ECON 5118 Macroeconomic Theory

with x t = (q t − q, k t − k) T , and[ ]1A =β − kF ′′ (k)φ−F ′′ (k).kφ1What do you expect the eigenvalues of A to be?4. Solve the money in utility problem in Chapter 8 withthe technique of dynamic programming.5. In this question we investigate the long-run fiscalstance in the U.S. as of 2013. You can do your ownresearch. Paul Krugman’s article in the New YorkTimes serves as a starting point.(a) Krugman claims that “the budget doesn’t haveto be balanced to put us on a fiscally sustainablepath.” Explain.(b) In view of equation (5.16) in the textbook, is$460 billion a sustainable deficit for the U.S. in2013?(c) “[R]ising health costs and an aging populationwill put the budget under growing pressure overthe course of the 2020s.” What would be thebest fiscal policy to address this long-run problem?Hints for SelectedAssignment and TestQuestionsChapter 2Question 5(a) With the given output capital ratio show that a =0.1(k ∗ ) 1−α . Then show thatF ′′ (k ∗ ) = α(α − 1)(0.1)/k ∗ = −0.002275.With equation (2.22) you can show thatU ′ (c ∗ )/U ′′ (c ∗ ) = −c ∗ = −0.4.Putting the values in the matrix A and find the twoeigenvalues. Is one of the absolute values less than 1?(b) You can show that the rescaling does not change theresult.Question 9The first-order conditions areβ s U ′ (c t+s − λ t+s = 0, s ≥ 0, (6)(−λ t+s 1 + φi )t+s+ µ t+s = 0, s ≥ 0, (7)k t+s[λ t+s F ′ (k t+s ) + φ ( ) ] 2 it+s2 k t+s−µ t+s−1 + (1 − δ)µ t+s = 0, s ≥ 1. (8)Substitute (6) into (7) and putting s = 0 gives(µ t = U ′ (c t ) 1 + φ i )t. (9)k tEquation (7), with s = 1, also implies thator, rearranging, becomesq t+1 = 1 + φ i tk t, (10)(it+1k t+1) 2= 1 φ 2 (q t+1 − 1) 2 . (11)Equation (2.35) can be obtained by substituting (6), (9),(10), and (11) into (8).9

Question 10In the long run, i = δk. Equation (2.33) implies thatIt follows thatNow (2.35) implies thati = 1 (q − 1)k.φq = 1 + φδ ≥ 1. (2.36)F ′ (k) = (1 + θ)q − (1 − δ)q − 1 (q − 1)22φ(= θ + δ + φδ θ + δ )2Question 11≥ θ + δ.Using the first-order Taylor approximation at q and(2.36),Thus(q t+1 − 1) 2 ≃ (q − 1) 2 + 2(q − 1)(q t+1 − q)= φ 2 δ 2 + 2φδ[q t+1 − (1 + φδ)]= 2φδq t+1 − 2φδ − φ 2 δ 2 .(1 − δ)q t+1 + 12φ (q t+1 − 1) 2 = q t+1 − δ − φδ22 .Putting this into (2.35) and setting c t+1 = c t = c givesq t = βFrom (2.37) and (2.36),which implies that][q t+1 + F ′ (k t+1 ) − δ − φδ2 . (2.38a)2F ′ (k) = θ + δ + φδθ + φδ22= θq + δ + φδ22 ,δ + φδ22 = F ′ (k) − θq.Putting this into (2.38a) gives[q t = β q t+1 + 1 − β ]β q + β [F ′ (k t+1 ) − F ′ (k)] ,which implies thatq t − q = β(q t+1 − q) + β [F ′ (k t+1 ) − F ′ (k)] . (2.38)Chapter 3Question 1Consider the case thatF t (K t , N t ) = A t (α k K ρ t + α n N ρ t ) 1/ρ , 0 ≠ ρ < 1.(a) It is straight forward to show that F t is linearly homogeneous.(b) f t (k t ) = A t (α k k ρ t + α n ) 1/ρ .