely small, e.g., have zero human caial in order for aggregae human caial o be finie. Neverheless his may be a useful absracion. To ry o give some inuiion, look a a oulaion consising of N (where N is very big) individuals. To calculaed heir aggregae human caial we can divide he oulaion ino grous of equal size, indexed by i ={,...}. Then we figure ou he average level of human caial er caia in each grou, h i. For higher we ge a finer division of he oulaion and for each we choose we ge a sequence of h i for i ={,...}. have drawn wo figures of h i for = 5 and 5 below. Noe ha if we increase, h i will change for each i bu i will kee is aroximae order of magniude consan n he sense ha he average over all i remains consan when increase bu he variance increases. 5

11 John Hassler, Sring 993 The amoun of human caial in each grou is hen: N h i, (3.0) So o ge oal human caial we sum over he grous: H = i= N h i (3.) This is equal o he are of all he bars in he figures, noing ha he base of each bar is N/. We can also comue how much an increase in he average human caial level for one grou i affecs aggregae human caial. is going o be: dh dh i = N (3.) Thus we see ha: dh dh i = N (3.3) decreases when increases (he ariion becomes finer). Suose we increase very much, sooner or laer hen N=. Afer ha we canno go on in realiy since hen we have o sar sliing eole (we could in fac divide eole ino ars and le he ars share equally he human caial of he individual, in his case i doesn', however, make much sense o hink of increasing he human caial of a ar of an individual). On he oher hand, if N is large we can increase very much before we run ino his roblem and we may be forgiven if we forge ha consrain. f we he le go o infiniy we have: lim N dh lim dh i i i= = 0 N = lim = 0 i h N h di (3.4) n some cases, like he one in P-se 3, i is much easier o do calculaions wih going o infiniy. Then we may formally disregard he exernaliy when oimizing for he individual. For large bu finie his is only on aroximaion. Noa bene he connecion o small firms and erfec markes. 6

12 John Hassler, Sring 993 Problem Se 4 Endogenous Growh Problem Growh from R&D 4 Consider he following version of an R&D model, similar o wha you saw in class. consiss of he following ars. Labor R&D Secor Many small R&D labs Use labor o do research Makes a discovery wih some insananeous robabiliy When a discovery is made he firm ges a aen ha i sells unil some oher firm makes a new and beer discovery Paen nermediae goods secor Sold under monoolisic comeiion. Needs aen bough from R&D secor Final good secor Sold under erfec comeiion. Buys J inus X j nermediae goods The roducion in he final goods secor is done according o: Y J = A X α, (4.) j j= where we have normalized he number of erfecly comeing firms o one (sic!). A is he roduciviy level which will increase over ime due o he work of he R&Dsecor. The final goods secor buys inermediae goods X j from a fixed number J monoolisically comeing firms roducing according o: X j = L j (4.) n order o be able o roduce, he inermediae goods roducing firms mus buy a aen (he same for all firms) from an R&D secor. n he laer we have a lo of small 4 Adaed from Aghion and Howi, and Helman.

13 John Hassler, Sring 993 R&D labs who do research. When hey do research each firm makes a new discovery wih insananeous robabiliy λz, where λ is a arameer and Z is he amoun of researcher (labor) engaged. The wage in boh secors using labor is he same. When a new discovery is made he invening firm ges a aen ha i can sell o all inermediae goods roducing firms. hus has a monooly which lass unil some oher firm makes a new discovery. A new aen increases he roduciviy A o γa. The ime index will from here on denoe he number of new discoveries from he sar of he world. Thus; A A + = γ (4.3) Quesions. The final good roducing firm ays he marginal roduc value for he inermediae goods. Normalizing he final goods rice o one. Calculae he demand (rice funcion) he monoolisic inermediae goods roducers face. Solve he saic rofi maximizaion roblem of he inermediae goods firms for a given wage w. Exress your resul in a funcion given he rice of he inermediae good as a marku on he good. nver his funcion o ge w as a funcion of arameers and X j. Find a funcion for he rofi π as a funcion of arameers and X j. The R&D firm ha has he curren aen is able o ge all rofis in he inermediae goods secor. The value of a aen is hus he discouned resen value of hese rofis unil a new aens is found. Les find his using he following ses:. Simlify by normalizing J o uniy. Using r o denoe he discoun rae and π as he flow of rofis unil he new aen comes wha is he value of he aen if he sell lengh of a aen acually is T?. New aens comes wih Poison robabiliy λz. This means ha he robabiliy λzτ of a sell of lengh T (he ime unil a new aen is found) is λze Se u he formula for he sum (inegral) of he discouned values of he rofis for all ossible sell lenghs. 3. Show ha his can be wrien as: π. (4.6) r + λz

