Homework #3 - Answers. Not Due, Ever Just FYI

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1 Page 1 of 10 No Due, Ever Jus FYI 1. The equaions below represen he Open Economy Model of Mankiw s Chaper 5, which deermines he endogenous variables Y, C, I, NX, and ε in erms of he exogenous variables K, L, r*, G, and T. Producion funcion: Y = F( K, L) (1) Consumpion: C = C( Y T ) (2) Invesmen: I = I(r*) (3) Ne Expors: NX = NX (ε) (4) Goods marke equilibrium: Y = C + I + G + NX (5) Here he I( ) and NX( ) funcions are boh downward sloping, F( ) is increasing in boh K and L, and C( ) is increasing in disposable income wih 0<MPC<1. a. Use he equaions of he model and he known properies (slopes) of he funcions o deermine he effec of an increase in axes, T, on he real exchange rae, ε. (By use he equaions I mean oally differeniae hem and solve for dε in erms of d T.) Ans: Holding consan all exogenous variables excep T, nohing changes in equaions (1) and (3), so hese may be ignored, and dy=di=0. Differeniaing wha remains, we ge: dc = C dt dnx = NX dε 0 = dc + dnx Where C' and NX' are he slopes of he respecive funcions. This is solved as follows: d ε = ( 1/ NX ) dnx = (1/ NX )( dc) = (1/ NX ) C dt Since N X < 0 and C '> 0, d T > 0 dε < 0. Tha is, a rise in axes causes he real exchange rae o depreciae. (In case he above is oo abbreviaed, he full procedure for solving he model his way is as follows. Sar by differeniaing he enire sysem of equaions oally wih respec o all exogenous and endogenous variables: dy = F dk F dl) K + L

2 Page 2 of 10 dc = C dy C dt ) di = I dr * dnx = NX dε dy = dc + di + dg + dnx Then solve his sysem of linear equaions reaing he differenials (dy, d K, ec.) as variables. Se he differenials of hose exogenous variables ha are no changing o zero (in his case everyhing excep d T ), and solve for he changes in whaever endogenous variables you are ineresed in (in his case dε).) b. Use a represenaion of he model in erms of a diagram or diagrams o work ou he effec of a major new invesmen opporuniy (which causes a higher level of invesmen for every value of he ineres rae) on ne expors. Assume ha before his change he economy is running a rade defici. (In his case, explain in words why a paricular curve or curves shifs, and hen use he diagrams o demonsrae how equilibrium will change as a resul.) Ans: The new invesmen opporuniy shifs he I(r) curve shown a he righ o he righ. Since he ineres rae is unchanged a r*, invesmen increases. Savings does no change, however, since i is fixed by he fixed levels of income and axes. So ne expors, which mus equal S I, r I(r) I(r)' S decrease. Since we sar wih a rade defici, and r* hus I>S, he rade defici becomes larger, as shown. c. Wrie a paragraph, using words only (no NX NX' equaions, graphs, or symbols), o explain o a policymaker of a counry ha is running a rade defici how a rise in he foreign price level will affec he size of his defici. I,S Ans: A rise in he foreign price level, if he exchange rae does no change, makes foreign goods relaively more cosly compared o domesically produced goods. This would simulae demand for expors, reduce demand for impors, and reduce he rade defici, if only he exchange rae would remain unchanged. Bu i won. Before his change, he rade defici was mached by a capial inflow, reflecing he fac ha your counry was spending more han is income and had o rely on capial from abroad. The rise in he foreign price level does nohing, in he long run, o reduce your counry s spending or o increase is income, so he capial inflow mus coninue unchanged. So if he rade defici were o decline, his would mean an excess demand for he counry s currency, which would herefore rise in value. In fac, for he exchange marke o remain in equilibrium, he currency mus appreciae by he same amoun ha he foreign price level has gone up, so as o leave relaive prices of domesic and foreign goods unchanged.

