# Econ 4721 Money and Banking Problem Set 2 Answer Key

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1 Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in ech period, ccording to N t nn t, where n.2. In ech period, young consumers re endowed with y 60 nd old consumers re endowed with 0 units of the single consumption good. Ech member of the genertions born in period nd lter hve the following utility function: u (c,t, c 2,t+ ) log c,t + β log c 2,t+ () with β 0.5. Members of the initil old genertion only live for one period nd hve utility u (c 0, ) log c 0,. The government expnds the money supply by fctor of z ech period, M t zm t. Assume tht z.5. The money creted ech period is used to finnce lump-sum subsidy of t+ goods to ech old person. () Solve for the (sttionry) Preto efficient lloction. The nswer should be two numbers ( ) c P O, c P 2 O. (b) Write the government s budget constrint in period t. (c) Define competitive equilibrium with money for this economy. (d) Solve for the rte of return of money (+ / ) nd the growth rte of the price level (p t+ /p t ) in sttionry equilibrium. The nswer should be two numbers. (e) Solve for the consumption lloction (c, c 2) nd lump-sum subsidy in sttionry equilibrium. The nswer should be three numbers. (f) Verify tht gents prefer the Preto efficient lloction to the competitive equilibrium lloction with infltion. (g) Illustrte the Preto efficient lloction ( ) c P O, c P 2 O nd the competitive equilibrium lloction (c, c 2) on the (c, c 2 ) plne. Your grph should lso include the fesibility line, the lifetime budget constrint, nd their indifference curves. Answer () (5 points) To find the Preto efficient lloction, we set up the following problem, explicitly incorporting the fct its solution will be sttionry: mx log c + β log c 2 (2) c,c 2 s.t. N t c + N t c 2 N t y (3)

2 Using the fct tht N t n N t, nd getting rid of N t terms, we rewrite constrint s: c + n c 2 y (4) You cn solve this problem in multiple wys. I use the quick solution formul I ve discussed in clss t some point: c P O c P O 2 So your nswer for prt () is ( ) c P O, c P 2 O (40, 24). (b) (5 points) The government budget constrint is very simple: y 40 (5) + β β 24 (6) + β y n (M t M t ) t N t (7) where the left hnd side (LHS) is the seignorge revenue, nd the RHS is the totl vlue of lump-sum rebtes tht re given to the people. (c) (5 points) We define the monetry equilibrium for this economy s follows: Definition. A competitive equilibrium with money for this economy would be n lloction ( ) c,t, c 2,t+ for every person born t time t, c 0, for the initil old, prices of money v t for ll t nd government trnsfers t for ll t such tht:. For every t, every person born t t, tking v t, v t+ nd t+ s given, chooses ( c,t, c 2,t+) to solve: 2. Initil old choose c 0, to solve, given v nd : mx log c,t + β log c 2,t+ (8) c,t,c 2,t+,m t c,t + vt m t y s.t. (9) c 2,t+ vt+m t + t+ mx c 0, log c 0, s.t. c 0, v m 0 + (0) 3. The government hs to pick t so tht it obeys its budget constrint for ll t : 4. All mrkets cler for ll t : v t+ (M t+ M t ) t+n t () N t c,t + N t c 2,t N t y (2) N t m t M t (3) where the first mrket-clering condition bove refers to the consumption mrket, nd the second refers to the money mrket. 2

3 (d) (5 points) To find the return rte on money, we use the sme trick s before. Strt with the money mrket clering condition: N t m t M t (4) nd multiply both prts by : N t ( m t ) M t (5) Now use the budget constrint for the first period of person s life to replce the expression in prentheses: N t (y c,t ) M t (6) nd you rrive t: N t (y c,t ) M t (7) Since t ws rbitrry, we cn write the sme expression for t +. Also, we focus on sttionry equilibrium, so c,t c for ll t, nd hence: We know tht /p t, so: + N t+(y c ) M t+ N t+ M t n N t(y c ) N t M t+ z M t 0.8 (8) + p t+ p t p t+ p t p t p t+ (9) (20) (e) (5 points) To solve for the sttionry CE lloction, you proceed s follows. First, you obtin the lifetime budget constrint, which will look s follows (I drop ll time subscripts since we tlk bout sttionrity here): c + c 2 y + (2) + + nd then you solve the individul gent s mximiztion problem. No mtter how you do it, you will rrive t something like this: c c 2 ( y + + β β + β ( vt+ v ) t + ) y + nd by (8) you cn simplify things s: c 2 ( ) (24) c 2 ( ) (25) (22) (23) 3

