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Macroeconomic Theory

A Dynamic General Equilibrium Approach

Michael Wickens

Princeton University Press

Princeton and Oxford

Questions

Questions to Accompany Chapter 2

(1) We have assumed that the economy discounts s periods ahead using thegeometric (or exponential) discount factor βs = (1 + θ)−s for {s = 0,1,2, . . . }.Suppose instead that the economy uses the sequence of hyperbolic discountfactors βs = {1,ϕβ,ϕβ2,ϕβ3, . . . }, where 0 < ϕ < 1.

(a) Compare the implications for discounting of using geometric and hyper-bolic discount factors.

(b) For the centrally planned model,

yt = ct + it,∆kt+1 = it − δkt,

yt = Akαt ,where yt is output, ct is consumption, it is investment, kt is the capitalstock, and the objective is to maximize

Vt =∞∑s=0

βsU(ct+s),

derive the optimal solution of the basic centrally planned model underhyperbolic discounting, and comment on any differences with the solutionbased on geometric discounting.

(2) Assuming geometric discounting βs , the utility function U(ct) = ln ct , andthe production function yt = Akt :

(a) derive the optimal long-run solution, and

(b) analyze the short-run solution

(3) In question (2), instead of eliminating the Lagrange multipliers λt whenderiving the optimal solution from the first-order conditions, obtain the jointsolution for {ct, kt, λt}. How would you interpret λt?

(4) Consider the CES production function

yt = A[αk1−(1/γ)t + (1−α)n1−(1/γ)

t ]1/(1−(1/γ)).

Show that the CES function becomes the Cobb-Douglas function as γ → 1.

(5) Consider the following centrally planned model with labor:

yt = ct + it,∆kt+1 = it − δkt,

yt = A[αk1−(1/γ)t + (1−α)n1−(1/γ)

t ]1/(1−(1/γ)),

2

where the objective is to maximize

Vt =∞∑s=0

βs[ln ct+s +ϕ ln lt+s]

and where yt is output, ct is consumption, it is investment, kt is the capitalstock, nt is employment, and lt is leisure (lt +nt = 1).

(a) Derive the long-run solutions for consumption, labor, and capital.

(b) What are the implied real interest rate and wage rate?

(c) Comment on the implications of having an elasticity of substitution be-tween capital and labor different from unity.

(6) Comment on the statement: “the saddlepath is a knife-edge solution; oncethe economy departs from the saddlepath it is unable to return to equilibriumand will instead either explode or collapse.”


(1) Rework the optimal growth solution in terms of the original variables, i.e.,without first taking deviations about trend growth.

(2) Consider the Solow–Swan model of growth for the production function Yt =A(eµtKt)α(eνtNt)1−α.

(a) Examine the solutions when technical progress is Hicks-neutral so thatµ = ν , and when technical progress is labor augmenting so that µ = 0.

(b) Show that the model has constant steady-state growth only when technicalprogress is labor augmenting.

(2) Compare the solutions for optimal growth when technical progress is Hicksneutral and labor augmenting.

(3) Suppose that technical progress is embodied in new investment so thatthe production function is Yt = (

∑∞v=0 eη(t−v)It−v)αN1−α

t and that there is nodepreciation of capital. Derive

(a) the solution to the Solow–Swan model;

(b) the optimal growth solution where the objective is to maximize

∞∑s=0

βs ln ct+s .

(4) Consider the following two-sector endogenous growth economy with twotypes of capital due to Rebelo (1991): physical kt and human ht . Both types ofcapital are required to produce goods output yt and new human capital iht . Themodel is

yt = ct + ikt ,∆kt+1 = ikt − δkt,∆ht+1 = iht − δht,

yt = A(φkt)α(µht)1−α,iht = A[(1−φ)kt]ε[(1− µ)ht]1−ε,

where ikt is investment in physical capital, φ and µ are the shares of physicaland human capital used in producing goods andα > ε. The economy maximizes

Vt =∞∑s=0

βsc1−σt+s

1− σ .

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(a) Assuming that each type of capital receives the same rate of return in bothactivities, derive

(i) the optimal long-run solution,

(ii) the rate of growth of each variable in steady state.

(b) How are these solutions affected when

(i) φ = µ,

(ii) φ = µ = 1,

(iii) α > ε?

(c) Hence summarize the implications of this model of growth.


(1) The household budget constraint may be expressed in different ways fromequation (4.2). Derive the optimal solution for consumption and compare thesewith the solution based on equation (4.2) for each of the following ways ofwriting the budget constraint:

(a) at+1 = (1 + r)(at + xt − ct), i.e., current assets and income assets notconsumed are invested.

(b) ∆at + ct = xt + rat−1, where the dating convention is that at denotes theend of period stock of assets and ct and xt are consumption and incomeduring period t.

(c) Wt =∞∑s=0

ct+s(1+ r)s =

∞∑s=0

xt+s(1+ r)s + rat , where Wt is household wealth.

(2) The representative household is assumed to choose {ct, ct+1, . . . } to maxi-mize

Vt =∞∑s=0

βsU(ct+s), 0 < β = 11+ θ < 1,

subject to the budget constraint∆at+1+ct = xt+rtat , where ct is consumption,xt is exogenous income, at is the (net) stock of financial assets at the beginningof period t, and r is the (constant) real interest rate.

(a) Derive the Euler equation and give an intuitive explanation of it.

(b) Assuming that r = θ and using the approximation

U ′(ct+1)U ′(ct)

= 1− σ∆ ln ct+1,

show that optimal consumption is constant.

