Faculty of Social Sciences - Økonomisk Institut ... of Social Sciences Adv. Macro 2 Exercises Week 7 2nd of November 2011 (week 7) Slide 1/40 Adv. Macro 2 Exercises Introduction Exercise

Faculty of Social Sciences

Adv. Macro 2 ExercisesWeek 7

2nd of November 2011 (week 7)Slide 1/40

Adv. Macro 2Exercises

Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Outline

1 Introduction

2 Exercise 3.13.1.13.1.23.1.3

3 Exercise 3.23.2.13.2.23.2.3

4 Exercise 3.33.2.13.2.23.2.3

5 Summing up

2nd of November 2011 (week 7) — Slide 2/40


Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Outline

1 Introduction

2 Exercise 3.13.1.13.1.23.1.3

3 Exercise 3.23.2.13.2.23.2.3

4 Exercise 3.33.2.13.2.23.2.3

5 Summing up



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Introduction• We have three exercises today:

• 3.1: On firm investment with perfect competition.• 3.2: On firm investment with imperfect competition.• 3.3: On money classical versus Keynesian views.

• No replay.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Outline

1 Introduction

2 Exercise 3.13.1.13.1.23.1.3

3 Exercise 3.23.2.13.2.23.2.3

4 Exercise 3.33.2.13.2.23.2.3

5 Summing up



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.1Consider a firm with the production function

Yt = F(Kt ,Lt )

where Yt , Kt , and Lt are output, capital input, and labor input per time unit attime t , respectively, while F is a neoclassical production function with CRS andsatisfying the Inada conditions. The increase per time unit in the firm’s capitalstock is given by

Kt = It −δKt δ > 0,

where It is gross investment per time unit at time t and δ is the capitaldepreciation rate. There is perfect competition in all markets and the realinterest rate faced by the firm is a constant r > 0.Cash flow at time t is

Rt = F(Kt ,Lt )−wt Lt − It −G(It )

where wt is the real wage and G(It ) is a capital installation cost functionsatisfying

G(0) = G′(0) = 0, G′′(I) > 0.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.2 aa) Set up the firm’s intertemporal production and investment problem as a

standard optimal control problem, given that the firm wants to maximize itsmarket value.

• Answer:

maxV0 =∫

∞

0(F(Kt ,Lt )−wt Lt − It −G(It ))e−rt dt s.t.

Lt ≥ 0, It free,

Kt = It −δKt ,

Kt ≥ 0, for all t ≥ 0.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.2 bb) Derive the first-order conditions and the transversality condition (TVC) for a

solution. Let the adjoint variable be denoted qt .

• Answer: From the current-value Hamiltonian we get:

H = F(K ,L)−wL− I−G(I) + q(I−δK )

∂H∂L

= FL(K ,L)−w = 0⇒ FL(K ,L) = w

∂H∂I

= −1−G′(I) + q = 0⇒ 1 + G′(I) = q

∂H∂K

= FK (K ,L)−qδ =−q + rq

and the transversality condition is

limt→∞

Kt qt e−rt = 0.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.2 c

∂H∂L

= FL(K ,L)−w = 0⇒ FL(K ,L) = w

∂H∂I

= −1−G′(I) + q = 0⇒ 1 + G′(I) = q

∂H∂K

= FK (K ,L)−qδ =−q + rq

limt→∞

Kt qt e−rt = 0.

c) What is the economic interpretation of qt ?

• Answer: qt can be interpreted as the shadow price (the marginal value tothe firm) of installed capital at time t along the optimal path.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.2 dd) Explain why F(Kt ,Lt ) = Lt F(kt ,1)≡ Lt f (kt ) where kt ≡ Kt/Lt and sign f ′

and f ′′.

• Answer Because of CRS. We have f ′ = FK (k ,1) > 0.So we also have f ′′ = FKK (k ,1) < 0.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.2 eFOC: FL(K ,L) = wProduction function on intensive form: F(Kt ,Lt ) = Lt f (kt )

e) Show that the optimal labor input is such that the capital-labor ratio at timet is an increasing function of wt .

• Answer: We can write

FL(Kt ,Lt ) =∂ [Lt f (kt )]

∂Lt= f (kt )+ f ′(kt )

−Kt

L2t

Lt = f (kt )− f ′(kt )kt ≡ϕ(kt ) = wt > 0.