4 To find, we use wht we lredy know: tht c c nd we lso know tht in equilibrium, the consumption mrket hs to cler: N t c + N t c 2 N t y c + n c 2 y, so we cn plug ll tht we know into the eqution bove: ( ) 6 60, from where 6. And hence c 45 nd c 2 8. (f) (5 points) Verifying tht PO lloction is better thn CE lloction is very esy. In PO lloction, guy gets 40 when young nd 24 when old. So his totl lifetime utility is: u P O log 40 + log (26) 2 In CE lloction, guy gets 45 when young nd 8 when old. Hence his totl lifetime utility is: u CE log log < u P O (27) Though this my seem like smll difference, we only cre bout the fct tht utility is smller t the PO lloction. (g) (5 points) I expect you to drw digrm similr to this one: 4

5 Problem 2 (30 points) This problem mimics Exercise 3.4 from the CF book. Consider the following modifiction of our overlpping genertions model with consumers living for 2 periods. As usul, individuls re endowed with y units of consumption good when young nd with nothing when old, nd the good is not storble. The government keeps the fit money stock constnt, i.e. z. The popultion in the economy grows t rte n >. In every period, the government imposes lump-sum tx ech young person for τ units of consumption good. The totl proceeds of the tx re then distributed eqully mong the old popultion in this period. We ssume tht the subsidy to the old is less thn they would be hppy to consume when old (so tht there re still resons for people to hold fit money). Do the following: () Write down the first nd second period budget constrints tht regulr person born in period t fces. (Hint: remember tht in every period there re more young people live thn old people.) Derive the lifetime budget constrint. (b) Derive the rte of return on fit money in sttionry monetry equilibrium. (c) Is the sttionry monetry equilibrium lloction Preto efficient? Discuss. (d) Does this government policy hve ny effect on consumers welfre? Explin. 5

6 (e) Does your nswer to prt (d) of this question chnge if we ssume tht the subsidy to the old exceeds the quntity they would prefer to consume? (f) Assume now tht tx collection is costly, so for every unit of consumption collected from the young, only 0.5 units end up being vilble for distribution to the old? Does your nswer to prt (d) chnge? Comment. Answer () (5 points) It is importnt to relize tht the totl tx revenue in this economy will be N t τ, nd tht these must be distributed mong N t old people. Remembering tht N t nn t, we get tht ech old guy will get trnsfer of nd hence the budget constrints re: The lifetime constrint N t τ N t nτ, c,t + m t y τ c 2,t+ + m t + nτ m t (c 2,t+ nτ) + c,t + c 2,t+ y τ + nτ + + (b) (5 points) Demnd for rel money blnces in sttionry equilibrium is so the money mrket clering condition implies nd hence + m t y c τ, N t (y c τ) M t We cn now plug this into the lifetime budget constrint: (c) (5 points) The fesibility condition for the economy is N t+(y c τ) M t+ N t+ n. N t(y c τ) N t M t c + n c 2 y τ + n nτ c + n c 2 y N t c + N t c 2 N t y c + n c 2 y nd since it coincides with the lifetime budget constrint, it is cler tht monetry CE lloction will be Preto efficient. Since the money stock does not grow over time, this should not be surprising.. 6

