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by: April Jerde

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Economic Theory ECON 712

April Jerde
UW
GPA 3.6

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This 56 page Class Notes was uploaded by April Jerde on Thursday September 17, 2015. The Class Notes belongs to ECON 712 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 28 views. For similar materials see /class/205154/econ-712-university-of-wisconsin-madison in Economcs at University of Wisconsin - Madison.

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Date Created: 09/17/15
Econ 712 Solow Growth Model TA Sang Yoon Tim Lee December 7 2006 Steady State Assume the aggregate production specification for the economy is Cobb Douglas Yt Kt AtLt1 0 lt x lt1 where Yt Yt is output Kt L is aggregate capital and labor and At is labor augmenting technology Hence AtLt is effective units of labor Assume L and At grow exogenously at rates 11 and g Lt Loent At Aoegt Solow assumes a constant fraction of output is saved or invested so that It sYt Recall the law of motion for capital from Rody39s class Kt At 1t 1 7 5Kt Kt At 7 Kt gm 7 am 50 taking At a 0 Kt SK 7 Q where Xt denotes the derivative of Xt X t wrt time Let kt Kt AtLt and yt Yt AtLt denote capital and output per effective unit of labor respectively Then yt k and kt KtAtLt Kt Kt Alt AtLt AtLt AtLt i SYt 7 HQ 7 AtLt ktn syting6kt sk fing6kt Hence in a steady state with It 0 5 1754 k ng6 So if we assume the economy is in a steady state output per capital is Atk Adsn g SPH t 1x 1x lnsi 171x 171x ln lnA0gt Mankiw Romer and Weil 1992 assume t 0 and A0 a e to obtain the empirically testable equation ngt Yt 1x 1x Inhj ia1imlnsilimng6e Speed of Convergence We can approximate the speed of convergence around the steady state as following The growth rate of yt 7 can be written as Blnyt i BlnfUQ i fkt at 7 at fkt t NH NJ Sfkt 11 g 5W Since yt f kt is strictly increasing we can take the inverse ie kt kyt Then the above equation changes to Blnyt 3t 7 Gm mwamgW At the steady state Gy 0 Hence taking a first order Taylor expansion around the steady state at cm 7 0mm 7 w 7 WWW 7 W 161 7 7501 g 6 y y where all the derivatives are evaluated at yt f By the inverse function theorem kyt fly andfor yt near f yt 7 yy m iff dlnf m lnyt 7 Inf so no 7 L 7L Lf 7 Gyt m y Sf 1k f y quotg51ny lny Now recall again from Rody39s class or micro that x in the production function corre sponds to the total share of output paid to capital owners ie 1x kt f kt yt Then we can further simplify the above to 31 m 175yungak717wng61nyrlny Alny7lnyt where A l 7 txn g 5 is called the quotspeed of convergence Solving this differential equation gives lnyt l7e Atlnye tlnyo lnyt7lny0 176Mlny7lny0 176 X lns7l7et lnng516Uy0 171x 171x which is another empirically testable equation Econ 712 Discussion Section 9122008 Laura A Dague 1 Basic ARMA Models To keep them straight just remember that AR stands for autoregressive7 so it looks like a regression7 of I on itself The MA stands for moving average7 and it is a weighted average of its eshocksi Models that use both are called ARMA models designated ARMAnumber of AR coef cients number of MA coef cients AR1 I 1111 e MA1 I 6 56271 ARUS I I 111171 121172 akItik 6 MAG 11 6 Jr letil 525t72 Jr zfzil ARMAUS 1 I I 111171 121172 akItik 6 616171 6167 2 ARMA Models in Lag Notation The lag operator L is de ned as Lkzt 117k Further we de ne the lag polynomial aLzt a0 a1L agL2 anLn t aozt alzt1 anhdl We can also write aL as 220 akLki You will often see this notation as we start working with in nite representationsi Refer to Sargent Chapter IX for more practice with lag operators and some details that we are skippingi In the general case we express ARMA models like this AR aLzt 6 MA I BLet ARMA aLzt BLet 21 Exercise You should be able to write the models above using the lag notationi 3 AR t0 MA An MA representation always exists Why However7 existence of an MA does not imply existence of an AR When is there a problem When we have an AR representation we can sometimes invert it to nd the fundamental MA When 31 Example Let7s begin with an AR2 process We want its MA representation In 041171 042172 6 or equivalently 17 a1L 7 a2L2zt e We can always nd complex numbers why A1 and A2 such that 17 a1L 7 ML2 717 AlL17 A2L By calculating directly and assuming we can we get I 17 AlL 117 AgL 1et ZMLWZAQ M j0 j0 22 Aih k ez 22 AiA kez7j j0 k0 j0 k0 Note you have to be careful when working with in nite series Above 1 used something called the Cauchy product Sargent gives you another way to work things out using a trick from algebra that you probably learned in calculus It goes like this 1 7 A1 2 17 AlL17 AQL A1 7 A217 AlL A1 7 A217 AgL implying 1 V 2 V A 7 7 J 739 It A1 A2 j 16 J A1 7 A2 FE 26 J m 1 j 2 j i 8 7 A2 1 A1 7 A2 A25t71 j0 4 Recall your Geometric Series l7m sure you already know this7 but just to refresh Will be useful all year n 61 6k 7 6n1 ik 176 Econ 712 Macroeconomic Theory First Exam University of Wisconsin Instructor Rody Manuelli October 29 2005 Instructions Please answer all questions If you get stuck in one section move to the next one Do not waste time on questions that you nd hard to solve Partial credit will be awarded if it is clear that you were approaching the prob lem in an essentially correct manner This is a closed book exam Students may bring one page both sides or two pages single sides of notes Please hand in the exam promptly at 12 Noon Each question is worth 50 points The point total for each section is indicated at the beginning of the section Look at these prices when deciding how to allocate your time If you believe that a question is wrong or poorly worded please make the minimal necessary changes to make it beautiful and well posed Of course unnecessary changes will result in a lower grade Please use one blue book for each question and write only on the right page The odd numbered page in a newspaper Please remember to put your name in each blue book Try to nish the required questions before you try the extra credit sections These are typically hard and carry no objective point value Good luck l 2 Q uest ions Problem 1 Endogenous Fertility Consider an economy populated by a large number of identical dynasties with utility functions given by U Zoning 0 lt 5 lt1 t0 where uc 01 9170 0 lt 0 lt 1 In this speci cation 0 is per capita consumption at time t and Ni is the number of members of generation t If each individual has 1 nt children the size of a dynasty satis es Nt1 S Nt1 Having children takes time Assume that each member of each generation is endowed with e units of time and that rearing 1n children requires b1n units of time Thus if a member of generation t has 1nt children the amount of labor effectively supplied to the market is t 6 7 We assume that this number cannot be negative We also rule out the possibility of a shrinking population Thus we consider the set of feasible ngs to be the interval 0 eb 7 1 Consider the planner s problem maXZB NMcz 20 subject to NtCt Nt1kt1 lt ZFkt7ZtNt 1 6ktNt7 0 lt h0N0 given where F is a standard ie twice differentiable increasing concave and homogeneous of degree one production function and the depreciation factor satis es 0 lt 6 lt 1 The planner mamimizes utility choosing sequences nt ht1Nt1f0 Z 15 points Describe the rst order condition of the planner s problem 2 15 points Assume that a steady state emists Consider economies that vary in the cost of rearing children ie b What does the model say about the impact of differences in b upon the capital labor ratio the wage rate and the interest rate in the steady state 5 10 points Co as far as you can providing conditions on eb that are nece essary for the steady state to exist 4 10 points Go as far as you can describing under what conditions a steady state emists If necessary make additional assumptions on the functions inuolued 5 Emtm Credit Go as far as you can showing under what conditions the steady state is unique What does this theory say about the relationship between pro7 ductiuity Z and population growth n Solution 2 Endogenous Fertility 1 Let the Lagrangian corresponding to the planner s problem be i tNtuct AtzFht e 7 b1 ntNt 17 6htNt NtCt Nt1kt1 MtNt1 t Nt1 If we ignore the non7negatiuity constraints the rst order conditions