(c) Does this mean that changes in income will have no effect on consump-tion? Explain.

(3) Derive optimal household consumption when the utility function reflectshabit persistence of the following forms:

(a) U(ct) = −(ct − γct−1)2

2+α(ct − γct−1);

(b) U(ct) = (ct − γct−1)1−σ

1− σ .

(4) Suppose that households have savings of st at the start of the period, con-sume ct but have no income. The household budget constraint is ∆st+1 =r(st − ct), 0 < r < 1, where r is the real interest rate.

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(a) If the household’s problem is to maximize discounted utility

Vt =∞∑s=0

βs ln ct+s ,

where β = 1/(1+ r):(i) Show that the solution is ct+1 = ct .

(ii) What is the solution for st?

(b) If the household’s problem is to maximize expected discounted utilityVt = Et

∑∞s=0 βs ln ct+s :

(i) Show that the solution is 1/ct = Et[1/ct+1].

(ii) Using a second-order Taylor series expansion about ct show that thesolution can be written as Et[∆ct+1/ct] = Et[(∆ct+1/ct)2].

(iii) Hence, comment on the differences between the nonstochastic andthe stochastic solutions.

(5) A household lives for two periods, t and t+1. The discount factor for periodtwo isβ = 1. It receives exogenous incomext andxt+1, wherext+1 is distributedN(xt,σ 2), but has no assets. Find the optimal level of ct

(a) if the utility function is quadratic, U(ct) = −12c

2t +αct , α > 0; and

(b) if the utility function is exponential, U(ct) = −e−αct , α > 0.

(c) What is the coefficient of relative risk aversion in case (b)?

(6) Suppose that firms face additional (quadratic) costs associated with theaccumulation of capital and labor so that firm profits are

Πt = Akαt n1−αt −wtnt − it − 1

2µ(∆kt+1)2 − 12ν(∆nt)

2,

where µ, ν > 0, the real wage wt is exogenous, and ∆kt+1 = it − δkt . If firmsmaximize the expected present value of the firm Et[

∑∞s=0(1+ r)−sΠt+s]:

(a) Derive the demand functions for capital and labor in the long run and theshort run.

(b) What would be the response of capital and labor demand to

(i) a temporary increase in the real wage in period t, and

(ii) a permanent increase in the real wage from period t?


(1) Suppose that the government finances expenditures by taxes on consump-tion at the rate τ and by debt. Starting from a position where the budget isbalanced and there is no government debt, analyze the consequences of

(a) a temporary increase in government expenditures in period t,

(b) a permanent increase in government expenditures from period t.

(2) Suppose that government expenditures gt are all capital expenditures andthe stock of government capital Gt is a factor of production. If the economy isdescribed by

yt = ct + it + gt,yt = Akαt Gt1−a,

∆kt+1 = it − δkt,∆Gt+1 = gt − δGt,

and the aim is to maximize∑∞s=0 βs ln ct+s , obtain the optimal solution.

(3) Suppose that government finances its expenditures through lump-sum taxesTt and debt bt but there is a cost of collecting taxes given by

Φ(Tt) = φ1Tt + 12φ2T 2

t , Φ′(Tt) � 0.

If the national income and government budget constraints are

yt = ct + gt +Φ(Tt),∆bt+1 + Tt = gt + rbt,

where output yt is exogenous, and the aim is to maximize Et∑∞s=0 βsU(ct+s)

for β = 1/(1+ r):(a) Find the optimal solution for taxes.

(b) Analyze the effects on taxes and debt of

(i) a temporary shock to government expenditures in period t,

(ii) a permanent increase in government expenditures.

(4) Assuming that output growth is zero, inflation and the rate of growth ofthe money supply are π , and that government expenditures on goods and ser-vices plus transfers less total taxes equals zt , analyze the consequences for thesustainability of the fiscal stance.

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(5) Consider an economy without capital with proportional taxes on consump-tion and labor that is described by the following equations:

yt = Anαt = ct + gt,gt + rbt = τct ct + τwt wtnt +∆bt+1,

U(ct, lt) = c1−σt

1− σ + γ ln lt.

(a) State the household budget constraint.

(b) If the economy seeks to maximize∑∞s=0 βsU(ct+s), derive the optimal

levels of consumption and employment for given gt , bt , and tax rates.

(c) Find the optimal rates of consumption and labor taxes by solving theassociated Ramsey problem.


(1)

(a) Consider the following two-period overlapping generations (OLG) model.People consume in both periods but work only in period two. The inter-temporal utility of the representative individual in the first period is

U = ln c1 + β[ln c2 +α ln(1−n2)+ γ lng2],

where c1 and c2 are consumption in periods one and two, n2 is work, andg2 is government expenditure in period two. Production in periods oneand two are

y1 = Rk1 = c1 + k2,

y2 = Rk2 + an2 = c2 + g2.

Find the optimal centrally planned solution.

(b) Now suppose that government taxes both labor and capital in period twoso that the private sector’s intertemporal budget constraint is

c1 + c2

R2= Rk1 + (1− τ2)an2

R2

and the government budget constraint is

g2 = τ2an2 + (R − R2)k2,

where R2 is the after-tax return to capital and τ2 is the rate of tax of laborin period two. Find the solution when taxes are preannounced. Why is thissolution not time consistent?

(c) Assume now that the government optimizes taxes in period two taking k2

as given. Show that the optimal labor tax in period two is zero, and findthe optimal rate of capital taxation in period two.

(d) Taking the solution in (c) as given, solve the consumer’s problem in periodone.

