ECON 5118 Macroeconomic Theory

Winter 2010

Test 1 January 29, 2010Answer ALL Questions Time Allowed: 1 hour 20 min

Attention: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Donot forget to put your name on the front page.

1. Suppose that the capital stock kt of an economy isgiven by the first-order difference equation

kt+1 = (1− δ)kt + it, t = 0, 1, 2, . . . ,

where it is investment in period t and 0 < δ < 1 isthe depreciation rate. Show that the capital stockin period t is given by

kt =∞�

s=1

(1− δ)sit−s.

2. Consider a simple economy with two goods, out-put yt and capital kt, with one aggregate householdand one aggregate competitive firm. The produc-tion function is given by yt = F (kt) in each periodt. Output can be consumed this period or investedfor the next period, that is, yt = ct + it. Thereforecapital accumulation is given by

∆kt+1 = it − δkt,

where δ is the rate of depreciation of capital.

(a) Derive the dynamic resource constraint.(b) Suppose the household’s preferences can be

represented by a stationary and separable util-ity function

∞�

s=0

βs log ct+s

where β = 1/(1 + θ) is a discount factor. Setup the intertemporal optimization problem.

(c) Derive the Euler equation of intertemporalconsumption.

3. Suppose that the household’s utility function ineach period t is

U(ct, lt) =c1−σt

1− σ+ log lt,

where ct is consumption lt is leisure time. The pro-duction function is

F (kt, nt) = Akαt n1−α

t ,

where nt is labour input and nt + lt = 1. The dy-namic resource constraint is

F (kt, nt) = ct + kt+1 − (1− δ)kt.

(a) Derive the Euler equation for intertemporalconsumption.

(b) Derive the relationship between labour supplyand consumption given the capital stock.

4. Suppose that in question 2 the production functionis

F (kt) = Akαt , 0 < α < 1.

Investment, however, requires installation cost of12φit/kt per unit, where φ > 0. Household utilityin each period t is given by

U(ct) = log ct.

(a) Set up the resource constraint and the opti-mization problem.

(b) Find Tobin’s q in each period. Is qt > 1?

5. (a) With the aid of a diagram explain what aPhillips curve is.

(b) Explain why monetarists such as Milton Fried-man argue against using monetary policy todrive unemployment below its neutral rate.

ECON 5118 Macroeconomic Theory

Winter 2010

Test 2 February 26, 2010Answer ALL Questions Time Allowed: 1 hour 20 min

Attention: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Youmay find the information in the Appendix useful. Donot forget to put your name on the front page.

1. Suppose that the production function of an economyis given by

Ft(Kt, Nt) = At(αkKρt + αnN

ρt )

1/ρ, 0 �= ρ < 1,

where Kt and Nt are capital and labour inputs re-spectively in period t.

(a) Determine whether Ft exhibits constant re-turns to scale.

(b) Express each function in per capita form.

(c) One of the key result of the Solow-Swan modelis that the larger the capital stock per capital,the lower the growth rate. Does this resulthold for the above production function?

2. Suppose that the aggregate production function inperiod t of a closed economy is given by

Yt = F (Kt, AtNt),

where At = (1−µ)t and F is strictly concave and ex-hibits constant returns to scale. The population Nt

grows at a constant rate n. Define effective labouras N#

t = AtNt.

(a) Show that output per effective labour, y#t =Yt/N

#t , can be expressed as

y#t = f(k#t ),

where k#t = Kt/N#t .

(b) Let household utility be U(Ct) = logCt in eachperiod and the discount factor is β = 1/(1 −θ). Derive the Euler equation for intertemporalconsumption.

(c) Find the steady-state solution for y#, k# andc#.

3. Suppose that 0 < ρ < 1. Show that

∞�

s=1

sρ(1− ρ)s−1 = 1/ρ.

4. Suppose that the household’s utility function ineach period t is

U(ct, lt) = log ct + η log lt, η > 0,

where lt is leisure time, nt is labour supply and nt+lt = 1. The budget constraint is

at+1 + ct = wtnt + xt + (1 + rt)at,

where at is financial asset, wt is the wage rate, andxt is an exogenous income in period t.

(a) Substitute the labour constraint nt+lt = 1 intothe utility function and set up the optimizationproblem.

(b) Derive the Euler equation for intertemporalconsumption.

(c) Derive the labour supply function in terms ofconsumption and the wage rate.

5. Consider a stock with price pt and pays dividendsdt in period t. Assume that consumers are risk-neutral and have a discount rate of θ = r, thus theymaximize

�∞s=0 ct+s/(1 + r)s. If consumers sell the

stock, it always happens after the company has paidthe dividends.

(a) Let xt be exogenous incomes and at be theamount of stock hold by the consumers in pe-riod t. Determinate the consumer budget con-straint.

