Suppose output is given by y = x +(k +εk)z +u, where z is some policy instrument controlled by the government and k is the expected value of the multiplier for that instrument. εk and u are independent, mean-zero disturbances that are unknown when the policymaker chooses z, and that have variances σ2 k and σ2 u. Finally, x is a disturbance that is known when zis chosen. The policy maker wants to minimize E[(y−y∗)2]. (a) Find E[(y− y∗)2] as a function of x, k, y∗, σ2 k, and σ2 u. (b) Find the first-order condition for z, and solve for z. (c) How, if at all, does σ2 u affect how policy should respond to shocks (that is, to the realized value of x)? Thus, how does uncertainty about the state of the economy affect the case for ‘‘fine-tuning’’? (d) How, if at all, does σ2 k affect how policy should respond to shocks (that is, to the realized value of x)? Thus, how does uncertainty about the effects of policy affect the case for ‘‘fine-tuning’’?
> Consider an economy consisting of a constant population of infinitely lived individuals. The representative individual maximizes the expected value of∞ t=0 u(Ct)/(1+ρ)t, ρ>0. The instantaneous utility function, u(Ct), is u(Ct) = Ct −θC2 t , θ>0. Assume t
> (a) Use an argument analogous to that used to derive equation (5.23) to show that household optimization requires b/(1 − t) = e−ρEt [wt(1 + rt+1)b/ wt+1(1−t+1)] (b) Show that this condition is implied by (5.23) and (5.26). (Note that [5.26] must hold in
> Suppose an individual lives for two periods and has utility lnC1 +lnC2. (a) Suppose the individual has labor income of Y1 in the first period of life and zero in the second period. Second-period consumption is thus (1+ r)(Y1 − C1); r, the rate of return,
> Consider the problem investigated in (5.16) (5.21). (a) Show that an increase in both w1 and w2 that leaves w1/w2 unchanged does not affect 1 or 2. (b) Now assume that the household has initial wealth of amount Z > 0. (i) Does (5.23) continue to hold? Wh
> Suppose the period-t utility function, ut, isut = lnct +b(1−t )1−γ/(1−γ), b >0, γ>0, rather than (5.7). (a) Consider the one-period problem analogous to that investigated in (5.12) (5.15). How, if at all, does labor supply depend on the wage? (b) Conside
> Let A0 denote the value of A in period 0, and let the behavior of ln A be given by equations (5.8) (5.9). (a) Express ln A1, lnA2, and ln A3 in terms of ln A0, εA1, εA2, εA3, A, and g. (b) In light of the fact that the expectations of the εA’s are zero,
> Consider the equation of motion for capital, Kt+1 = Kt+Kα t (AtLt)1−α−Ct −Gt −δKt. (a)( i) Show that ∂ln Kt+1/âˆ
> (a) If the ~ At’s are uniformly 0 and if lnY t evolves according to (5.39), what path does lnY t settle down to? (Hint: Note that we can rewrite [5.39] as lnY t−( n + g)t =Q + α[lnY t−1 −(n+g)(t −1)]+(1−α) ~ At, where Q ≡ α lnˆ s + (1−α)(A + ln ˆ + N )−α
> Consider the model of Section 5.5. Suppose, however, that the instantaneous utility function, ut, is given by ut = lnct+b(1−t )1−γ/(1−γ),b >0,γ>0, rather than by (5.7) (see Problem 5.4). (a) Find the first-order condition analogous to equation (5.26) tha
> Suppose that each worker must either work a fixed number of hours or be unemployed. Let CE i denote the consumption of employed workers in state i and CU i the consumption of unemployed workers. The firm’s profits in state i are therefore Ai F(Li)−[CE i
> Suppose technology follows some process other than (5.8) (5.9). Do st = ˆ s and t = ˆ for all t continue to solve the model of Section 5.5?
