the ramsey model - fudan...

The Ramsey Model

Jinfeng Ge

Fudan University

2 19th, 2013

Jinfeng Ge (Fudan University) The Ramsey Model 2 19th, 2013 1 / 37

Introduction

Ramsey model: differs from the Solow model only because it explicitlymodels the consumption and saving.

Beyond its use as a basic growth model, also a benchmark for manyarea of macroeconomics, such as the real business cycle theory, etc.

Introduce the modern macroeconomics concepts (DSGE framework,dynamic stochastic general equilibrium), such as rational expectation,dynamic general equilibrium.


Preferences, Technology and Demographics I

Infinite-horizon, continuous time.

Representative household with instantaneous utility function

u (c (t)) .

Assume that utility function is strictly increasing, concave, twicecontinuously differentiable with derivatives u′ and u′′, and satisfiesthe following Inada type conditions:

limc→0

u′ (c) = ∞, limc→∞

u′ (c) = 0.

Representative household represents set of identical households,L (0) = 1 and L (t) = exp (nt) L (0).


Preferences, Technology and Demographics II

Production function is represented by

Y (t) = F (K (t) , L (t)) ,

where the production function satisfied the properties assumed inSolow model. The only difference is that we assume A = 1.

Then we can define the parameters as following

y (t) =Y (t)

L (t), k =

Y (t)

L (t), c (t) =

C (t)

L (t),

y (t) = f (k (t)) = F (k (t) , 1) .


Preferences, Technology and Demographics III

The law of motion of capital

K (t) = I (t)− δK (t) = Y (t)− C (t)− δK (t) .

Rewrite the above equation as

k (t) = f (k (t))− c (t)− (δ + n) k (t) .

The budget constraint of household is

C (t) + K (t) = w (t) L (t) + r (t)K (t) ,

Rewrite the above equation

k (t) = w (t) + rk (t)− c (t)− nk (t) .


CRRA Utility Function I

CRRA (constant relative risk aversion) utility function is used quiteoften

u (c) =c1−θ − 1

1− θ, if θ > 0 and θ 6= 1

u (c) = ln (c) , if θ = 1

the Arrow-Pratt coefficient of relative risk aversion for atwice-continuously differentiable concave utility function u

mr = −u′′c

u′.

CRRA utility function gives the constant relative risk aversion θ.


CRRA Utility Function II

In the Ramsey model, there is no uncertainty, so risk aversion doesnot matter. But the intertemporal elasticity of substitution doesmatter. The intertemporal elasticity of substitution between s and tis defined as

mi = −d log (c ((s) /c (t)))

d log (u′ (c (s)) /u′ (c (t))),

as s → t,

mi = −d log (c ((s) /c (t)))

d log (u′ (c (s)) /u′ (c (t)))= − u′

u′′c=

1

mr,


Market Equilibrium Conditions

The capital market clearing demands

r (t) = FK (K (t) , L (t))− δ = f ′ (k (t))− δ.

The labor market clearing demands

w (t) = FL (K (t) , L (t)) = f (k (t))− k (t) f ′ (k (t)) .

Who are the demand and supply side on these two markets?


Transversality Condition

Transversality condition is

limt→∞

exp (−ρt) k (t) u′ (t) = 0.

How to understand this condition? What is the intuition?


Household Maximization I

Household maximize the discounted utility

V =∫ ∞

0e−ρtU (c (t)) dt

subject tok (t) = w (t) + rk (t)− c (t)− nk (t) .


The Maximum Principle I

How to solve this dynamic problem? We need use the maximumprinciple which state the conditions solving the problem.

Suppose that we are going to solve a problem like

max∫ T

0e−ρt f (x (t) , u (t) , t) dt s.t.

x (t) = g (x (t) , u (t) , t)

where x (t) is state variable, and u (t) is control variable.