(b) Set up the Bellman equation and show that inequilibrium

pt =dt+1 + pt+1

1 + r.

(c) Assume that there is no bubble in the stockmarket, that is, lims→∞ pt+s/(1 + r)s = 0.Solve the above difference equation for pt. In-terpret the result.

(d) Suppose that the dividends are constant for ev-ery period, that is, dt = d. What will happento the price of the stock if interest rate goesup?

Appendix

Geometric Series

For −1 < x < 1,

∞�

s=0

axs =a

1− x.

Also,

∞�

s=0

1

(1 + r)s=

1 + r

r,

∞�

s=1

1

(1 + r)s=

1

r.

Taylor Series

Let f(x) be a single variable function of x which is dif-ferentiable to any order s. The Taylor expansion of fcentred at a point x∗ is given by

f(x) =∞�

s=0

f (s)(x∗)

s!(x− x∗)s.

In the multivariate cases, suppose f : Rn → R is a C2

function. Then the second-order Taylor approximationof f(x) about a point x∗ is

f(x) � f(x∗)+∇f(x∗)T(x−x∗)+(x−x∗)T∇2f(x∗)(x−x∗),

where ∇f(x∗) and ∇2f(x∗) are the gradient and Hessianof f , both evaluated at the point x∗.

Logarithmic Approximation

For small values of x,

log(1 + x) � x.

For x around 1,log x � x− 1.

Let f(x) be any differentiable function. The log-linearapproximation of f(x) around a point x∗ is

f(x) � f(x∗) + x∗f �(x∗)x̂

where x̂ = log x− log x∗.

First-Order Difference Equation

Suppose that {yt} is a sequence of variables which sat-isfies the first-order difference equation

yt+1 = ayt + b, t = 0, 1, 2, . . . ,

where a and b are constant and −1 < a < 1. Then

limn→∞

yt+n =b

1− a.

For the non-constant coefficient equation

yt+1 = ayt + bt, t = 0, 1, 2, . . . ,

if a ≥ 1 or a ≤ −1, then

yt = −1

a

∞�

s=0

bt+s

as= −

∞�

s=0

bt+s

as+1.

On the other hand, if −1 < a < 1, then

yt =∞�

s=1

asbt−s.

“Sometimes our borrowers find themselves underwater.”

2

ECON 5118 Macroeconomic Theory

Winter 2010

Test 3 March 19, 2010Answer ALL Questions Time Allowed: 1 hour 20 min

Attention: Please write your answers on the answer

book provided. Use the right-side pages for formal an-

swers and the left-side pages for your rough work. You

may find the information in the Appendix useful. Do

not forget to put your name on the front page.

1. The primary deficit of the government of a closed

economy, expressed in real term as proportions of

output, is given by

dtyt

= (1 + πt+1)(1 + γt+1)bt+1

yt+1− (1 +Rt)

btyt. (1)

(a) Express the above equation as a first-order dif-

ference equation.

(b) Suppose that R < π + γ and that the

deficit/output ratio is constant, that is,

dt+n

yt+n=

dtyt, n = 1, 2, . . . ,

show that

limn→∞

bt+n

yt+n=

1

(1 + π)(1 + γ)− (1 +R)

dtyt

� 1

π + γ −R

dtyt

< ∞.

(c) Give an economic interpretation of the result.

2. Suppose the household budget constraint in real

terms is given by

(1 + τ c)ct + kt+1 + bt+1 = (1− τw)wtnt

+ [1 + (1− τk)rkt ]kt + (1 + rbt )bt.

where bt is bond investment, kt is capital stock, wtnt

is labour income, rkt and rbt are rates of return on

capital and bonds, τ c, τw, and τk are tax rates on

consumption, wages, and capital respectively. The

household utility function in each period is

U(ct, lt) =c1−σt

1− σ+ η

�l1−σt

1− σ

�,

Table 1: Temporary Tax Cut

Period GBC

t− 1 : gt−1 +Rbt = Tt−1

t : gt−1 +Rbt = Tt−1 +∆Tt +∆bt+1

t+ 1 : gt−1 +R(bt +∆bt+1) = Tt−1 +∆Tt+1 +∆bt+2

t+ 2 : gt−1 +Rbt = Tt−1

where lt is leisure. Time constraint is given by nt +

lt = 1.

(a) Set up the utility maximization problem.

(b) Find the marginal rate of substitution between

leisure and consumption.

(c) What is the effect of a higher income tax rate

τw on labour supply?

3. Suppose the government cuts taxes in period t(∆Tt < 0) to boost the economy by bond financ-

ing. It then increase taxes in next period to restore

the fiscal balance. Everything is back to normal in

period t + 2. Table 1 show the dynamics of the

government budget constraint.

(a) Find ∆bt+1,∆bt+2, and ∆Tt+1 in terms of the

tax cut ∆Tt.