are C u cz At km 1 A 5At111 5 ZFkHt1717 nt Mt AZF2H1 Nt1 i M Atkt1 5U0t1 At11ZFHt171kt1 1 WW1 CH1 Mz11 WHO where Ht E litt is the capital labor ratio Let 6 1 p 1 2 At the steady state if one exists per capita quantities are constant and pop7 ulation is growing at a constant rate It follows from 1a and 1b that p6 2Fin1 2 Under standard conditions on F this equation is satis ed by a unique capital7 labor ratio M This ratio is independent of the number of children per house7 hold It follows that the wage rate de ned as w FH1 the interest rate r 2Fkn 1 7 6 are also independent of b 5 A necessary condition for emistence of a solution and hence for emistence of a steady state is that the effective discount factor tNt be less than one At the steady state Nt1 Thus tNt lt1 if and only if 61 n lt 1 Since 1 n lt eb a sufficient condition for emistence is 1 blt 4 Using 1c in 1d and imposing that the Lagrange multipliers be constant we obtain ZFgk1k 62FH7116k 70 ZFik11 Since feasibility implies and uc 0 u c 1 7 07 it follows that the equilibrium n solues 17 61nzFiH71 7 1 f yam 1 7 6 0 Check the algebra It looks like if 6 17 mzmm lt 1 0 2137037 1 i 5 5 1H6 b there is one steady state with positiue n Problem 3 Pricing Forward Contracts and Paintings Consider an economy populated by a large number of identical households with utility function giuen by EdZB Mcmaa 0 lt 6 lt1 t0 where 0 is consumption of a nondurable good and at is consumption of art The function u is twice differentiable increasing and strictly concaue if necessary you may assume that it satis es the Inada condition In this economy equilibrium per capita consumption is giuen by the emogenous stochastic process If it helps you may assume that there are a number of trees per household each dropping diuidends dit and that 0 dit Assume that the per capita supply of art is a Z Z5points Let aforward contract with strike price K be a contract that requires its holder to buy one unit of a particular asset newt period we only consider one period forward contracts at price K Consider the forward contract that requires its holder to buy one one period bond newt period at price K It is claimed that the price of such contract today qf is giuen by qiu 3231 K3131 where R131 is the price at time t of a jeperiod bond It is also claimed that the price of a forward contract to purchase one period bonds at price K J periods in the future is QKJ Rdilt 7 KB Discuss these claims 2 15 points Consider a tree that drops fruit according to the stochastic process dt It is claimed that the price of a Jiperiod forward contract that will deliuer a share of that tree J periods into the future at price K is 500 p 7 K33 where pt is the current price of a share 5 15 points It is claimed that if the utility function is separable ie if we assume that MC a M0 Ma for some nice functions ucc and uaa that art prices will be independent of consumption because the marginal utility of art is independent of the marginal utility of consumption and the supply or art is red 4 5 points Assume that at time t it is learned that at time t J the per capita supply of art will increase to 1 ya y gt 0 Go as far as you can analyzing the impact of such an announcement on art prices today Do not assume that the utility function is separable 5 Ezrtm Credit Go as far as you can analyzing the impact of the announcement described in section 4 upon short term interest rates Solution 4 Pricing Forward Contracts and Paintings The deriuation of the rst order conditions is standard 1 The deriuation of the rst order conditions is standard There are many ways of pricing these forward contracts One can use the price kernel ofArrow securities or directly treat them as one period shares with random return In either case it follows that u CH1 l 313 KN w 93 Since U Ctj 1E 6 tb uct the result follows A similar argument shows that 3 jt QHJ Rdilt 7 KB Using the standard pricing formulas the price of the forward contract qu satis es 39 UKCH 5 J E J M gt 6 A W Since the price of a share satis es uCtPt Etbuct1pt1b 5EtUCt1dt17 repeated substitution implies that 7 j UKCHJ 7 71 PH KN 5E4 uct Ptgl KB 139 1593 51Etu0tj17tjl Z kEtUCtkdtkl 191 Using this formula in the formula for the price of the forward contract implies j QSU Pt KR Z kEtUCtkdtkl 191 It follows that the claim is not true A standard deriuation treating the payoff of art uaua as its diuidend and pricing it as a regular stock or simply writing down the budget constraint shows that the price of art satis es ULCtPat Etbuhct1pat1i EthWtHN It follows by repeated substitution that pat If u aatj u aa7 this formula simpli es to 6 MW pat 7 15 which implies that in good times art prices increase art prices and con sumption are positiuely correlated Note that the model does not have clear implications about the relationship between art prices and interest rates 6 4 The general asset pricing formula with noneseparable preferences is DO 3u0t at aat a 1E 17 J J pt t auct7atact Ifu is strictly concaue then an increase in any future at will decrease 8uctj atj8atj and hence lower art prices Note that any changes in the stock of art that will take place at t J will leaue the process 8uctkatkBctkg unchanged It follows that see section I R will not change for h 17 2 J 7 1 Econ 712 Discussion Section 9182008 Laura A Dague 1 The Law of Iterated Projections1 or what it means for a sequence to be rational The LIP comes in handy quite often Add it to your bag of tricks Consider projecting the object pro y lX T which is itself a projection on the information set T i The LIP states pm pm y le Tl lTl pmily m 2 Why do all Epsilons have the same variance Formally we have assumed that the scalar stochastic process It is zeromean and second order stationary with nite autocovariance so forma y 1EIt 0 and 2covztztj 0j lt 00 Take note that this especially applies when j0 so that varzt DOES NOT depend on t It is constant over time Now 00 varzt vaTZ ajet v j0 agva et a va ekl aiva ekk We also know that 00 vaTzt1 vaTZajet1j j0 Subtracting them and imagine subtracting them for all lags and you can con vince yourself that we need all variances of epsilons to be the same Often you will see this type of analysis built out of a white noise process which has constant variance by de nition 3 Autocovariance function of MA or AR2 Don7t be confused by the distinctions between theory and estimation Theory tells us we can talk about Hilbert spaces and get a moving average represen 1Look at Sargent p 228229 for how to prove it 2Note I have been using some standard examples which are available in any time series book You might nd John H Cochrane7s lecture notes called Time Series for Macroeconomics and Finance Although the machinery we are using is slightly different you might nd this to be a useful reference for learning I did tation From the clata7 we get the autocovariance function by estimating co variances We can then get the coefficients of the MA representation from the autocovariance function using the following equation m2 JEWZMZ I 31 Example 1 MA1 I e 9671 Then 07 0 varzt 1 92 0751 Eztzt1 9062 0750 Eztztj 07Vj 2 2 32 Example 2 MA 2 I 6 916271 t926272 Then mo wan 1 6 05 0751 Eztzt1 911 92 0752 Eztzt2 92062 0750 Eztztj 07Vj 2 3 We can recursively guess that the autocovariance of MA00 is the following 00 002k Zg wg j0 33 Example 3 AR1 1 151171 6 The MA representation of this is I 1 L71 z ZWEL 39 j0 So7 using the MA00 we have 2 m 239 024516 Uxkae iqbg j0 4 Some Past HW Solutions It has not yet been revealed whether you will be provided with solutions I provide you with the following solution suggestions from the professor given to all students in 2007 41 HW1 Q1 Let y denote the N11 vector of observations on the dependent variable Suppose there are K regressors with associated NIK regressor matrix X Since X is full rank the columns of the matrix generate a Hilbert space of dimension N 7 K call this space G The subspace is a subset of RN By the Hilbert space decomposition theorem 3E such that RN H 69 EHJE For any y by the Hilbert space projection theorem Hg 6 H such that y g hgJE Let g X6 this is without loss of generality we are working with data all of which are bounded Since 9 E GX e Xy 7 X6 0 Since X is full rank X X is invertible therefore 6 X X 1Xy which is the OLS estimator 42 HW2 Q1 Suppose that I aLet