> Consider the model of Section 5.5. Assume for simplicity that n=g=A=N=0. Let V(Kt,At), the value function, be the expected present value from the current period forward of lifetime utility of the representative individual as a function of the capital sto
> Consider the model of Section 5.3 without any shocks. Let y∗, k∗, c∗, and G∗ denote the values of Y/(AL), K/(AL), C/(AL), and G/(AL) on the balanced growth path; w∗ the value of w/A; ∗ the value of L/N; and r∗ the value of r. (a) Use equations (5.1) (5.4
> Suppose that Y(t) =K(t)α[(1−aH)H(t)]β, H(t)=BaHH(t), and K(t)=sY(t). Assume 0
> Consider the following model with physical and human capital: where aK and aH are the fractions of the stocks of physical and human capital used in the education sector. This model assumes that human capital is produced in its own sector with its own p
> Consider the model of Section 4.1 with the assumption that G(E)=eφE. Suppose, however, that E, rather than being constant, is increasing steadily: E(t) = m, where m > 0. Assume that, despite the steady increase in the amount of education people are getti
> Consider the model in Problem 4.5. (a) What are the balanced-growth-path values of k and h in terms of sk, sh, and the other parameters of the model? (b) Suppose α = 1 3 and β = 1 2. Consider two countries, A and B, and suppose that both sk and sh are tw
> Suppose output is given by Y(t)=K(t)αH(t)β[A(t)L(t)]1−α−β, α>0, β>0, α + β
> Suppose the production function is Y = Kα(e φEL)1−α,0
> Suppose output in country i is given by Yi = Ai Qie φEiLi. Here Ei is each worker’s years of education, Qi is the quality of education, and the rest of the notation is standard. Higher output per worker raises the quality of education. Specifically, Qi i
> Suppose there are a large number of firms, N, each with profits given by F(eL)−wL, F(•) > 0, F(•) < 0. L is the number of workers the firm hires, w is the wage it pays, and e is workers’ effort. Effort is given by e =min[w/w∗,1], where w∗ is the ‘‘fair w
> Consider the Tabellini Alesina model in the case where α can only take on the values 0 and 1. Suppose that there is some initial level of debt, D0. How, if at all, does D0 affect the deficit in period 1?
> Suppose there are three voters, 1, 2, and 3, and three possible policies, A, B, and C. Voter 1’s preference ordering is A, B, C; voter 2’s is B, C, A; and voter 3’s is C, A, B. Does any policy win a majority of votes in a two-way contest against each of
> If the tax rate follows a random walk (and if the variance of its innovations is bounded from below by a strictly positive number), then with probability 1 it will eventually exceed 100 percent or be negative. Does this observation suggest that the tax-s
> Consider the Barro tax-smoothing model. Suppose there are two possible values of G(t) GH and GL with GH > GL. Transitions between the two values follow Poisson processes (see Sections 7.4 and 11.2). Specifically, if G equals GH, the probability per unit
> Consider the Barro tax-smoothing model. Suppose that output, Y, and the real interest rate, r, are constant, and that the level of government debt outstanding at time 0 is zero. Suppose that there will be a temporary war from time 0 to time τ. Thus G(t)
> Consider an individual who lives for two periods. The individual has no initial wealth and earns labor incomes of amounts Y1 and Y2 in the two periods. Y1 is known, but Y2 is random; assume for simplicity that E[Y2]=Y1. The government taxes income at rat
> By definition, the budget deficit equals the rate of change of the amount of debt outstanding: δ(t)≡ D(t). Define d(t) to be the ratio of debt to output: d(t) ≡ D(t)/Y(t). Assume that Y(t) grows at a constant rate g > 0. (a) Suppose that the deficit-to-o
> Consider the model of crises in Section 13.9, and suppose T is distributed uniformly on some interval [μ−X,μ+X], where X > 0 and μ−X ≥0. Describe how, if at all, each of the following developments affects the two curves in (R,π) space that show the deter
> Consider the same setup as in Problem 13.14. Suppose, however, that there is an initial level of debt, D. The government budget constraint is therefore D +M i=1 Gi = MT. (a) How does an increase in D affect the Nash equilibrium level of G? (b) Explain in
> Suppose the economy consists of M >1 congressional districts. The utility of the representative person living in district i is E +V(Gi)− C (T ). E is the endowment, Gi is the level of a local public good in district i, and T is taxes (which are assumed t
> Suppose that in the Shapiro Stiglitz model, unemployed workers are hired according to how long they have been unemployed rather than at random; specifically, suppose that workers who have been unemployed the longest are hired first. (a) Consider a steady
> Consider the model in Section 13.6. Suppose an international agency offers to give the workers and capitalists each an amount F > 0 if they agree to reform. Use analysis like that in Problem 13.12 to show that this aid policy unambiguously raises the pro
> Consider the model in Section 13.7. Suppose, however, that if there is no reform, workers and capitalists both receive payoffs of −C rather than 0, where C ≥0. (a) Find expressions analogous to (13.37) and (13.38) for workers’ proposal and the probabilit
> Consider the Alesina Drazen model. Describe how, if at all, each of the following developments affects workers’ proposal and the probability of reform: (a) A fall in T. (b) A rise in B. (c) An equal rise in A and B.