Define the current value Hamiltonian as

H (t) = f (x (t) , u (t) , t) + λ (t) g (x (t) , u (t) , t)


The Maximum Principle II

Necessary conditions:∂H (t)

∂u (t)= 0,

x (t) =∂H (t)

∂λ (t),

λ (t) = ρλ (t)− ∂H (t)

∂x (t).


Household Maximization II

Write down the current value Hamiltonian

H (t) = u (c (t)) + λ (t) (w (t) + rk (t)− c (t)− nk (t)) .

Two first order conditions are

u′ (c (t)) = λ (t) ,

λ (t) = ρλ (t)− ∂H (t)

∂k (t)= λ (ρ + n− r (t)) .

Then we can obtain one of two key differential equations

u′′ (c (t))

u′ (c (t))

d (c (t))

dt= ρ + n− r (t) .

with CRRA utility function, we obtain

c (t)

c (t)=

r (t)− ρ− n

θ.


Steady State

Another key differential equation is

k (t) = f (k (t))− c (t)− (δ + n) k (t) .

Define the steady state as c = 0 and k = 0.

With two differential equations, we obtain

f (k∗)− c∗ − (δ + n) k∗ = 0,

f ′ (k∗)− δ− ρ− n = 0.

Then

k∗ =

(α

δ + ρ + n

) 11−α

.

c∗ =

(1− α (δ + n)

δ + ρ + n

)k∗α.


The Modified Golden Rule

Recall that in the Solow model, the Golden rule requires that thesaving rate is α and the steady state capital stock satisfies

f ′ (k∗) = δ + n.

From the previous analysis, we know that in the steady state, thesaving rate is

α (δ + n)

δ + ρ + n.

The saving rate is smaller than α, and converge to α when ρ goes to1. And f ′ (k∗) = δ + n+ ρ capital stock is smaller than the Goldenrule level.

The modification of the Golden rule is that the capital stock isreduced from the Golden rule level because of the time preference.

This rule implies that ultimately the marginal productivity of capital,and thus real interest rate is determined by the rate of timepreference and n.


Dynamics I

To study dynamics, we use the following phase diagram.Introduction to Modern Economic Growth

c(t)

kgold0k(t)

k(0)

c’(0)

c’’(0)

c(t)=0

k(t)=0

k*

c(0)

c*

k

Figure 8.1. Transitional dynamics in the baseline neoclassical growth model.

accumulate continuously until the maximum level of capital (reached with zero consumption)

k > kgold. Continuous capital accumulation towards k with no consumption would violate the

transversality condition. This establishes that the transitional dynamics in the neoclassical

growth model will take the following simple form: c (0) will “jump” to the stable arm, and

then (k, c) will monotonically travel along this arm towards the steady state. This establishes:

Proposition 8.4. In the neoclassical growth model described above, with Assumptions 1,

2, 3 and 40, there exists a unique equilibrium path starting from any k (0) > 0 and converging

to the unique steady-state (k∗, c∗) with k∗ given by (8.21). Moreover, if k (0) < k∗, then

k (t) ↑ k∗ and c (t) ↑ c∗, whereas if k (0) > k∗, then k (t) ↓ k∗ and c (t) ↓ c∗ .

An alternative way of establishing the same result is by linearizing the set of differential

equations, and looking at their eigenvalues. Recall the two differential equations determining

the equilibrium path:

k (t) = f (k (t))− (n+ δ)k (t)− c (t)

322


Dynamics II

Equilibrium is determined by two differential equations:

k (t) = f (k (t))− c (t)− (δ + n) k (t) .

c (t)

c (t)=

f ′ (k (t))− δ− ρ− n

θ.

More over, we have an initial condition k (0) > 0, also atransversality condition

limt→∞

exp (−ρt) k (t) u′ (t) = 0.


Dynamics III

Here is a recipe which show us how to do analysis using phase diagram.

Use two differential equations, form two lines using conditions c = 0and k = 0. Then show the direction of change in the four domains.

There are three possible path, only one is the equilibrium path. Couldyou prove it?