3 Page 3 of 10 Only hen will he ne demand for foreign currency o buy foreign goods equal he ne supply of foreign currency ha is accompanying he capial inflow. 2. In homework #2 quesion 4, you used he closed economy long-run model of Mankiw s Chaper 3 o find he effecs of he model s exogenous variables on is endogenous variables. Which of he answers ha you gave for he closed economy would be changed in he open economy model? Ans: The answers for he closed economy were as follows: Effecs on: Due o: Y W r C I K L G T The answers for Y and W are unchanged, since hese depend solely on he producion funcion. The answers for C are also unchanged, for essenially he same reason. The answers for r and I are all now changed: hey are all zero, since r=r* and changes only for a change in r*, and since I depends only on r. 3. The following is a version of Mankiw s unemploymen model, expanded slighly o make explici how U and E change over ime. Labor force: L = E + U (1) Unemploymen dynamics: U & = du / d = se fu ( = E& ) (2) Combine hese equaions o express U & in erms only of U and exogenous variables. Then graph his funcion, on he axes below. From his graph, explain wheher, or under wha circumsances, he unemploymen rae will converge over ime o he equilibrium rae described by Mankiw. Ans: Subsiuing E=L U from (1) ino (2), we ge U & = sl ( s + f ) U which is graphed here. Since U & > 0 o he lef of poin A and U & < 0 o he righ of poin A, U will change over ime as shown by he arrows, converging on poin A a which he unemploymen rae is u=u/l=s/(s+f). U & sl s s + A f L U

4 Page 4 of Using Mankiw s framework for explaining seady-sae unemploymen in erms of raes of job separaion, s, and job finding, f, indicae how you would expec ha one, boh, or neiher of hese raes would be affeced by he following, and also how as a resul he seady-sae unemploymen rae, u, would change. Explain. a. A reducion in family sizes makes i more difficul for he unemployed o be suppored by family members. Ans: The unemployed will be less relucan o ake undesirable jobs, knowing ha hey have less suppor a home, and hey will herefore become re-employed more quickly. If his is all ha happens, s does no change, f rises, and consequenly u falls. I is also possible ha boh employers and employees will be more relucan o iniiae separaions, so ha s falls, reinforcing he fall in u. b. Lawsuis for wrongful dismissal make firms relucan o fire workers. Ans: Firms will keep workers longer, which lowers he rae of separaion, bu hey may also become more selecive in hiring workers in order o be more sure ha hey won wan o fire hem laer. So f may fall oo. If so, hen he effec on unemploymen is ambiguous. c. The inerne makes i easier for workers o find and apply for jobs. Ans: This jus speeds up he process and herefore increases f, reducing u. d. A one-ime wave of corporae mergers causes an unusual number of workers o be laid off. Ans: This will lay off a lo of workers emporarily, bu i should no aler he rae a which workers coninue o be separaed from jobs, or unemployed workers o find hem. Thus here is no change in s or f, and herefore no change in he longrun equilibrium rae of unemploymen. (Of course, in he shor run unemploymen presumably rises, bu he model here is abou he long run.) 5. The following is a version of he Solow Growh Model in discree ime wih an explici Cobb-Douglas funcional form for he producion funcion. For simpliciy, he populaion of his counry does no grow Y = K L Producion funcion L =100 Consan populaion K+ 1 K = I 0. 05K Consan rae of depreciaion, 0.05 I = S = s Y Consan rae of savings ou of income, s a. Saring wih a consan savings rae, s=0.1, show ha he long-run levels of income and he capial sock in his economy are Y=200 and K=400. (In his case,

5 Page 5 of 10 because he populaion does no grow, he seady sae involves a saionary capial sock and income.) Wha is he corresponding level of consumpion, C=Y- S, in his long-run equilibrium? Ans: In he seady sae wih K +1 =K, we mus have I=0.05K. Subsiuing for I=sY=s(K )=10sK 0.5 we ge 10sK 0.5 = 0.05K, or 200sK 0.5 =K. Squaring his and canceling a K, his simplifies o Thus K = 40,000s 2 = 40,000(0.01) = 400 Y = = (20)(10) = 200 C = (1 0.1)(200) = 180 b. Saring now a ime 0 in he long-run equilibrium from par (a), suppose ha he economy coninues in his equilibrium for 10 periods, bu hen, saring in period 11 he savings rae rises from 0.1 o 0.2 and says here forever. Calculae he levels of Y, K, and C, for =11,12,,200. Graph each of hese variables over ime (saring a ime 0 so ha you can beer see he change). (You ll find i easies o use a spreadshee for hese calculaions and graphs.) The complee resuls are oo big o include here, bu he firs 20 periods afer he increase in he savings rae are shown in he able below, followed by he graph for all 200 periods. L K Y S C