> Suppose there aretwoperiods.Governmentpolicywillbecontrolledbydifferentpolicymakers in the two periods. The objective function of the period-t policymaker is U + αt[V(G1)+V(G2)], where U is citizens’ utility from their private consumption; αt is the weig
> Consider the Tabellini Alesina model in the case where α can only take on the values 0 and 1. Suppose, however, that there are 3 periods. The period-1 median voter sets policy in periods 1 and 2, but in period 3 a new median voter sets policy. Assume tha
> Consider the Tabellini Alesina model in the case where α can only take on the values 0 and 1. Suppose that the amount of debt to be issued, D, is determined before the preferences of the period-1 median voter are known. Specifically, voters vote on D at
> Assume, as in Problem 12.2, that prices are completely unresponsive to unanticipated monetary shocks for one period and completely flexible thereafter. Assume also that y= c −ar and m − p = b +hy−ki hold each period. Suppose, however, that the money supp
> Consider a discrete-time model where prices are completely unresponsive to unanticipated monetary shocks for one period and completely flexible thereafter. Suppose the IS equation is y = c −ar and that the condition for equilibrium in the money market is
> Consider a discrete-time version of the analysis of money growth, inflation, and real balances in Section 12.1. Suppose that money demand is given by mt − pt = c−b(Etpt+1 − pt), where m and p are the logs of the money stock and the price level, and where
> Suppose that output is given by y = yn +b(π − πe), and that the social welfare function is γy−aπ2/2, where γ is a random variable with mean γ and variance σ2 γ . πe is determined before γ is observed; the policymaker, however, chooses π after γ is known.
> Describehoweachofthefollowingaffectsequilibriumemploymentandthewage in the Shapiro Stiglitz model: (a) An increase in workers’ discount rate, ρ. (b) An increase in the job breakup rate, b. (c) A positive multiplicative shock to the production function (t
> Suppose a policymaker is in office for two periods. Output is given by (12.63) each period. There are two possible types of policymaker, type 1 and type 2. A type-1 policymaker, which occurs with probability p, maximizes social welfare, which for simplic
> Suppose inflation is determined as in Section 12.8. Suppose the government is able to reduce the costs of inflation; that is, suppose it reduces the parameter a in equation (12.64). Is society made better or worse off by this change? Explain intuitively.