(e) Compare U in this case with case (b).

(2) Consider the following two-period OLG model in which each generation hasthe same number of people, N . The young generation receives an endowmentof x1 when young and x2 = (1 + φ)x1 when old, where φ can be positive ornegative. The endowments of the young generation grow over time at the rateγ. Each unit of saving (by the young) is invested and produces 1 + µ units ofoutput (µ > 0) when they are old. Each of the young generation maximizes

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ln c1,t + (1/(1+ r)) ln c2,t+1, where c1,t is consumption when young and c2,t+1

is consumption when old.

(a) Derive the consumption and savings of the young generation and theconsumption of the old generation.

(b) How do changes in φ, γ, and µ affect these solutions?

(c) If φ = µ how does this affect the solution?


(1) An open economy has the balance of payments identity

xt −Qxmt + r∗ft = ∆ft+1,

where xt is exports, xmt is imports, ft is the net holding of foreign assets, Q isthe terms of trade, and r∗ is the world rate of interest. Total output yt is eitherconsumed at home ch

t or is exported, thus

yt = cht + xt.

Total domestic consumption is ct ; yt and xt are exogenous.

(a) Derive the Euler equation that maximizes∑∞s=0 βs ln ct+s with respect to

{ct, ct+1, . . . ;ft+1, ft+2, . . . }, where β = 1/(1+ θ).(b) Explain how and why the relative magnitudes of r∗ and θ affect the steady-

state solutions of ct and ft .

(c) Explain how this solution differs from that of the corresponding closedeconomy.

(d) Hence comment on whether there are any benefits of trade in this model.

(e) Obtain the solution for the current account.

(f) What are the effects on the current account and net asset position of apermanent increase in xt?

(2) Consider two countries which consume home and foreign goods cH,t andcF,t . Each period the home country maximizes

Ut = [c(σ−1)/σH,t + c(σ−1)/σ

F,t ]σ/(σ−1)

and has an endowment of yt units of the home produced good. The foreigncountry is identical and its variables are denoted with an asterisk. Every unit ofa good that is transported abroad has a real resource cost equal to τ so that, ineffect, only a proportion 1− τ arrives at its destination. In the home economyPH,t is the price of the home good and PF,t is the price of the foreign good. P∗H,tand P∗F,t are the corresponding prices in the foreign country where all prices aremeasured in terms of a common unit of world currency.

(a) If goods markets are competitive, what is the relation between the fourprices and how are the terms of trade in each country related?

(b) Derive the relative demands for home and foreign goods in each country.

(c) How is this affected if yt = y∗t ?

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(d) For yt = y∗t obtain the implications of transport costs for

(i) the terms of trade, and

(ii) the openness of the economy (i.e., the relative expenditures on homeand foreign goods)?

(3) Suppose the model in question (2) is modified so that there are two periodsand intertemporal utility is

Vt = U(ct)+ βU(ct+1),

where ct = [c(σ−1)/σH,t + c(σ−1)/σ

F,t ]σ/(σ−1). Endowments in the two periods are ytand yt+1. Foreign prices P∗H,t and P∗F,t and the world interest rate are assumedgiven. The first-period budget constraint is

PH,tyt + B = PH,tcH,t + PF,tcF,t = Ptct,where the general price level is Pt and B is borrowing from abroad at the foreignreal interest rate r∗. The second period budget constraint is

PH,t+1yt+1 − (1+ r∗)B = PH,t+1cH,t+1 + PF,t+1cF,t+1 = Pt+1ct+1.

(a) Derive the optimal solution for the home economy, including the domesticprice level P .

(b) Show that if the foreign good is always imported, its home price is PF =P∗F /(1− τ).

(c) Derive the domestic real interest rate r and examine whether real interestparity exists.

(d) How does ct depend on r?

(4) Suppose the “world” is compromised of two identical countries. Each countryconsumes home and foreign goods and maximizes

Vt =∞∑s=0

βs(cαH,t+sc

1−αF,t+s)1−σ

1− σsubject to its budget constraint. Expressed in terms of home’s prices, the homecountry budget constraint is

PH,tcH,t + StPF,tcF,t +∆Bt+1 = PH,tyH,t + rtBt,where cH,t is consumption of home produced goods, cF,t is consumption of for-eign produced goods, PH,t is the price of the home country’s output, which isdenoted yH,t and is exogenous, PF,t is the price of the foreign country’s output,Bt is the home country’s borrowing from abroad expressed in domestic cur-rency which is at the world rate of interest rt , and St is the nominal exchangerate.

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(a) Using an asterisk to denote the foreign country equivalent variable (e.g.,c∗H,t is the foreign country’s consumption of domestic output), what are thenational income and balance of payments identities for the home country?

(b) Derive the optimal solution for the home country, including its generalprice level Pt , taking the foreign country—its output, exports, and prices—and the exchange rate as given.

(c) What is the corresponding solution for the foreign country?

(d) For the home country analyze the effects on consumption, the currentaccount and the net foreign asset position of shocks to home output,foreign exports and the exchange rate.

(e) Compare the effects on the sustainability of the current account of tem-porary and permanent shocks to these variables.

(f) How would your answers differ if domestic and foreign goods were perfectsubstitutes?

(5) For the model described in question (2), suppose that there is world centralplanner who maximizes the sum of individual country welfares:

Wt =∞∑s=0

βs[(cαH,t+sc1−α

F,t+s)1−σ

1− σ + [(c∗H,t+s)α(c

∗F,t+s)1−α]1−σ

1− σ].

