(a) Assume that the entrepreneur is willing to undertake the project, and analyze the problem from the point of view of the outside investor.

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1 Problem Se # Soluions Course Macro IV TA: Todd Gormley, gormley@mi.edu Disribued: November 9, 004 Due: Tuesday, November 3, 004 [in class]. Financial Consrains (via Cosly Sae Verificaion) Consider an economy composed of enrepreneurs and ouside invesors. Boh ypes are risk neural and can always inves heir wealh in ouside capial markes and earn an expeced gross reurn. Each enrepreneur has wealh w, where w is disribued uniformly beween zero and wo among he enrepreneurs. Each enrepreneur also has he opion o underake a projec ha requires an indivisible invesmen of and has an i.i.d. reurn of x U[0, x ]. Ouside invesors have los of wealh bu no access o projecs. They are willing o lend money o enrepreneurs if heir expeced reurn from lending money is no exceeded by he reurn o heir ouside opion. However, ouside invesors canno verify he reurns from he projec unless hey pay a fixed cos c. Assume ha he conrac beween he invesor and enrepreneur akes he form of a deb conrac: he enrepreneur pays a reurn D o he ouside invesor whenever he can do so. When he canno afford o pay, he ouside invesor pays he verificaion cos c and akes all he profis. Tha is: If x D, he invesor ges D and he enrepreneur ges x D If x < D, he invesor ges x c (which may be negaive) and he enrepreneur ges 0 There is no bargaining in his model. Invesors are perfecly compeiive, so he enrepreneurs will never offer more han necessary o ge he financing ( w) ha hey need o do he projec. (a) Assume ha he enrepreneur is willing o underake he projec, and analyze he problem from he poin of view of he ouside invesor. i. Firs, find he invesor s expeced gain if she invess in he projec. Wha are he expeced verificaion coss of he invesor? Expeced eurn, (D) is given by: x D D x c ( D) = dx+ D x dx 0 x D D cd D ( ) = D + x x D cd D ( ) = D x Noice ha a higher deb paymen D has a number of effecs. A higher D increases he reurn o he invesor in good saes bu also reduces he

2 probabiliy of he enrepreneur being able o repay her deb. Thus, a higher D also increases he expeced verificaion coss, cd /x, of he invesor. ii. Graph his expeced reurn as a funcion of D and show graphically how he equilibrium value D will be chosen. ( D) is a parabola ha has x-inerceps a D = 0 and D = ( x c) as shown below. The equilibrium value D will be chosen such ha he expeced reurn of he invesor jus equals her opporuniy coss. i.e. ( D ) = ( w). As shown in he graph, his can occur a wo poins, bu we know ha compeiion among he invesors will ensure ha he lowes D will be chosen. Why is his rue? Well, we know ha a any D> D, anoher invesor will find i opimal o underbid he oher invesors by lowering he D he requess from he enrepreneur. This underbidding will coninue unil we hi D and invesors are jus indifferen beween lending and no lending. (D) ( w) D x D iii. Wha is he sign of he derivaive D / w? Inerpre I is easy o see from our graph ha he derivaive is negaive. Thus, he larger amoun of wealh posed by he enrepreneur, he smaller amoun he will have o pay back he invesor via D. iv. Under wha circumsances will here be no lending? There will be no lending in his model, if here doesn exis a D such ha ( D) = ( w). This will happen if ( w) > ( D) for all D. Three poenial causes could be:. The wealh of he borrower, w, is very low.. The ouside opporuniy cos of funds is very high. 3. Verificaion coss, c,are very high. 4. Or, he expeced reurn, x, is low.

3 (b) Taking D as given (as seen in par (a, ii), i is a funcion of he model parameers). i. Wrie down he condiion under which he enrepreneur is willing o underake he projec. Call his he enrepreneurs [I] consrain. Don subsiue D ou of your equaion. Taking D as given, he expeced reurn o he enrepreneur is: x D x D dx x x D x D x D + x D The enrepreneur will ake he projec if his expeced reurn exceeds his opporuniy cos of w. D x D + w ii. Now use he equilibrium condiion for D found in par (a,ii) o express he enrepreneur s [I] consrain in erms of x, he expeced reurn of he projec,, he ouside reurn, and he expeced cos of verificaion for he bank found in par (a,i). The equilibrium condiion for D found in par (a,ii) is: D cd D = ( w) x D cd D + =( w) x Plugging his ino he enrepreneurs [I] consrain, we have: cd x 0 x cd x + x Assuming ha he invesor is willing o lend, his or her expeced reurn mus exceed he ouside opporuniy cos and he expeced cos of monioring in order for he projec o be underaken. (c) Using your answer from par (b), show ha here are projecs implemened in an efficien economy ha are no implemened here. Which 3