e fundamental By the law of iterated projections Itlt7j Itlt7j7k 777jvrtlt7j7kint7j Since H2719 Ht7j67777j E E where the Hilbert space E is implicitly de ned by Ht e E 69 HELij Therefore m v is a linear combination of 611 171416 Thus m v is MAk l The coef cients on the 7s that build up 77 are the corresponding as of the MA representation 43 HW2 Q2 Let I pt1 at It is immediate that It tk pkzt To see this compute zqkl and apply the law of iterated projections Therefore It tk 7 It tk1 k 1 k pzikiptjkiy HOWEVER 11106 Izik prt7k71et7k7 Itlt7k 1tlt7k71 P 5719 44 HW2 Q3 The cases It e and It k are observationally equivalent ie a realization of the X process cannot distinguish them Any draw of 6 can be asserted to be the value of k The case I 6 has the property that it does not obey the law of large numbers ie it is an example of a nonergodic process 45 HW2 Q4 Since the two processes possess identical autocovariance functions they must possess identical Wold moving average representations This means va39rzt 7 It tk varyt 7 yt kk since the terms in the variance expressions are func tions of the respective Wold MA representations Econ 712 Spectral Analysis TA Sang Yoon Tim Lee December 8 2006 The rigorous math behind the following is quite involved and mentioned in the lecture notes So instead of rigorous proofs I39ll just give a very heuristic explanation of how to get to the spectral density function hopefully it is intuitive and as you39ll notice the order of presentation will be almost exactly opposite of the lecture notes you39ll see why A lot of the explanation is based on Chapter 6 7 of quotThe Analysis of Time Series An Introductionquot Sixth Edition by Chris Chatfield Very good book I only found this book recently please don39t get mad for my not having told you earlier in the semester Representing a Time Series with Cycles The main reason that spectral analysis seems ambiguous is because we are more used to thinking of a time series in terms of auto regressive or moving average representations ln spectral analysis we want to think in terms of cycles Recall the identity 6399 cosQ isir167 so the exponent represents the sum of a quotrealquot cycle and quotimaginaryquot cycle Also recall that any complex number on the complex plane can be represented as A6 where A is the amplitude and 6 is the angle So a cycle through time can be represented as xt Aeith9 where a is some fixed frequency ie the number of radians per unit time Of course xt will be complex which doesn39t make sense but just hang on for a while Also this process is still deterministic but we39ll change it so that it is stochastic shortly Of course we are not interested in only such simple cycles eg we expect stock prices 1 to have a weekly cycle monthly cycle seasonal cycle and so on So given an actual series xt a better way to represent it in terms of only cycles will be xt i AkeiaijQk39 k1 The series above are not stationary they39re not even stochastic so we introduce random ness The natural way to introduce randomness is to assume that the Ak39s or Qk39s or both are random For now let39s assume that these random numbers are chosen such that xt is second order stationary On the other hand since elmt9 6961th 7 we can suppress the phase part into the amplitude ie where ak Ake9k But why stop here Indeed by sending K a 00 we would obtain 71 xt e39wawdw 7139 where aw is some random function that assigns coefficients to em for each w The rea son that we only have to integrate from 7739 to 739 is because we are looking at discrete series try thinking about it But now this looks kind of familiar It is exactly the spectral representation theorem we learned in class The details are in the lecture notes but all that theorem is saying is that there exists a random function dzw equivalent to awdw in the above case Further more the lecture notes show what properties this random process must satisfy if we want to ensure that the time series is real Remember the only reason we are using complex numbers here is for notational convenience since using complex numbers we can express the amplitude and phase at once as demonstrated above Deriving the Spectral Density Let39s focus on the spectral representation 71 xt e39wdzw 771 2 Note that by representing a time series in terms of cycles all the randomness is now coming from the random function dzw Now recall the argument starting on page 4 of lecture notes 5 The autocovariance function 0k of xt is I39m skipping steps see lecture notes for derivation 0k 13M eiwtkdzwW eiwkEltdzltwgtWgt Now define dPw Edzwdzw Existence of this function is ensured by the Wiener Khintchine Theorem This is what we call the spectral distribution function and if the derivative of this function wrt w exists we get the spectral density function dPw Since this satisfies 0k eikawdw 771 which is just the inverse Fourier transform of the autocovariance function we find that 00 2 000671141 700 f w 2739 k the Fourier transform of the autocovariance function where w E 7 7T 71 Hence instead of starting with the definition of the spectral density as in the lecture notes by starting from a cyclical representation of a time series we arrive at the definition of the spectral density Hopefully this way it is more natural to see how the spectral density is related to the autocovariance In particular as learned in class at k 0 altogtai ww ew so the meaning of the spectral density function is straightforward it tells us how each random coefficient attached to each frequency contributes to the overall variance of xt 3 Asset Pricing Notes Chi Tat Wong October 2007 Abstract Lecture notes literally from Econ 712 Fall 2007 Use at your own risk MODEL 1 Model 2mm mastsfmm t0 et is random and f O7 Vk implies Ctt thkt5t et is the source of randomness xi gt et e 58 7 et is a Markov process New 6 Aleneme Ham 6 Ala Example 1 Md 5t1 51 p5t5t175t N N 5H1 getet17 at is a sequence of Mid 711 For flt6t76t1 7 El5t1l6tl 6fetede E gem a El96t1 wet Agltegtfltetagtdet If at 6 gen ame Eig5t1 let 296jf66j 11 Expected utility M00 52f6t76jul0t1 619 52 j1 0 bi pfst1 2 qt ej 2t ej S wealth at t At Stock j1 Arrow security lottery Ct1 6139 bt2 5139 Pf15t2 5139 S yt1 5139 Rt1bt1 m1 pal an dt15jl 2 an M an of Winning ticket in a complete market Let j t1 5139 5f 57 5139 At1 5139 MODEL 2 The lagrangian 2 5139 CH1 5139 bt2 5139 Pia 5139 St2 5139 271 Qt1 esZt155 n M 025771 5139 u at 6 2 f at 61 At1 61 gt 6quot Rt1bt1 pm 61 dm 619 871 At wealtht 7 Gt 7 bi 7 pfstH 7 2 qt 61 2t 61 j1 FOCs ct u At bi 7A 3446 2 At 61 f at ej 0 discounted expected MU j1 5H1 1 Atpf 52f 57 5139 At1 5139 13211 5139 dt1 519 0 j1 Z 5139 1 MJ 5139 5At1 51 575 0 cz16j f676ju cz16jMde 0 bt1 1 u 6Rt1 2 U CH1 519 f 5157 6139 I Rt1EtJ uI 025771 St1 u 00p BE 1 cm 1771 dim Z 6139 U 0 Q 6139 51 cm 619 f 67 6139 6 C7544 qt 6 u at 61 Non random economies at 71 She 07 6 6H71 7 u at 7 mHu CH1 Etltgltetogt 7 296f6t76 Egef6t76d6 Fem 7 Pltet1sewetgt Etaemu 7 glteFltetde 5 p5 6d 16 Rt1ej E t MODEL 3 u 0 6E u CH1 Bill I Et l CH1 Rt1l If u gt 0 R211 7 3H1 must have some state 7126 Remark 7 Nonstochastic economy 1 5 Tt1 1321 Rt1 In general X 7 Y two random variables CW XX E X E00 Y EYl EXY 7 Let X u 91 Y 3Z1 Rt17 Et XY 0 Et X Et Y 0011 XY 0 Etulct1B1 Rt1 Conditional Variance GOOd time Rt1 l7 Ct1 l U CH1 l Bond R2111 qt 61 no arbitrage 221 I 91 51 f 5h 51 7 6E u ct1 5 7 u 0 u 0 Assume pure discount bond R l From bi t1 n k Ctbt1 Pf5t12 1t5jzt ejZPt7tiLt7ti F1 1 w 71 L057ltiPttipfstdt H o Lt7ti AzPOZIWi zgtxz16jf6n jPt 17ti 6139 j1 Remark Riskifree is related to payoff not price In this model price is endogeneous u comm 6Eu cz1Pt1ti This is stock With no dividend check St1 FOC U Emil 1305 i 1715 1 5Etzgt1tudct0 X1 inilqwiyrk backward M 00130215 1 51E 1 02 6mm cm P 252 i at The term structure of interest rate 1mm Ptti 1 1 wot 1 17 tz i EfUrl OO MODEL 4 if u is bounded Insert diagram here Case I 0 is iid ct at M CH1 Eu CH1 quotBad timesquot 0 is low 11 gt E 11 0511 Temporary shock 7 borrowing Case II Growth 0511 yt ct 7H1 is iid 0179 110 17 0 1 0 9 P tt1 6E 0597391 1 1 Ptt1 3W yt gt1 grow1ng t t1 Permanent shock 7 no borrowing More general process will be in between permanent and temporary shock Lucas critique Ct 5 yt1 yf t17 111 t1 N N M570 7 G gt t 0Z7 37 17 0t 17 0tg vi 1nUt1 N N pmag MODEL 5 c 1 0092 mpg Ptt 1 7 u P 1515 1 Et 11 0511 