Question: Consider an economy with technological progress but

> 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

> 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

> 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(

> Consider the following variant on the model of noise-trader risk in equations (10.15) (10.23). There are three periods, denoted 0, 1, and 2. There are two assets. The first is a safe asset in perfectly elastic supply. Its rate of return is normalized to

> (a) Show that in the model analyzed in equations (10.15) (10.23) of Section 10.4, the unconditional distributions of Ca 2t and Cn 2t are not normal. (b) Explain in a sentence or two why the analysis in the text, which uses the properties of lognormal dis

> Consider the model of Section 10.2 with a different friction: there is no cost of verifying output, but the entrepreneur can hide fraction 1−f of the project’s output from the investor (with 0 ≤ f ≤ 1). Thus the entrepreneur can only credibly promise to

> Consider the model of investment under asymmetric information in Section 10.2. Suppose that initially the entrepreneur is undertaking the project, and that (1 + r)(1−W) is strictly less than RMAX. Describe how each of the following affects D: (a) A small

> Consider the model of Section 10.1. Suppose, however, that there are M households, and that household j’s utility is Vj = U(C1) + βs jU(C2), where βs j > 0 for all j and s. That is, households may have heterogeneous preferences about consumption in diffe

> Consider deposit insurance in the Diamond Dybvig model of Section 10.6. (a) If fraction φ>θof depositors withdraw in period 1, how large a tax must the government levy on each agent in period 1 to be able to increase the total consumption of the non with

> Consider the Diamond Dybvig model described in Section 10.6, but suppose that ρR < 1. (a) In this case, what are ca∗ 1 and cb∗ 1 ? Is cb∗ 1 still larger than ca∗ 1 ? (b) Suppose the bank offers the contract described in the text: anyone who deposits one

> Consider modeling the noise traders in the model of equations (10.15) (10.23) of Section 10.4 in terms of shocks to the quantity they demand of the risky asset rather than to their expectations of the price of the asset. Specifically, suppose the demand

> Consider the steady state of the Diamond Mortensen Pissarides model of Section 11.4. (a) Suppose that φ =0. What is the wage? What does the equilibrium condition (11.70) simplify to? (b) Suppose that φ =1. What is the wage? What does the equilibrium cond

> Consider the model of Section 10.1. Assume that utility is logarithmic, that β =1, and that there are only two states, each of which occurs with probability one-half. In addition, assume there is only one investment project. It pays RG in state G and RB

> Let H denote the stock of housing, I the rate of investment, pH the real price of housing, and R the rent. Assume that I is increasing in pH, so that I =I(pH), with I(•) > 0, and that H =I − δH. Assume also that the rent is a decreasing function of H: R=

> Consider the model of investment in Sections 9.2 9.5. Suppose it becomes known at some date that there will be a one-time capital levy. Specifically, capital holders will be taxed an amount equal to fraction f of the value of their capital holdings at so

> Consider the model of investment in Sections 9.2 9.5. Describe the effects of each of the following changes on the K = 0 and q = 0 loci, on K and q at the time of the change, and on their behavior over time. In each case, assume that K and q are initiall

> Consider the Romer model of Section 3.5. For simplicity, neglect the constraint that LA cannot be negative. Set up the problem of choosing the path of LA(t) to maximize the lifetime utility of the representative individual. What is the control variable?

> Consider the social planner’s problem that we analyzed in Section 2.4: the planner wants to maximize∞ t=0 e−βt[c(t)1−θ/(1−θ)]dt subject to k(t)=f (k(t))−c(t)−(n+g)k(t). (a) What is the current-value Hamiltonian? What variables are the control variable,

> Consider an individual choosing the path of G to maximize∞ t=0 e−ρt− a 2G(t)2dt, a > 0, ρ>0.Here G(t) is the amount of garbage the individual creates at time t; for simplicity, we allow for the possibility that G can be negative. The individual’s creatio

> The major feature of the tax code that affects the user cost of capital in the case of owner-occupied housing in the United States is that nominal interest payments are tax-deductible. Thus the after-tax real interest rate relevant to home ownership is r

> Corporations in the United States are allowed to subtract depreciation allowances from their taxable income. The depreciation allowances are based on the purchase price of the capital; a corporation that buys a new capital good at time t can deduct fract

> Consider the analysis of the effects of uncertainty about discount factors in Section 9.7. Suppose, however, that the firm finances its investment using a mix of equity and risk-free debt. Specifically, consider the financing of the marginal unit of capi

> Describe how each of the following affects steady-state employment in the Diamond Mortensen Pissarides model of Section 11.4: (a) An increase in the job breakup rate, λ. (b) An increase in the interest rate, r. (c) An increase in the effectiveness of mat

> Consider a firm that is contemplating undertaking an investment with a cost of I. There are two periods. The investment will pay off π1 inperiod1and π2 inperiod2. π1 is certain, but π2 is uncertain. The firm maximizes expected profits and, for simplicity

> Consider the model of investment with kinked adjustment costs in Section 9.8. Describe the effect of each of the following on the q =0 locus, on the area where K = 0, on q and K at the time of the change, and on their behavior over time. In each case, as

> Consider the model of investment under uncertainty with a constant interest rate in Section 9.7. Suppose that, as in Problem 9.10, π(K) = a −bK and that C(I) = αI 2/2. In addition, suppose that what is uncertain is future values of a. This problem asks y

> Suppose that π(K)=a −bK and C(I)= αI 2/2. (a) What is the q =0 locus? What is the long-run equilibrium value of K? (b) What is the slope of the saddle path? (Hint: Use the approach in Section 2.6.)