Since F satisfies the Inada conditions, this equation has a solution k > 0.Since ϕ′(k) =−kf ′′(k) > 0, we have that ϕ(·) can be inverted.Thus, kt = ϕ−1(wt )≡ k(wt ) where

k ′(wt ) =1

ϕ′(k(wt ))=− 1

k(wt )f ′′(k(wt ))> 0.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.3 aFOC: 1 + G′(I) = q

Suppose, from now, that wt = w for all t ≥ 0, where w is a positive constant. Letthe corresponding optimal capital-labor ratio be denoted k .

a) Show that the optimal investment level, It , can be written as an implicitfunction of qt .

• Answer: We have G′(I) = q−1It follows that: I = G′−1(q−1)≡M(q) where

M(1) = 0

G′′(M(q))M ′ = 1↔M ′ =1

G′′> 0



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.3 bb) Construct a phase diagram for the (K ,q) dynamics, assuming that a

steady state with K > 0 exists. Let the steady state value of K be denotedK ∗.

• Answer:Locus 1: K = I−δK = M(q)−δK .From K = 0 we get M (q) = δK .We have that dq

dK |K =0 > 0 because M ′ > 0.We have that K = 1→ q = 1.

Locus 2: q = (r + δ)q−FK (K ,L) = (r + δ)q− f ′(k).

From q = 0 we get q = f ′(k)r+δ

= q∗.

Existence of steady state requires f ′(k)r+δ

> 1.

Arrows and full diagram explained on white board.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.3 c+dc) For an arbitrary K0 > 0, indicate in the diagram the movement of the pair

(Kt ,qt ) along the optimal path.

d) In a new diagram draw the time profiles of qt , It and Kt . Comment on why,in spite of the marginal productivity of capital in the steady state exceedingr + δ, there is no incentive to increase K above K ∗.

• Answer: q remains on its constant level. There is convergence to the(K ∗,q∗) steady state.

• Normally we have f ′(k)

= r + δ in steady state, but this implicitly impliesq = 1 and thus is the case of no installation costs.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.3 ee) It is common to call K ∗ the “desired capital stock”. Find an algebraic

expression for the desired capital stock.

• Answer I∗ = M

(f ′(k)r+δ

)= δK ∗↔ K ∗ = 1

δM

(f ′(k)r+δ

)



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.1.3 ff) Show that optimal net investment, In

t ≡ It −δKt , equals δ(K ∗−Kt ). In thisnet investment rule you should recognize a principle from introductorymacroeconomics. What is the name of this principle? Comment.

• Math answer: Int ≡ It −δKt = I∗−δKt = δ(K ∗−Kt ).

• Interpretation answer: Capital adjustment principle.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Outline

1 Introduction

2 Exercise 3.13.1.13.1.23.1.3

3 Exercise 3.23.2.13.2.23.2.3

4 Exercise 3.33.2.13.2.23.2.3

5 Summing up



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.1 IConsider a firm supplying its own differentiated good in the amount yt per timeunit at time t. The production function is

yt = K αt L1−α

t 0 < α < 1

where Kt and Lt is capital and labor input at time t .The nominal wage and the nominal general price level in the economy faced bythe firm are constant over time and exogenous to the firm. So the real wage isan exogenous positive constant, w . The demand, yd , for the firm’s output isperceived by the firm as given by

yd = p−ε Y e

nε > 1

where p is the price set in advance by the firm (as a markup on expectedmarginal cost), relative to the general price level in the economy, n is the givenlarge number of monopolistically competitive firms in the economy, Y e is theexpected overall level of demand, and ε is the (absolute) price elasticity ofdemand.We assume that p is kept fixed within the time horizon relevant for thisanalysis.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.1 IIThe increase per time unit in the firm’s capital stock is given by

Kt = It −δKt δ > 0 K0 > 0 given,

where It is gross investment per time unit at time t and δ is the capitaldepreciation rate. We assume that p is high enough to always be above actualmarginal cost so that it always pays the firm to satisfy demand. Then cash flowat time t is

Rt = pyd −wLt − It −G(It ),

where G(It ) is a capital installation cost function satisfying

G(0) = G′(0) = 0, G′′(I) > 0.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.2 a

yt = K αt L1−α

t 0 < α < 1

yd = p−ε Y e

nε > 1

a) Given yt = yd , find Lt as a function of Kt .

• Answer: Lt =(K−α

t yd) 1

1−α ≡ L(Kt ,yd

)



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.2 bb) Given that the real interest rate faced by the firm is a constant r > 0, set up

the firm’s intertemporal production and investment problem as a standardoptimal control problem, given that the firm wants to maximize its marketvalue.