10 003447 Pt7t1 BE 11 0t1yt1iin 5E 1 00 yf t1ut1in 10 003447 Pt7t1 51 007 yziw Et t1Ut1in e s t gggge v t gzgi Assume Et1 and Ut1 are independent 1 Ptt1 1 0tn 1yf 1ewn2gewfngg 1 02 02 111Rz1 111377 gt11nyz77 711n170z7wg7772f7m7772f7 1 G a 012 1ngt77 gt711nyz7717 4wg 7772 77MU 7772 5 y 2 2 71110750 50 Now suppose 1 7 0t 17 0t6 vi Ut1 U 012 07 g 17 G 2 1nRt1 50 511nyt621ny 7 R 1 t Lucas critique 60 61 52 no use after policy change regime change no use in policy analysis Lucas Tree economy U Ct 6Rt1Etuct1i 1 P 1515 jEt 7 period bond 2 u Et 11 0511 pf1 dt1 Price of stock 3 i M 0 rrow s securi ies 900 7 uctfcz A t 4 iv Yield 7 Rewrite 3 7 Optimal capital structure and price earning s ratio 7gt pricing options K d1t7 39 quot 7dkt Xi 0 Edm 191 From 4 qltcltXtX gt 7 6 ltme MODEL 6 From 3 and get rid of pi Imp mittCH1 u CH1 Z 51E Etj U 02w dtji t TEt ETA U CTDPSH T EVE W 025jdtji t E 5TH CTMTH j1 40 by transversality condition 5 7 DO 39 u 07541 Pt Z M u at 3ny 39 5 11 T Xzj j p 2 Et dt P t t Expectational theory j1 From 5 M8 51E Et dtij jCovt dm PltttjEtltdtJ j00w Market price of risk x H H M8 x H H If u 110 1110 Risk neutral Gout 0 Two trees dim dgm Et 0 E 012 E 0151 Gout Covtct1d r1 gt 0 Govt cthfH lt 0 Expectational theory only concern lst moment 00 Et dtfij Et dig P 2626 7 7 7 0 12 j p394 133 2nd moment I A I B Covt wwj 7dt lt07 Gout u 1 7dtj gt0 df df17df27 39 39 One period bond 170707n 7Ptt1 Ball n1 Constant coupon bond 7171717 jiperiodbond 00 1 07 7Pttj MODEL If XHZ X and XZH X 1 unit t 2 component effect u c u 0 XMD A I A I II I ABWBWJWXmXWQLX dX Two period ArrowiDebreu price 2 II 7 2ucXH A I A I II q ltth i 6 ucXt XfXz7XfX7X dX 2 0 XMD A2 II 6 u c f Xi X II 7 ju 0 XMD Aj II 91XZ7X 6 u XhX U CXt 39 A Z A jmd j Xtj fm Xty Xm dXtj Z dtj Xtj 97 Xtme39 dXtj j1 All the price can be expressed in terms of Arrow security 73 XHj arbitrary payout Pf f jEt MEt 7 tj Xtj 11 If B bonds are issued A dm dtj Tm Xtj B A DO 39 u A 5 E 1E Jd Pt F1 5 t u t1 3 5 DO 39 u at I A P Z jEt 155 dtj t 7 tj Xtj B j1 V dzj Option Gives the holder the right to buy before devidends are paid out a share at price 13 strike price V0971 7 vXt1 IXt7Xt1dXt1 x 7 P5 Xt1 dt1 Xt1 13 Z 0 if Xt1 E A 0 ifx em MODEL 7 vlt2 tz31313lt157quot15pf VXt713 A 17211 Xt1 dt1Xt1 7 13 QXt Xt1dXt1 pal Xm dt1 Xm 7 23 qltXt7Xt0dXt1 X pt1 Xz1 dz1 Xt1 7 p QXt Xt1dXt1 An V p211 dt1qXt7Xt1dXt1 139Xt7Xt1dXt1 X S X PM 17 pt1 dt1 17 9Xt7Xt71dXt1 An 33 1313 15715 1 V0913 pfi pt1 dt1 7 13 q Xt Xt1dXt1 A B For sure get 17 115 Econ 712 Macroeconomic Theory First Exam University of Wisconsin Instructor Rody Manuelli October 16 2004 Instructions Please answer all questions If you get stuck in one section move to the next one Do not waste time on questions that you nd hard to solve Partial credit will be awarded if it is clear that you were approaching the prob lem in an essentially correct manner This is a closed book exam Students may bring one page both sides or two pages single sides of notes Please hand in the exam promptly at 12 Noon Each question is worth 50 points The point total for each section is indicated at the beginning of the section Look at these prices when deciding how to allocate your time If you believe that a question is wrong or poorly worded please make the minimal necessary changes to make it beautiful and well posed Of course unnecessary changes will result in a lower grade Please use one blue book for each question and write only on the right page The odd numbered page in a newspaper Please remember to put your name in each blue book Good luck 1 2 Q uest ions Problem 1 Habit Persistence Consider an economy populated by a large nume ber of households with utility functions giuen by Z tuewzwt 0 lt6 lt1 2 0 t0 where u R a R is twice differentiable increasing and strictly concaue if necessary you may assume that it satis es the Inada condition The uariable 0 is indiuidual consumption and 2t is a measure of lagged consumption To be precise 1 i 66jct1j 0 3 6C 3 1 M8 22 x H o It follows that alternatiuely it is possible to describe the law of motion for 2t as Zt1 Z 17 Sela 0 In this setting 2t is a measure of habit persistence as it implies that the marginal utility of any giuen leuel of consumption decreases the higher the leuel of past con sumption The technology in this economy is standard and giuen by fltkt7 1 510k Q where the functions f is strictly concaue increasing and satis es Inada conditions Assume 6 6 01 Z 15 points Let the planner maximize the utility of the representatiue agent subject to all the feasibility constraints Argue that under some condition on gt6C an interior steady state ewists and is unique Describe the condition that 41566 has to satisfy 1 10 points What does the model say about the impact of crossecountry differ ences in how much people care about past consumption ias measured by b7 on the steady state output per worker as ZUpoints De ne a competitiue equilibrium in which in each period households trade at least one period bonds capital consumption and inuestment goods Assume that consumption is taped at the rate 739 that is the cost ofpurchasing 0 units of consumption is 1 TC The reuenue produced by this tam is rebated in a lumpesum fashion to the households 4 15 points Economist A argues that current consumption produces an emtere nality in the sense that it lowers the marginal utility of future consumption Giuen this heshe suggests that a tad on consumption with proceeds rebated to the consumer in lumpesum fashion will guarantee that the steady state of the competitiue equilibrium of this economy will coincide with the steady state of the planner s problem and that the tam rate that attains this equality of the two steady states minimizes the ualue of 2t at the steady state Go as far as you can analyzing this claim Nate Section 5 requires you to describe the steady state uersion of the competitiue equilibrium with constant consumption taxes in this economy You need not need to show that euery competitiue equilibrium conuerges to a steady state It suffices to assume that a steady state emists and deriue its properties Solution 2 Sketch The Lagrangian corresponding to the planner s problem ige noring the nonenegatiuity constraints is E ZBQLMC Wt Atfkt1 510k Ct kt1l t0 Mth 1 5J3 Oil The rst order conditions are C I 1N0 Wt At 0t kt1 I A 5At11 5k f kt1l7 Zt1 I 0 59t11 5c 5U 0t1 gtZt17 and the feasibility constraints at equality Note that the consumer would like to choose 2t as small as possible since this lowers utility it follows that the law of motion of this uariable must hold as an equality In a steady state all uariables and Lagrange multipliers are constant Thus at a steady state if one exists it follows that P5k f WL 0 fh76kh 662 0 Emistence of a unique strictly positiue uector that satis es the rst two equations follows from standard conditions on the production function It remains to check that 0 7 Z gt 0 Howeuer this is equiualent to 01 7 gboc gt 0 A necessary and sufficient condition is that zboc lt 1 The model implies that the steady state leuel of output per worker is independent of 4b In a competitive equilibrium households solue max 2 rue 7 2 t0 subject to 1 TCt xi bt1 S win 72 Jr Rtbt Ut7 Zt1 Z 1 6amp2 Ct kt1 S 1 610k 1 T113 HitRjleTH 2 0 n g 1 where ut is a transfer receiued from the gouernment Firms solue maXCt pmt wtquot 721 subject to 91 S FU h where Fh1 and F is homogeneous of degree one An equilibrium is an allocation cf If1 nf fim aprice system R1 p2 111 and a sequence of bond holdings bt l such that Z Giuen prices households mawimize utility 2 Giuen prices rms mamimize pro ts 5 Markets clear The rst order conditions for utility mamimization are eualuated at the steady state W 7 M 1 7 02 M 5AE151 7 L A MR 602 0 Giuen that r from the rm s problem and that feasibility holds it follows that pm f W 0 fh76kh Thus the steady state of a competitiue equilibrium coincides with the solution of the planner s problem for all tax rates The reason for this is simple In the steady 4 state habit persistence does not play any role in determining consumption since the marginal rate of substitution is giuen by the discount factor Moreouer with inelastic labor supply consumption taxes are not distortionary For this to be the case they would have to create a wedge between current and future consumption Howeuer a constant