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Lecture 2 The Centralized EconomyEconomics 5118 Macroeconomic Theory

Kam Yu

Winter 2013

Outline1 Introduction

2 The Basic DGE Closed Economy

3 Golden Rule Solution

4 Optimal SolutionThe Euler EquationInterpretationStatic EquilibriumDynamicsAlgebraic Analysis

5 Real Business Cycle DynamicsTechnology ShocksGolden Rule Revisited

6 Labour in the Basic Model

7 Investmentq-Theory

Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 2 / 45

Introduction

The Production Economy

Recall a static production economy

E = {(U i , ei , θij ,Y j)|i ∈ I, j ∈ J }

There are n goods and services.

Each household i ∈ I has continuous, strongly increasing, and strictlyquasi-concave utility function U i : Rn

+ → R+ and is endowed withei ∈ Rn

+.

Each competitive firm j ∈ J has a production set Y j that is compactand strongly convex.

θij is the share of household i in firm j .


Introduction

Really Nice Results from Microeconomic Theory

1 A Walrasian equilibrium exists: there is a price vector p∗ such that∑i∈I

di (p∗,mi (p∗)) =∑j∈J

y j(p∗) +∑i∈I

ei .

That is, all n markets of goods and services clear.

2 FWTE: The Walrasian equilibrium allocation is Pareto efficient.

3 SWTE: Any desirable Pareto efficient allocation (x, y) can beachieved as a Walrasian equilibrium allocation after a suitable incometransfer program between households.

4 With full information on preferences and technology, communism andcapitalism achieve the same outcome in a static economy.


Introduction

To Make Things Really Simple . . .

There are two goods, n = 2, called capital k and output y .

The households only consume one good, effectively we can considerthem as one big household (|I| = 1)

One aggregate competitive firm (|J | = 1).

Of course the ownership share become θ11 = 1.

This is a bit too simple. So instead of static equilibrium, we studythis economy over time, t = 1, 2, . . .


The Basic DGE Closed Economy

The Ramsey Model

In period t, the endowment is the capital stock kt . The firm producesoutput yt using capital as input:

yt = F (kt) (2.3)

Output yt is divided into two parts, consumption ct and investment it :

yt = ct + it . (2.1)

This is called the national income identity, or the resource constraint.

Investment it is saved as capital for next period. In production, thefirm consumes only part of the capital stock, δkt , where δ is calledthe depreciation rate. Capital stock in period t + 1 is therefore

∆kt+1 = kt+1 − kt = it − δkt . (2.2)


The Basic DGE Closed Economy

Dynamic Resource Constraint

The last three equations gives

F (kt) = ct + ∆kt+1 + δkt . (2.4)

Like the static model, our objective is to maximize utility derived fromconsumption, not output (we are not communists!) But we have aproblem. Should we maximize

1 utility in each period, treating every period as equally important,(golden rule) or,

2 the present value of total utility of all present and future periods,using an appropriate discount factor? (optimal solution)


Golden Rule Solution

Golden Rule — The Steady State

The dynamic resource constraint (2.4) can be written as

ct = F (kt)− kt+1 + (1− δ)kt . (2.5)

The steady state is attained when all variable are the same in allsubsequent periods, i.e., ct = c and kt = k , t = 1, 2, . . . . Then (2.5)becomes

c = F (k)− δk . (2.6)

The necessary condition for maximization is

∂c

∂k= F ′(k)− δ = 0, (2.7)

which means that marginal product is equal to the depreciation rate whenconsumption is maximized in the steady state.


Golden Rule Solution

Golden Rule Steady State

16 2. The Centralized Economy

F'(k)

!

k# k

Figure 2.1. The marginal product of capital.

y

c# +! k#

! k#

k

kk#

!

! !

F(k)

max c = c# = F(k#) " k#

Figure 2.2. Total output, consumption, and replacement investment.

ct + #kt + 1

#max c = c

kk#

F(k) " k!

$c/$k = F'(k) " = 0!

Figure 2.3. Net output.

occurs where the slope of the tangent is zero. At this point net investment F !(k)" ! 0.=

We can now find the sustainable level of consumption. This occurs when the capital stock is constant over time, implying that !k 0 and that net invest-= ment is zero. The maximum point on the line is then the maximum sustainable level of consumption c#. This requires a constant level of the capital stock k#. This solution is known as the golden rule.