• Answer:

maxV0 =∫

∞

0(pyd −wL

(Kt ,y

d)− It −G(It ))e−rt dt s.t.

It free

Kt = It −δKt ,

Kt ≥ 0, for all t ≥ 0.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.2 cc) Derive the first-order conditions and the transversality condition (TVC) for a

solution.

• Answer: H = pyd −wL(Kt ,yd

)− It −G(It ) + q(I−δK ),

The first-order conditions are:

∂H∂I

= −1−G′(I) + q = 0⇒ 1 + G′(I) = q

∂H∂K

= −w∂L∂Kt−qδ =−q + rq

and the transversality condition is

limt→∞

Kt qt e−rt = 0



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 aa) Show that the optimal investment level, It , can be written as an implicit

function of qt .

• Answer: We have G′(I) = q−1It follows that: I = G′−1(q−1)≡M(q) where

M(1) = 0

G′′(M(q))M ′ = 1↔M ′ =1

G′′> 0



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 bb) Construct a phase diagram for the (K ,q) dynamics, assuming that a

steady state with K > 0 exists. Let the steady state value of K be denotedK ∗.

• Answer:Locus 1: K = I−δK = M(q)−δK .From K = 0 we get M (q) = δK .We have that dq

dK |K =0 > 0 because M ′ > 0.We have that K = 1→ q = 1.

Locus 2:

We have ∂L∂K =− α

1−αK−

α1−α−1 (yd

) 11−α =− α

1−α

(yd

K

) 11−α ≡ η(K )

We have from FOC −w ∂L∂Kt−qδ =−q + rq↔ q = (r + δ)q + wη(K )

From q = 0 we get q =−wη(K )r+δ

= wr+δ

α

1−α

(yd

K

) 11−α

We have limq=0,K→0 q = ∞ and limq=0,K→∞ q = 0.

Arrows and full diagram explained on white board.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 cc) For an arbitrary K0 > 0, indicate in the diagram the movement of the pair

(Kt ,qt ) along the optimal path.

• Answer: Standard with K predetermined and q as a jump variable.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 dd) Express the level of gross investment in steady state, I∗, as a function of

K ∗.

• Answer: 0 = M(q∗)−δK ∗↔ I∗ = M(q∗) = δK ∗



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 ee) By curve shifting in the phase diagram, sign ∂I∗/∂r and ∂I∗/∂yd -

• Answer: I∗/∂r < 0 and ∂I∗/∂yd > 0.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 fI∗/∂r < 0 and ∂I∗/∂yd > 0

f) Comment by relating to the signs of the partial derivatives in a standardIS-LM model.

• Answer: They are the usual signs. Our analysis is founded on (1)imperfect competition leading to falling demand in price, (2) sticky prices.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 gg) By curve shifting in the phase diagram, sign ∂K ∗/∂w . Comment.

• Answer: ∂K ∗/∂w > 0. This is the result of factor substitution in a costminimization problem.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Outline

1 Introduction

2 Exercise 3.13.1.13.1.23.1.3

3 Exercise 3.23.2.13.2.23.2.3

4 Exercise 3.33.2.13.2.23.2.3

5 Summing up



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.3.1Consider the standard textbook Keynesian money demand hypothesis,

Mdt = Pt ·L(Yt , it ) LY > 0,Li < 0

where Pt > 0 is the general price level (in terms of money), L(·) is a real moneydemand function (“L” for liquidity), Yt is aggregate production per time unit, andi > 0 is the nominal interest rate on short-term bonds. Let time be continuousand define the short-term real interest rate by

rt = it −πt

where πt ≡ Pt/Pt (the inflation rate). Since asset markets move fast, it is naturalto assume that the money market clears continuously, i.e.,

Mt = Pt ·L(Yt , it ),

for all t ≥ 0. Let the growth rate of any positive variable x be denoted gx .



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.2 aMd

t = Pt ·L(Yt , it ) LY > 0,Li < 0

Abstracting from fluctuations around the trend, suppose that gM = µ and gY = γ,both constant, and that the real interest rate has a constant trend level, r .

a) Briefly, interpret the signs of the partial derivatives of the liquidity demandfunction L(·).

• Answer LY > 0: Need for transactions (more money lowers transactiontime).

• Answer Li < 0: Opportunity cost of holding money.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.2 bSpecify the liquidity demand function as L(Yt , it ) = Yt/V (it ), where V ′(i) > 0.

b) Show that in this case a constant long-run inflation rate, π, is consistentwith the model and that this π satisfies a simple equation where also µ andγ enter. Relate to your empirical knowledge.