tad rate does not distort the choice between present and future consumption as both are taped at the same rate Problem 3 Housing Prices Consider an economy populated by a large number of identical households with utility function giuen by Ed 6 uczhz 0 lt 6 lt1 t0 where 0 is consumption of a nondurable good and hi is consumption of housing The function u is twice differentiable increasing and strictly concaue if necessary you may assume that it satis es the Inada condition In this economy there are I trees per household with each of them producing d units of nondurable consumption at t Assume that 0 2 d Each household trades in shares to all the trees bonds of all maturities a full set of ArroweDebreu state contingent securities and housing One unit of housing costs qt units of nonedurable good Thus the total cost of a house of size ht i5 Itht In equilibrium the supply of housing per household is giuen by a stochastic process hi 1 10 points Go as far as you can describing how to price trees and houses as a function of the emogenous stochastic processes ditht 2 15 points Assume that cabliaPiQ um 7 0 lt a lt 1 0 gt 0 and that ht h gt 0 Go as far as you can deriuing the implications of the model for house prices How is qt related to the ualue of the stoch market Nate de ne the ualue of the stock market as pt 2 pit where pit is the price of tree i as 15 points Assume that cabliaPiQ um 7 0 lt a lt 1 0 gt 0 and that ht pct u gt 0 Go as far as you can deriuing the implications of the model for house prices How is qt related to R l il ithe collection ofprices ofjeperiod bonds 4 10 points Let the function u be arbitrary and let the stochastic processes dn htH be yiuen Assume that households do not own the houses they liue in they rent houses from deuelopers That is in euery period they pay a certain amount of rent rt Go as far as you can to deriue the equilibrium per period rent and the price of a house Does the stochastic process for qt in this enuironment di er from the one you computed in 1 Assume that the utility function and the stochastic processes are the same Solution 4 Sketch Z The household solues gt0 max Edi tuct ht t0 subject to gt0 I 0i cht1 Z R lbjt Zpitsit1 qzx 2z 5W1 X F1 i1 gt0 I E 2 311113th Z Sit1ltpit dit 2271053717 t 9th F1 i1 This budget constraint is standard and we follow the conuention that in period t the household sells its house and receiues qtht and then it purchases the house that it will liue in newt period The rst order conditions include among others Uc027 htMt 5Etuc0t17ht1 1t1 Uh0t17 ht17 Uc0ty him 5Etuc0t17 ht1Piz1 dit17 uc ah 39 Bill 7 61Et 075 Ha ucct7 ht Repeated substitution in the rst equation implies that the price of the house is giuen by Met ht w u C ht qt jEt hlt 1 4a j1 This empression can also be written as DO 39 uc0tj7 htj uh0tj7htj q 61E t t uc0tht uc0tj7htj Since the price of a tree satis es w uc 0 sh pit ZBJEZ 1 htJditj 7 F1 M02 z w 93 v it follows that houses are priced like stocks trees with a diuidend equal to the marginal rate of substitution between consumption of nondurables and housing Using the specific utility function and the assumption that the stock of housing is constant it follows that qti jEt Efe Since the ualue of the stock market is w Meme 5 1E Ct 39 7 Pt t ucltct7h 1 it follows that the ualue of housing is a constant proportion of the ualue of the stock market More precisely the model implies that 51704 Qt Pt 04 In this case simple calculations show that gt0 79 Qt Z jEt 7 a7 04 11 M 1 04 00 71 E R It 06 1 1t If the household rents the house then its budget constraint is giuen by gt0 I Ct rtht Z RjTtlet Zpitsit1 qxt x ztxt x dx j1 i1 X gt0 I E 2 311113th Z Sit1ltpit dit 2271053717 07 F1 i1 where rt is the per period rent It is immediate to uerify that in an interior solution uhcz ht nude ht Now since a house is identical to a tree that drops diuidends giuen by rt the standard pricing formula for trees as applied to houses is 7 DO 39 Cth htj Q 7 51E uxcty ht 7 tj 7 DO jE ucctj7htjuhctj7htj qt 6 t Uc0tfh uc0tj7htj 7 which is the formula deriued in 1 Thus in this economy housing prices are independent of the structure of ownership Econ 712 Macroeconomic Theory Second Exam 1 University of Wisconsin Instructor Rody Manuelli December 11 2003 Instructions Please answer all questions If you get stuck in one section move to the next one Do not waste time on questions that you nd hard to solve Partial credit will be awarded if it is clear that you were approaching the prob lem in an essentially correct manner This is a closed book exam Students may bring one page both sides or two pages single sides of notes Please hand in the exam promptly at 11 AM Each question is worth 50 points The point total for each section is indicated at the beginning of the section Look at these prices when deciding how to allocate your time If you believe that a question is wrong or poorly worded please make the minimal necessary changes to make it beautiful and well posed Of course unnecessary changes will result in a lower grade Suggestion Do not try the extra credit sections until you nished the regular exam They are more challenging than the regular questions and they are there so that you can show off Please use one blue book for each question and write only on the right page The odd numbered page in a newspaper Please remember to put your name in each blue book Good luck l 2 Q uest ions Problem 1 Aid Programs and Unemployment Consider a model in which uni employed workers draw wage offers from the distribution Once ajob is accpeted it lasts foreuer and the wroker cannot quit In this economy there is a program in place designed to help low income and unemployed workers The progran works as follows If the worker is unemployed he receiues a wage equal to b If he is employed the bene t is reduced at the rate k for employed indiuiduals This for a worker who participates in the program income is w 7 b w 0 unemployed yp 7 b 1 7 kw w gt 0 employed Participation in the program is uoluntary and an indiuidual may choose to quit the program after obseruing his wage draw The indiuidual cannot quir the job It is clear that an indiuidual will leaue the program ifw gt ypw It follows that a worker will leaue the program ifw gt if where if bk since at this point ypu7 if Ignore how this program is nanced ie ignore the gouernment budget constraint N 10 points Describe the indiuidual search problem Argue that the optimal strat egy is of the reservation wage uariety Denote the reseruation wage by Z 1 15 points Consider the case in which the endogenously determined reseruae tion wage Z is such that Z S if Co as far as you can characterizing the effect of increasing b on the reservation wage and the auerage duration of uneme ployment Describe the impact of an increase in b on the probability of leauing unemployment and on the probability of leauing the program b3 15 points Consider the case in which the reseruation wage Z is such that Z gt if Co as far as you can characterizing the effect of increasing b on the reseruation wage and the auerage duration of unemployment Describe the impact of an increase in b on the probability of leauing unemployment and on the probability of leauing the program v 10 points What accounts for the differences in the results you obtained in 2 and 9 Ezrtm Credit Co as far as you can describing the effects of an increase in k This looks harder and the answer may depend on where in a particular interual lies the reseruation wage Problem 2 Social Security and Growth with Finite Lifetimes Consider a model in which indiuiduals liue for two periods Let c be consumption of time j good by an indiuidual of generation t Each indiuidual is endowed with one unit of labor when young and zero when old Indiuiduals inelastically supply their one unit of labor when 2 young and they saue in the form of purchases of real capital when they are old Thus a representative agent of generation t solues the following problem max 1110 6111c1 subject to t 0 Jr St wt7 S S St1 7 t17 Girl where st is saying and 1 rt1 is the rate of return between periods t andt1 Since there is no population growth we normalize the size of each generation to one The initial old simply mamimize their second period consumption Aggregate output in this economy Y satis es y AKthl a 0 lt a lt1 Assume that capital