Optimal Solution

The Optimization Problem

In the optimal solution the present value of current (period t) and future(t + s) utility is maximized:

maxct+s ,kt+s+1

∞∑s=0

βsU(ct+s)

subject to F (kt+s) = ct+s + kt+s+1 − (1− δ)kt+s ,

where β = 1/(1 + θ) and θ > 0 is called the social discount rate. TheLagrangian is

Lt =∞∑s=0

{βsU(ct+s)

+ λt+s [F (kt+s)− ct+s − kt+s+1 + (1− δ)kt+s ]}, (2.8)

where λt+s is the Lagrange multiplier in period t + s.


Optimal Solution

Necessary Conditions

The first-order conditions are

∂Lt∂ct+s

= βsU ′(ct+s)− λt+s = 0, s ≥ 0, (2.9)

∂Lt∂kt+s

= λt+s [F ′(kt+s) + 1− δ]− λt+s−1 = 0,

s ≥ 1, (2.10)

with the resource constraint

F (kt+s) = ct+s + kt+s+1 − (1− δ)kt+s , s ≥ 0,

and the transversality condition

lims→∞

βsU ′(ct+s)kt+s+1 = 0. (2.11)


Optimal Solution The Euler Equation

The Euler Equation

Eliminating λt+s and λt+s−1 in (2.10) using (2.9) gives

βsU ′(ct+s)[F ′(kt+s) + 1− δ] = βs−1U ′(ct+s−1), s ≥ 0.

For s = 1 this can be written as

βU ′(ct+1)

U ′(ct)

[F ′(kt+1) + 1− δ

]= 1. (2.12)

This is called the Euler equation, which is the corner stone of dynamicoptimization problems in consumption.


Optimal Solution Interpretation

Interpretation of the Euler Equation

The Euler equation reflects the intertemporal substitution of consumptionbetween two consecutive periods. Consider periods t and t + 1:

Vt = U(ct) + βU(ct+1).

Using the implicit function theorem, the slope of the indifference curve inthe (ct , ct+1) space is called the marginal rate of time preference:

dct+1

dct= − U ′(ct)

βU ′(ct+1). (2.13)



Interpretation continued

The budget constraint in period t and t + 1 can be written respectively as

kt+1 = F (kt) + (1− δ)kt − ct ,

ct+1 = F (kt+1)− kt+2 + (1− δ)kt+1.

Using the chain rule to differentiate ct+1 with respect to ct , we get

dct+1

dct= −[F ′(kt+1) + 1− δ]. (2.14)

This is the slope of the intertemporal production possibility frontier(IPPF). Equating (2.13) and (2.14) gives the Euler equation. That is, atthe optimal point (c∗t , c

∗t+1), the indifference curve of the household is

tangent to the IPPF.



Graphical Interpretation

! !

! !

21 2.4. Optimal Solution

ct + 1

max ct+1

c * t + 1

t

Vt = U(ct) + !U(ct + 1)

c* max ct 1 + rt + 1

Figure 2.4. A graphical solution based on the IPPF.

2.4.4 Graphical Representation of the Solution

The solution to the two-period problem is represented in figure 2.4. The upper curved line is the indi!erence curve that trades o! consumption today for con-sumption tomorrow while leaving Vt unchanged. It is tangent to the resource constraint. The lower curved line represents the trade-o! between consump-tion today and consumption tomorrow from the viewpoint of production, i.e., it is the IPPF. It touches the indi!erence curve at the point of tangency with the budget constraint. This solution arises as in equilibrium equations (2.13) and (2.14), and (2.16) must be satisfied simultaneously so that

! ddcct+t

1 !!

Vconst. = F "(kt+1)+ 1 ! ! = 1 + rt+1 = !

"c"ct+t

1 !!

IPPF .

The net marginal product F "(kt+1)!! rt+1 can be interpreted as the implied =real rate of return on capital after allowing for depreciation. An increase in rt+1

due, for example, to a technology shock that raises the marginal product of capital in period t + 1 makes the resource constraint steeper, and results in an increase in Vt , ct , and ct+1.