• Answer:πt = π→ it = r + i ≡ i→Mt = Pt L(Yt , it ) = Pt Yt/V (it ) = Pt Yt/V (i)↔gM = gP + gY ↔ gP = π = gM −gP = µ− γ

• Fits well.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.2 cRecall that the (income) velocity of money is defined as Pt Yt/Mt . Classicalmonetary theory (the Quantity Theory of Money) claims that for given monetaryinstitutions the velocity of money is a constant, V , and thus independent of thenominal interest rate.

c) What is the prediction implied by the classical theory concerning thelong-run inflation rate, given µ and γ? Comment in relation to the aboveKeynesian result.

• Answer: We get the same result: gP = π = gM −gP = µ− γ.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.2 c

Estimation of logMt = α + βy logYt + βr rt + εt yields

βy = 0.868 (0.070) βr =−0.094 (0.018)

d) Discussion the relationship between the data and our theories.2nd of November 2011 (week 7) — Slide 35/40


Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 aWe now consider a short time interval, say a month, where the level of themoney supply is practically constant, apart from level shifts implemented by thecentral bank, through open market operations as part of its monetary policy.Suppose the money supply in this way shifts from M to M ′ > M.

a) According to the classical monetary theory (which is in fact more or lessshared by the ”monetarists”, like Milton Friedman, and also to some extentby the ”new classical” theorists like Robert Lucas), which variable willrespond in the short run and how?

• Answer: Only prices.



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 bWe now consider a short time interval, say a month, where the level of themoney supply is practically constant, apart from level shifts implemented by thecentral bank, through open market operations as part of its monetary policy.Suppose the money supply in this way shifts from M to M ′ > M.

b) Now answer a) from the point of view of Keynesian theory where Pt is”sticky” in the short run?

• Answer: M ↑⇒ i ↓⇒ r ↓ (because π is sticky)⇒ Y d ↑⇒ Y ↑ and N ↑



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Exercise 3.2.3 cc) What problems could there be with using the Keynesian theory in a real

crisis?

• Answer: The liquidity trap, zero lower bound of the nominal interest rate.Countered by QE, operation Twist and rising inflation expectations.

• Krugman quote (link)Usually we worry about a deflationary trap, which comes about as follows: suppose that theeconomy is depressed, that as a result prices begin falling, and interest rates are up againstthe zero lower bound. Then as deflationary expectations take hold, the real interest rate riseseven as the nominal rate stays pinned at zero - and this rising real rate helps keep theeconomy depressed. Japan has been in this trap for a long time.

But here’s what I’ve been thinking: we don’t have to get all the way to actual deflation for

something like this to take hold. The key point is that long-term interest rates, which are what

matter for spending, are effectively bounded some ways above zero. The reason is option

value: the short rate could move up, but it can’t go down, so the yield curve has to be

upward-sloping. Indeed, Japan’s 10-year rate is still 1 percent even though the short rate has

been zero for many years and is likely to stay zero for years to come. (Yes, this is Keynes on

liquidity preference).


http://krugman.blogs.nytimes.com/2011/09/23/the-low-inflation-trap-slightly-wonkish/


Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Outline

1 Introduction

2 Exercise 3.13.1.13.1.23.1.3

3 Exercise 3.23.2.13.2.23.2.3

4 Exercise 3.33.2.13.2.23.2.3

5 Summing up



Introduction

Exercise 3.1

3.1.1

3.1.2

3.1.3

Exercise 3.2

3.2.1

3.2.2

3.2.3

Exercise 3.3

3.2.1

3.2.2

3.2.3

Summing up

Summing up• Seeing the firms investment problem as a standard optimal control

problem.• Solving this problem.• Perfect competition leads to the simple capital adjustment principle.• Imperfect competition and sticky prices is compatible with a textbook

IS-LM model.• Cost minimization for given demand leads to factor substitution when

relative factor prices change.

• We repeated the basis ideas on money.• Long run: π = µ− γ can be derived from both the Quantity Theory of

money (constant V ) and money demand separable in the velocity and aconstant long-run real interest rate. Evidence suggests that velocity (V )varies considerably over time and that the money demand function israther stable.

• Short run: In classical theory there is money neutrality, in Keynesian theorythere are not. The liquidity trap can be a problem in practice.


Documents

Faculty of Social Sciences - Økonomisk Institut ... of Social Sciences Adv. Macro 2 Exercises Week 7 2nd of November 2011 (week 7) Slide 1/40 Adv. Macro 2 Exercises Introduction Exercise