depreciates fully Assume that there is one or a large number of rms that rent capital from households and hire labor to maximize pro ts 1 10 points De ne an equilibrium for this economy Go as far as you can dise playing a di erence equation that the equilibrium must satisfy 1 10 points Argue that there is a unique steady state capital per worker Is it the case that at the equilibrium steady state capital per worker the marginal product of capital is greater than one as 20 points Consider now the impact of a social security system Each indiuidual of generation t fincluding the initial oldi receiue bt1 units of consumption when old and are tamed at the rate 739 when they are young Thus the indiuidual optimization problem is maxlnc 6111c1 subject to 1 7 Twt St17 t1 bt1 c 8t S Girl 3 These social security payments are nanced with tames Thus the gouernment budget constraint is bi th Argue that an increase in 739 will lead to a decrease in the steady state leuel of capital per worker 4 10 points Are there parameterizations of this economy basically uectors A 04 such that the introduction of a small social security system ie small 739 is a Pareto improvement ouer the laissezefaire equilibrium when comparing steady state utilities Is it possible that such a policy change results in a Pareto supee rior allocation This means that indiuiduals of all generations are better off including the initial old Note You may assume that before the introduction of the social security system the economy is at the steady state 9 Emtm Credit Consider now a social security system that inuests in special gouernment issued bonds These bonds pay a rate of return equal to 1 qt where qt 3 rt that is gouernment bonds have potentially a lower rate of return than priuate assets In this case indiuiduals understand that their bene ts depend on their contributions ie each indiuidual belieues that bt1 th1 qt Go as far as you can analyzing the e ect of this social security regime when qt rt Argue that ifqt lt rt a gouernment that inuests the proceeds from the social security tad purchasing physical capital can use the social security surplus to nance a policy of transfers to the young Go as far as you can describing the equilibrium in the case qt 1 7 tart 0 lt 4b lt 1 Econ 712 Macroeconomic Theory First Exam University of Wisconsin Instructor Rody Manuelli October 20 2007 Instructions Please answer all questions If you get stuck in one section move to the next one Do not waste time on questions that you nd hard to solve Partial credit will be awarded if it is clear that you were approaching the prob lem in an essentially correct manner This is a closed book exam Students may bring one page both sides or two pages single sides of notes Please hand in the exam promptly at 12 Noon Each question is worth 50 points The point total for each section is indicated at the beginning of the section Look at these prices when deciding how to allocate your time If you believe that a question is wrong or poorly worded please make the minimal necessary changes to make it beautiful and well posed Of course unnecessary changes will result in a lower grade Please use one blue book for each question and write only on the right page The odd numbered page in a newspaper Please remember to put your name in each blue book Try to nish the required questions before you try the extra credit sections These are typically hard and carry no objective point value Good luck l Problem 1 Interest Rates and Housing Prices Consider an economy popue lated by a large number of identical dynasties with utility functions given by gt0 U 2 Emma Mag 0 lt 5 lt1 t0 where uc and uh are strictly increasing and strictly concaue functions In this contewt 0 denotes consumption of nonedurable eg food goods while ht is the stock of durables eg houses at time t Each individual is endowed with one tree Each tree drops fruit a nonedurable good according to the process The supply of houses is red ie ht h7 for all t This economy is open to international trade and borrowing and lending in nonedurable consumption Housing is nonetradable The gross world interest rate in units ofnonedurable consumption is denoted 1rf In all sections assume that at t 0 this country s foreign debt denominated in units of nonedurable consumption is 0 Z 10 points Assume that 0 E for all t7 and that 1 r 6 1 Determine the price of a house and the equilibrium rent 1 10 points Suppose that the domestic tree drops a sequence of dividends satisfying 1 5Z tcz5 20 What is the relationship between house prices and domestic output of none durables and the trade balance as 15 points Assume now that 0 E but that at t 0 there is a unanticipated shock to world interest rates After the shock interest rates are given by t 1 gt5 1 151727mT 17 t 6 1 tT1T2 Go as far as you can describing the impact of the shock to interest rates on housing prices ie compare housing prices at t 0 before and after the shock to interest rates v 15 points Consider the economy from the previous section Go as far as you can describing the dynamic behavior of housing prices fromt 0 to in nity What does the model say if anything about the relationship between housing prices and contemporaneous interest rates What does the model imply about the relationship between housing prices and the trade balance Note In sections 5 and 4 you may assume that u is 01 91 7 0 if necessary Solution 2 Sketch w as Let s rst determine the equilibrium consumption sequence 1 Let At be the Lagrange multiplier associated with the feasibility constraint The rst order conditions of the consumer s problem in clude u cz At 2 phtAt 6t1pht1 Uht1 It follows given that u ct u 57 that the price of the average house satis es 5 Nb ph 1 7 6 u c In order to determine the rental price q observe that an investor has two options he can buy a bond or he can buy a house rent it for a period and then sell it The two projects have to offer the same rate of return Let 6 1 1p Thus 1pM7 Ph or Pph I u h Hp q UTE Denote the endowment sequence by et Since the relevant budget constraint is given by 260 thtltht1 7 hi 2639 t0 t0 t0 and DO 00 2 re 2 re t0 t0 then the optimal choice of the consumer is ht h and 0 E The equilibrium is as in the previous case In this case there is no connection between the trade balance igiuen by the difference between E 7 eti and house prices The rst order condition corresponding to the optimal choice of bonds is 51 6 1u c1 t 01 Mam t2 T 1 we T Wat It follows that the sequence is decreasing and it conuerges afterT periods In equilibrium the budget constraint implies that T 6 t gt0 T 6 t gt0 lt gtcmlt gt 20 7 z tT1 t0 7 z tT1 3 v Satisfaction of the budget constraint implies that 00 gt E and 0TH 0L lt 5 Form the Euler equation corresponding to the equilibrium choice of housing seruices we get that 7 w jwl h quot215 we puma The housing price formula is just a special case of the general asset pricing formula for a stock The general formula in this nonestochastic world is u 00 INCH J 1 d p t 1716 Ct t17 where dt are diuidends measured in units of consumption In the case of the house the diuidend is just the marginal utility u h which eualuated in terms of consumption good not in utility is just u hu ctj at time tj The result then follows Since we argued that 00 gt Ethen the time zero price after the shock is higher than before that is 1 u h 1 u h gt pitco p we The effect of a decrease in interest rates is to induce a maybe large jump in house prices The analysis follows easily from the asset pricing formula in the preuious sec tion Since 7 1 u h phi pu cz and is decreasing and conuerges after T periods then house prices are decreasing as well until period T After that period they stabilize at 1 u h phL EUTCL which is lower than the before shock prices The shock to interest rates make house prices initially ouershoot and then they settle at a permanently lower leuel The effect of interest rates is ambiguous In the rst period a lower interest rate created an appreciation of the stock of houses In subsequent periods house prices decrease euen though interest rates remain low Finally when interest rates stabilize at the old leuel house prices stabilize as well but at a lower leuel In the rst few periods this economy is running a trade de cit as 00 gt E and house prices increase in period 0 and