> ConsidertheKrugmanmodelofSection12.7.Assumetheeconomyisina steady state starting in period 3 and that i1 =0. (a) Suppose i2 =0. (i) How, if at all, does an increase in M2, holding M1 and M∗ fixed, affect P1? Explain. (ii) How, if at all, does an increase
> Consider the Krugman model of Section 12.7. Assume that i1 = 0 and that the economy is in steady state starting in period 2. Suppose, however, that y1 (the value of y in period 1) need not equal y∗ (the value of y starting in period 2). How, if at all, d
> Consider the Krugman model of Section12.7. Assume the economy is in a steady state of the type described in that section starting in period 2. Suppose, however, that prices are completely sticky in period 1, so that P1 = P, and that it is output rather t
> Suppose inflation is described by the accelerationist Phillips curve, π(t)= λy(t),λ>0, and that output is determined by a simple IS curve, y(t) =− b[i(t)−π(t)], b > 0. Initially, the central bank is setting the nominal interest rate at a strictly positiv
> Suppose the economy is described by linear IS and money-market equilibrium equations that are subject to disturbances: y=c−ai+ε1, m− p =hy−ki+ε2, where ε1 and ε2 are independent, mean-zero shocks with variances σ2 1 and σ2 2, and where a, h, and k are po
> Consider the system given by (12.41). (a) What does the system simplify to when φπ = 1? What are the eigen values of the system in this case? Suppose we look for self-fulfilling movements in ~ y and π of the form πt = λtZ, ~ yt = cλtZ, |λ|≤1. When φπ = 1
> Consider the model of Section 12.4. Suppose, however, the aggregate supply equation, (12.16), is πt = πt−1 +α(yt−1 − yn t−1 )+επ t , where επ is a white-noise shock that is independent of εIS and εY. How, if at all, does this change to the model change e
> Summers (1988, p. 386) states, ‘‘In an efficiency wage environment, firms that are forced to pay their workers premium wages suffer only second-order losses. In almost any plausible bargaining framework, this makes it easier for workers to extract conces
> Consider an economy where money is neutral. Specifically, assume that πt = mt and that r is constant at zero. Suppose that the money supply is given by mt =kmt−1 +εt, where ε is a white-noise disturbance. (a) Assume that the rational-expectations theory
> Suppose that aggregate supply is given by the Lucas supply curve, yt = yn+b(πt−πe t ),b > 0,andsupposethatmonetarypolicyisdetermined by mt =mt−1+a+εt, where ε is a white-noise disturbance. Assume that private agents do not know the current values of mt o
> Suppose you want to test the hypothesis that the real interest rate is constant, so that all changes in the nominal interest rate reflect changes in expected inflation. Thus your hypothesis is it =r + Etπt+1. (a) Consider a regression of it on a constant
> Suppose that instead of adjusting their real money holdings gradually toward the desired level, individuals adjust their expectation of inflation gradually toward actual inflation. Thus equations (12.80) and (12.81) are replaced by m(t)=Cexp(−bπe(t)) and
> Suppose that money demand is given by ln(M/P) = a −bi+ lnY, and that Y is growing at rate gY. What rate of inflation leads to the highest path of seignorage?
> Suppose the relationship between output and inflation is given by yt = yn +b(πt −Et−1πt), where b >0 and where Et−1 denotes the expectation as of period t−1. Suppose there are two types of politicians, ‘‘liberals’’ and ‘‘conservatives.’’ Liberals maximiz
> Suppose the relationship between unemployment and inflation is described by πt =πt−1−α(ut − un ) + εS t , α>0, where the εS t ’s are i.i.d., mean-zero disturbances with cumulative distribution function F(•). Consider a politician who takes office in peri
> Consider the situation analyzed in Problem 12.19, but assume that there is only some finite number of periods rather than an infinite number. What is the unique equilibrium? Data from Problem 12.19: Consider a policymaker whose objective function is ∞
> Consider the following model of income determination. (1) Consumption depends on the previous period’s income: Ct =a+bYt−1. (2) The desired capital stock (or inventory stock) is proportional to the previous period’s output: K∗ t = cY t−1 . (3) Investment
> Consider the situation described in Problem 12.19. Find the parameter values (if any) for which each of the following is an equilibrium: (a) One-period punishment. πe t equals ˆ π if πt−1 = πe t−1 and equals b/a otherwise; π = ˆ π each period. (b) Severe