(b) State the Ricardian Equivalence Theorem.

4. The table on the next page lists the fiscal positions

of some selected countries in 2010. Answer each of

the following questions for Greece:

(a) Does the country has a primary deficit or sur-

plus?

(b) Does the country’s long-run fiscal stance sta-

ble or unstable in view of the difference equa-

tion (1)?

Repeat the above analysis for Italy and Canada.

Which of these three countries will likely to have a

fiscal crisis in the near future? Explain.

5. Suppose the household budget constraint in real

terms is given by

(1 + πt+1)bt+1 − bt + (1 + πt+1)mt+1 −mt + ct

= xt +Rtbt,

where bt is bond investment, mt is cash holding,

ct is consumption, xt is exogenous income, πt is the

inflation rate, and Rt is the nominal interest rate on

bonds in period t. The household utility function in

each period is represented by

U(ct,mt) = c1−αt mα

t .

(a) Set up the utility maximization problem.

(b) Derive the Euler equation.

(c) Derive the money demand function.

(d) What is the effect of a higher interest rate on

money demand?

Appendix

Geometric Series

For −1 < x < 1,

∞�

s=0

axs=

a

1− x.

Also,

∞�

s=0

1

(1 + r)s=

1 + r

r,

∞�

s=1

1

(1 + r)s=

1

r.

Logarithmic Approximation

For small values of x,

log(1 + x) � x.

For x around 1,

log x � x− 1.

Let f(x) be any differentiable function. The log-linear

approximation of f(x) around a point x∗ is

f(x) � f(x∗) + x∗f �

(x∗)x̂

where x̂ = log x− log x∗.

First-Order Difference Equation

Suppose that {yt} is a sequence of variables which sat-

isfies the first-order difference equation

yt+1 = ayt + b, t = 0, 1, 2, . . . ,

where a and b are constant and −1 < a < 1. Then

limn→∞

yt+n =b

1− a.

For the non-constant coefficient equation

yt+1 = ayt + bt, t = 0, 1, 2, . . . ,

if a ≥ 1 or a ≤ −1, then

yt = −1

a

∞�

s=0

bt+s

as= −

∞�

s=0

bt+s

as+1.

On the other hand, if −1 < a < 1, then

yt =∞�

s=1

asbt−s.

2

ECON 5118 Macroeconomic Theory

Winter 2010

Final Exam April 22, 2010Answer ALL Questions Time: 9:00 a.m. – 12:00 p.m.

Attention: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Youmay find the information in the Appendix useful. Re-member to put your name on the front page.

1. Write a short critical essay on the differences be-tween the dynamic general equilibrium model andthe traditional Keynesian model. How does the NewKeynesian economics relate to the two approaches?

2. Suppose that installation cost of each unit of capitalin a centralized economy is 1

2φit/kt, where φ > 0.The resource constraint is therefore

F (kt) = ct +

�1 +

φit2kt

�it, φ ≥ 0. (1)

The central planner’s problem is to maximize�∞s=0 β

sU(ct+s) subject to the resource con-straint (1) and the capital accumulation equation

kt+1 = it + (1− δ)kt. (2)

(a) Set up the optimization problem and derive thefirst-order conditions.

(b) Define Tobin’s q in period t as qt = µt/λt,where λt and µt are the Lagrange multipli-ers for the constraints (1) and (2) respectively.Show that

it+s =1

φ(qt+s − 1)kt+s, s ≥ 0.

What is the condition for investment to takeplace in each period?

3. Suppose that a strictly concave production functionYt = Ft(Kt, Nt) exhibits constant returns to scale.

(a) Show that in per capita terms, the functioncan be expressed as yt = ft(kt).

(b) Show that the growth rate of capital per personis

∆kt+1

kt� ∆Kt+1

Kt− ∆Nt+1

Nt= γ − n,

where γ and n are the growth rates of capitaland labour respectively.

4. The representative household’s problem is

max∞�

s=0

βs c1−σt+s

1− σ, σ > 0,

subject to the budget constraints

∆at+s+1 + ct+s = xt+s + rt+sat+s, s = 0, 1, . . . ,

where ct, at, xt, and rt are consumption, asset, ex-ogenous income, and interest rate respectively inperiod t and β is the discount rate.

(a) Identify the state variable(s) and the controlvariable(s).

(b) Set up the Bellman equation.

(c) Find the necessary conditions for optimization.

(d) Derive the Euler equation for intertemporalconsumption.

5. Consider the lump-sum taxation model where thecentral planer maximizes household utility

∞�

s=0

βsU(ct+s, gt+s)

subject to the resource constraint

F (kt) = ct + kt+1 − (1− δ)kt + gt

and the government budget constraint gt = Tt.

(a) Identify the choice variables and the state vari-able.

(b) Set up the Bellman equation.