(a) What are the constraints in this problem?

(b) Derive the optimal world solution subject to these constraints where out-puts and the exchange rate are exogenous.

(c) Comment on any differences with the solutions in question (2).

(d) What are the effects of shocks to home and foreign outputs and to theexchange rate?


(1) Suppose that the nominal household budget constraint is

∆Bt1 +∆Mt+1 + Ptct = Ptxt + (1+ rt)Bt,where ct is consumption, xt is exogenous income, Bt is nominal bond holding,Mt is nominal money balances, Pt is the general price level, mt = Mt/Pt , and ris the real rate of return.

(a) Derive the real budget constraint.

(b) Comment on whether this implies that money is not be super-neutral inthe whole economy.

(c) If households maximize

Vt =∞∑s=0

βsU(ct+s ,mt+s),

where the utility function is

U(ct,mt) = [cαt m1−αt /αα(1−α)1−α]1−σ

1− σ ,

obtain the long-run demand for money.

(2) Consider an economy in which money is the only financial asset, and supposethat households hold money solely in order to smooth consumption expendi-tures. The nominal household budget constraint for this economy is

Ptct +∆Mt+1 = Ptyt,where ct is consumption, yt is exogenous income, Pt is the price level, and Mtis nominal money balances.

(a) If households maximize∑∞s=0 βsU(ct+s), derive the optimal solution for

consumption.

(b) Compare this solution with the special case where β = 1 and inflation iszero.

(c) Suppose that xt is expected to remain constant except in period t + 1when it is expected to increase temporarily. Examine the effect on moneyholdings and consumption.

(d) Hence comment on the role of real balances in determining consumption.

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(3) Suppose that in the economy described in question (1) there is a governmentwhich consumes a random amount gt = g+et , where et is an independently andidentically distributed random shock with zero mean. The government imposesa lump-sum tax of Tt = g and balances its budget with the use of money. yt isnow determined by the national income identity yt = ct + gt .

(a) If households maximize Et∑∞s=0 βs ln ct+s , derive the optimal solutions for

consumption and money holding.

(b) Comment on how money shocks affect consumption and money holding.

(c) Is money super-neutral in this economy?

(4)

(a) Comment briefly on the “classical dichotomy” in macroeconomics thatnominal shocks have no long-run effect on real variables.

(b) Consider the nominal household budget constraint

Pt+1kt+1 +∆Mt+1 + Ptct + PtTt = wtnt + (1+ rt)Ptkt,where c is consumption, k is the capital stock, n is employment, M isnominal money balances, T is real taxes, P is the general price level, w isthe real wage rate, and r is the real rate of return on capital.

(i) Derive the real household budget constraint.

(ii) What does this reveal about the effect of inflation on real variablessuch as consumption?

(c) Suppose that the government’s budget constraint tax is Tt = mt − (1 +πt+1)mt+1, where T < 0 and m is real money.

(i) Interpret T .

(ii) Combine the household and government budget constraints to elim-inate m and obtain the consolidated constraint.

(d) If the economy’s production function F(kt,nt) is homogeneous of degreeone and factors are paid their marginal products so that Fk,t −δ = rt andFn,t = wt ,

(i) show that F(kt,nt) = (rt + δ)kt +wtnt ,(ii) and that in the long-run c = rk+wn.

(e) Discuss the implications of these results for the “classical dichotomy.”

(5) Suppose that some goods c1,t must be paid for only with money Mt and therest c2,t are bought on credit Lt using a one-period loan to be repaid at the startof next period at the risk-free nominal rate of interest R. The prices of thesegoods are P1t and P2t . If households maximize

∑∞s=0(1+ R)−sU(ct+s), where

U(ct) =cα1,tc

1−α2,t

αα(1−α)1−α ,and income yt is exogenous:

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(a) Find the optimal solutions for c1,t , c2,t , Mt , and Lt .

(b) Explain the interest sensitivities of money and credit.

(c) Suppose that households had the freedom to choose to use either moneyor credit for all purchases. How would this affect the solution?

(6) Consider the following demand for money function which has been used tostudy hyperinflation:

mt − pt = −α(Etpt+1 − pt), α > 0,

where Mt = nominal money, mt = lnMt , Pt = price level, and pt = lnPt .

(a) Contrast this with the usual demand function for money, and commenton why it might be a suitable formulation for studying hyperinflation?

(b) Derive the equilibrium values of pt and the rate of inflation if the supplyof money is given by

∆mt = µ + εt,where µ > 0 and Et[εt+1] = 0.

(c) What will be the equilibrium values of pt if

(i) the stock of money is expected to deviate temporarily in period t+1from this money supply rule and take the value m∗

t+1;

(ii) the rate of growth of money is expected to deviate permanently fromthe rule and from period t + 1 grow at the rate υ?

(7) Consider the effect of inflation on front-end loading.

(a) Assume an economy with zero inflation. Households take out a loan for nperiods. Each period they pay interest on the loan at the nominal interestrate R (also the real interest rate r ). The loan is not repaid until the end.What is the present value of the cost of this loan using the discount rateR?

(b) Suppose instead that the economy has an inflation rate of π > 0. House-holds now pay each period at the nominal interest rateR = r+π . Calculatethe present value of the cost of the loan using the new discount rate R.

(c) Suppose that each period households take out a loan to pay the additionalinterest charges due to inflation and repay these loans in the final periodtogether with the original loan. Recalculate the present value of the costof the loan.