> (a) Convergence. Let yi denote log output per worker in country i. Suppose all countrieshavethesamebalanced-growth-pathleveloflogincomeperworker, y∗. Suppose also that yi evolves according to dyi(t)/dt=− λ[yi(t)− y∗]. (i) What is yi(t) as a function of y
> Briefly explain whether each of the following statements concerning a cross country regression of income per person on a measure of social infrastructure is true or false: (a) ‘‘If the regression is estimated by ordinary least squares, it shows the effec
> Suppose the true relationship between social infrastructure (SI) and log income per person (y) isyi =a +bSIi+ei. There are two components of social infrastructure, SIA and SIB (with SIi = SIA i +SIB i), and we only have data on one of the components, SIA
> Consider the model of Section 4.1 with the assumption that G(E) takes the form G(E)=e φE. (a) Find an expression that characterizes the value of E that maximizes the level of output per person on the balanced growth path. Are there cases where this value
> Suppose that policymakers, realizing that monopoly power creates distortions, put controls on the prices that patent-holders in the Romer model can charge for the inputs embodying their ideas. Specifically, suppose they require patent holders to charge δ
> Consider the model of Section 3.5. Suppose, however, that households have constant-relative-risk-aversion utility with a coefficient of relative risk aversion of θ. Find the equilibrium level of labor in the R&D sector, LA.
> Suppose that output is given by equation(3.22), Y(t)=K(t)α [A(t)L(t)]1−α; that L is constant and equal to 1; that K(t)=sY(t); and that knowledge accumulation occurs as a side effect of goods production: A(t)=BY(t). (a) Find expressions for gA(t) and gK(t
> Consider the model of Section 3.3 with β +θ>1 and n > 0. (a) Draw the phase diagram for this case. (b) Show that regardless of the economy’s initial conditions, eventually the growth rates of A and K (and hence the growth rate of Y) are increasing contin
> Consider a policymaker whose objective function is ∞ t=0 βt(yt −aπt2/2), where a >0 and 0
> Suppose the economy is described as in Problem 7.1, and assume for simplicity that m is a random walk (so mt = mt−1 +ut, where u is white noise and has a constant variance). Assume the profits a firm loses over two periods relative to always having pt =
> Consider the model of Section 3.3 with β +θ =1 and n =0. (a) Using (3.14) and (3.16), find the value that A/K must have for gK and gA to be equal. (b) Using your result in part (a), find the growth rate of A and K when gK = gA. (c) How does an increase i
> Consider the economy described in Section 3.3, and assume β + θ 0. Suppose the economy is initially on its balanced growth path, and that there is a permanent increase in s. (a) How, if at all, does the change affect the gA = 0 and gK = 0 lines? How, if
> Consider the economy analyzed in Section 3.3. Assume that θ + β 0, and that the economy is on its balanced growth path. Describe how each of the following changes affects the gA=0 and gK =0 lines and the position of the economy in (gA,gK) space at the mo
> Consider two economies (indexed by i =1,2) described by Yi(t)=Ki(t)θ and Ki(t)=siYi(t), where θ>1. Suppose that the two economies have the same initial value of K, but that s1 > s2. Show that Y1/Y2 is continually rising.
> Which of the following possible regression results concerning the elasticity of long-run output with respect to the saving rate would provide the best evidence that differences in saving rates are not important to cross-country income differences? (1) A
> Assume that there are two sectors, one producing consumption goods and one producing capital goods, and two factors of production: capital and land. Capital is used in both sectors, but land is used only in producing consumption goods. Specifically, the
> Consider a variant of the model in equations (3.22) (3.25). Suppose firm i’s output is Yi(t) = Ki(t)α[A(t)Li(t)]1−α, and that A(t) =BK(t). Here Ki and Li are the amounts of capital and labor used by firm i and K is the aggregate capital stock. Capital an
> Consider the model of Section 3.5 with two changes. First, existing knowledge contributes less than proportionally to the production of new knowledge, as in Case 1 of the model of Section 3.2: A(t) = BLA(t)A(t)θ,θ 0. (Consistent with this, assume that ut
> (a) Show that (3.48) follows from (3.47). (b) Derive (3.49).