then decrease from period one on Since in the long run the country has a trade surplus ie 0L lt 5 it must be the 4 case that there is some t lt T such that after t the trade balance is positiue Howeuer from t to T7 house prices are still decreasing In summary the model does not predict any simple relationship between interest rates the trade balance and house prices Problem 3 Public Goods Taxes and Congestion Consider an economy pope ulated by a large number of identical dynasties with utility functions giuen by mime 0lt6 lt1 t0 where uc is a strictly increasing and strictly concaue function Production in this economy requires capital and labor as well as a gouernment prouided good This good denoted g7 can be interpreted as infrastructure and is subject to congestion If a rm employs h units of capital and n units of labor its output is 45 yFltkn 0lt gtlt1 9 FIn where FIn is interpreted as the auerage leuel of output per rm Thus holding g constant an increase in output by other rms reduces this rm s productiuity due to congestion From the point of uiew of an individual rm there are constant return to scale doubling the amount of priuate inputs holding g and the output of other rms constant doubles priuate output Capital euolues according to kt1 S 1 510k 192 and feasibility requires that Ctgtkt S 34 and as usual the leuel of initial capital is giuen Z 15 points Consider the planner s problem Note that the planner understands that the true production function is giuen by y Fkn1 g and chooses g as well as other inputs optimally Describe the condition that determines the optimal steady state capital per worker 2 Z5points Consider now a competitiue equilibrium in which the planner chooses g optimally and nances g with lumpesum tames Is the steady state capital per worker optimal in the sense of coinciding with the planner s choice in the preuious section 5 20 points Consider now a competitive equilibrium in which the planner uses a tad on capital income to nance g Assume that tax reuenue equals gouernment spending no public debt To be precise letr be the rentalprice of capital then tax reuenue is Trh7 where 739 is the tam rate chosen by the gouernment Let and g739 be the equilibrium leuels of the capital stock per worker and the publicly prouided good Go as far as you can nding the tam rate if any that has the property that the steady state equilibrium ualues of capital per worker and the public good coincide with the planner s choice More precisely is there a tax rate 739 such that W k1 97 9 where hg are the steady state ualues from the solution to the planner s problem Solution 4 Sketch Z The planner s rst order conditions include 1 l FWt lJl q gfily which implies that for all t7 9 zilWmm The Euler equation for capital is W00 u cz11 5k 1 Fkt17m1 gf1Fkkz17mm Using the optimal choice ofg in the preuious equation we get that W00 5U Ct11 5k1 17 Fkkt17nt1 Thus if we let 6 1 p 17 the steady state capital labor ratio satis es p 6i lt1 gt l f k7 where E Fh 1 2 In a competitiue equilibrium rms choose their capital stock so that the marginal product equals the rental price Thus letting qt denote the rental price at time t7 the rst order condition is Q Fkat lt y 6 us while the equality of rates of return condition requires that Rt W17 6k qu Since taxes are lumpesum no margin is distorted Given that all rms are iden tical and that the government uses the e icient rule to determine government spending ie it sets gt bllinWty t it follows that the equilibrium marginal product of capital satis es qt Fkkt7nt 1i In the steady state the interest rate equals the discount factor and the equilibrium capitalelabor ratio denoted h satis es p 6i gt 1 f l Comparing this condition with the planner s version it is clear that h gt 16 The problem is that the private sector fails to internalize the fact that an empansion of private output requires more public goods and this is costly Even though taxes raise su icient revenue this is not enough as they do not signal the true cost of additional capital In an equilibrium with taxes the appropriate version of the condition that de termines the optimal choice of capital is TFImIgt Fafh In the steady state the no arbitrage condition requires 1515k17 1 Qt Fkkt77lt 7 I 45 pm 1 mm Thus the optimal tax rate if it exists solves Tfkk 45 p6 1Tf k k fW and it must be such that the level of government spending is e icient requires This Tf WW 9 bl1 fk 7 There is no 739 that can satisfy both equations after all we have two equations in one unknown It is possible to show modulo some algrbraic mistake that a necessary condition for a tax that satis es the budget constraint to satisfy the Euler equation is that k is consistent with WWW Wow 7 7 f W but this in turn implies that the tad on capital is 100 In this case consumers will choose to inuest zero and this cannot be an equilibrium If the youernment has access to a mix of lump sum and capital tames then it is possible to support the rst best To see this note that the youernment can choose the ratio of the publicly prouided good to priuate output optimally ie 9 mm W z and then pick the tad rate so that p 6k 17 Tf k gt 1 which requires that 739 4b Thus the right Pigouuian tax internalizes the emtra cost of an additiona unity of capital but lump sum tames are still required to raise the appropriate amount of resources Econ 712 Macroeconomic Theory First Exam University of Wisconsin Instructor Rody Manuelli October 18 2003 Instructions Please answer all questions If you get stuck in one section move to the next one Do not waste time on questions that you nd hard to solve Partial credit will be awarded if it is clear that you were approaching the prob lem in an essentially correct manner This is a closed book exam Students may bring one page both sides or two pages single sides of notes Please hand in the exam promptly at 12 Noon Each question is worth 50 points The point total for each section is indicated at the beginning of the section Look at these prices when deciding how to allocate your timell If you believe that a question is wrong or poorly worded please make the minimal necessary changes to make it beautiful and well posed Of course unnecessary changes will result in a lower grade Suggestion Do not try the extra credit sections until you nished the regular examl They are more challenging than the regular questions and they are there so that you can show off Please use one blue book for each question and write only on the righ page The odd numbered page in a newspaper Please remember to put your name in each blue book Good luck 1 2 Problem 1 Productivity in the Capital Goods Industry and Growth Consider Questions an economy populated by a large number of households with utility functions given by i tidct 0lt5lt17 20 where u is twice di erentiable increasing and strictly concaue if necessary you may assume that it satis es the Inada condition There are two goods in this economy consumption and investment Feasibility is completely described by C S ZlFlU t n12 90 S 22F2k2t7 22 hi 3 176htsct k1 k2 S kt n1 n2 S 1 The notation is standard 0 is consumption at time t act is the output of new capital goods at t and hit are the quantities of capital labor allocated to sectori at time t It is assumed that the functions Fl i 17 2 are twice di erentiable concave and homogeneous of degree one Note that capital and labor are fully malleable and can be costlessly reallocated across sectors N 93 v 9 9 10 points De ne a competitive equilibrium for a closed economy in which households buy new capital goods from the capital producing rms and rent old capital to the rms in both sectors 10 points Go as far as you can showing that a steady state emists 5 points Go as far as you can analyzing the e ect of an increase in the prof ductiuity of the capital producing sector ie an increase in 22 on the price of new capital goods relative to consumption goods in the steady state 5 points Go as far as you can analyzing the e ect of an increase in the produce tiuity of the capital producing sector ie an increase in 22 on the capitalelabor ratio in each of the two sectors in the steady state 10 points Go as far as you can analyzing the e ect of an increase in the prof ductiuity of the capitalproducing sector ie an increase in 22 on employment in each of the two sectors in the