14 John Hassler, Sring Now we should look for an equilibrium. There is free enry o he R&D-secor so execed aggregae flows of rofi is zero in his secor. The flow of rofis is π + λz and he cos is w r + λ Z Z. Use he resource consrain ha Z+L =. Now you + have enough informaion o se u he zero rofi condiion as an equaion only arameers, X and X +. Se u his difference equaion. 5. Show ha he difference equaion has a seady sae a: * α6( r + λ) X = λ α) + γα6 (4.) Do comaraive saics (see wha haens o average growh when arameers are changed) and inerre wih inuiion. Problem Growh and Governmen Sending Here here is a inu G o he roducion funcion ha is only rovided by he governmen, like infra-srucure services. The roducion funcion of he economy is hus aken o be: α y = k g α. (4.5) Governmen sending on G is financed via a fla income ax so: g =τ y. (4.6) The reresenaive consumer maximize a CES uiliy funcion: σ c max ρ e d σ > 0 c σ 0 α α s. k= ( τ ) k g c (4.7) rt lim k e = 0 T T Quesions. Wha is he marginal roduc of caial faced by he consumer. 3

15 John Hassler, Sring 993. For a given and consan ax rae solve he consumer roblem and ge a growh rae of he economy γ. Wrie γ as a funcion of σ, τ, α and ρ. (4.5)(4.6)3. Find an exression for d γ dτ resuls. and ry o sign i. And ry o give some inuiion for your 4. Now le here be a cenral benevolen lanner. Assume ha we can find an oimal ah where g/k is consan (is his reasonable?). Show ha he roblem of he cenral lanner can be wrien as: σ c max e c, τ σ 0 s. k= ( ττ ) k c lim k e T T ρ rt d α/( α) = 0 σ > 0 (4.) Solve he roblem and see if he cenrally lanned economy grows faser or slower han he marke economy. Wha is he inuiive reason for your resul. (4.)5. Discuss he relevancy of he model. 4

16 John Hassler, Sring 993 Problem Se 5 The Overlaing Generaions Model Quesion. Consider an overlaing generaions model in discree ime, where eole work and earn a wage w in he firs eriod and are reired consuming heir savings s lus ineres in he second. A generaion born a ime consumes c when young and d + when old. The hus solve he roblem: max 4Uc ( ) + ( + θ) Ud ( ) + 9 c, d+ s.. c + s = w d = s ( + r ) + + (5.) There is roducion wih consan reurns o scale. Perfec markes ensures ha: r = f '( k) w = f ( k ) r k (5.) The savings in eriod consiues he caial for he nex generaion o work wih in eriod +. Since he oulaion grows a rae n we have: k n s + ( + ) = (5.3). Solve he roblem of he consumer for a general uiliy funcion. Exress your resul as a savings funcion only deending on k, k + and arameers. Give me an equaion ha imlicily defines a seady sae. Can we say anyhing abou he sabiliy of his seady sae?. Now assume log uiliy. Find a savings funcion. Show ha i does no deend on k + and exlain why. 3. Now assume Cobb-Douglas roducion so f ( k)= k α. (5.7) Use he savings funcion o ge a difference equaion which you solve for a seady sae and find a condiion for sabiliy.

