Macroeconomics 1. Ali Shourideh. Final Exam

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1 Macroeconomic 1 Ali Shourideh Final Exam Problem 1. A Model of On-he-Job Search Conider he following verion of he McCall earch model ha allow for on-he-job-earch. In paricular, uppoe ha ime i coninuou and worker - employed or unemployed dicoun he fuure a rae ρ. A worker ha ha a job eparae from her job a rae η. An unemployed worker find a job a rae λ 0 - he wage i hen drawn from a diribuion F w) which i coninuou. Moreover, a worker who ha a job ge an opporuniy o draw a new wage offer a rae λ 1 - again from he ame diribuion F w). The worker hen decide wheher o accep hi new offer or no. A uual, aume ha he flow value of unemploymen i b; he worker are rik-neural and canno ave. a. Le V w) be he value of a job a hand ha pay he wage w. Le U be he value of unemploymen. Wrie he Bellman equaion for U and V w). Soluion. The value funcion are given by ρv w) = w + λ 1 max {V w ) V w), 0} df w ) + η U V w)) 1) ρu = b + λ 0 max {V w) U, 0} df w) ) b. Calculae he reervaion wage, w, for an unemployed worker a a funcion of he value funcion V w). Under wha condiion, w < b hold? Provide an inuiion for hi reul. Soluion. For w, we mu have ha U = V w ). Subracing, ) from 1) evaluaed a w, we have w = b + λ 0 λ 1 ) max {V w) V w ), 0} df w) When λ 0 < λ 1, he above how ha w < b. Inuiively, when λ 1 > λ 0, acceping a job give a beer opion value o a worker han being unemployed - a higher probabiliy of a new draw which alway increae he value for a worker. Thi implie ha an unemployed worker i willing o ake a job a a wage lower han unemploymen benefi in order o obain hi beer opion value. c. Wha i he reervaion wage for an employed worker? Soluion. Since V w) i an increaing funcion of w, he reervaion wage of an employed worker i w - he wage of he job he i currenly employed in. d. How doe an increae in unemploymen benefi, b, affec he job-o-job urnover? Show your reul and explain inuiively. 1

2 Soluion. We have ρ + η) V w) = w + λ 1 ρ + η) V w) = 1 λ 1 V w) 1 F w)) V 1 w) = ρ + η + λ 1 1 F w)) w V w ) V w) df w ) + ηu Therefore, w = b + λ 0 λ 1 ) V w) V w ) df w) w = b λ 0 λ 1 ) V w) V w ) d 1 F w)) w = b + λ 0 λ 1 ) 1 F w)) V w) dw w = b + λ 0 λ 1 ) w 1 F w) ρ + η + λ 1 1 F w)) dw We have w b w b = 1 λ 0 λ 1 ) w b 1 + λ 0 λ 1 ) 1 F w )) = 1 ρ + η + λ 1 1 F w )) w ρ + η + λ0 1 F w )) = 1 b ρ + η + λ 1 1 F w )) 1 F w ) ρ + η + λ 1 1 F w )) Thi implie ha w > 0. A rie in he reervaion wage lead o a lower urnover a le people b accep job and herefore le people move beween job a well.

3 Problem. A Sochaic AK economy Conider an economy in which producion i done only wih capial and oupu i given by Y = A K where A i an i.i.d. ochaic proce and i diribued according o he c.d.f. F A). Capial depreciae a rae δ. Suppoe ha he economy exhibi a repreenaive agen and ha preference of uch hypoheical agen i given by β u C ) where u ) i a CRRA uiliy funcion given by { c 1 γ γ 1 1 γ u c) = log c γ = 1 =0 a. Define a Compeiive Equilibrium for hi economy auming ha rading occur in equenial ae marke. { Soluion. A compeiive equilibrium for hi economy i a equence of allocaion C ), X ), K +1 ), B a well a price {p ), r ), q )} where = A and: 1. Houehold olve: ubjec o p ) C ) + X ) + max β π ) u C )) =0 +1 q ) B ) r ) K 1 ) + p ) B ) 1 δ) K 1 ) + X ) = K +1 ). Firm olve 3. Marke clear max p ) A ) k r ) k k B ) = 0 ) = K ) 1 K f C ) + X ) = A ) K 1 ) Noe ha if A ha a coninuou diribuion, he above really hould be re-wrien wih inegral over inead of um. 3

4 b. Sae he Fir Welfare Theorem and ue i o formulae a compeiive equilibrium a a oluion o a planning problem. Soluion. According o he FWT, a compeiive equilibrium mu be pareo opimal. A a reul, i mu olve he following planning problem max β π ) u C )) ubjec o =0 C ) + K +1 ) = A ) K 1 ) + 1 δ) K 1 ) c. Formulae he planning problem problem recurively. You are required o ue only one ae variable. Soluion. Bellman equaion i given by V Y ) = max u C) + β ubjec o V A + 1 δ) K ) df A ) C + K = Y d. Solve he funcional equaion aociaed wih hi recurive problem and find he value funcion and policy funcion uing a gue and verify mehod. Wha aumpion on he fundamenal hould be made o ha he oluion o hi problem exi and i unique? Soluion. We gue ha V Y ) = B Y 1 γ. Wih hi gue, he opimizaion in he FE above 1 γ become Y K ) 1 γ A + 1 δ) K ) 1 γ max + βb df A ) K If we le x = K /Y, hen olving he above opimizaion i equivalen o 1 x) 1 γ A + 1 δ) x) 1 γ max + βb df A ) x Taking FOC, we have 1 x) γ = βbx γ E A + 1 δ) 1 γ ) γ x = βbe A + 1 δ) 1 γ 1 x For each B he above ha a unique oluion. We can herefore, replacing in he objecive in he FE and have v Y ) = C)1 γ A + βb + 1 δ) K ) 1 γ df A ) = Y 1 γ 1 x) 1 γ + βbx 1 γ E A + 1 δ) 1 γ = Y 1 γ 1 x) 1 γ + x 1 x) γ = Y 1 γ 1 x) γ = βbx γ E A + 1 δ) 1 γ Y 1 γ 4