4 enrepreneurs will no be able o sar a projec in he economy wih posiive verificaion coss? In he efficien economy, he projecs will always be underaken if heir expeced reurn exceeds he ouside opporuniy cos, i.e. x. In he economy wih verificaion coss, however, an enrepreneur will only inves if and only if: cd x + x Now, he expeced reurn of he projec mus exceed he ouside opporuniy cos by he amoun he enrepreneur expecs he ll have o pay he bank for is expeced verificaion coss, cd /x. From earlier, we know ha D is decreasing in wealh. Therefore, here will now exis a cuoff poin where individuals wih insufficien wealh will be unable o ake he projec now. [Noe: I ve implicily assumed x, c are such ha invesors are willing o lend for any given w. Technically, we also have o check ha he lenders acually wan o lend for all hese w.] If c = 0, we reurn o he resul of he efficien economy, and he wealh of he enrepreneurs will no maer for wheher a projec is underaken or no.. Amplificaion and Persisence (via Kiyoaki and Moore) Consider an economy wih wo ypes of agens: farmers and gaherers. There is a coninuum of each ype. There are also wo goods: an ordinary nondurable produc (frui) and a durable producive asse (land). The oal supply of land is equal o K. The farmer has consan reurns o scale echnology: he uses k unis of ime land o produce ak unis of ime + frui. The farmer is also is also subjec o he flow of funds consrain, which implies his invesmen expendiure is financed by his oupu and ne borrowing: q( k k) + b = ak+ b () where is one plus he real ineres rae, q + is he land price in erms of frui a ime +, and b is he value of deb underaken a ime. For simpliciy, you should assume ha each farmer is always eager o expand (due o heir grea enjoymen of farming), bu faces he following credi consrain: b q k () + (a) Combine equaions () and () o prove he following condiion: k = a q k q + + q ( ) b (3) This is sraighforward. The credi consrain will bind exacly by our assumpion ha he farmers always wan o borrow more. So plugging ino () using 4

5 q k b = + We hen have: + ( ) + = + q k k b ak q k Wih a lile rearranging, we ge equaion (3). i) Why can we inerpre µ = q q+ / as he amoun of down paymen necessary per uni of capial purchased? Farmers always wan o buy as much capial as possible by our assumpions. Hence hey will borrow he maximum amoun hey can o do so. q is he price per uni of capial hey mus pay, and q + / is he maximum amoun hey can borrow per uni of capial purchased (because i is he mos hey can promise o pay back omorrow). Thus, µ = q q+ / is he amoun lef over per uni of capial purchased ha he farmer mus pay for direcly. i.e. i is his/her down paymen. ii) How do we inerpre he expression inside he bracke? The expression inside he bracke is he farmer s ne worh. ( a+ q) k is he amoun of frui he farmer receives a ime. He receives a reurn a for each uni of land he has, k, and he receives a price q for selling he land a ime. (Implicily, he farmers all sell heir exising land in each period of ime before purchasing new land.) He mus hen pay back he b he borrowed las period. iii) Why is k / aposiive? This derivaive is posiive because an increase in a increases he farmer s ne worh, and he or she uses his increase o purchase more land. (b) Consider equaion (3), suppose q and q + increase by %. Explain how his changes he necessary down paymen and ne worh of he farmer and how each change impacs farmers land demand. Which effec is sronger when ak < b? This clearly increases he farmer s down paymen by % also. (.0) q (.0) q / =.0µ + The ne worh of he farmer increases since he now receives a higher price for he land he has. The increase in he necessary down paymen causes he farmer o purchase less land. The is easy o see in equaion (3). However, he increase in ne worh induces he farmer o purchase more land. The wo effecs work in opposie direcions. 5

6 Plugging ino equaion (3), we have: k = ( a (.0) q) k (.0) q + b + (.0) q qk k = + [ akb] > k q + q q + (.0) q when ak < b, he new amoun of land, k, is greaer. i.e. he increased demand because of higher ne worh exceeds he decrease in demand because of he higher necessary down paymen. Now consider he gaherers who use a decreasing reurns o scale echnology, such ha k unis of ime land o produce Gk ( ) unis of ime +frui. They do no face any borrowing consrain and will maximizes he expeced discouned consumpion of frui wih discoun facor / <. Land marke equilibrium implies k + k = K. (c) Use he land marke equilibrium condiion and he FOC of he gaherer s maximizaion problem o prove he following marke clearing condiion: q q G K k ( ) + = The gaherer s maximizaion problem is: ( ) max q k G k qk k + + The gaherer expecs o receive a reurn of G( k ) from his land nex period along wih he revenues of selling i, q+ k nex period. He discouns his fuure reurn by /, and subracs off he cos of he land, qk. Taking he FOC, we have: ( ) q+ G k q 0 + = earranging and plugging in for k = K k, we have our soluion. i) Wha is he sign of G/ k? Explain Since G > 0, i is easy o see ha an increase in he farmer s use of land (higher k ), reduces he use of land by gaherers and increases he marginal produc of heir land. i.e. G/ k >0 ii) Given his condiion and holding fuure prices consan, how will oday s land prices respond o an increase in? (No mah) k 6