17 John Hassler, Sring Now le's use his model o comare wo ension sysems, he fully funded and he ay as you go. Firs we ake he ay as you go sysem. Le he governmen ax he young a rae τ and immediaely ay he receis o he currenly old. Assuming log uiliy, erfec foresigh and Cobb-Douglas echnology solve for he seady sae caial sock k* as a funcion of τ. Wha is he sign of he derivaive of k* wih resec o τ a τ=0? 5. Do he same as under 4. bu now assume ha he ax is used o buy caial which is used as rivae caial in he roducion. The ax receis lus ineres is hen given back o he old generaion nex eriod. The old ge ensions equal o he ax hey aid lus ineres. Solve for he seady sae and he sign of he derivaive of k* wih resec o τ a τ=0. 6. Wihou doing any calculaions, comare he effecs of he wo ension sysems in a small oen economy so ha he ineres rae is given from abroad a r*. f '( k ) = r k = f ' ( r ) * * 7. Go back o he closed economy wihou ensions and axes. Now les sudy he effecs of a emorarily exra big generaion ("fyrioaliserna"). Using he difference equaion for k you found under 3., show he dynamic effecs on k, w and r if n is exra big a bu hen goes back o normal. Follow he economy a leas a few eriods afer. Bonus quesion. Analyze he welfare of he big generaion comared o is followers.

18 John Hassler, Sring 993 Aendix o P-Se 5 A Simle Examle wih Deah Probabiliy Here is an examle ha shows ha i is no he chance of deah bu raher he birh of new generaions ha breaks Ricardian Equivalence in an overlaing generaions model. Le's assume ha here is a robabiliy o die which is, ha r=θ=0, and ha here is a erfec caial marke so if one saves and don die one ges a reurn of / on he savings. The individual hen maximizes max U c c, c 8 s.. c w c = + U c τ τ The budge consrain for he governmen is hen G = τ + τ. n he second eriod here are eole around, so he axrecies are τ. Then a balanced ax shif saisfies τ + τ = 0 τ = τ Subsiue his ino he budge consrain of he individual and we ge c w c = w c = w c = τ + τ ( τ + τ ) τ + ( τ ) ( τ + τ ) τ τ So he he chance o ge away wih some ax aymens is exacly balanced by ha higher axes have o be aid if one survives..e., execed ax aymens are indeenden of he deah robabiliy. Why is he seady sae caial sock lower in a Pay-as-You-Go Sysem? n he answer o Problem se 5 we found, as in Blanchard and Fisher, ha he seady sae caial sock was lower in he Pay-as-you-go sysem. They inerre his as a 3

19 John Hassler, Sring 993 resul of he ransfer srucure of he sysem. There is, however, anoher inerreaion ha does no sress he srucural differences. 5 n he exerimen we looked a we inroduced he Pay-as-You-Go-Sysem by immediaely saring he ransfer scheme from young o old. This means ha he generaion ha is old when he scheme begins ges an unearned ension. This is he clue o he effec. n an acuarially fair sysem, he budge consrain of each generaion is unchanged and hus also heir consumion. The governmen doesn' consume so aggregae consumion and roducion is unchanged as well he seady sae caial sock. This shows ha i is no he Pay-as-You-Go srucure ha is he clue o he effecs bu raher he acuarial unfairness. n he exerimen in Problem Se 5 as well as in Blanchard and Fisher, he caial sock is lower wih he Pay-as-You-Go sysem of he following reason; he inroducion of he sysem means ha he currenly old ake resources from heir descendans. is his iner-generaional ransfer ha gives he resul no he Pay-as-You-Go srucure. 5 See Lindbeck, Assar (99), "Klarar vi ensionerna?", SNS, Sockholm. 4

20 John Hassler, Sring 993 Problem Se 6 Consumion under uncerainy Quesion. Consider an agen who lives for wo eriods and has he same uiliy funcion in boh eriods bu discouns uiliy in he second wih +θ. She faces a consan ineres rae r and her income in eriod one is y. n eriod wo her income ~ y is random wih mean y. She maximizes execed uiliy seen from eriod.. Define he maximizaion roblem of he consumer and show ha he firs order condiion is r U'( c ) = + U'( c ) +θ E 7 (6.) where E defines he execaions oeraor. Will equaion (6.) be saisfies also in a mulieriod model? (f you can, define a value funcion and use he Bellman equaion o rove he answer. Oherwise make an informed guess and rovide some heurisic argumens.). Now assume quadraic uiliy U = αc βc. (6.6) Find he FOCs. Solve for he consumion c of he consumer. How does he consumion deend on he degree of income risk? Exlain your resul and discuss he usefulness of his uiliy secificaion. 3. Using he quadraic uiliy funcion, le's do a Hall-ye consumion ime series regression. c = µ + φc + ε (6.8) Wha is he rue regression coefficien? Will OLS on (6.8) give you unbiased and consisen esimaes of? Wha haens if you include more RHS variables ha are known in eriod -, e.g., c -?