5 which ha he form a he gueed value. If we e he above equal o B Y 1 γ, we have 1 γ βx γ E A + 1 δ) 1 γ = 1 x = βe A + 1 δ) 1 γ) 1 γ and B = 1 x) γ = 1 βe A + 1 δ) 1 γ) 1 ) γ γ Therefore, in order for hi oluion o be meaningful, we mu have ha βe A + 1 δ) 1 γ < 1. The policy funcion ha he form K = xy = x A + 1 δ) K e. Uing he policy funcion calculaed above, calculae he average growh rae of hi economy - auming ha he daa come from he aionary diribuion of he model. The reul hould be expreed in erm of he fundamenal parameer of he model a well a variou momen of he diribuion F ). Soluion. The growh rae of he economy i given by 1 + g +1 = A +1K +1 A K = A +1x A + 1 δ) K A K = x A +1 A + 1 δ) A Since hock are i.i.d., hi average growh rae i given by A + 1 δ 1 + g = xe A E A = βe A + 1 δ) 1 γ) 1 γ A + 1 δ E A E A In wha follow, aume ha δ = 1 and ha F A) = A 0 1 e log x+σ σx π / log A) Tha i, A i diribued ) according o a log-normal diribuion whoe mean i A and i variance i e σ 1 A. f. Wha i he average growh rae of he economy? How doe growh inerac wih rik, i.e., how doe he average growh rae of he economy depend on σ? Explain your anwer inuiively. σ dx 5

6 Soluion. From above, we have Noe ha log A N Therefore, 1 + g = βe A 1 γ) 1 γ E A log A σ, σ ). Therefore, E A 1 γ = E e 1 γ) log A 1 γ)γ 1 γ) log A+ σ = e 1 + g = β 1 1 γ A γ e 1 γ)σ Therefore, an increae in rik σ lead o an increae in growh when γ < 1 and a decreae in growh when γ > 1. The inuiion for hi reul can be een by rewriing he Euler equaion a follow: u C ) = βe A +1 u C +1 ) = β {E A +1 E u C +1 ) + Cov A +1, u C +1 ))} An increae in rik, ha wo effec: 1. i increae he average marginal uiliy E u C +1 ). Thi i becaue marginal uiliy i a convex funcion of conumpion and an increae in rik raie i average value - i increae he probabiliy of ail even. The conumer would like o inve more o inure hemelve again he rik of uch even a a precauion.. i decreae he correlaion beween marginal uiliy and reurn; ha i, i make i more negaive - uual rikaverion. When γ < 1, he precauionary moive dominae while when γ > 1, he rik-averion moive dominae. Thu for low value of γ, an increae in rik lead o higher invemen and higher growh while for high value of γ an increae in rik lead o low invemen and growh. g. Calculae he price of a rik-free real bond, P f. Wha i he rik-free rae in hi economy. Soluion. The price of a rik-free bond i given by The reurn on he ae i given by P f = βe u C +1 ) u C ) = βe 1 x) xa +1 A K ) γ 1 x) A K ) γ = βx γ E A γ +1 = β βe A 1 γ) γ γ E A γ = E A γ E A 1 γ R f = 1 P f = E A1 γ E A γ 6

7 h. Calculae he price, P, of holding a firm ha own he phyical capial and inve from he oupu ha i produce and how ha P = K ; Hin: Thi i an ae wih a dividend proce given by D = A K K +1. Wha i he reurn on uch an ae? Soluion. We have u C +1 ) P = βe u C ) D +1 + P +1 1 C γ = βe β D =+1 = βe =+1 +1 = βe A +1 K +1 + βe =+ +1 = βe A +1 K +1 + β 1 = βe 1 C γ β A K K +1 ) 1 C γ β A K 1 β 1 C γ K ) ) E β 1 β C γ A K K = A +1 K +1 + β 1 E β C 1K γ By he Euler equaion, we have E 1 β C γ 1 =+ A = 1 E 1 E =+ β C γ 1 β C γ 1 A 1 = 0 β 1K E 1 β C γ 1 A 1 = 0 β C γ 1 ) A 1 where he la equaliy follow from law of ieraed expecaion. We herefore have P = K +1 + β 1 ) E A 1 = K +1 which complee he claim. The reurn on he capial i given by R k +1 = P +1 + D +1 P = A +1K +1 K + + K + K +1 = A +1 7

8 i. Calculae he equiy premium a he difference beween average reurn on phyical capial and he rik-free rae. How doe hi depend on rik-averion, γ, and variance of he hock, σ. Explain inuiively. Soluion. The equiy premium i defined a E R k R = EA E A 1 γ E A γ = EAE A γ E A 1 γ E A γ = Ae γ log A σ )+ γ σ e 1 γ) e γ log A σ ) + γ σ = A1 γ e γ1+γ)σ A 1 γ e 1 γ)γσ A 1 γ e γ1+γ)σ = A 1 e γσ log A )+ σ 1 γ) σ Thi i an increaing funcion of γ and σ. A σ rie, he rik inheren in holding phyical capial increae and a a reul inveor will require a higher rae of reurn. The ame logic hold for γ; a houehold become more rik-avere hey demand a higher average rae of reurn in order o hold he capial ock. 8