6 Page 6 of K Y C Period c. Wha are he long-run equilibrium levels of Y, K, and C for he economy wih his new higher savings rae? Does i appear from your soluion in par (b) ha hese variables are approaching heir new long-run values over ime? Ans: Using s=0.2 in he resul derived in par (a), K = 40,000(0.2) 2 = 40,000(0.04) = Y = = (40)(10) = 400. C = (1-0.2)(400) = 320. Yes, he numbers in (b) did approach hese values. d. From your soluions in pars (a), (b), and (c), how many periods does i ake for each of he variables Y, K, and C o move half way from he firs long-run equilibrium o he second? How many periods does i ake for consumpion o surpass is level of he old long-run equilibrium? Ans: Seady sae Y changes from 200 o 400, and i ges half way here, o 300, afer 29 periods (in period 39, since he process sared in period 11). Seady sae K changes from 400 o 1600, and reaches is half-way poin, 1000, afer 36 periods. Consumpion changes from 180 o 320, reaching half way (250) afer 34 periods. Consumpion falls a firs, from 180 o 160, because of he increased saving. I rises back o acually in period 7, and from hen on rises even higher.

7 Page 7 of The following are he equaions of he Solow Growh Model wih echnical progress: Y = F( K, EL) Producion funcion (consan reurns o scale) L & = dl / d = nl Consan rae of populaion growh, n E & = de / d = ge Consan rae of echnical progress, g K = dk / d = I δk Consan rae of depreciaion, δ I = S = sy Consan rae of savings ou of income, s Here, E represens efficiency unis of labor per worker, and one can hink of he model as deermining oupu, wages, ec. per efficiency uni. Then, even if for example he wage per efficiency uni is consan over ime, since workers are acquiring more efficiency unis a he growh rae g, heir wage per worker will be growing. a. Le k=k/el be he capial-labor raio wih labor in efficiency unis, and use he model o derive is equaion of moion: k & = sf ( k) ( n + δ + g) k where f ( k) = F( k,1) = Y / EL is oupu per efficiency uni. Ans: Differeniaing k we ge k& = k K& K E& L& sy δk = E L K g n = sy / EL δ g n = K / EL sf ( k) ( n + δ + g) k Muliplying hrough by k yields he resul. b. In a seady sae of his model, wha are he growh raes of he following variables? Y Y/L I C/L=(Y S)/L n+g g n+g g K/EL 0

8 Page 8 of 10 c. Saring from a seady sae in his model, suppose ha a some ime 0 he rae of echnical progress now suddenly and permanenly rises, from g 0 o g 1. Find he new seady sae in he firs diagram below, and hen, on he axes below ha, coninue drawing an approximae ime pah for per capia consumpion, C/L, over ime afer 0. (To make your life easier, I ve labeled he verical axis wih he logarihm of per capia consumpion, ln C/L, because when a variable grows a a consan rae, is logarihm follows a sraigh line.) f(k) (n+δ+g)k (n+δ+g 1 )k (n+δ+g 0 )k f(k) sf(k) ln C/L k 1 * k 0 * k g 1 g 0 0 Per capia consumpion grows a he consan rae g 0 before ime 0. A ime 0 echnology begins o grow a he faser rae g 1, bu his does no immediaely change anyhing. Over ime, however, he curren level of invesmen no longer keeps up wih he faser growing efficiency unis of labor, so ha k, and herefore Y/EL and C/EL, boh decline. They approach he new lower seady sae,