(c) The growth rate of capital per person γ is decreasingin k t ifdγ= − sy [tdk t kt2 1 − k ]t dy t< 0,y t dk twhich requires the capital elasticity of output to be lessthat one, i.e.,k t dy t< 1.y t dk tThis condition holds if f t (k t ) is straightly concave. Youshould be able to show thatandf ′′f ′ t(k t ) = α k A t k ρ−1t (α k k ρ t + α n ) 1 ρ −1 ,t (k t ) = α k A t (ρ − 1)k ρ−2t (α k k ρ t + α n ) 1 ρ −1[]11 −1 + α n /(α k k ρ .t )The second derivative f ′′t (k t ) is negative since ρ < 1.Question 4(a) By definitiony # t = 1N # tF (K t , N # t ) = F (k # t , 1) = f(k # t ).(b) Assuming that A t = (1 + µ) t and N t = (1 + n) t N 0 ,it can be shown that N # t = (1 + η) t N 0 , where 1 + η =(1 + µ)(1 + n). The dynamic resource constraint isf(k # t ) = c # t + (1 + η)k # t+1 − (1 − δ)k# t .Instantaneous utility becomes (setting N 0 = 1)log C t = log[(1+η) t c # t ] = log c # t +t log(1+η) ≃ log c # t +tη.The optimization problem ismax∞∑c # t+s ,k# t+s+1 s=0[ ]β s log c # t+s + η(t + s)subject to f(k # t ) = c # t + (1 + η)k # t+1 − (1 − δ)k# t .10

The problem can be solved by the Lagrangian methodor by the Bellman equation. The Euler equation isQuestion 6c tβ c# t−1c # [f ′ (k # t ) + 1 − δ] = 1 + η.tWrite the Euler equation as( ) [ σ ( ) ]−(1−α) ct+1kt+1= β 1 + αA− δ .h t+1Take log on both sides, apply the log approximation, anduse the fact thatQuestion 7k t+1=αh t+1 1 − α .The following comments may be helpful: 3Pivot Capital Management points toChina’s incremental capital-output ratio(ICOR), which is calculated as annual investmentdivided by the annual increase inGDP, as evidence of the collapsing efficiencyof investment. Pivot argues that in 2009China’s ICOR was more than double itsaverage in the 1980s and 1990s, implying thatit required much more investment to generatean additional unit of output. However, it ismisleading to look at the ICOR for a singleyear. With slower GDP growth, because ofa collapse in global demand, the ICOR rosesharply everywhere. The return to investmentin terms of growth over a longer period is moreinformative. Measuring this way, BCA Researchfinds no significant increase in China’sICOR over the past three decades.Moreover, if we put aside the effect of depreciation, ψ t isthe reciprocal of the marginal product of capital, whichis decreasing in K. This relationship gives rise to thedownward sloping curve in chart 4 of the above quotedarticle. As the chart indicates, China is still in an earlystage of economic development.Question 8It follows from the capital accumulation equation thatγ = ∆k# t+1k # t= 1 [s(k # t ) α−1 − δ + η ].1 + η1 + η3 “Not just another fake,” Economist, January 14, 2010.In the steady state γ = 0, which implies that the steadystatevalue of k # t becomes( ) 1/(1−α) sk #∗ =. (12)η + δFrom the optimal growth model, the optimal saving rateand the steady-state effective capital areands =α(η + δ)ση + δ + θ(k #∗ α=ση + δ + θ) 1/(1−α)respectively. Setting σ and θ to zero, we haves =k #∗ =α(η + δ), (13)δ( α) 1/(1−α). (14)δSubstituting the saving rate in (13) into (12), the effectivecapital becomes the same as in (14). This meansthat the Solow-Swan model is a special case of the optimalgrowth model when the social discount rate θ becomeszero.Chapter 