Appropriate notion of saddle-path stability: consumption is thecontrol variable, and c (0) is free: it has to adjust to satisfytransversality condition. Since c(0) can jump to any value, thereexists a one-dimensional manifold tending to the steady state on thestable arm.


The role of expectation

The Euler equation gives the rate of change of consumption as a functionof variables at the current moment. It may be interpreted as thathouseholds need not to form expectations of future variables in makingtheir consumption/saving decisions and that the assumption of rationalexpectation is not required. However, from the intertemporal budgetconstraint that the household can not plan without knowing the entirepath of both the wage and interest rate. Expectations are thus crucial tothe allocation of resources in the market economy. The Euler equationonly determines the rate of change, not the level of consumption.


Efficiency

Question: is the market equilibrium Pareto optimal? The answer isyes.

How to show this point, we need solve the socially planner problem.


Social Planner’s problem I

Assume that there is social planer maximize the society’s utility

max∫ ∞

0e−ρtu (c (t)) dt

The budget constraint faced by socially planner is the capitalaccumulation function

k (t) = f (k (t))− (δ + n) k (t)− c (t) .

Solve this problem, we obtain the same differential equations. Couldyou show it? Then we know the market equilibrium is Pareto optimal.


Social Planner’s problem II

The Hamilton is

Ht = u (c (t)) + λ (t) [f (k (t))− ((δ + n) k (t)− c (t))]

Then we obtain the same differential equations as the competitiveequilibrium.

Then the math tell us competitive equilibrium gives the sameallocation as the social planner problem.


The Ramsey Model in Discrete Time I

Economically, nothing is different in discrete time.

Mathematically, a few details need to be sorted out.

Sometimes discrete time will be more convenient to work with, andsometimes continuous time.


The Ramsey Model in Discrete Time II

let us form the household’s problem

maxCt

∞

∑t=0

βtu (Ct)

s.t. Ct +Kt+1 = wtLt + rtKt

Then the lagrange function is

∞

∑t=0

βt [u (Ct) + λt (wtLt + rtKt − Ct −Kt+1)]

First order condition of Ct and Kt+1

λt = u′ (Ct) ,

λt = βrt+1λt+1.

The Euler equation is

u′ (Ct) = βrt+1u′ (Ct+1) .


Intuition of the Euler Equation

Suppose that consumer decrease consumption ε in period t, then theutility loss in period t is

u′ (Ct) ε.

ε is put into investment. Then extra income rt+1ε is realized in thenext period and increased utility is

βrt+1u′ (Ct+1) ε.

In the equilibrium, the loss and gain should be the same, therefore

βrt+1u′ (Ct+1) ε = u′ (Ct) ε.


Conclusion

Open the black box of capital accumulation and analyze the tradeoffbetween investment and consumption.

By constructing a dynamic general equilibrium framework, it pavesthe way for further analysis of capital accumulation, human capitaland endogenous technological progress.

Any new insights about the sources of cross-country incomedifferences and economic growth relative to the Solow growth model?No.


Global Imbalance

What is Global Imbalance

Basic facts of Global Imbalance.

Reason?


Global Imbalance-facts of current accounts 1980-2012Figure 1: Global Imbalances: Current Accounts.

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

1980 1984 1988 1992 1996 2000 2004 2008 2012

% of World GDP

U.S. Europe Japan Oil Producers Emerging Asia ex-China China

Financial Crisis Asian Crisis

Notes: Oil Producers: Bahrein, Canada, Kuweit, Iran, Lybia, Nigeria, Norway, Mexico, Oman, Russia, Venezuela,

Saudi Arabia; Emerging Asia ex-China: Indonesia, Korea, Malaysia, Philippines, Singapore, Taiwan, Thailand.

Europe: European Union. Source: IMF World Economic Outlook

Figure 2: Global Imbalances: World Interest Rates.