2.4.5 Static Equilibrium Solution

We now return to the full optimal solution and consider its long-run equilibrium properties. The long-run equilibrium is a static solution, implying that in the absence of shocks to the macroeconomic system, consumption and the capital stock will be constant through time. Thus ct c#, kt k#, !ct 0, and !kt 0= = = =for all t. In static equilibrium the Euler equation can therefore be written as

#U "(c#) U "(c#)

[F "(k#)+ 1 ! !] = 1,

implying that 1F "(k#) =#+ !! 1 = !+ $.


Optimal Solution Static Equilibrium

Steady-State SolutionIn the steady state (long-run), ct = c∗ and kt = k∗ for all t. The Eulerequation becomes

βU ′(c∗)

U ′(c∗)

[F ′(k∗) + 1− δ

]= 1,

or, with β = 1/(1 + θ),

F ′(k∗) = 1/β + δ − 1 = δ + θ. (2.21)

From the resource constraint we have

c∗ = F (k∗)− δk∗. (2.22)

Comparing with the golden rule solution, where F ′(k#) = δ, the long-runcapital stock is at a lower level. That is,

c∗ < c# and k∗ < k#.


Optimal Solution Static Equilibrium

Comparing Golden Rule and Optimal Solution


F'(k)

! + "

!

k* k# k

Figure 2.5. Optimal long-run capital.

y

F(k)

k*

k*

k# k

k!

! !

! "+c * + k*

c# + k#!

!

Figure 2.6. Optimal long-run consumption.

The solution is therefore di!erent from that for the golden rule, where F !(k) = !. Figure 2.1 is replaced by figure 2.5. This shows that the optimal level of capital is less than for the golden rule. The reason for this is that future utility is discounted at the rate " > 0.

The implications for consumption can be seen in figures 2.5 and 2.6. In fig-ure 2.5 the solution is obtained where the slope of the tangent to the production function is !+ ". As the tangent must be steeper than for the golden rule, this implies that the optimal level of capital must be lower. Figure 2.6 shows that this entails a lower level of consumption too. Thus c" < c# and k" < k#.

We have shown that discounting the future results in lower consumption. This may seem to be a good reason for not discounting the future. To see what the benefit of discounting is we must analyze the dynamics and stability of this solution.


Optimal Solution Dynamics

Linear Approximation

So far we have established two dynamic relations between two consecutiveperiods, the Euler equation and the resource constraint:

βU ′(ct+1)

U ′(ct)

[F ′(kt+1) + 1− δ

]= 1,

∆kt+1 = F (kt)− δkt − ct . (2.17)

The relation between ct+1 and ct can be better seen by taking a first-orderTaylor approximation of U ′(ct+1) about ct :

U ′(ct+1) ' U ′(ct) + U ′′(ct)∆ct+1.

The Euler equation becomes, with U ′(ct)/U′′(ct) ≤ 0,

∆ct+1 = − U ′(ct)

U ′′(ct)

[1− 1

β[F ′(kt+1) + 1− δ]

]. (2.18)



Dynamics of Consumption and Capital

In the steady state, F ′(kt+1) = F ′(k∗) = δ + θ and so

∆c = − U ′(c∗)

U ′′(c∗)

[1− 1

β[δ + θ + 1− δ]

]= 0.

Two conclusions:

1 When k > k∗, F ′(k) < F ′(k∗) and by (2.18) ∆c < 0.

2 When k < k∗, F ′(k) > F ′(k∗) and by (2.18) ∆c > 0.

From the resource constraint (2.17),

1 When ct > F (kt)− δkt , then ∆k < 0

2 When ct < F (kt)− δkt , then ∆k > 0



Phase Diagrams

! "


c t + !

k t +

1

!ct + 1 = 0

!ct + 1 > 0 !ct + 1 < 0

k* kt

Figure 2.8. Consumption dynamics.

A complication is that both equations are nonlinear. We therefore consider a local solution (i.e., a solution that holds in the neighborhood of equilib-rium) obtained through linearizing the Euler equation by taking a Taylor series expansion of U !(ct+1) about ct . This gives

U !(ct+1) " U !(ct)+!ct+1U !!(ct).

Hence U !(ct+1) U !!!ct

U !! ! 0,U !(ct)

" 1 + U ! +1, U !

and U ! 1!ct . (2.18)+1 = U !! 1 #

![F !(kt+1)+ 1 # "] Thus we have two equations that determine the changes in consumption and capital: equations (2.17) and (2.18).