steady state 10 points Consider now an economy that can trade in goods with the rest of the world but that it cannot borrow or lend Thus this economy can import and emport both consumption and capital goods but trade must be balanced Let the internationalprice of the consumption good be 1 and the price of the inuestment good be pk Describe the steady state gt2 Emtm Credit Consider the economy described in section 6 Assume that at time 0 it is learned that there will be an decrease in the price of capital goods at time T a Go as far as you can emploring the impact of this announcement on con sumption and output in the short run If you cannot get de nitiue results emplain the source of the ambiguity b Go as far as you can emploring the impact of this announcement on short and long interest rates If you cannot get de nitiue results explain the source of the ambiguity Note In your answer to all the sections you may assume that the equilibrium is interior Solution to Problem 1 Sketch The representative household solves the follow ing problem max in 0 1 Ctbia tbil HHsz subject to wtrthtRtbt t01 176htxt t01 0 C 90 bt1 kt1 11m BTUCleTJrl 7 T400 ho gt 0 and b0 giuen7 9790 kt1l Z 0 0 0 t 0 1 It is easier for me to consider the case of two rms one in the consumption sector and the other in the capital goods sector Of course given constant returns to scale this is without loss of generality The rm in the consumption sector solves maxct 7 wtnlt 7 rthlt 2 subject to C S 21F1k1t7n1t7 The representative rm in the capital goods sector solves max 90 wtht 721 3 subject to 90 S 22F2k2t7n2t De nition 2 A recursiue competitive equilibrium is a collection of price sequences Hui rildpi leilly t 0 1 an allocation llcflylrfly kit7kamp7ni7n zll t 01 and a sequence of bond holdings 511 such that 1 Giuen prices the allocation and the sequence 151 solue I with hfth t h utility marimization 2 Giuen prices the allocation solues and pro t marimization 5 The allocation is feasible market clearing 4 b5 b0 0h5 ho gt 0 is qiuen Given that it is assumed that the solution is interior it is immediate to derive from the rms rst order conditions 21F1k1tnu 7 4 ZlFrlUfltynlt wt 5 pht22F192k2t7n2t 7 6 me2Ffo2t nZt wt 7 The relevant rst order conditions for the household is this is corresponds to the optimal choice of capital pietyCt Bl 6pkt1ulct1 UCt17 t1l 8 If a steady state exists then these equations iand the feasibility constraintsi must be satis ed for constant values of all the variables Thus 8 and 6 imply 16l176T3922F1920i271 7 Where Is h is the capital labor ratio This equation uniquely pins down HS Using 4 7 it follows that W10 WARS 10 where 1 ngdan is a strictly decreasing function Thus 10 pins down Hf Feasibility implies that If lilnl H2lt1i n1 6h 22F2H211 7 n1 4 These two equations imply that 22F2H 1 76H 7 0 l 12 22F2H 1 76H 61 7 7 7 n17 and h is given by 11 The aggregate level of consumption is 0 21F1Hf1nf 13 while the price of capital goods is ZlFlOiLl may 14 p1 It follows that a steady state exists and it is unique Consider now the effect of an increase in 22 From 9 it follows that ngdZZ gt 0 Next equation 10 implies that dlildZZ gt 0 Given that 9 implies that 22FH l is constant the increase in H1 implies that the price of capital goods decreases Finally the impact on n1 is ambiguous as it depends on the elasticity of substitution If this is an open economy and the international price is pk the equilibrium is the same as in the closed economy case if the solution is interior ie pk pg Problem 3 New Technologies and Asset Prices Consider an economy popue lated by a large number of households with utility functions giuen by Enigma o lt 6 lt1 t0 where u is twice di erentiable increasing and strictly concaue if necessary you may assume that it satis es the Inada condition Initially in this economy there is only one technology iwhich we label technology 1 Output of the consumption good using this technology satis es 91 S 31Fa1t7n1t7 where alt is the amount of land used at time t and nu is the number of workers hours allocated to this technology It is assumed that F is twice di erentiable homogeneous of degree one concaue and its has strictly decreasing marginal productiuities of the two factors In addition assume that Thinjlerxa n 00 Land and labor are in wed supply We normalize the aggregate quantities to one In this economy indiuiduals trade famong other assetsi bonds of all maturities shares in farms ie land and labor Howeuer agents are free to trade any assets eg options they want At time 0 it is learned that at time T gt 0 a new technology will become auailable This technology is completely described by 2 92 E Z n22 where z is a stochastic process with the property that 23 gt 21Fn1 1 The announcement speci es the distribution of fort 2 T but not the real izations You may assume that the actual values of 2 will not be known until time t N 1 93 v 9 10 points Go as far as you can describing the equilibrium levels of output land prices wages and interest rates at time 0 before the announcement 20 points Describe the impact of the announcement on a R1701 the price ofj period bonds forj 1 andj Tk7 k gt 0 Emplain your ndings b The empected wage rate E0wt for t 0 andt T k7 k gt 0 Emplain your ndings c The ualue of the stock market aggregate ualue of land at t 0 10 points Emplain the economic forces that drive four ndings in 2 In partice ular in what sense iif anyi can the stock market be used as an indicator of the future health of this economy 10 points Suppose that is a sequence of random variables Assume that at time t euery agent in the economy obserues a signal which we denote act which is positively correlated with 2 the rst ualue of the technology More speci cally at time t euery indiuidual knows the distribution of 2 conditional on sot Discuss and explain whether these signals will a ect a Prices of bond of short maturity b Prices of bonds of long maturity c Stock land prices Emtm Credit Suppose that technology 2 is described by 92 S ZfFa2t7n2t7 23 gt 21 Go as far as you can emploring the e ects of this new technology relative to the linear technology described above on a The ualue of land b The empected wage rate E0wt fort 0 and t T k7 k gt 0 Solution to Problem 1 Sketch The standard rst order conditions for pricing bonds and stocks are 51E U Ct R 1 J 15 t u u Ctjdtjf and 00 E t pt 5 me It follows that to determine asset prices all we need is to determine the equilibrium consumption allocation Consider the pre announcement equilibrium It is clear that nu l as there is only one technology It follows that 0 21F11 d 21152071 111 1Fn11 Given these equilibrium quantities asset prices at t 0 are R1701 jy 5 1 Z F 1 1 p 17 6 a 7 Let s consider next the equilibrium allocation after period T Since the marginal product of labor must equal the wage in both technologies workers are mobile it follows that w 7 21mm tltT t T 23 21Fn1n1t t2 T7 where the condition 22 21Fnln1t determines uniquely nu It follows from our assumptions that 0 lt nu lt 1 That n1 cannot be zero follows from the assumption of unbounded marginal product of labor while n1 lt l is a direct consequence of 22 gt 21Fquotl7 Thus for t 2 TuN1t 2 111 while they are equal for t lt T Dividends land rents are of i 21Fa11 tltT t T 21Fa1n1t t T It follows that Z S dt with strict inequality for t 2 T this is due to nu lt 1 while consumption is 21F11 tltT Ct max 21F1n 2517 n t T In this case we have that E 2 05 with a strict inequality for t 2 T What is the effect on asset prices For j lt T 131701 j 12701 as consumption is unchanged However for j 2 T 51 2 cj and 1301 lt 131701 see 15 Finally the post announcement price of land is T71 00 r N N 130 Z 6quotle112 WM j1 jT 00 which is clearly lower since 51 2 01 and de S dj for j 2 T In this economy a drop in the stock market is good news as it signals the advent of a new more productive technology Finally if a signal 90 is available it is clear that o It will not affect the price of short term bonds they are constant o It will affect the price of long bonds in so far as it affects the conditional expectation of the marginal utility of consumption Formally it will affect long bond prices since is affects El nmam 21Fln gig1 7 l 90 o It will affect stock prices for the same reason as above

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