Ramsey-Cass-Koopmans model

Joanna Siwińska-Gorzelak, Ph.D.

WNE UW

Plan

• Introduction

• Mathematical tools

• Solution

• Conclusion

• Note that the slides contain only the main elements; a lot of explanations is discussed during the lecture, but is not included in the slides

Introduction

• The RCM model is in fact a modified Solow model, however….

• ….in the Solow model the savings rate was given, which is a drawback of the model (although in the long run turns out to be quite a reasonable assumption)

• RCK model assumes that the savings rate is chosen by households in the optimization process (does the savings rate varies with income?)

• The economy is still closed

• To ease the whole lecture, assume that there is no technical progress, or A is a constant

Introduction

• Assume a constant growth rate of the population, equal to n

• We will omit the time subscripts (instead of xt we will have write x)

nt

teNN

0

Households

• Households supply 1 unit of labour , in exchange for a wage, they consume and save

• The time horizon household is infinite (intergenerational altruism)

• The representative household maximizes utility, where c is the consumption per head:

• 𝑈 = 𝑡=0∞

𝑢 𝑐𝑡 𝑒𝑛−𝜌 𝑡 𝑑𝑡 ρ>0; n>0; oraz ρ>n

• 𝑢′ 𝑐 > 0; 𝑢′′ 𝑐 < 0

• lim𝑐→0

𝑢′ 𝑐 = ∞; lim𝑐→∞

𝑢′ 𝑐 = 0;

The functional form of the utility function

• Let’s assume that the household’s instantaneous utility function takes the form of constant relative risk aversion utility:

• 𝑢 𝑐𝑡 =𝑐𝑡1−𝜃

1−𝜃; 𝜃 > 0

• …where Ѳ gives the willingness to accept deviations from smooth consumption pattern; bigger Ѳ implies a lower willingness to deviate from equal consumption across time

Household’s budget contraint

• The present value of total lifetime consumption can not be higher that the present value of the lifetime income

• 𝑡=0∞

𝑒−𝑟𝑡𝑐𝑡𝐿𝑡𝑑𝑡 ≤ 𝑡=0∞

𝑒−𝑟𝑡𝑤𝑡𝐿𝑡𝑑𝑡

• Using the formula:

• 𝐿𝑡= 𝐿0 𝑒𝑛𝑡; and setting L0 =1

• We get

• 𝑡=0∞

𝑒(𝑛−𝑟)𝑡𝑐𝑡𝑑𝑡 ≤ 𝑡=0∞

𝑒(𝑛−𝑟)𝑡𝑤𝑡𝑑𝑡

Maximisation problem

• The Lagrangian is:

• 𝐿 = 𝑡=0∞

𝑒(𝑛−𝜌)𝑡𝑐𝑡1−𝜃

1−𝜃𝑑𝑡 + 𝜆[𝑘0 + 𝑡=0

∞𝑒(𝑛−𝑟) 𝑡𝑤𝑡 −

𝑡=0∞

𝑒(𝑛−𝑟) 𝑡𝑐𝑡]

• The first order conditions (FOC) are:

• (1) 𝑑𝐿

𝑑𝑐𝑡= 0

• (2) 𝑑𝐿

𝑑𝜆= 0

Maximisation problem

• FOC (1)

•𝑑𝐿

𝑑𝑐𝑡= 𝑒 𝑛−𝜌 𝑡 (1−𝜃)𝑐𝑡

−𝜃 (1−𝜃)

(1−𝜃)2+ 𝜆(−𝑒 𝑛−𝑟 𝑡)=0

• 𝑒 𝑛−𝜌 𝑡𝑐𝑡−𝜃 = 𝜆𝑒 𝑛−𝑟 𝑡

• Take logs:• 𝑛 − 𝜌 𝑡 − 𝜃 ln 𝑐𝑡 = ln 𝜆 + (𝑛 −r)t

• Take time derivatives

• 𝑛 − 𝜌 − 𝜃ሶ𝑐

𝑐𝑡= (𝑛 − 𝑟)

• And finally:

•ሶ𝑐

𝑐𝑡= −

𝑛−𝑟−𝑛+𝜌

𝜃=

𝑟−𝜌

𝜃

Euler’s Equation

• This equation gives the pattern of consumption over time

• Intuitively, higher r is an incentive to save, hence to decrease consumption today in return for the „reward” of higher consumption tomorrow.

• Higher time discount implies that the future consumption is valued less

• The bigger the difference between the two, the more incentive there is to save (decrease today’s consumption).

r

c

c

Firms• Neoclassical productin function

• Firms maximise profits

• We know that this implies:

• In per capita terms

),( NKFY

drR

RKwNNKFet

rt

),(0

wMPNdN

dF

RMPKdK

dF

kkfkfN

KkfNkf

dN

dY

kfN

kfNdK

dY

L

KNfkNfY

)()()()(

)(1

)(

)()(

2

Capital accumulation

tt

tt

kdRra

ka

)(

kndckfk )()(

The Solution

kndckfk

orazrdRc

c

)()(

11

Hard to say, what this means….

Phase diagram

• Let’s define, when c and k are not growing and then let’s draw this..

kndyckndcyk

rc

c

)()(0

0

Phase diagram

Graph copied from the slides of dr hab. Marcin Kolasa http://web.sgh.waw.pl/~mkolas/EARF/Ramsey.pdf

0

c

0

k

Phase diagram

n

żewiemy

ndkfdk

dc

kndkfc

gdyk

dkf

gdyc

G

,

)(0

)()(

0

)(

0

• How do we know that the „stable consumption” line crosses the bell shaped „stable capital” line to the left of the peak?

• From calculations…

.. the amount of capital which defines the line of „stable consumption" must be less than the capital, which determines the maximum consumption, if the capital is stable, whichis kG

Phase diagram - dynamics

kG

Concusions

• A country reaches the steady-state

• The level of capital in the steady-state is determined by ρ

• When the country is converging to the steady state, the saving rate will change

• This is determined by the shape of the saddle path, which in turn depends on Θ

• Low Θ – we do not care for smoothing consumption over time -capital grows rapidly, the economy is moving quickly to steady state

• High Θ - we want to smooth consumption over time – we consume relatively large, capital grows slowly, the economy tends to slow the steady state

Patience as the key to growth?

• Discount rate (or patience) determine our willingness to save

• How high is the discount rate?

• Big research, see for example: http://www.nyu.edu/econ/user/bisina/FredLoew.pdf

• This research is also trying to find the determinants of the discount rate. It seems that among them are:

– Income

– Selected inventions (printing press)

– Culture (individualism)

– Religion (reincarnation)

Estimated discount rate

Weing, Reiger, Hens, 2016

• Which offer would you prefer?• A. a payment of $3400 this month• B. a payment of $3800 next month

Patience and development

• „Patience, or the inverse of the time preference rate, is a central variable in theoretical models of economic growth. In the Ramsey–Cass–Koopmans growth model with exogenous technical progress and an endogenous saving rate, more patient countries have a higher steady state capital stock and higher output per worker.”

• Hubner & Vannoorenberghe, 2015 http://www.sciencedirect.com/science/article/pii/S0165176515004139

Endogenity?

• Patience may be endogenous with respect to income

• Instrument – language; how does the country’s language differentiate between the present and the future?

• Language is related to patience, but not to growth

• The results are similar – patience, instrumented by language - affects GDP per capita

Grammar & savings (Chen, 2013)