(d) Compare the expenditure flows made by the households each period in-cluding the final period for these three cases, and hence comment on thereal effects of inflation.

(e) Banks are often reluctant to make loans to pay extra interest charges.What is the likely effect of this on the real cost of inflation?


(1) Consider an economy that produces a single good in which householdsmaximize

Ut =∞∑s=0

βs[

ln ct+s +φ ln lt+s + γ lnMt+sPt+s

]

subject to the nominal budget constraint

Ptct +∆Bt+1 +∆Mt+1 = Ptdt +Wtnt + rBt,where c is consumption, l is leisure, n is employment (n + l = 1), W is thenominal wage rate, d is total real firm net revenues distributed as dividends, Bis nominal bond holdings, R is the nominal interest rate, M is nominal moneybalances, and P is the price level. Firms maximize the present value of nominalnet revenues

Πt =∞∑s=0

(1+ r)−sPt+sdt+s ,

where dt = yt−wtnt , the real wage iswt = Wt/Pt , and the production functionis yt = Atnαt .

(a) Derive the optimal solution on the assumption that prices are perfectlyflexible.

(b) Suppose that, following a shock, for example, to the money supply orto technological progress At , firms are able to adjust their price withprobability ρ and otherwise price retains its previous value. Examine theconsequences for the dynamic behavior of the economy.

(c) Suppose instead that wages adjust to shocks with probability ρ. Reexam-ine the consequences for the dynamic behavior of the economy.

(2) Rework the analysis of price determination under imperfect competitionwith N = 2 and with total household consumption obtained by aggregatingusing the aggregator

ct = ct(1)φct(2)1−φ

φφ(1−φ)1−φ .

(3) Rework the analysis of price determination with intermediate goods withN = 2 and with the final output related to intermediate inputs through

yt = yt(1)φyt(2)1−φ

φφ(1−φ)1−φ .

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(4) Consider the model for pricing with intermediate inputs. The demand for afirm’s output was shown to be

yt(i) =(Pt(i)Pt

)−φyt.

Its profits can be written as

Πt(i) = Pt(i)yt(i)− Ct(i),where its total cost is

Ct(i) = φ− 1φ

Pt(i)yt(i).

(a) Find the optimal price Pt(i)∗ if the firm maximizes profits period byperiod while taking yt and Pt as given.

(b) If instead the firm chooses a price which it plans to keep constant for allfuture periods and hence maximizes

∑∞s=0(1+ r)−sΠt+s(i), show that the

resulting optimal price can be written as

Pt(i)# = φ(1+ r)(φ− 1)r

∞∑s=0

(1+ r)−sMCt+s(i),

where MCt(i) is marginal cost.

(c) What is this price if expressed in terms of Pt(i)∗?

(d) Hence comment on the effect on today’s price of anticipated future shocksto demand and costs.

(5) Suppose in question (4) that the firm seeks to maximize

∞∑s=0

(1+ r)−sΠt+s(i)

but may reset its price each period with probability ρ.

(a) What is the optimal price in period t expressed in terms of Pt(i)∗?

(b) Comment on the differences between this price and Pt(i)#.

(6) Consider an economy with two sectors i = 1,2. Each sector sets its pricefor two periods but does so in alternate periods. The general price level in theeconomy is the average of sector prices: pt = 1

2(p1,t + p2,t). In the period theprice is reset it is determined by the average of the current and the expectedfuture optimal price: p#

t = 12(p

∗t +Etp∗t+1), hence pt = 1

2(p#t +p#

t−1). The optimalprice is assumed to be determined by p∗t − pt = φ(wt − pt), where wt is thewage rate.

(a) Derive the general price level if wages are generated by ∆wt = et , whereet is a zero-mean i.i.d. process.

(b) How does the price level respond in period t to an anticipated shock inwages in period t + 1?


(1)

(a) Suppose that a consumer’s initial wealth is given byW0, and the consumerhas the option of investing in a risky asset which has a rate of return r anda risk-free asset which has a sure rate of return f . If the consumer maxi-mizes the expected value of a strictly increasing, concave utility functionU(W) by choosing to hold the risky versus the risk-free asset, and if thevariance of the return on the risky asset is V(r):

(i) Find an expression for the risk premium ρ that makes the consumerindifferent between holding the risky or the risk-free asset.

(ii) What is the risk-premium if E(r) = f ?

(b) Suppose that the consumer’s utility function is the hyperbolic absoluterisk aversion (HARA) function

U(W) = 1− σσ

[αW

1− σ + β]σ, α > 0, β > 0, 0 < σ < 1.

(i) Explain how absolute risk aversion differs from relative risk aversion.

(ii) Discuss how the magnitude of the risk premium varies as a functionof the parameters α, β, and σ .

(2) Suppose there exists a representative risk averse consumer who derivesutility from current and future consumption as

U = U(ct)+ βEtU(ct+1),

where 0 < β < 1 is the consumer’s subjective discount factor, and the single-period utility function has the form

U(ct) = c1−σt − 1

1− σ , σ � 0.

The consumer receives an exogenous income of yt in period t, and yt+1 inperiod t + 1. While yt is known at date t, yt+1 is random. The consumer cansave by purchasing shares in a stock or by holding a risk-free one-period bond.The ex-dividend price of the stock is given by Pt in period t. The stock pays arandom stream of dividends from period t + 1 given by {Dt+1+i}∞i=0. The bondsells for Pft in period t and has a payoff of one in period t + 1.

(a) Find an expression for the stock price that must hold at the consumer’soptimum. Interpret this expression.