> In the model of delegation analyzed in Section 12.8, suppose that the policymaker’s preferences are believed to be described by (12.69), with a> a, when πe is determined. Is social welfare higher if these are actually the policymaker’s preferences, or if
> Consider the model of Section 3.2 with θ
> Consider the Ramsey model with Cobb Douglas production, y(t) = k(t)α, and with the coefficient of relative risk aversion (θ) and capital’s share (α) assumed to be equal. (a) What is k on the balanced growth path (k∗)? (b) What is c on the balanced growth
> Derive an expression analogous to (2.40) for the case of a positive depreciation rate.
> Describe how each of the following affects the c = 0 and k = 0 curves in Figure 2.5, and thus how they affect the balanced-growth-path values of c and k: (a) A rise in θ. (b) A downward shift of the production function. (c) A change in the r
> Piketty (2014) argues that a fall in the growth rate of the economy is likely to lead to an increase in the difference between the real interest rate and the growth rate. This problem asks you to investigate this issue in the context of the Ramsey Cass K
> Consider a household with utility given by (2.2) (2.3). Assume that the real interest rate is constant, and let W denote the household’s initial wealth plus the present value of its lifetime labor income (the right-hand side of [2.7]). Find the utility m
> Assume that the instantaneous utility function u(C) in equation (2.2) is lnC. Consider the problem of a household maximizing (2.2) subject to (2.7). Find an expression for C at each time as a function of initial wealth plus the present value of labor inc
> (a) Suppose it is known in advance that at some time t0 the government will confiscate half of whatever wealth each household holds at that time. Does consumption change discontinuously at time t0? If so, why (and what is the condition relating consumpti
> Consider an individual who lives for two periods and whose utility is given by equation (2.43). Let P1 and P2 denote the prices of consumption in the two periods, and let W denote the value of the individual’s lifetime income; thus the budget constraint
> Consider a Diamond economy where g is zero, production is Cobb Douglas, and utility is logarithmic. (a) Pay-as-you-go social security. Suppose the government taxes each young individual an amount T and uses the proceeds to pay benefits to old individuals
> Consider the steady state of the model of Section 11.4. Let the discount rate, r, approach zero, and assume that the firms are owned by the households; thus welfare can be measured as the sum of utility and profits per unit time, which equals yE−(F+V)c+b
> Suppose that in the Diamond model capital depreciates at rate δ, so that rt = f (kt)−δ. (a) How, if at all, does this change in the model affect equation (2.60) giving kt+1 as a function of kt? (b) In the special case of logarithmic utility, Cobb Dougla
> Suppose Yt = F(Kt,AtLt), with F(•) having constant returns to scale and the intensive form of the production function satisfying the Inada conditions. Suppose also that At+1 = (1 + g)At, Lt+1 =(1+n)Lt, and Kt+1 = Kt +sYt −δKt. (a) Find an expression for
> Consider the Diamond model with logarithmic utility and Cobb Douglas production. Describe how each of the following affects kt+1 as a function of kt: (a) A rise in n. (b) A downward shift of the production function (that is, f (k) takes the form Bkα, an
> (a) Consider the Ramsey Cass Koopmans model where k at time 0 (which as always the model takes as given) is at the golden-rule level: k(0) = kGR. Sketch the paths of c and k. (b) Consider the same initial situation as in part (a), but in the version of t
> Problem 2.11: (a) At time 0, the government announces that it will tax investment income at rate τ from time 0 until some later date t1; thereafter investment income will again be untaxed. (b) At time 0, the government announces that from time t1 to some
> Consider the policy described in Problem 2.10, but suppose that instead of announcing and implementing the tax at time 0, the government announces at time 0 that at some later time, time t1, investment income will begin to be taxed at rate τ. (a) Draw th
