Question: Consider the model of Section 10.1.

> 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(

> 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

> 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

> Consider an individual who lives from 0 to T, and whose lifetime utility is given by U =T t=0 u(C(t))dt, where u(•) > 0,u(•) < 0. The individual’s income is Y0 +gt for 0 ≤ t < R, and 0 for R ≤ t ≤ T. The retirement age, R, satisfies 0 < R < T. The intere

> Consider the new Keynesian Phillips curve with indexation, equation (7.76), under the assumptions of perfect foresight and β=1, together with our usual aggregate demand equation, yt =mt −pt. (a) Express pt+1 in terms of its lagged values and mt. (b) Cons

> Consider a continuous-time version of the Taylor model, so that p(t)=(1/T )T τ=0 x(t−τ)dτ, where T is the interval between each firm’s price changes and x(t−τ) is the price set by firms that set their prices at time t−τ. Assume that φ =1, so that p∗ i (t

> Consider an economy like that of the Caplin Spulber model. Suppose, however, that m can either rise or fall, and that firms therefore follow a simple two-sided Ss policy: if pi − p∗ t reaches either S or −S, firm i changes pi sothat pi−p∗ t equals 0. As

> Consider the experiment described at the beginning of Section 7.4. Specifically, a Calvo economy is initially in long-run equilibrium with all prices equal to m, which we normalize to zero. In period 1, there is a one-time, permanent increase in m to m1.

> Repeat Problem 7.4 using lag operators. Data from Problem 7.4: Consider the Taylor model with the money stock white noise rather than a random walk; that is, mt =εt, where εt is serially uncorrelated. Solve the model using the method of undetermined co

> Consider the Taylor model with the money stock white noise rather than a random walk; that is, mt =εt, where εt is serially uncorrelated. Solve the model using the method of undetermined coefficients.

> Consider the Taylor model. Suppose, however, that every other period all the firms set their prices for that period and the next. That is, in period t prices are set for t and t +1; in t +1, no prices are set; in t +2, prices are set for t+2 and t+3; and

> Suppose there are two sectors. Jobs in the primary sector pay wp; jobs in the secondary sector pay ws. Each worker decides which sector to be in. All workers who choose the secondary sector obtain a job. But there are a fixed number, Np, of primary-secto

> Consider the efficiency-wage model analyzed in equations (11.12) (11.17). Suppose, however, that fraction f of workers belong to unions that are able to obtain a wage that exceeds the nonunion wage by proportion μ. Thus, wu = (1+μ)wn, where wu and wn den

> Moulton Corporation engaged in the following seven transactions during December, Year 12, in preparation for opening the business on January 1, Year 13. We continue with data for Moulton Corporation in Chapter 3, Problem 3.22. You will not need some of t

> Express the following transactions of Winkle Grocery Store, Inc., in journal entry form. If an entry is not required, indicate the reason. You may omit explanations for the journal entries. The store: (1) Receives $30,000 from John Winkle in return for 1

> Assume that during Year 14, Inheritance Brands, a U.S. manufacturer and distributor, engaged in the following five transactions. Inheritance Brands applies U.S. GAAP and reports its results in millions of U.S. dollars ($). You may round to one significan

> Assume that during Year 15, Bullseye Corporation, a U.S. retailer, engages in the following six transactions. Bullseye Corporation applies U.S. GAAP and reports its results in millions of U.S. dollars ($). Do not be concerned that after these transaction

> GAAP classifies items on the balance sheet in one of the following ways: Asset (A) Liability (L) Shareholders’ equity (SE) Item that would not appear on the balance sheet as conventionally prepared under GAAP. (N/A) Using the abbreviations, indicate the

> GAAP classifies items on the balance sheet in one of the following ways: Asset (A) Liability (L) Shareholders’ equity (SE) Item that would not appear on the balance sheet as conventionally prepared under GAAP. (N/A) Using the abbreviations, indicate the

> When will a firm’s fiscal year differ from a calendar year?