21 John Hassler, Sring Now le's look a some oher uiliy funcions. Firs le he individual have he following uiliy funcion. αc e Uc ()= α (6.) Find he degree of absolue risk aversion of his individual. Now, se for simliciy r and T o zero and le he second eriod income be disribued according o ~ % y + ε, wih robabiliy 0.5 y = & ' y ε, wih robabiliy 0.5 (6.) Solve for c. Define s(h) as recauionary savings, i.e. he difference beween he consumion wih no risk and wih risk. s( ε) c ε = 07 c. (6.3) s recauionary savings increasing in he variance? (Hin; look a s'(h).) s i increasing in income? 5. Now le Uc () = ln() c (6.5) Find he degree absolue risk aversion of his individual, how does i deend on income. Solve for c. s recauionary savings increasing in he variance? s i increasing in income?

22 John Hassler, Sring 993 Problem Se 7 Asse ricing and erm srucure in he Lucas (978) model Quesion. Consider a large amoun of idenical agens who live for ever on a closed island wih ale rees on i (no very aealing). Each individual owns one ree which every year gives y ales. y is sochasic bu he same for all individuals a given year and nohing can be done o affec he harves nor can ales be sored beween years. The rice of ales is normalized o. There is a erfec markes for risk free bonds of mauriies from o K eriods. They give one SEK a mauriy and k, is he rice of a bond in ime eriod ha maures a +k. The agens have a ime addiive uiliy funcion wih he consumion of ales each eriod, c, as he only argumen, ime is discree and heir ime reference is +T.. Define he maximizaion roblem of he consumer.. Use he following arbirage argumen o derive an Euler equaion for c and c +k : f he consumion ah of he individual is oimal she should neiher gain nor loose execed uiliy by aking dc of curren consumion, invesing i in he risk free bond wih a mauriy of k and hen using he roceeds o buy ales in +k. Formalize his argumen and use i o derive he Euler equaion; firs for k= and hen for any sricly osiive k. 3. Do you hink ha he reurn of holding a year bond beween and + and he reurn on holding a wo year bond one year and he selling i, will be he same and/or he same in execaion?.e., is, %, + K K, = & KE K ', +,? Why, or why no?

23 John Hassler, Sring Le he uiliy funcion in each eriod be given by α c Uc ( )= α (7.4) and assume he he harvess follow ln( y ) ln( y ), = ρ + ε (7.5) wih H i.i.d. N(0,V ) and -<U<. f all agens are he same, all agens ge he same harves in a given eriod and ales are non-sorable, wha is he consumion of each individual? Follow he difference equaion you hen can se u for c forward and show ha: ln( c ) = N ρ ln( c ), σ + + ρ =N ρ ln( c ), σ + k d k k k k ρ 4 9 (7.6) ρ Now remember he roeries of he log-normal disribuion, in aricular ha if ln(x) is N(m,V ), hen a 4 9 E x = e am a 6 + σ (7.7) Now find he execaion of he marginal uiliy a +k if we know c. (7.6)5. Use your answers o calculae,k. s anyone going o buy or issue hese bonds? Then find he yield o mauriy defined in a coninuously comounding way as: log( k, ). (7.0) k Make sure you undersand his way of defining yield o mauriy. Ofen we hear ha he rices of financial asses follow random walks, do hey here? Why, or why no? 6. For a one year bond, how does he yield deend on D, V and U. How does i deend on c, for osiive, negaive and zero values of U? Why?

24 John Hassler, Sring Assume ha U is osiive, use he exression for he yield o mauriy o draw a figure wih some aroximae yield srucures for normal, bad and good curren harvess. 3