These equations confirm the static-equilibrium solution as when ct c$ and=kt = k$, we have !ct+1 = 0, !kt+1 = 0, and F !(k$) = "+#. From equation (2.18) we note that when k > k$ we have F !(k) < F !(k$), and therefore F !(k)+1#" < F !(k$) + 1 # ". It follows that if k k$ we have !c 0, i.e., consumption is = = constant, and if k > k$ then !c < 0, i.e., consumption must be decreasing. By a similar argument, if k < k$ then !c > 0 and consumption is increasing. Thus, !c " 0 for k # k$. This is represented in figure 2.8.

The dynamic behavior of capital is determined from equation (2.17). When ct # F(kt+1)# "kt we have !kt+1 " 0. This is depicted in figure 2.9. Above the curve consumption plus long-run net investment exceeds output. The capital stock must therefore decrease to accommodate the excessive level of consump-tion. Below the curve there is su!cient output left over after consumption to allow capital to accumulate.

Combining figures 2.8 and 2.9 gives figure 2.10, the phase diagram we require. Note that this applies in the general nonlinear case and is not a local approx-imation. The optimal long-run solution is at point B. The line SS through B is known as the saddlepath, or stable manifold. Only points on this line are attain-able. This is not as restrictive as it may seem, as the location of the saddlepath


k

!kt + 1 < 0

c t + !

k t +

1

!kt + 1 > 0

c < F(k) " k#

c > F(k) " k#

!k = 0 c = F(k) " k#

Figure 2.9. Capital dynamics.

ct + !kt + 1

#c

* c

!k = 0

k* k# k

A

S

S

B

Figure 2.10. Phase diagram.

is determined by the economy, i.e., the parameters of the model, and could in principle be in an infinite number of places depending on the particular values of the parameters. The arrows denote the dynamic behavior of ct and kt . This depends on which of four possible regions the economy is in. To the northeast, but on the line SS, consumption is excessive and the capital stock is so large that the marginal product of capital is less than ! + ". This is not sustainable and therefore both consumption and the capital stock must decrease. This is indicated by the arrow on SS. The opposite is true on SS in the southwest region. Here consumption and capital need to increase. As the other two regions are not attainable they can be ignored. The economy therefore attains equilibrium at the point B by moving along the saddlepath to that point. At B there is no need for further changes in consumption and capital, and the economy is in equilib-rium. Were the economy able to be o! the line SS—which it is not and cannot be—the dynamics would ensure that it could not attain equilibrium. When there are two regions of stability and two of instability like this the solution is called a saddlepath equilibrium.

2.4.7 Algebraic Analysis of the Saddlepath Dynamics

An algebraic analysis of the dynamic behavior of the economy may be based on the two nonlinear dynamic equations describing the optimal solution, namely,



The Saddle Path


k

!kt + 1 < 0

c t + !

k t +

1

!kt + 1 > 0

c < F(k) " k#

c > F(k) " k#

!k = 0 c = F(k) " k#


ct + !kt + 1

#c

* c

!k = 0

k* k# k

A

S

S

B






Optimal Solution Algebraic Analysis

Linear Approximation of the Euler EquationThe Euler equation is a non-linear equation in ct+1, ct , and kt+1. For analgebraic solution we need to linearize it. So let

f (x) =U ′(ct+1)

U ′(ct)

[F ′(kt+1) + 1− δ

],

where x = [ct+1 ct kt+1]T. The first-order Taylor approximation of fabout x∗ = [c∗ c∗ k∗]T is

f (x) ' f (x∗) +∇f (x∗)T(x − x∗)

=U ′(c∗)

U ′(c∗)[F ′(k∗) + 1− δ]

+U ′′(c∗)

U ′(c∗)[F ′(k∗) + 1− δ](ct+1 − c∗)

− U ′(c∗)

[U ′(c∗)]2U ′′(c∗)[F ′(k∗) + 1− δ](ct − c∗)

+U ′(c∗)

U ′(c∗)F ′′(k∗)(kt+1 − k∗)



f (x) ' F ′(k∗) + 1− δ

+U ′′(c∗)

U ′(c∗)[F ′(k∗) + 1− δ](ct+1 − ct)

+ F ′′(k∗)(kt+1 − k∗).