20

(b) Find an expression for the bond price that must hold at the consumer’soptimum. Interpret this expression.

(c) Derive an expression for the risk premium on the stock that must hold atthe consumer’s optimum. Interpret this expression.

(3)

(a) What is the significance of an asset having the same payoff in all states ofthe world?

(b) Consider a situation involving three assets and two states. Suppose thatone asset is a risk-free bond with a return of 20%, a second asset has aprice of 100 and payoffs of 80 and 200 in the two states, and a third assethas payoffs of 100 and 0 in the two states. Find

(i) the prices of the implied contingent claims in the two states,

(ii) the price of the third asset.

Additionally, answer the following question.

(iii) What is the risk premium associated with the second asset?

(4) Consider the following two-period problem for a household in which thereis one state of the world in the first period and two states in the second period.Income in the first period is 6; in the second period it is 5 in state one whichoccurs with probability 0.8, and is 10 in state two. If instantaneous utility isln ct and the rate of time discount is 0.2 derive

(a) the optimal levels of consumption in each period,

(b) the state prices,

(c) the stochastic discount factors,

(d) the risk-free “rate of return” (i.e., rate of change) to income in period two,

(e) the “risk premium” for income in period two.

(5) Investors form a portfolio consisting of two risky assets which has the returnrp = xr1 + (1 − x)r2, where ri denote the returns on two risky assets andx denotes the fraction invested in asset 1. The returns on these assets arenormally distributed as ri ∼ N(µi,σ 2

i ) and Cov(r1, r2) = ρσ1σ2.

(a) Find the value of x that minimizes the variance of the portfolio return rp.

(b) Derive the mean-variance portfolio frontier under the assumption that therisky assets are

(i) uncorrelated,

(ii) perfectly negatively correlated,

(iii) perfectly positively correlated.

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(c) If E(r1) = 30% and E(r2) = 20%, the variance–covariance matrix for thetwo risky assets is

Σ =[

0.0081 0

0 0.0025

].

What is the optimal portfolio?

(d) Suppose that an investor A chooses a portfolio consisting of 75% in asset1 and 25% in asset 2 while another investor B chooses a different portfoliowith 50% in asset 1 and 50% in asset 2.

(i) Treating each investor’s portfolio as the “market portfolio,” whatbetas will each investor calculate for asset 1?

(ii) Consequently, which of the following is true and why?

∗ Investor A will require a higher rate of return on asset 1 thaninvestor B.

∗ They will both require the same rate of return.

∗ Investor B will require a higher rate of return on asset 1 than willinvestor A.


(1) A household with the utility function

U(ct) = (ct − γct−1)1−σ

1− σwhich maximizes Et

∑∞s=0 βsU(ct+s) can either invest in equity with a return of

rt or a risk-free one-period bond with return rft . Derive

(a) the optimal consumption plan,

(b) the equity premium, and

(c) the optimal consumption plan if there is only a risk-free asset.

In addition, comment on your findings.

(2)

(a) A household with the utility function U(ct) = ln ct , which maximizesEt∑∞s=0 βsU(ct+s), can either invest in a one-period domestic risk-free

bond with nominal return Rt , or a one-period foreign currency bond withnominal return (in foreign currency) of R∗t . If the nominal exchange rate(the domestic price of foreign exchange) is St derive

(i) the optimal consumption plan,

(ii) the foreign exchange risk premium.

(b) Suppose that foreign households have an identical utility function but adifferent discount factor β∗, what is their consumption plan and their riskpremium.

(c) Is the market complete? If not:

(i) What would make it complete?

(ii) How would this affect the two risk premia?

(3)

(a) If in question (1) there are only two-period bonds, what is

(i) the optimal consumption plan,

(ii) the risk premium for the domestic household?

(b) How would your answers change if instead that there is a one-perioddomestic risk-free bond but only a two-period foreign bond?

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(4) Let St denote the current price in dollars of one unit of foreign currency; Ft,Tis the delivery price agreed to in a forward contract; r the domestic interest ratewith continuous compounding; r∗ the foreign interest rate with continuouscompounding.

(a) Consider the following two portfolios:

(i) one long forward contract plus an amount of cash equal toFt,Te−r(T−t);

(ii) an amount e−r∗(T−t) of the foreign currency.

Find the value of the forward exchange rate Ft,T .

(b) Suppose that the foreign interest rate exceeds the domestic interest rateat date t, r∗ > r . What is the relation between the forward and spotexchange rates?

(c) A one-year long forward contract on a non-dividend-paying stock is en-tered into when the stock price is $40 and the risk-free interest rate is 10%per annum with continuous compounding.

What are the forward price and the initial value of the forward contract?

(d) Six months later, the price of the stock is $45 and the risk-free interestrate is still 10%. What are the forward price and the value of the forwardcontract?

(5) Consider a Vasicek model with two independent latent factors z1t and z2t .The price of an n-period bond and the log discount factor may be written as

pn,t = −[An + B1nz1t + B2nz2t],

mt+1 = −[z1t + z2t + λet+1],

and the factors are generated by

zi,t+1 − µi = φi(zit − µi)+ ei,t+1, i = 1,2.

(a) Derive the no-arbitrage condition for an n-period bond and its risk pre-mium.

(c) Express the yield on an n-period bond and its risk premium in terms ofthe yields on one- and two-period bonds.

(d) Derive an expression for the k-period ahead forward rate on an n-periodbond.

(e) Suppose that the second factor is an observable variable, how would thisaffect your answers to parts (a) and (b)?


