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 point estimate of 5 with a standard error of 2; (2) a point estimate of 0.1 with a standard error of 0.01; (3) a point estimate of 0.001 with a standard error of 5; (4) a point estimate of−2 with a standard error of 5. Explain your answer.
> 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 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 po
> 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.
> 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
> Consider a Ramsey Cass Koopmans economy that is on its balanced growth path. Suppose that at some time, which we will call time 0, the government switches to a policy of taxing investment income at rate τ. Thus the real interest rate that households face
> Consider N firms each with the constant-returns-to-scale production function Y= F(K,AL), or (using the intensive form) Y=ALf(k). Assume f0, f< 0. Assume that all firms can hire labor at wage wA and rent capital at cost r, and that all firms have the same
> Consider the setup described in Problem 2.19. Assume that x is zero, and assume that utility is constant-relative-risk-aversion with θ 1?
> There are two ways in which the Diamond and Samuelson models differ from textbook models. First, markets are incomplete: because individuals cannot trade with individuals who have not been born, some possible transactions are ruled out. Second, because t
> Consider the static search and matching model analyzed in equations (11.71) (11.75). Suppose, however, that the matching function, M(•), is not assumed to be Cobb Douglas or to have constant returns. Is the condition for the decentralized equilibrium to
> Suppose that the old individuals in period 0, in addition to being endowed with Z units of the good, are each endowed with M units of a storable, divisible commodity, which we will call money. Money is not a source of utility. (a) Consider an individual
> Suppose, as in the Diamond model, that Lt two-period-lived individuals are born in period t and that Lt = (1 + n)Lt−1. For simplicity, let utility be logarithmic with no discounting: Ut =ln(C1t)+ln(C2t+1). The production side of the economy is simpler th
> Derive equation (1.50).
> (a) In the model of convergence and measurement error in equations (1.39) and (1.40), suppose the true value of b is −1. Does a regression of ln(Y/N)1979 − ln(Y/N)1870 on a constant and ln(Y/N)1870 yield a biased estimate of b? Explain. (b) Suppose there
> Consider a Solow economy on its balanced growth path. Suppose the growth accounting techniques described in Section 1.7 are applied to this economy. (a) What fraction of growth in output per worker does growth accounting attribute to growth in capital pe
> One view of technological progress is that the productivity of capital goods built at t depends on the state of technology at t and is unaffected by subsequent technological progress. This is known as embodied technological progress (technological progre
> Go through steps analogous to those in equations(1.29) (1.32) to find how quickly y converges to y∗ in the vicinity of the balanced growth path.
> Consider the same setup as at the start of Problem 1.10: the economy is described by the assumptions of the Solow model, except that factors are paid their marginal products and all labor income is consumed and all other income is saved. Show that the ec
> Consider Problem 1.10. Suppose there is a marginal increase in K. (a) Derive an expression (in terms of K/Y,δ, the marginal product of capital FK, and the elasticity of substitution between capital and effective labor in the gross production function F(•
> This question asks you to use a Solow-style model to investigate some ideas that have been discussed in the context of Thomas Piketty’s recent work (see Piketty, 2014; Piketty and Zucman, 2014; Rognlie, 2015). Consider an economy described by the assumpt
> Consider the model of Section 11.4. (a) Use equations (11.65) and (11.69), together with the fact that VV =0 in equilibrium, to find an expression for E as a function of the wage and exogenous parameters of the model. (b) Show that the impact of a rise i
> Assume that both labor and capital are paid their marginal products. Let w denote ∂F(K,AL)/∂L and r denote [∂F(K,AL)/∂K]−δ. (a) Show that the marginal product of labor, w, is A[ f(k)−kf (k)]. (b) Show that if both capital and labor are paid their margina
> Suppose that investment as a fraction of output in the United States rises permanently from 0.15 to 0.18. Assume that capital’s share is 1 3 . (a) By about how much does output eventually rise relative to what it would have been without the rise in inve
> Find the elasticity of output per unit of effective labor on the balanced growth path, y∗, with respect to the rate of population growth, n. IfαK(k∗) = 1 3 , g = 2%, and δ = 3%, by about how much does a fall in n from 2 percent to 1 percent raise y∗?
> Consider a Solow economy that is on its balanced growth path. Assume for simplicity that there is no technological progress. Now suppose that the rate of population growth falls. (a) What happens to the balanced-growth-path values of capital per worker,
> Suppose that the production function is Cobb Douglas. (a) Find expressions for k∗, y∗, and c∗ as functions of the parameters of the model, s, n, δ, g, and α. (b) What is the golden-rule value of k? (c) What saving rate is needed to yield the golden-rule
> Consider an economy with technological progress but without population growth that is on its balanced growth path. Now suppose there is a one-time jump in the number of workers. (a) At the time of the jump, does output per unit of effective labor rise, f
> Describe how, if at all, each of the following developments affects the break-even and actual investment lines in our basic diagram for the Solow model: (a) The rate of depreciation falls. (b) The rate of technological progress rises. (c) The production
> Suppose that the growth rate of some variable, X, is constant and equal to a > 0 from time 0 to time t1; drops to 0 at time t1; rises gradually from 0 to a from time t1 to time t2; and is constant and equal to a after time t2. (a) Sketch a graph of the g
> Use the fact that the growth rate of a variable equals the time derivative of its log to show: (a) The growth rate of the product of two variables equals the sum of their growth rates. That is, if Z(t)=X(t)Y(t), then Z(t)/Z(t)=[X(t)/X(t)]+[Y(t)/Y(t)]. (b
> This problem asks you to show that with some natural variants on the approach to modeling agency riskin Problem10.7, consumption is not linear in the shocks, which renders the model intractable. (a) Consider the model in Problem 10.7. Suppose, however, t
> Consider the model of Section 11.4. Suppose the economy is initially in equilibrium, and that y then falls permanently. Suppose, however, that entry and exit are ruled out; thus the total number of jobs, F + V, remains constant. How do unemployment and v
> Consider Problem 10.6. Suppose, however, that the demand of the period-0 noise traders is not fully persistent, so that noise traders’ demand in period 1 is ρN0+N1,ρ0. They have no initial wealth. (a) Consider first period 1. (i) Consider a representativ
> Consider the previous problem. For simplicity, assume A0 = 0. Now, however, there is a third type of agent: hedge-fund managers. They are born in period 0 and care only about consumption in period 2. Like the sophisticated investors, they have utility U(