It follows that the first-order Taylor approximation of the Euler equation is

β[F ′(k∗) + 1− δ +

U ′′(c∗)

U ′(c∗)[F ′(k∗) + 1− δ]∆ct+1

+ F ′′(k∗)(kt+1 − k∗)]' 1.

Using F ′(k∗) = δ + θ (2.21) and rearranging gives

(ct+1 − c∗) = (ct − c∗)− F ′′(k∗)U ′(c∗)

(1 + θ)U ′′(c∗)(kt+1 − k∗). (2.23)



Linear Approximation of the Resource Constraint

Recall the resource constraint

∆kt+1 = F (kt)− δkt − ct . (2.17)

Use a first-order Taylor approximation for F (kt), (2.17) becomes

kt+1 − kt ' F (k∗) + F ′(k∗)(kt − k∗)− δkt − ct .

Using (2.21), c∗ = F (k∗)− δk∗ (2.22), and rearranging gives

kt+1 − k∗ ' F (k∗) + (δ + θ)(kt − k∗)− δkt − ct − k∗ + kt

= F (k∗)− δk∗ + δkt + θkt − θk∗ − δkt − ct − k∗ + kt

= −(ct − c∗) + (1 + θ)(kt − k∗). (2.24)



Back to the Euler Equation

Substitute (2.24) into (2.23), we have

(ct+1 − c∗) = (ct − c∗)

− F ′′(k∗)U ′(c∗)

(1 + θ)U ′′(c∗)[(1 + θ)(kt − k∗)− (ct − c∗)]

or

(ct+1 − c∗) =

[1 +

F ′′(k∗)U ′(c∗)

(1 + θ)U ′′(c∗)

](ct − c∗)

− F ′′(k∗)U ′(c∗)

U ′′(c∗)(kt − k∗) (2.23a)



Linear Dynamical System

The linearized Euler equation (2.23a) and resource constraint (2.24) canbe expressed in matrix form as[

ct+1 − c∗

kt+1 − k∗

]=

[1 + F ′′U′

(1+θ)U′′ −F ′′U′

U′′

−1 1 + θ

] [ct − c∗

kt − k∗

].

This is a two-dimensional linear dynamical system

xt+1 = Axt .

with xt = (ct − c∗, kt − k∗)T. The system converges to the steady-state ifthe absolute values of the two eigenvalues of the matrix A are both lessthan 1. See Devaney (2003, p. 173–179) for details. In particular, theoptimal solution will give the saddle path depicted in Figure 2.10.


Real Business Cycle Dynamics

The Business Cycle

An economy is constantly impacted by shocks.

Shocks can be temporary or permanent, anticipated or unanticipated.

Real business cycle theory focuses on technology shocks (innovations).

After a shock the economy follows the saddle path and converges tothe new steady-state equilibrium.

During the adjustment periods the optimality assumption ismaintained.

Stabilization policy may be useful in the presence of marketimperfections.


Real Business Cycle Dynamics Technology Shocks

Adjustment Process for a Positive Technology Shocks

For a permanent shock,

1 Marginal product shifts from F ′0 to F ′1.

2 Optimal long-run capital stock raised from k∗0 to k∗1 . Equilibriumpoint moved from A to B.

3 At time t = 0, capital is fixed at k∗0 . Consumption jumps from c ′0 toc ′1, i.e., from point A to C on the new saddle path.

4 Consumption and capital stock converge over time to the newsteady-state equilibrium at point B.

5 Results: consumption and capital both increase.

For a temporary shock, k∗ remains the same. Consumption is temporaryadjusted to absorb the shock.


Real Business Cycle Dynamics Technology Shocks

Effects of a Positive Technology Shock

28

F'(k)

! "+ A

* k1* kk0

Figure 2.11. The e!ect on capital of a positive technology shock.

B

2. The Centralized Economy

F1 '

F0 '

k1* k*

A

B

C

c t + #

k t + 1

#k = 0

#c = 0

' c1 ' c0

k0

Figure 2.12. The e!ect on consumption of a positive technology shock.