7 (d) (5 points) The txtion-subsidy policy hs no welfre effects, becuse the tx nd subsidy cncels out from the lifetime budget constrint. So if the policy is cnceled, nothing would hppen. The only exception would occur if the tx ws lrge enough. (e) (5 points) If the tx is sufficiently lrge, people cnnot choose the optiml bundle they would choose in the bsence of the tx. For low vlues of τ, n individul cn freely choose the optiml bundle (if trnsfers do not bring up consumer s second-period income to c 2, he cn use money to get extr consumption). However, for lrger vlue of τ the individul cnnot hold the level of rel blnces to chieve the optiml point, becuse the trnsfer policy gives more to people when old thn they would ever hve chosen to purchse for themselves using money. (If we llowed people to hold negtive mounts of money, this problem would go wy. However, this won t work in equilibrium ll consumers re identicl nd jointly they cnnot borrow from nyone else except themselves.) The individul must settle for lower vlue of c nd becuse of this, the high vlue for the tx dversely ffects individul welfre. In this cse, ll second period consumption is finnced by the government trnsfer. (f) (5 points) In this cse, the second period subsidy will be 0.5nτ nd the lifetime constrint will chnge to c,t + c 2,t+ y τ + nτ, + + so in sttionry equilibrium c + n c 2 y τ 2. Thus the budget set would lie strictly within the fesible set. The monetry equilibrium could not chieve the Preto efficient lloction. In this cse, the tx/trnsfer system (due to its inefficiency) would hve negtive impct on welfre. It would be better to eliminte the tx/trnsfer system. Individuls cn provide for their own second-period consumption through fit money holdings. Problem 3 (35 points) There re two countries, USA (lbeled ) nd Chin (lbeled b). Ech country is described by our stndrd overlpping genertions model with consumers tht live for two periods. As usul, individuls in US re endowed with y units of consumption good when young nd with nothing when old, nd the good is not storble. (For Chin, the corresponding quntities re y b nd zero, respectively). Popultion in US grows ccording to the following lw of motion: Nt n Nt. Similrly, for Chin we hve Nt b n b Nt. b Ech country hs its own currency, dollrs nd renminbi. The supply of dollrs follows the following eqution: Mt z Mt. And the supply of renminbi follows: Mt b z b Mt. b As usul, we re interested in sttionry monetry equilibri. Do the following: 7

8 () Suppose ech country implements foreign currency controls. Derive the rel rte of return on dollrs nd on renminbi. (b) Explin why the exchnge rte e t hs to be equl to the rtio of vt nd vt b. Suppose tht popultion in Chin grows fster thn in US nd tht both countries expnd their money supply t the sme rte. Will the US dollr pprecite or deprecite over time ginst the renminbi? Explin. (c) Suppose tht ll foreign currency controls were lifted. Demonstrte tht we will not be ble to determine the exchnge rte e t ny more. Discuss. For the rest of the problem, we ssume tht n n b nd tht N N b 00. We will lso ssume tht M M b 600 nd tht the initil money stock is divided eqully mong the initil old genertions in ech country (so z z b ). In ddition, we ssume tht every young person wnts to hold rel money blnces tht re worth 8 units of consumption (so y c y b c b 8). Finlly, the exchnge rte is fixed t ē 2, so dollr cn be exchnged for 2 renminbis. There re no foreign currency controls. (d) Find the vlue (mesured in goods) of dollr nd of renminbi. Wht is the consumption of n old person? (Hint: use the globl money mrket clering condition from prt (c)). (e) Suppose tht every initil old person in US nd in Chin decides to reduce his holdings of Chin s money by 2 renminbi. Every person turns 2 renminbi to the Chinese monetry uthorities wishing to exchnge it for dollr. Assume tht the US monetry uthorities re cooperting with their Chinese counterprts nd re willing to print s mny new dollrs s necessry. Wht will hppen to the totl money stock of dollrs nd renminbi? How will the vlues of ech currency chnge? (f) Now suppose tht the sme sitution hppens s in prt (e), but the US monetry uthorities refuse to cooperte. Insted, the Chinese government decides to honor its pledge for the fixed exchnge rte by txing every old Chinese citizen eqully. Wht will be the vlues of ech currency? How mny goods must ech old Chinese person be txed? How much does n old US citizen consume, nd how much does the old Chinese citizen consume? Who benefits from this policy? Answer () (5 points) With foreign currency controls, ech country hs its own money mrket clering condition. In US we get: which implies: v t+ v t vt N t (y c ) Mt, N t+ (y c ) M t+ N t (y c ) M t N t+ Mt Mt+ Nt n z. 8