4Question 61. Let I n = ∑ ns=1 sρ(1 − ρ)s−1 .2. Find (1 − ρ)I n .3. Derive I n − (1 − ρ)I n = ρI n .4. Find lim n→∞ ρI n .Question 10(a) The budget constraint of the household isp t ∆a t+1 + c t = x t + d t a t ,where x t is the exogenous income and a t is the numberof shares owned, both in period t. The price of ∆a t+1is p t because it is bought before the dividend is paid inperiod t + 1. The Bellman equation is{v t (a t ) = max c t + 1c t 1 + r v t+1The necessary conditions are1 − 11 + rλ t+11 + r( )}xt + d t a t + p t a t − c t.p tλ t+1p t= 0, (15)p t + d tp t= λ t , (16)11

where λ t = v ′ t(a t ). Equation (15) implies that λ t+1 =p t (1 + r) and λ t = p t−1 (1 + r). Substituting these intoequation (16) gives the result.(b) The first-order difference equation in p t isThe solution isp t+1 = (1 + r)p t − d t+1 .p t =∞∑s=1d t+s(1 + r) s , (17)which means the price of the stock in period t is the sumof the present values of future dividends. Notice that nomarket bubble means that the transversality conditionis satisfied.(c) It is obvious from equation (17) that an increase ininterest rate will cause the price of the stock to go down.Chapter 5Question 1For n = 1, and from the table provided, the GBCs atperiods t and t−1 imply that ∆b t+1 = ∆g t . Now assumethat equation (5) holds for periods up to t + n − 2. Fromthe GBC in period t + n − 1,where∆b t+n = ∆g t {(1 + R) + R[(1 + R) + (1 + R) 2+ · · · + (1 + R) n−2 ]}= ∆g t {(1 + R) + Rσ}, (18)σ = (1 + R) + (1 + R) 2 + · · · + (1 + R) n−2 .Multiplying both sides by 1 + R and subtracting to getσ = 1 R [(1 + R)n−1 − (1 + R)].Substitute σ into (18) and the result follows.Question 4The Bellman equation isv(k t ) = maxc t,g t{U(c t , g t ) + βv(F (k t ) − c t − g t + (1 − δ)k t )}.The necessary conditions for dynamic optimization are∂v∂c t= U c,t − βλ t+1 = 0,∂v∂g t= U g,t − βλ t+1 = 0,∂v∂k t= βλ t+1 [F ′ (k t ) + 1 − δ] = λ t ,where λ t = v ′ (k t ). The Euler equation isβU c,tU c,t−1[F ′ (k t ) + 1 − δ] = 1,and the marginal rate of substitution between privateand government consumptions isMRS = U c,tU g,t= 1.In the steady state F ′ (k t ) = δ + θ.Question 5(b) The MRS between leisure and consumption isηc σ tl σ t= (1 − τ w )w t1 + τ c (19)(c) Using l t = 1 − n t , equation (19) becomes[ η(1 + τ c ] 1/σ)n t = 1 −(1 − τ w c t .)w tTherefore n t is decreasing in τ w .Chapter 8Question 1Since c t = m t for all t, by choosing m t+1 in period t,c t+1 becomes a state variable in period t + 1. Thereforethe control variable is m t+1 and the state variables arec t and b t . The utility maximization problem becomesmax β t U(c t ),m t+1subject to the transition equation[ ] []ct+1m=t+1.b t+1 [x t + (1 + R t )b t ]/(1 + π t+1 ) − m t+1The Bellman equation is{v(c t , b t ) = max U(c t ) + βv(m t+1 ,m t+1}[x t + (1 + R t )b t ]/(1 + π t+1 ) − m t+1 ) .Let ∇v(c t , b t ) = λ t = [λ c,t λ b,t ] T , the necessary conditionsareβ(λ c,t+1 − λ b,t+1 ) = 0, (20)and[U ′ (c t )0]+β[ ] [ ] [0 0λc,t+1 λc,t=0 (1 + R t )/(1 + π t+1 ) λ b,t+1λ b,t].(21)12

