Noah Williams Economics 312. University of Wisconsin Spring Midterm Examination Solutions 1 FOR GRADUATE STUDENTS ONLY

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1 Noah Williams Ecoomics 32 Departmet of Ecoomics Macroecoomics Uiversity of Wiscosi Sprig 204 Midterm Examiatio Solutios FOR GRADUATE STUDENTS ONLY Istructios: This is a 75 miute examiatio worth 00 total poits. Questio is worth 40 poits, all other questios are worth 30 poits. ANSWER QUESTION the choose TWO of followig THREE questios. DO NOT ANSWER ALL OF THE QUESTIONS. If you do, your grade will be based o the LOWEST of the questios. I order to get full credit, you must give a clear, cocise, ad correct aswer, icludig all ecessary calculatios. Notes ad books will ot be permitted. Explai your aswers clearly ad use graphs whe helpful. ANSWER THIS QUESTION 40 poits. Cosider a variatio o the Solow model where the savigs rate is variable istead of costat. I particular suppose that as usual output is produced competitively via a Cobb-Douglas productio fuctio: Y K α N α, ad the populatio grows at the costat rate > 0. Suppose that there is o depreciatio or productivity growth δ g 0, ad that total savigs is give by S t sry t, where: sr sr φ where we itroduce the costats s > 0 ad φ > 0. Thus sr is icreasig i r, so that higher real iterest rates iduce higher savigs rates. a Solve for the steady state equilibrium per-worker quatities of capital, output, ad cosumptio as a fuctio of α, φ, s, if φ < α. α Solutio: As usual, we itroduce the otatio k K/N, y Y/N k α. The K sry From firm s optimizatio problem Kui Huag, khuag32@wisc.edu k K N K Ṅ N N srkα k sr φ k α k r MPK αk α

2 Thus the fudametal equatio of the Solow Model is The steady state is at k 0, where k sα φ k α+φα k sα k φ sα y k α φ [ sα c sry φ +φ α α +φ α +φ ] sα φ α +φ α b What are the effects of a icrease i the iterest elasticity of savigs φ o the per-worker quatities of capital, output, ad cosumptio? Cosider both the short-ru ad log-ru effects, ad cotiue to assume φ < α eve after the α icrease. Solutio: First cosider the log-ru effects of a icrease i φ o k, y, ad c, we just eed to check the sig of k, y, c. φ φ φ Defie uφ sαφ, vφ +φ α, the Thus k uφ vφ e vφluφ k φ uφ vφ e vφluφ e vφluφ vφluφ uφ vφ v φluφ + vφ u φ uφ Sice u φ sαφ l α, v φ α+φ 2, we have k φ sα φ +φ α sαφ 2 l α + φ + If l α > sαφ l, k > 0, +φ φ k NEW > k, ynew > y ; if l α < sαφ l, k < 0, +φ φ k NEW < k, ynew < y. + φ α l α 2

3 Similarly, c φ If if sα φ α +φ α sαφ α l 2 α + φ + + φ α l α sα φ +φ α sαφ l 2 α + φ + + φ α l α sαφ l 2 α + φ + sα φ α + φ α l α +φ α sα φ α sαφ sα φ α l + φ α + φ + l α +φ α sα φ +φ α +φ +φ l sαφ l sαφ + l α α + l α α sα φ +φ sα φ > 0, c φ > 0, c NEW > c ; +φ < 0, c < 0, φ c NEW < c. +φ α As for the short ru, capital, output ad cosumptio will coverge to the ew steady state. We ca also draw the graph as follows. Suppose 0 < α <, ad gφ sry sα φ k α+φα is the actual ivestmet, k is the break-eve ivestmet, the g φ sα φ k α+φα l αk α. Whe αk α > k < α α, g φ > 0, the ew actual ivestmet lies above the old oe; Whe αk α < k > α α, g φ < 0, the ew actual ivestmet lies below the old oe. c Now suppose φ α. How does this chage the model? What are the effects of α a icrease i the savig fractio s o capital ad output ow? 3

4 Solutio: If φ α α, the k sα α α k The steady state is at k 0, where k sα α α y k α sα α α A icrease i s icreases k ad y i the log ru, ad i the short ru, capital ad output will coverge to the ew steady state. ANSWER TWO OF THE FOLLOWING THREE QUESTIONS 30 poits each 2. Cosider the optimal growth model with ielastic labor supply, ad for simplicity assume that there is o populatio or productivity growth. Household prefereces are: The capital evolutio equatio is: β t uc t t0 α K t+ δ K t + F K t C t. Suppose the ecoomy is iitially i the steady state ad the there is a uexpected ad permaet icrease i household patiece, so β icreases or θ /β falls. a What are the log-ru effects of this chage o cosumptio ad capital? Solutio: The Lagragia fuctio is L β t uc t + λ t [ δ K t + F K t C t K t+ ] t0 By solvig the FOCs with respect to C t ad K t+, we ca get the Euler Equatio: u C t βu C t+ + F K t+ δ where β +θ. I Steady State, C t C t+, from the Euer Equatio, we get the C 0 curve F K t+ δ + θ I Steady State, K t K t+, from the capital evolutio equatio, we get the K 0 curve 4