(1) The Buiter and Miller (1981) model of the exchange rate may be describedby the following equations:

yt = α(st + p∗t − pt)− β(Rt −∆pt+1 − rt)+ gt + γy∗t ,mt − pt = yt − λRt,∆pt+1 = θ(yt −ynt )+π#

t ,∆st+1 = Rt − R∗t ,

where y is output, yn is full employment output, g is government expenditure,s is the log exchange rate,R is the nominal real interest rate, r is the real interestrate,m is log nominal money, p is the log price level, π# is target inflation, andan asterisk denotes the foreign equivalent.

(a) Derive the long-run and short-run solutions for output, the price level andthe exchange rate.

(b) Hence comment on the effects of monetary and fiscal policy.

(c) Suppose that the foreign country is identical and the two countries com-prise the “world” economy. Denoting the corresponding world variable asx̄t = xt + x∗t and the country differential by x̃t = xt − x∗t ,

(i) derive the solutions for the world economy and for the differencesbetween the economies;

(ii) analyze the effects of monetary and fiscal policy on the world econ-omy.

(2) Suppose that the “world” is compromised of two identical countries. Eachcountry consumes home and foreign goods and maximizes

Vt =∞∑s=0

βs[(cαH,t+sc1−α

F,t+s)1−σ

1− σ + γ lnMt+sPt+s

]

subject to its budget constraint and uncovered interest parity. Expressed interms of home’s prices, the home country budget constraint is

PH,tcH,t + StPF,tcF,t +∆Mt+1 +∆Bt+1 = PH,tyH,t + rtBt,where cH,t is consumption of home produced goods, cF,t is consumption offoreign produced goods, PH,t is the price of the home country’s output, which isdenoted yH,t , PF,t is the price of the foreign country’s output, Pt is the domesticlevel, Mt is domestic currency, and Bt is the home country’s borrowing fromabroad expressed in domestic currency and is at the world rate of interest rt andSt is the nominal exchange rate. Output and the supply of money are exogenous.

25

(a) Using an asterisk to denote the foreign country equivalent variable (e.g.,c∗H,t is the foreign country’s consumption of domestic output andM∗

t is theforeign country’s currency), derive the optimal solution for both countriesand for the world interest rate and the exchange rate.

(b) Analyze the effects of shocks to home and foreign outputs and moneysupplies.


(1) Consider the following characterizations of the IS–LM and DGE models.

IS–LM:

y = c(y, r)+ i(y, r)+ g,m− p = L(y, r).

DGE:

∆c = − 1σ(r − θ) = 0,

y = c + i+ g,y = f(k),∆k = i,fk = r ,

where y is output, c is consumption, i is investment, k is the capital stock, gis government expenditure, r is the real interest rate, m is log nominal money,and p is the log price level.

(a) Comment on each equation and on the underlying approaches to macro-economics.

(b) Discuss the usefulness of each model for analyzing the short and the longrun.

(c) Comment on the implications for the effectiveness of monetary and fiscalpolicy in the two models.

(2)

(a) How might a country’s international monetary arrangements affect itsconduct of monetary policy?

(b) What other factors might influence the way it carries out its monetarypolicy?

(3) The Lucas–Sargent proposition is that systematic monetary policy is ineffec-tive. Examine this hypothesis using the following model of the economy due toBull and Frydman (1983):

yt = α1 +α2(pt − Et−1pt)+ut,dt = β(mt − pt)+ vt,∆pt = θ(p∗t − pt−1),

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where y is output, d is aggregate demand, p is the log price level, p∗ is themarket clearing price, m is log nominal money, and u and v are zero-meanmutually and serially independent shocks.

(a) Derive the solution for output and prices.

(b) Ifmt = µ+mt−1+et , where et is a zero-mean serially independent shock,comment on the effectiveness of

(i) an unanticipated shock to money in period t,

(ii) a temporary anticipated shock to money in period t,

(iii) a permanent anticipated shock to money in period t.

(c) Hence comment on the Lucas–Sargent proposition.

(4) Consider the following model of the economy:

yt = −β(Rt − Etπt+1 − r),πt = Etπt+1 +αyt + et,Rt = γ(Etπt+1 −π∗),

where et is a zero-mean serially independent shock.

(a) What are the long-run and short-run solutions for πt and yt?

(b) How would the behavior of inflation, output and monetary policy be af-fected by

(i) a temporary shock et ,

(ii) an expected shock et+1?

(c) Suppose that the output equation is modified to

yt = −β(Rt − Etπt+1 − r)− θet,where et can be interpreted as a supply shock. How would the behaviorof inflation, output, and monetary policy be affected by a supply shock?

(5) Suppose that a monetary authority is a strict inflation targeter attemptingto minimize E(πt −π∗t )2 subject to the following model of the economy:

πt = aRt + zt + et,where a = α + εt , E(zt) = z + εzt , and εat and εzt are random measurementerrors of α and z, respectively; εat , εzt , and et are mutually and independentlydistributed random variables with zero means and variances σ 2

z , σ 2a , and σ 2

e .

(a) What is the optimal monetary policy

(i) in the absence of measurement errors,

(ii) in the presence of measurement errors?

(b) What are the broader implications of these results for monetary policy?

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(6) A highly stylized model of an open economy is

pt = αpt−1 + θ(st − pt),st = Rt + Rt+1,

where pt is the price level, st is the exchange rate, and Rt is the nominal interestrate. Suppose that monetary policy aims to choose Rt and Rt+1 to minimize

L = (pt − p∗)2 + β(pt+1 − p∗)2 + γ(Rt − R∗)2,

where pt−1 = Rt+2 = 0.