From (20) we haveλ c,t+1 = λ b,t+1 so that λ c,t = λ b,t . (22)The first line of (21) implies thatλ c,t = U ′ (c t ) so that λ c,t+1 = U ′ (c t+1 ). (23)By (22), the second line of (21) can be written asβ(1 + R t )1 + π t+1λ c,t+1 = λ c,t .Substitute (23) into the above, we get the Euler equationQuestion 2βU ′ (c t+1 )U ′ (c t )1 + R t= 1.1 + π t+1If r = R − π > 0, the budget constraint implies thatb t + m t = 11 + R∞∑s=0( ) s 1 + π(c t+s − x t+s + Rm t+s ).1 + R(24)Putting c t+s = c t and m t+s = m t for s = 1, 2, . . . , into(24) givesb t +m t = 1 + r c t1 + R r − 11 + R∞∑s=0x t+s(1 + r) s + R1 + R1 + rm t ,rwhere have used 1/(1 + r) = (1 + π)/(1 + R) and resultsfrom geometric series. Now assume that 1 + R ≃ 1 + rfor low inflation rate, we haveb t + m t = c tr − 11 + r∞∑s=0x t+s(1 + r) s + R r m t.Using r = R − π and rearranging yields the result.Question 3(b) The Euler equation isMoney demand is( ) η ct m t+1 1 + R t+1β= 1.c t+1 m t 1 + π t+1m t =η c t,1 − η R twhich implies that a higher interest rate reduces moneydemand.Question 7LetThenI =∞∑sx s = x + 2x 2 + 3x 3 + · · · .s=0Proceed to show thatxI = x 2 + 2x 3 + 3x 4 + · · · .I − xI =x1 − x ,and solve for I. In (8.22), result for the m t term isstraight forward. For the µs term, let x = α/(1+α) andapply the above result to get αµ.Question 10The following argument may be helpful.1. If preferences are homothetic, U can be expressedas a linearly homogeneous function.2. Apply the Euler theorem (part 2) to U. That is, ifwe let x = (c, l, m), then∇ 2 Ux = 0.3. Show that the last row of the above matrix multiplicationmeans thatU cm,t c t + U lm,t l t + U mm,t m t = 0.4. Now show that the first-order condition of the government’sproblem with respect to m t becomesV m,t = (1 + µ)U m,t − µU lm,t = 0.5. Since U is separable in l t and m t , U lm,t = 0.6. Conclude that U m,t = 0.Chapter 9Question 2The first-order conditions for the final-good producer’sprofit maximization problem areα i P( yx i) 1/φ− p i = 0, i = 1, . . . , N,where φ = 1/(1−ρ) is the elasticity of substitution. Theconditional demand function for input i is thereforex i =(αi Pp i) φy. (9.15a)13

Tests2012W, Test 2, Question 3Since c t−1 is given in period t, it is effectively a statevariable. So let y t = c t−1 . The problem becomesmaxc t∞ ∑t=1β t U(c t , y t ),subject to the transition equation[ ] [ ]yt+1 c= t.w t+1 w t − c tThe Bellman equation isLetv(y t , w t ) = maxc t{U(ct , y t ) + βv(c t , w t − c t ) } .∇v(y t , w t ) =The necessary conditions areand[ ]λy,t.λ w,tU c,t + β(λ y,t+1 − λ w,t+1 ) = 0, (37)[ ]Uy,t+ β0Putting (38) into (37) to getso that[ ] [ ] [ ]0 0 λy,t+1 λy,t= . (38)0 1 λ w,t+1 λ w,tU c,t + βU y,t+1 − λ w,t = 0,λ w,t = U c,t + βU y,t+1 . (39)Push forward one period, we haveλ w,t+1 = U c,t+1 + βU y,t+2 . (40)Substitute (39) and (40) into the second line of (38) andreplace y t with c t−1 , we obtain the Euler equationβ ∂U(c t+1, c t )∂c t+1+ β 2 ∂U(c t+1, c t )∂c t= ∂U(c t, c t−1 )∂c t+ β ∂U(c t, c t−1 )∂c t−1.c○2013 The Pigman Inc. All Rights Reserved.16