(c) Derive the necessary conditions from the Bell-man equation.

(d) Find the Euler equation, the marginal rate ofsubstitution of household consumption to gov-ernment services, and the steady-state condi-tion.

6. (a) Write a short essay on the three functions ofmoney.

(b) Explain what the Friedman rule of money sup-ply is.

7. Suppose that the household budget constraint isgiven by

(1+πt+1)(bt+1+mt+1)− bt−mt+ ct+T (ct,mt)

= xt +Rtbt, (3)

where T (ct,mt) ≥ 0 is a transaction cost of con-sumption, with

T (0,m) = 0, Tc ≥ 0, Tcc ≥ 0, Tm ≤ 0, Tmm ≥ 0.

Households maximize�∞

s=0 U(ct+s) subject to theconstraint (3).

(a) Derive the Euler equation of intertemporalconsumption.

(b) Use the first-order conditions to show that

Tm,t+1 = −Rt+1. (4)

(c) Suppose that in the steady state ct = c,mt =m,Rt = R,πt = π, and rt = θ. Show that thebudget constraint (3) and equation (4) can beexpressed as

c+ πm+ T (c,m)− x− θb = 0, (5)

Tm(c,m) +R = 0. (6)

(d) Use equations (5) and (6) to show that

∂m

∂c= − 1 + Tc

π + Tm> 0.

8. A necessary condition for the central bank to setthe optimal rate of inflation is

(1 + µ)Um,t + µ(Ucm,tct − Ulm,tnt + Umm,tmt) = 0,

where the household utility function U(ct, lt,mt) isincreasing and concave in consumption ct, leisure lt,and real money balance mt. The labour supply con-straint is nt = 1− lt. Suppose that U is homotheticand separable in lt and mt.

(a) Apply the Euler theorem to U and show that

Ucm,tct + Ulm,tlt + Umm,tmt = 0.

(b) Show that the Friedman rule is optimal.

9. An aggregate firm produces the final good y with aCES technology

y =

�N�

i=1

αixρi

�1/ρ

, 0 �= ρ < 1,N�

i=1

αi = 1,

where xi is the ith intermediate input with pricepi. Denote P as the price of the final good andφ = 1/(1− ρ) as the elasticity of substitution.

(a) Show that the demand function for input i is

xi =

�αiP

pi

�φ

y.

(b) Assuming a competitive market for the finalgood market with zero profit, express P interms of pi and φ.

(c) Each intermediate good is produced by a singlefirm with a production function

xi = Kαi L

1−αi , 0 < α < 1,

where Ki and Li denote capital and labour in-puts of firm i with nominal market prices Prand Pw (so that r and w are the real pricesrelative to the final good price P ). Show thatprofit maximizing price of firm i given the finalgood price and demand is

pi =PC

ρ,

where C = c(r, w) is the unit cost in producinginput i.

10. Consider an economy consisting of many identityfirms. In any period only a fraction ρ of firms canset new prices, with the firms chosen at random.The objective of the firms is

minp#t

∞�

s=0

(1− ρ)sEt(p#t − p∗t+s)

2,

where p∗t is the profit maximizing price in period t.

(a) Show that the optimal solution is

p#t = ρ∞�

s=0

(1− ρ)sEtp∗t+s.

2

(b) Express p#t recursively in term of p∗t andEtp

#t+1.

(c) Show that

p#t − pt = ρ(p∗t − pt)+ (1− ρ)Et[p#t+1− pt+1]

+ (1− ρ)Etπt+1.

(d) Show that the average (log) price in period t is

pt = ρ∞�

s=0

(1− ρ)sp#t−s.

Appendix

Geometric Series

For −1 < x < 1,

∞�

s=0

axs =a

1− x.

Also,

∞�

s=0

1

(1 + r)s=

1 + r

r,

∞�

s=1

1

(1 + r)s=

1

r.

Logarithmic Approximation

For small values of x,

log(1 + x) � x.

For x around 1,log x � x− 1.

Let f(x) be any differentiable function. The log-linearapproximation of f(x) around a point x∗ is

f(x) � f(x∗) + x∗f �(x∗)x̂

where x̂ = log x− log x∗.

First-Order Difference Equation

Suppose that {yt} is a sequence of variables which sat-isfies the first-order difference equation

yt+1 = ayt + b, t = 0, 1, 2, . . . ,

where a and b are constant and −1 < a < 1. Then

limn→∞

yt+n =b

1− a.

For the non-constant coefficient equation

yt+1 = ayt + bt, t = 0, 1, 2, . . . ,

if a ≥ 1 or a ≤ −1, then

yt = −1

a

∞�

s=0

bt+s

as= −

∞�

s=0

bt+s

as+1.

On the other hand, if −1 < a < 1, then

yt =∞�

s=1

asbt−s.

“Of course money is in my utility function.”

3