2.5.2 Permanent Technology Shocks

A positive technology shock increases the marginal product of capital. This is depicted in figure 2.11 as a shift from F ! to F1

! . As ! + " is unchanged, the 0

equilibrium optimal level of capital increases from k"0 to k"1 . The exact dynamics of this increase and the e!ect on consumption is shown

in figure 2.12. A positive technology shock shifts the curve relating consump-tion to the capital stock upwards. The original equilibrium was at A, the new equilibrium is at B, and the saddlepath now goes through B. As the economy must always be on the saddlepath, how does the economy get from A to B? The capital stock is initially k"0 and it takes one period before it can change. As the productivity increase raises output in period t, and the capital stock is fixed, consumption will increase in period t so that the economy moves from A to C, which is on the new saddlepath. There will also be extra investment in period t. By period t + 1 this investment will have caused an increase in the stock of capital, which will produce a further increase in output and consumption. In period t + 1, therefore, the economy starts to move along the saddlepath—in geometrically declining steps—until it reaches the new equilibrium at B. Thus a

28

F'(k)

! "+ A

* k1* kk0

Figure 2.11. The e!ect on capital of a positive technology shock.

B

2. The Centralized Economy

F1 '

F0 '

k1* k*

A

B

C

c t + #

k t + 1

#k = 0

#c = 0

' c1 ' c0

k0

Figure 2.12. The e!ect on consumption of a positive technology shock.

2.5.2 Permanent Technology Shocks

A positive technology shock increases the marginal product of capital. This is depicted in figure 2.11 as a shift from F ! to F1

! . As ! + " is unchanged, the 0

equilibrium optimal level of capital increases from k"0 to k"1 . The exact dynamics of this increase and the e!ect on consumption is shown

in figure 2.12. A positive technology shock shifts the curve relating consump-tion to the capital stock upwards. The original equilibrium was at A, the new equilibrium is at B, and the saddlepath now goes through B. As the economy must always be on the saddlepath, how does the economy get from A to B? The capital stock is initially k"0 and it takes one period before it can change. As the productivity increase raises output in period t, and the capital stock is fixed, consumption will increase in period t so that the economy moves from A to C, which is on the new saddlepath. There will also be extra investment in period t. By period t + 1 this investment will have caused an increase in the stock of capital, which will produce a further increase in output and consumption. In period t + 1, therefore, the economy starts to move along the saddlepath—in geometrically declining steps—until it reaches the new equilibrium at B. Thus a


Real Business Cycle Dynamics Golden Rule Revisited

Dynamics of the Golden Rule

We looked at the steady-state equilibrium of the golden rule:

F ′(k#) = δ,

c# = F (k#)− δk#.

What is the dynamics from the initial stage to the steady state?

Take the optimalsolution model and setθ → 0.

Then β → 1 in theEuler equation.

Resource constraintunchanged.

Point B approaches A.


k

!kt + 1 < 0

c t + !

k t +

1

!kt + 1 > 0

c < F(k) " k#

c > F(k) " k#

!k = 0 c = F(k) " k#


ct + !kt + 1

#c

* c

!k = 0

k* k# k

A

S

S

B






Real Business Cycle Dynamics Golden Rule Revisited

Stability of the Golden Rule

The book says the golden rule has a unstable equilibrium. But . . .

Golden rule is a special case of optimal solution, with social discountrate θ = 0.

There is no inherent instability in the golden rule, unless we insist onsetting c# = F (k#)− δk# in all time.

Shocks can be accommodated by adjusting consumption to be on thesaddle path.

The true value of θ is an empirical question.


Labour in the Basic Model

Work and Leisure

Assumptions:

Households choose between labour time, nt and leisure time lt .

Total time normalized to one: nt + lt = 1.

The utility function U(ct , lt) is increasing and concave, withUc > 0,Ul > 0,Ucc ≤ 0, Ull ≤ 0,Ucl = 0.

The production function F (kt , nt) satisfies the Inada conditions. Thatis, Fk > 0, Fkk ≤ 0,Fn > 0,Fnn ≤ 0,Fkn ≥ 0, limk→∞ Fk = 0,limk→0 Fk =∞, limn→∞ Fn = 0, limn→0 Fn =∞.

Resource constraint:

F (kt , nt) = ct + kt+1 − (1− δ)kt .

Labour constraint:nt + lt = 1.



OptimizationThe Lagrangian is

Lt =∞∑s=0

{βsU(ct+s , lt+s)

+ λt+s [F (kt+s , nt+s)− ct+s − kt+s+1 + (1− δ)kt+s ]

+ µt+s [1− nt+s − lt+s ]}.