7 q will rise because of he increased demand from farmers. This comes direcly ou of his condiion. (d) Now le s pu all he pieces ogeher and analyze he impac of a one-ime, emporary, upward shock o he produciviy of farmers, a, a ime. Jus give - senence explanaions for each par below. i) Describe he direc impac on land demanded by farmers a ime. The immediae impac is an increase in he farmer s ne worh. This allows farmers o increase heir demand for land. ii) How does he demand change affec he price of land and cause amplificaion? The increased demand by farmers causes he price o rise using our marke clearing condiion found in par (c). This can cause amplificaion because his increases farmer s ne worh by raising he value of his exising sock of land. This addiional collaeral allows him o borrow and purchase more land a ime. (Noe: I m assuming ha he posiive impac from he increase in ne worh exceeds he negaive impac from he higher down paymen ha is now necessary. In equilibrium his will acually be rue.) iii) Why does he shock persis and affec farmer s ne worh and demand for land omorrow (afer he shock is gone)? When farmers have more land oday, hey will produce and sell more land omorrow. This implies persisence because heir ne worh omorrow is also higher han i oherwise would have been, and hey will be able o borrow more and buy more land han if he shock had never happened. iv) Why do hese fuure impacs furher amplify he shock oday? Well, if demand for land is higher omorrow, our marke clearing condiion ensures ha he price of land omorrow is also higher. This relaxes he farmer s borrowing consrain oday furher amplifying he iniial impac of he shock! 3. Banks and Bank uns (via Diamond and Dybvig) Assume here is a coninuum of individuals ha are each endowed wih one uni of currency. There are hree ime periods, = 0,,. A = 0, individuals have wo opions wih regards o how hey can inves heir money. They can eiher suff i in heir maress, where i ges a reurn equal o, or hey can inves i in a long-erm projec ha yields a reurn = 4 in period wo. For example, in individual ha invess an amoun I will receive 4I in period wo, and have I suffed under he maress. However, individuals always have he opion of wihdrawing heir money from he long-erm projec early in period one a a penaly. If hey wihdraw early, hey only receive a reurn L = /4 in period, raher han he reurn = 4 in period. 7

8 A ime =, a fracion π = / of he individuals receive a liquidiy shock. These individuals are impaien and only value consumpion in period one. The fracion π individuals ha do no receive a liquidiy shock are paien and only value consumpion in period wo. A ime = 0, each individual has an equal chance of being hi by he liquidiy shock. Assume ha individuals do no discoun he fuure, so ha heir ex-ane expeced uiliy is given by, U = πu( c) + ( π ) u( c), where c and c is he consumpion period and respecively, and uc () = / c. (a) Assume here are no markes available o individuals, so ha individuals mus simply inves on heir own. Given ha he individual has invesed an amoun I a ime = 0, wha will be he opimal levels of consumpion, c,, if: c i) he individual receives a liquidiy shock (i.e. is impaien) c = I + LI = (3/ 4)I c = 0 ii) he individual does no receive a liquidiy shock (i.e. is paien) c = 0 c = I + I = + 3I (b) Wha is he opimal level of invesmen, I? Given I, wha is he ex-ane expeced uiliy of an individual? Explain in - senences why boh paien and impaien individuals regre heir iniial invesmen decision ex-pos in period afer heir ype is realized. The individual s maximizaion problem is given by: The FOC is hus, ( ) + π u( c) max πu c ( ) I max I ( ( ) I) ( + I) ( + 3I) = ( ( 3 4) I) 4 + 3I = 3 I I = /9 ( ( ) I) + ( + I) ( ) Thus, he ex-ane expeced uiliy is given by: U= - 6 /3 6 3 U= U=- 0 ( ( )) ( + ) = 0 8