5 C F K δk Whe θ /β falls, the K 0 curve is uaffected, while the C 0 curve shift to the right. Thus the C ad K both icrease i the log ru. We ca draw the graph as follows: b What happes to cosumptio ad capital at the time of the chage? Solutio: At the time of the chage, C jumps dow to the saddle path, while K is uchaged at that time. After that, both C ad K move alog the saddle path to the ew steady state. 3. Cosider a two period problem where a cosumer has prefereces over cosumptio i the two periods give by: log c + β log c. She has o iitial assets ad has icome y i the first period y i the secod, pays taxes et of beefits T i the first ad T i the secod, ad ca borrow ad led s at iterest rate r, thus givig the two flow costraits: c + s y T, c + rs + y T. The govermet fiaces spedig through taxes ad borrowig: G T + B, G + + r G B T, where the govermet borrows at a lower rate tha households: r G < r. a Solve for the aget s optimal cosumptio choices c ad c ad savigs s. 5

6 Solutio: The preset value budget costrait is c + c + r y T + y T + r yp V The Lagragia fuctio is L log c + β log c + λ y P V c c + r FOCs: λ c β λ c + r We ca get the cosumptio Euler equatio c β + r c Combied with the preset value life time budget costrait, we have c y T + y T + β + r c β + r y T + y T + β + r β s + β y T y T + β + r b Now suppose that the govermet cuts taxes i the curret period, so T falls by some amout, but govermet spedig is uchaged. Thus future taxes must rise to pay back the pricipal ad iterest o the deficit this policy creates. How does this affect the cosumer s optimal choices? Solutio: The govermet preset value budget costrait is T + T + r G G + G + r G The ew tax is T T, while govermet spedig is uchaged. So T T + + r G The ew life time icome for the cosumer is ỹ P V y T + y T + r y T + y T + r + + rg + r y P V + r rg + r 6

7 Sice r G < r, we have ỹ P V i both periods. > y P V. Thus the cosumer will icrease cosumptio 4. Cosider the two-period dyamic geeral equilibrium model, which we ca depict graphically as i class with equilibrium i the labor market labor supply ad demad ad the goods market output supply ad demad. Suppose the ecoomy is iitially i equilibrium, ad the a ew govermet program is aouced. This program will make public ifrastructure ivestmets i the curret period that will be fuded by lump sum tax reveue ad will icrease future productivity. That is, the program combies a icrease i G today oly, ot G as well with a icrease i z i the future. As i class, assume that the respose of labor supply to iterest rates is small. a What effect will the program have o cosumptio ad ivestmet demad, ad thus o output demad? Solutio: The program combies a icrease i G today with a icrease i z i the future. Distiguish betwee shift of the demad, supply curve ad chage of the equilibrium result. If we just cosider a icrease i G today, the icrease i curret or future taxes reduces household wealth, thus Cosumptio demad C d shifts to the left, but by less tha the amout G shifts to the right, ad Y d r C d r + I d r + G shifts to the right. If we just cosider a icrease i z i the future, sice z F K K, N icreases, I d r shifts to the right, Y d r C d r + I d r + G shifts to the right. I sum, C d shifts to the left, but by less tha the amout G shifts to the right, I d r shifts to the right, ad Y d r C d r + I d r + G shifts to the right. b What effect will this program have o labor supply ad labor demad? How will the program affect the output supply curve? Solutio: If we just cosider a icrease i G today, the icrease i curret or future taxes reduces household wealth, thus leisure falls ad so labor supply N s icreases shifts to the right, ad output supply Y s r icreases shifts to the right. There is o effect o labor demad curve N d. If we just cosider a icrease i z i the future, there is o direct effect o labor market oly through chage i r, if r icreases, N s shifts to the right, hece o direct effect o Y s r. There is o effect o labor demad curve N d. I sum, N s shifts to the right, there is o effect o labor demad curve N d, Y s r shifts to the right. c What will be the et equilibrium effects o output, iterest rates, employmet, ad wages? Solutio: We ca draw the equilibrium effects i the two graphs below. The equilibrium output icreases, iterest rate icreases, employmet icreases, wages decrease. 7

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GRADE 11 NOVEMBER 2012 MATHEMATICS P1

GRADE 11 NOVEMBER 2012 MATHEMATICS P1 Provice of the EASTERN CAPE EDUCATION NATIONAL SENIOR CERTIFICATE GRADE 11 NOVEMBER 01 MATHEMATICS P1 MARKS: 150 TIME: 3 hours This questio paper cosists of 14 pages, icludig a iformatio sheet ad a page