The first-order conditions are∂Lt∂ct+s

= βsUc,t+s − λt+s = 0, s ≥ 0, (2.25)

∂Lt∂lt+s

= βsUl ,t+s − µt+s = 0, s ≥ 0, (2.26)

∂Lt∂nt+s

= λt+sFn,t+s − µt+s = 0, s ≥ 0, (2.27)

∂Lt∂kt+s

= λt+s [Fk,t+s + 1− δ]− λt+s−1 = 0, s ≥ 1, (2.28)



Key Results

1 Euler Equation:

βUc,t+1

Uc,t[Fk,t+1 + 1− δ] = 1. (2.29)

2 Eliminating λt+s and µt+s from the first three first-order conditionsgives, for s = 0,

Ul ,t = Uc,tFn,t . (2.30)

This means that if the household provides an extra unit of workingtime, marginal product is Fn,t . Marginal utility gain of consuming thisextra output is Uc,tFn,t . This should be equal to the marginal utilityof leisure.



Steady-State Equilibrium

1 Setting Uc,t+1 = Uc,t = Uc and Fk,t+1 = Fk in the Euler equationgives

Fk = θ + δ.

2 Consumption c , labour n, and leisure l can be solved by the resourceconstraint, labour constraint, and (2.30).

3 The short-run solutions for ct and kt are the same as before.



Wage Rate and Rate of Return to CapitalAssume that technology exhibits constant returns to scale. Then theproduction function F (kt , nt) is linearly homogeneous. Applying Eulertheorem to F gives

F (kt , nt) = Fn,tnt + Fk,tkt . (2.31)

With two factors of production, this means national product is equal tonational incomes. Real wage rate wt and return to capital rt are thereforegiven by

wt = Fn,t , rt = Fk,t − δ.Equation (2.31) can be written as

F (kt , nt) = wtnt + (rt + δ)kt .

The wage rate is given by

wt =F (kt , nt)− (rt + δ)kt

nt.


Investment

Time to Build and Installation Costs of CapitalCapital stock takes time to adjust, but so far we have assumed thatinvestment is instantaneous with no installation cost. In practice, capitalinvestment takes time to build and additional resources are needed fordesign and installation.

Capital investment like wind turbines needstime and resources to design and build.


Investment q-Theory

Costs of Installation

Suppose that installation cost of each unit of capital is 12φit/kt , where

φ ≥ 0. The resource constraint becomes (abstract from labour and leisure)

F (kt) = ct +

(1 +

φit2kt

)it , φ ≥ 0. (2.32)

The Lagrangian of the optimization problem is

Lt =∞∑s=0

{βsU(ct+s)

+ λt+s

[F (kt+s)− ct+s − it+s −

φi2t+s

2kt+s

]+ µt+s [it+s − kt+s+1 + (1− δ)kt+s ]

}.


Investment q-Theory

First-Order Conditions

∂Lt∂ct+s

= βsU ′(ct+s)− λt+s = 0, s ≥ 0,

∂Lt∂it+s

= −λt+s

(1 +

φit+s

kt+s

)+ µt+s = 0, s ≥ 0,

∂Lt∂kt+s

= λt+s

[F ′(kt+s) +

φ

2

(it+s

kt+s

)2]

−µt+s−1 + (1− δ)µt+s = 0, s ≥ 1,

Define the Tobin’s q in period t as qt = µt/λt . Then the second equationcan be written as

it+s =1

φ(qt+s − 1)kt+s , s ≥ 0. (2.33)

Therefore investment takes place only if qt+s > 1.


Investment q-Theory

Tobin’s q

Tobin’s q can be interpreted as the ratio of market value of one unitof investment to its cost.

(Exercise) Using the first-order conditions and setting s = 1, we get

F ′(kt+1) =U ′(ct)

βU ′(ct+1)qt − (1− δ)qt+1 −

1

2φ(qt+1 − 1)2. (2.35)

Four equations, (2.32), (2.33), (2.35), and the capital accumulationequation

kt+1 = it + (1− δ)kt (2.32a)

can be used to solve for four unknowns, ct , kt , it , and qt .