9 Page 9 of 10 however, a which oupu and herefore C grow a he faser rae n+g 1 >n+g 0. In effec, hey approach he lower bu seeper seady-sae growh pah ha hey would have already been on if he capial sock had somehow fallen insananeously a ime 0 o is new seady-sae level. d. Augmen he model now o include he Quaniy Equaion from Chaper 4 o deermine he price level. Assuming ha velociy of money is consan a V 0 and ha he cenral bank expands he money supply a he consan annual rae m, wrie an expression for he seady-sae rae of inflaion, π = P & / P, in erms of any or all of m, s, n, δ, and g. Ans: The quaniy equaion is MV=PY which, since V is consan, implies ha π = P & / P = ( M& / M ) ( Y& / Y ). From he growh model, Y grows a a rae equal o populaion plus echnology, or n+g. Thus π = m n g. e. How will an increase in he rae of echnical progress, wih oher parameers consan (as in par c) affec he rae of inflaion in par (d)? Do you hink his would acually happen? Ans: From par (d), a rise in g reduces he rae of inflaion. Because i increases he growh rae of GDP, i increases he need for money. Wih no change in m, prices canno rise as rapidly. In fac, however, i is almos cerain ha if real income were o rise more rapidly, he Fed would provide for ha need by increasing he rae of moneary expansion. 7. While he rae a which capial acually depreciaes depends primarily on echnology and is probably no sensiive o public policy, he rae a which firms claim ha i depreciaes is very much a maer of policy. Since i is usually in a firm s ineres o exaggerae he rae of depreciaion so as o inflae is coss for ax purposes and reduce is ax he governmen has rules regarding how rapidly firms can depreciae heir capial. Recognizing his, describe wha you would expec o be he effecs on he performance of a growing economy of increasing he governmen limi on how rapidly firms can depreciae heir capial. Tha is, in he broad conex of he Solow growh model (bu allowing, of course, for he presence of a governmen), describe wha you would expec o happen, afer such an increase, o he levels of savings and invesmen, and also o he growh rae of he economy, boh iniially and over ime. (I m no asking you o model his formally, bu only o apply he insighs ha you ve goen from our various long-run models, including he Solow model, o his siuaion.) Ans: Increasing he limi on claimed depreciaion will no change how rapidly capial acually depreciaes, so in spie of appearances, his policy will no aler he (n+δ)k line in he graph of he Solow model.

10 Page 10 of 10 Wha he increase does do is o permi firms o pay less in axes, hus increasing he funds ha eiher hey or heir sockholders have available for oher purposes. Therefore, one effec of his policy change is ha of a ax cu, increasing disposable income and herefore privae savings. Wha his does o naional savings, however, depends on how he governmen responds. If he ax cu is no accompanied by any change in governmen spending, hen public saving falls by he full amoun of he ax cu. Since he privae secor saves only a fracion of any increase in disposable income, while he governmen is dis-saving all of i, i follows ha naional savings goes down. In he Solow model, his means a shif downward in he sf(k) curve. (One could insead assume, perhaps, ha he governmen runs a balanced budge and herefore reduces spending by he full amoun of he los axes. In ha case, naional savings does rise, and all of he answers below would be reversed.) A second effec ha one migh expec would be o increase he real rae of reurn on physical capial, compared o oher asses, since physical capial is subjec o depreciaion and permis his sor of ax dodge, while oher asses (e.g. human capial) do no. Thus we migh expec on his accoun for invesmen o be increased. However, in he Solow model i is really savings, no invesmen, ha consrains growh, since invesmen is adjused auomaically (hrough he loanable funds marke and he ineres rae, presumably) o equal savings, and savings depends only on income, no on he ineres rae. So if he change in depreciaion rules were o increase desired invesmen relaive o savings, his would jus drive up he ineres rae o offse i, leaving invesmen ulimaely jus equal o savings. In he end, hen, i seems ha his policy effecs he economy primarily as a ax cu. I reduces naional savings, and herefore invesmen, because he governmen reduces is saving by more han he increase by he privae secor (or i increases boh savings and invesmen if he governmen runs a balanced budge). From here on, he Solow model ells us direcly wha o expec in he way of effecs on growh. This downward shif in he savings curve causes an immediae decrease in invesmen, which is hen below ha needed o mach (rue) depreciaion and populaion growh. The counry s capial sock per person herefore begins o shrink. Thus he policy causes a lower growh rae. However, over ime, he decumulaion of capial drives is marginal produc up, and he growh rae rises back up. In he end, as always in he Solow model, he dampening of he growh rae is only emporary, and he growh rae reurns o is prior value equal o populaion growh. Income per capia has gone down, however, and perhaps consumpion per capia if he economy had no already surpassed is golden rule level. (Noe ha he depreciaion rae referred o by he golden rule is he rue one, and i is no alered by his change in accouning rules.)