9 Similr steps cn be pplied to Chin, nd we will obtin: v b t+ v b t nb z b. (b) (5 points) We must hve e t v t vt b, (28) for people to be willing to hold both currencies in equilibrium. Otherwise, if e t v b t > v t, everyone prefers to hold renminbi insted of dollrs, nd if e t v b t < v t, everyone prefers to hold dollrs insted of renminbi. The evolution of exchnge rte is given by: e t+ e t v t+ vt+ b vt vt b v t+ vt b vt+ b vt n z z b n b, nd if z z b nd n b > n, we hve e t+ e t n n b <, so US dollrs deprecite over time ginst renminbis. Becuse of currency controls, demnd for renminbi grows fster thn demnd for dollrs, nd so every dollr bill becomes reltively less vluble over time. (c) (5 points) Now there will be single worldwide money mrket clering condition: v t M t + v b t M b t N t (y c ) + N b t ( y b c b ), nd given eqution (28), we cn replce v t with e t v b t : e t vt b Mt + vt b Mt b Nt (y c ) + Nt b ( y b c b ) (29) v b t [ et Mt + Mt b ] N t (y c ) + Nt b ( y b c b ), nd we re left with single eqution with two unknowns, v b t nd e t, so there re infinitely mny solutions. The intuition is strightforwrd: in world like this dollrs nd renminbis re perfect substitutes, just like dollr bills printed in Cliforni nd dollr bills printed in Minnesot. So we cnnot determine both the vlues of individul currencies nd their exchnge rtes simultneously. (d) (5 points) Given ll the numbers, nd using eqution (29), we cn obtin: N t (y c ) + N b t ( y b c b ) 00 (8) + 00 (8) 3600, nd since M t M b t 600, we hve v t v b t nd since v t e t v b t, we hve 2v b t + v b t 6 v b t 2 9

10 nd then v t 2 (2) 4. Every old person consumes c 2 v t m t + v b t m b t 4 (3) + 2 (3) 8 units. (We know tht m t m b t 3 since the money supply is divided mong old people eqully.) (e) (5 points) This illustrtes coopertive stbiliztion of exchnge rtes. Since every old person surrenders 2 renminbi, nd there re 200 old people in totl, there re 400 renminbis surrendered. The exchnge rte is fixed t e 2, so people will demnd 200 dollrs in exchnge. Thus the new money supplies re: M t M b t , nd we cn use the sme world money mrket clering condition (29) to determine v t nd v b t : 2v b t (800) + v b t (200) 3600 v b t 2 nd hence v t 4 nd nothing chnges. So coopertive stbiliztion works s complete insurnce for people ginst fluctutions in currency mrkets. Every old person now consumes units, so their welfre is not ffected. c 2 v t m t + v b t m b t 4 (4) + 2 () 8 (f) (0 points) Updte: This prt contins n error. Do not grde it, insted, give everyone 0 points for it. Now we re in world where Chin unilterlly defends the fixed exchnge rte. As before, every old person surrenders 2 renminbi, nd there re 200 old people in totl, so there re 400 renminbis surrendered. The exchnge rte is fixed t e 2, so people will demnd 200 dollrs in exchnge. The Chinese government hs to rise v t (200) of txes in order to honor its pledge. Nturlly, Chin cn only tx its own citizens, so every old Chinese person hs to py tx of τ 200v t 00 2v t. The totl stock of renminbi flls to 200, s in the previous prt. The stock of dollrs is now unffected, however. So the sme condition (29) now implies nd since the exchnge rte is fixed t e t 2, we hve 2v b t (600) + v b t (200) 3600 v b t , v t 2v b t

11 Now the old in US nd Chin consume different mounts. US citizens py no tx, so their consumption is c 2 v t m t + v b t m b t 36 7 (4) () nd they re clerly better-off reltive to the sitution in which both countries cooperted. The vlue of their dollr holdings go up nd they py no tx for this to hppen. Chinese consumers, however, re worse off c b 2 v t m t + v b t m b t τ 36 7 (4) () 36 7 (2) nd you my notice tht consumption of Chinese old went down by precisely the sme mount s the increse of US old consumption.

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