Investment q-Theory

Long-Run Solution

In the steady state, (2.32a) implies that i = δk. Also, (2.33) implies that

i =1

φ(q − 1)k .

Therefore1

φ(q − 1) = δ,

orq = 1 + φδ ≥ 1. (2.36)

Equation (2.35) gives (exercise)

F ′(k) = θ + δ + φδ

(θ +

δ

2

)≥ θ + δ. (2.37)

No installation cost means φ = 0 so that q = 1 and F ′(k) = θ + δ asbefore. With φ > 0, the steady-state capital stock given by (2.37) is lower.


Investment q-Theory

Short-Run Dynamics

To see the dynamics of the model we need to linearize (2.35) about thesteady-state solution with the following steps (exercise):

1 Take the first-order Taylor approximation of (qt+1 − 1)2 about thesteady-state value q.

2 Put the result in (2.35).

3 Let ct+1 = ct in the steady state.

4 Use (2.36) and (2.37) to show that

δ +φδ2

2= F ′(k)− θq.

5 Combining the results to get

qt − q = β(qt+1 − q) + β[F ′(kt+1)− F ′(k)]. (2.38)


Investment q-Theory

Solution

Equation (2.38) is a first-order difference equation in qt − q. The forwardsolution is

qt − q =∞∑s=1

βs [F ′(kt+s)− F ′(k)].

Therefore qt can be seen as the present value of future marginal productsof the investment in period t. Eliminating it using (2.32a) and (2.33) gives

1

φ(qt − 1)kt = kt+1 − (1− δ)kt ,

or, using q = 1 + φδ,

(qt − q + φ)kt = φkt+1. (2.40a)

This equation with (2.38) form the dynamic interaction between kt and qt .


Investment q-Theory

Dynamics of qt and kt

The linearized versions of equations (2.38) and (2.40a) are

(1− β)(qt − q)−βF ′′(k)(kt − k)

= β∆qt+1 + βF ′′(k)∆kt+1, (2.39)

kt(qt − q) = φ∆kt+1. (2.40)

In the steady state ∆qt+1 = ∆kt+1 = 0, (2.39) becomes

kt − k =θ

F ′′(k)(qt − q) (2.41)

Since F ′′(k) < 0, kt is negatively related to qt .


Investment q-Theory

A Permanent Positive Productivity Shock

The line ∆q = 0 is from(2.41).

The line ∆k = 0 is from(2.40).

The two lines originalintersects at A.

Productivity shock shifts the∆q = 0 line.

Economy jumps to B on thesaddle path and converges toC .


q

B

!k = 0 1 + "#

A

!q = 0

C

k

Figure 2.13. Phase diagram for q.

where k is the steady-state level of kt . Thus, as Fkk < 0, in steady state kt is negatively related to qt through

!kt ! k (qt ! q). (2.41)= Fkk

2.7.1.3 The E!ect of a Productivity Increase

The dynamic behavior of kt and qt can be illustrated by considering the e!ect of a permanent increase in capital productivity. In figure 2.13 the line !q 0=depicts the long-run relation between kt and qt given by equation (2.41). This was derived from equation (2.39) by setting !kt+1 !qt+1 0. The line !k 0= = =gives the long-run equilibrium level of kt and is obtained from equation (2.40) by setting !kt+1 0. Before the productivity increase these two lines inter-= sected at A. Note that at this initial equilibrium qt 1. Following the produc-=tivity increase there is a “jump” increase in qt so that qt > 1. This induces a rise in investment above its normal replacement level "k. Initially, kt remains unchanged and so the economy moves to point B. New investment increases the capital stock each period until the economy reaches its new long-run equilib-rium at C by moving along the saddlepath from B. At this point qt is restored to its long-run equilibrium level of one and the equilibrium capital stock, output, and consumption are permanently higher.

2.7.2 Time to Build

An alternative way of reformulating the basic model that results in more general dynamics is to assume that it takes time to install new investment. Kydland and Prescott (1982) were the first to incorporate this idea from neoclassical invest-ment theory into their real-business-cycle DGE macroeconomic model (see also Altug 1989).

Taking account of time-to-build e!ects results in a respecification of the capi-tal accumulation equation (2.2). Consider two ways of doing this. Suppose, first, that investment expenditures recorded at time t are the result of decisions to invest ist made earlier. Moreover, suppose that a proportion #i of recorded


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