9 Boh ypes of individuals are ex-pos unhappy. The paien individuals will have wished hey saved everyhing. This would give a payoff of 4 in period and a uiliy of /4 which is beer han his curren uiliy of 3/5. The impaien will have wished hey saved nohing, allowing hem o consume in period and ge a uiliy of which is beer han his curren uiliy of 6/5. (c) Now suppose an ex-pos financial marke exiss where individuals can rade bonds a ime =. Each bond coss p unis of goods a ime =, and he bond pays uni of goods a ime =. Assume all individuals inves an iniial amoun I = /. i) Wha is he aggregae demand and supply of bonds a =? I ( π ) if p < p Aggregae demand = ( π )( I),0 if p = 0 if p > Aggregae supply = [ π ] ii) Wha is he equilibrium price p? 0 if p< L 0, I if p = L π I if p > L From par (i), we see ha demand = supply when p = /4. iii) How much do impaien individuals consume in each period? c = I + pi = c = 0 iv) How much does a paien individual consume in each period? c = 0 I c = + I = p (d) When ex-pos financial markes exis, wha is he ex-ane expeced uiliy of individuals? Compare his wih par (b). Are individuals beer off? And, do individuals now have any regres abou heir iniial invesmen decision? The expeced ex-ane uiliy is given by: πu c ( ) + ( π) u( c) 5 () = 4 8 9

10 Comparing his o par (b), i is clear individuals are beer off now. Moreover, we see ha individuals do no have any ex-pos regres. (e) Now suppose ha when ypes are revealed in period, his informaion is publicly observable. Suppose here exiss a social planner ha individual s enrus all of heir endowmen o a ime 0. The social planner will pay impaien individuals c in period and paien individuals c in period. i) Solving he social planner s problem, wha is c and c? The social planner s problem is given by he following: ( ) ( ) ( ) max πu c + ( π) u c s.. πc + π c = c, c Using a Langrangian (wih muliplier λ ) he FOCs are as follows: c c = λ = λ Thus, we have, c = c c = c Plugging his back ino he budge consrain, we solve for he soluion: c = 4/3 c = 8/3 ii) How much does he social planner inves? (i.e. wha is I?) The social planner simply saves ( ) π c 8 = = /3 4 3 iii) Wha is an individual s ex-ane expeced uiliy now? ( ) ( π ) ( ) U = πu c + u c U = + = iv) Why is he social planner able o improve he individual s ex-ane uiliy relaive o ha found in par (d)? The social planner improves welfare because i is able o provide insurance o he individuals. I is always he case ha individuals wihou a liquidiy shock (i.e. paien) are beer off han hose ha realize a liquidiy shock a ime. The social planner jus provides insurance agains he shock. 0

11 (f) Now suppose an agen s ype is privae informaion, and he social planner can only offer a conrac coningen only an individual s announcemen of his or her ype a ime. (i.e. she canno condiion he conrac on oher agens announcemens). Furhermore, a ime, she mees each agen once wih he meeing order randomly deermined. If individual s repor honesly, can he social planner offer he same conrac as in par (e)? Is i opimal for an individual o repor honesly when everyone else does? Explain in - senences how his planner can be inerpreed as a bank. If individual s repor honesly, han he social planner can offer he same conrac. I is easy o see ha individuals will always wan o be hones. An impaien person would never repor as paien since ha would yield uiliy. A paien person would never repor as impaien since his would enail a drop in consumpion from 8/3 o 4/3. The inerpreaion of he planner as a bank wih a sequenial service consrain is sraighforward. The bank akes deposis a ime, and i promises o payou a small reurn a ime for hose ha need o wihdraw heir funds early. Those ha wihdraw a ime ge a larger reurn. I does his by keeping some funds on hand for impaien individuals and by invesing he res. (g) Suppose all agens fear a bank run, and each agen repors o he bank a ime as being impaien. How many individuals will ge paid by he bank before i runs ou of money in period? Given his, explain why his bank run can be an equilibrium i.e. why is i opimal for a paien individual o run on he bank when he/she expecs a bank run? If everyone repors, he bank mus conver he I = /3 i invesed in he illiquid long-erm projec a ime. Because of his, is reurn on hese funds will only be /. Combining his wih he I = / 3 ha he bank kep in liquid asses, i has a oal of 3/4 asses o pay ou in period. Dividing his by he promised firs period payoff of 4/3, we see ha only he firs 9/6 of individuals making claims in period ge paid before he bank runs ou of cash o make paymens. I is now clear why a bank run is an equilibrium. When a paien individual expecs a bank run, his expeced uiliy of waiing o wihdraw in period is negaive infiniy (zero consumpion) because he bank will be bankrup by hen. Therefore, he or she is beer off running on he bank also where he or she has a probabiliy 9/6 of geing a payou of 4/3 before he bank collapses. (h) Suppose he bank implemens a policy of only paying he firs π individuals ha show up a ime, and he res will ge paid a ime. (i.e. i suspends converibiliy). Will his eliminae he bank run as an equilibrium? Yes, his will eliminae he bank run equilibrium. A paien person will never have an incenive o run because he knows he bank will never collapse and hence he bank-run equilibrium is no longer susainable.