(a) Find the time consistent solutions for Rt and Rt+1. (Hint: first find Rt+1

taking pt and Rt as given.)

(b) Find the optimal solution by optimizing simultaneously with respect toRt and Rt+1.

(c) Compare the two solutions and the significance of γ.

(7) Consider the following model of Broadbent and Barro (1997):

yt = α(pt − Et−1pt)+ et,dt = −βrt + εt,mt = yt + pt − λRt,rt = Rt − Et∆pt+1,

yt = dt,

where et and εt are zero-mean mutually and serially correlated shocks.

(a) Derive the solution to the model

(i) under money supply targeting,

(ii) inflation targeting.

(b) Derive the optimal money supply rule if monetary policy minimizes

Et∞∑s=0

(pt+s − Et+s−1pt+s)2

subject to the model of the economy.

(c) What does this policy imply for inflation and the nominal interest rate?

(d) Derive the optimal interest rate rule and compare the behavior of inflationunder the two types of monetary policy.

(e) How would these optimal policies differ if monetary policy was based ontargeting inflation instead of the price level?

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(8) Consider the following model of an open economy:

πt = µ + βEtπt+1 + γxt + eπt,xt = −α(Rt − Etπt+1 − θ)+φ(st + p∗t − pt)+ ext,

∆st+1 = Rt − R∗t + est,where eπt , ext , and est are zero-mean mutually and serially independent shocksto inflation, output and the exchange rate.

(a) Derive the solution to the model.

(b) What is the optimal monetary policy if the monetary authority is a strictinflation targeter with the objective of minimizing Et

∑∞i=0βi(πt+i−π∗t+i)2?

(c) How would optimal monetary policy respond to an anticipated futureincrease in the world interest rate R∗t+1?


(1) Consider a variant on the basic real-business-cycle model. The economy isassumed to maximize

Et∞∑s=0

βsc1−σt+s

1− σ

subject to

yt = ct + it,yt = Atkαt ,

∆kt+1 = it − δkt,lnAt = ρ lnAt−1 + et,

where yt is output, ct is consumption, it is investment, kt is the capital stock,At is technical progress, and et ∼ i. i.d.(0,ω2).

(a) Derive the optimal solution.

(b) Log-linearize this solution about its steady state and show that the result-ing model can be expressed as a first-order VAR in ln ct − ln c̄ and lnkt −ln k̄, where ln c̄ and ln k̄ are the steady-state values of ln ct and lnkt .

(c) If, in practice, output, consumption, and capital are nonstationary I(1)variables:

(i) Comment on why this model is not a useful specification.

(ii) Suggest a simple respecification of the model which would improveits usefulness.

(d) In practice, we find that output, consumption, and capital also have inde-pendent sources of random variation.

(i) Why is this not compatible with this model?

(ii) Suggest possible ways in which the model might be respecified toachieve this.

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(2) Consider the real-business-cycle model defined in terms of the same vari-ables as in question (1) with the addition of employment, nt :

Ut = Et∞∑s=0

βs(ct+s − γnφt+s)1−σ − 1

1− σ ,

yt = ct + it,yt = Atkαt n1−α

t ,

∆kt+1 = it − δkt,lnAt = ρ lnAt−1 + et,

where et ∼ i. i.d.(0,ω2).

(a) Derive the optimal solution.

(b) Log-linearize this solution about its steady state and express the resultingmodel as a VAR.

(c) What is the implied dynamic behavior of the real wage and the real interestrate?

(3) Consider two countries which produce an identical good and have identicalpreferences and technology. In the home country households in each countrymaximize

Et∞∑s=0

βs(cγt+sl

1−γt+s )1−σ

1− σsubject to their budget constraint

ct +∆Bt+1 = dt +wtnt + rBt,where c is consumption and l is leisure, n = 1− l is work, w is the wage rate,d is dividend earnings, and B is a one-period bond paying interest r . Firmsmaximize the present-value of the firm

Et∞∑s=0

(1+ r)−sdt+s(yt+s − it+s −wt+snt+s),

where net revenues are distributed as dividends so that dt = yt − it − wtntand y is output, i is investment, and capital k satisfies ∆kt+1 = it − δkt . Theproduction function is yt = Atkat n

1−αt , where At is technical progress which

satisfies lnAt = ρ lnAt−1+εt , where εt is a zero-mean i.i.d. variable. Governmentspends g and pays for this with taxes on output so that its budget constraintis gt = τtyt . The foreign country is the same but its variables are denoted withan asterisk.

(a) Derive the optimal solution assuming that each country is small and takesthe other as given.

(b) Show how domestic technology shocks affect the behavior of the solution.

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(c) In the world economy the sum of domestic and foreign bonds is zero.Hence

(i) derive the balance of payments for each country assuming interestparity;

(ii) analyze the effects of foreign technology shocks on the domesticeconomy.

References

Broadbent, B., and R. J. Barro. 1997. Central bank preferences and macroeonomicequilibrium. Journal of Monetary Economics 30:17–43.

Buiter, W., and M. Miller. 1981. Macroeconomic policy and international competitive-ness. Oxford Economic Papers 33:143–75.

Bull, C., and R. Frydman, R. 1983. The derivation and interpretation of the Lucas supplyfunction. Journal of Money Credit and Banking 83:82–95.

Rebelo, S. 1991. Long-run policy analysis and long-run growth. Journal of PoliticalEconomy 99:500–21.

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