-3

-2

-1

0

1

2

3

4

5

6

7

1980 1984 1988 1992 1996 2000 2004 2008

percent

world-short real US-long real 10-year TIPS

Notes: world-short real: ex-post 3-month real interest rate for the G-7 countries (GDP weighted). US-long real: 10

year yield on U.S. Treasuries minus 10-year expected inflation. 10-year TIPS: yield on inflation indexed 10-year

Treasuries. Source: Global Financial Database, IMF International Statistics, OECD Economic Outlook, Survey of

Professional Forecasters

53

Source: Gourinchas-Rey (2013)


Global Imbalance-global current accounts distribution

Blanchard and Milesi Ferretti, 2009Source: Blanchard and Milesi Ferretti, 2009


Global Imbalance and internal saving and investment I

CA=S-I.Saving and Investment Trends (in percent of domestic GDP)

Source Blanchard and Milesi Ferretti, 2009Source: Blanchard and Milesi Ferretti, 2009


Global Imbalance and internal saving and investment II

Saving and Investment Trends (in percent of domestic GDP)

EUR deficit: Greece, Ireland, Italy, Portugal, Spain, UK, Bulgaria, Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Romania, Slovak Republic, Turkey, UkraineEUR surplus: Austria, Belgium, Denmark, Finland, Germany, Luxembourg, Neth, Sweden,Switzerland.

Source: Blanchard and Milesi Ferretti, 2009


Allocation Puzzle

Figure 3: Average productivity growth and capital inflows between 1980 and 2000.

AGO

ARG

BEN

BGD

BOL

BRA

BWA

CHL

CHN

CIV

CMR

COG

COL

CRICYP

DOMECU

EGYETH

FJI

GAB

GHAGTM

HKG

HND

HTI

IDN IND

IRN

ISR

JAM

JOR KEN

LKA

MARMEX

MLI

MOZ

MUS

MWI

MYS

NER

NGA

NPL

PAK

PAN

PER

PHLPNG

PRYRWA

SEN

SGP

SLV

SYR

TGO

THA

TTO

TUN

TUR

TWN

TZA

UGA

URY

VEN

ZAF KOR

MDG

−10

−5

05

10

15

Capital In

flow

s (

perc

ent of G

DP

)

−4 −2 0 2 4 6Productivity Growth (%)

Source: Gourinchas and Jeanne (forth.). Sample of 68 developing economies

Figure 4: G-7 Cross Border Assets and Liabilities (percent of world GDP)

0%

50%

100%

150%

200%

250%

1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010

G7 and BRIC Cross Border Assets and Liabili5es (% of world GDP)

G-‐7 BRIC (Brazil, India, Russia, China)

Source: Lane and Milesi-Ferretti (2007a) updated to 2010. Gross external assets and liabilities, scaled by world

GDP.

54

Source: Gourinchas and Jeanne (forthcoming). Sample of 68 developingeconomies


Why is it a puzzle?

With the Euler equation

βrt+1u′ (Ct+1) = u′ (Ct) .

We can conclude that interest rate is higher if growth rate is higher.Capital should flow into China, but capital flow is reversed!


The Two-country Metzler diagram I

Slide 2-30

Global Imbalances Figure 11: Global Imbalances & the two-country Metzler diagram.

I*

S*

r

S*,I*

I

S

r

S,I



The Two-country Metzler diagram II

Slide 2-29

Global Imbalances Figure 10: The Two-country Metzler diagram.

I*

S*

r

S*,I*

I

Sr

S,I



China’s Saving Rate Puzzle

The Chinese Savings Puzzle?

-5

0

5

10

15

20

25

30

35

40

45

50

55

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Savings % of GDP CA % of GDP Investment % of GDP

Savings, Investment and Current Account in China


Some Answers

Internal financial friction leads to lacking of safe assets. For example:Bernanke(2005), Caballero et al (2008).

Song et al (2011): internal financial friction and misallocation.

Precautionary saving.

Marriage market competition Wei et al (2011).


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