> Suppose the costs of adjustment exhibit constant returns in κ and κ. Specifically, suppose they are given by C(κ/κ)κ, where C(0) = 0, C(0) = 0, C(•) > 0. In addition, suppose capital depreciates at rate δ; thus κ(t) = I(t)−δκ(t). Consider the representat

> Consider a firm that produces output using a Cobb Douglas combination of capital and labor: Y=KαL1−α,0

> Consider the setup of the previous problem without the assumption that lims→∞ Et [Pt+s/(1+r)s]=0. (a) Deterministic bubbles. Suppose that Pt equals the expression derived in part (b) of Problem 8.8 plus (1+r)tb, b > 0. (i) Is consumers’ first-order condi

> Consider a stock that pays dividends of Dt in period t and whose price in period t is Pt. Assume that consumers are risk-neutral and have a discount rate of r; thus they maximize E[∞ t=0 Ct/(1+r)t ]. (a) Show that equilibrium requires Pt = Et[(Dt+1 + Pt+

> Consider the two-period setup analyzed in Section 8.4. Suppose that the government initially raises revenue only by taxing interest income. Thus the individual’s budget constraint is C1 +C2/[1+(1− τ)r] ≤ Y1 +Y2/[1+(1− τ)r], where τ is the tax rate. The g

> Suppose that Ct equals [r/(1+r)]{At +∞ s =0 Et [Yt+s]/(1+r)s}, and that At+1 =(1+r )(At +Yt −Ct). (a) Show that these assumptions imply that Et[Ct+1]=Ct (and thus that consumption follows a random walk) and that ∞ s=0 Et [Ct+s]/(1 + r )s =At + ∞ s=0 Et [

> In the setup described in Problem 11.10, suppose that w is distributed uniformly on [μ−a,μ+a] and that C V, and rejects it if ˆw

> Suppose instantaneous utility is of the constant-relative-risk-aversion form, u(Ct)=C1−θ t /(1−θ),θ>0. Assume that the real interest rate, r, is constant but not necessarily equal to the discount rate, ρ. (a) Find the Euler equation relating Ct to expect

> In the model of Section 8.2, uncertainty about future income does not affect consumption. Does this mean that the uncertainty does not affect expected lifetime utility?

> Actual data do not give consumption at a point in time, but average consumption over an extended period, such as a quarter. This problem asks you to examine the effects of this fact. Suppose that consumption follows a random walk: Ct = Ct−1 +et, where e

> The average income of farmers is less than the average income of non-farmers, but fluctuates more from year to year. Given this, how does the permanent-income hypothesis predict that estimated consumption functions for farmers and non farmers differ?

> Consider the following seemingly small variation on part (b) of Problem 8.16. Choose an N, and define e ≡ 200/N. Now, assume that Y can take on only the values 0,e,2e,3e,...,200, each with probability 1/(N+1). Likewise, assume that C can only take on the

> Consider the dynamic programming problem that leads to Figure 8.4. This problem asks you to solve the problem numerically with one change: preferences are logarithmic, so that u(C) = lnC. Specifically, it asks you to approximate the value function by val

> Consider an individual who lives for three periods. In period 1, his or her objective function is lnc1 +δ lnc2 +δ lnc3, where 0

> Consider an individual who lives for two periods and has constant-absolute risk-aversion utility, U =−e−γC1 − e−γC2,γ>0. The interest rate is zero and the individual has no initial wealth, so the individual’s lifetime budget constraint is C1 +C2 = Y1 +Y2

> Suppose that the utility of the representative consumer, individual i, is given by T t=1 [1/(1+ρ)t](Cit/Zit)1−θ/(1−θ), ρ>0,θ>0, where Zit is the ‘‘reference” level of consumption. Assume the interest rate is constant at some level, r, and that there is n

> Suppose that, as in Section 8.2, the instantaneous utility function is quadratic and the interest rate and the discount rate are zero. Suppose, however, that goods are durable; specifically, Ct =(1−δ)Ct−1 + Xt, where Xt is purchases in period t and 0≤ δ

> Consider a worker searching for a job. Wages, w, have a probability density function across jobs, f (w), that is known to the worker; let F(w) be the associated cumulative distribution function. Each time the worker samples a job from this distribution,

> Consider an economy with two possible states, each of which occurs with probability one-half. In the good state, each individual’s consumption is 1. In the bad state, fraction λ of the population consumes 1−(φ/λ) and the remainder consumes 1, where 0

> Suppose the only assets in the economy are infinitely lived trees. Output equals the fruit of the trees, which is exogenous and cannot be stored; thus Ct = Yt, where Yt is the exogenously determined output per person and Ct is consumption per person. Ass