growth theory: review - university of southampton

Growth AccountingThe Solow Model

Savings behaviourThe One Sector Growth Model

Growth Theory: ReviewLecture 1.1, Exogenous Growth

Topics in Growth, Part 2

June 11, 2007

Lecture 1.1, Exogenous Growth 1/76 Topics in Growth, Part 2



Growth Accounting: Objective and Technical FrameworkTFP residualDecomposing growth in GDP per workerResults

Growth Accounting: Objective 2

◮ Many factors play role to determine output in a country

◮ Certainly, size of the labour force and capital stock do◮ But also, education, government regulation, weather,...

◮ Any theory of economic growth chooses which of thesefactors to emphasize as

◮ sources of GDP growth within countries◮ explanation for differences in levels/growth rates across

countries

◮ Growth accounting:tool to evaluate relative importance of such factors →Theory & Policy Implications





Technical framework 3

◮ Ignore the demand side for now

◮ Carefully specify the supply side

◮ Inputs: capital, K , and labour, L

◮ Output, Y

◮ State of technology, A





Technology, F 4

Y = F (K , AL) where

Y = output

K = capital (input / factor)

L = labour (input / factor)

A = state of technology

H = AL = effective labour

Assumptions◮ Marginal products positive and diminishing◮ Constant returns to scale





Marginal products 5

◮ Marginal product of labour

◮∂F∂L = FL > 0 positive

◮∂

2F∂L2 = FLL < 0 and diminishing

◮ Marginal product of capital

◮∂F∂K = FK > 0 positive

◮∂

2F∂K 2 = FKK < 0 and diminishing





Constant returns to scale (CRS) 6

F (λK , AλL) = λF (K , AL) for λ > 0

◮ Implications of CRS

◮ Size (of firms) does not matter → representative firm

◮ Euler’s theorem: Factor payments exhaust the output

◮ See example





Cobb-Douglas production function 7

F (K , AL) = K α(AL)1−α 1 > α > 0

◮ CRS◮ Positive and diminishing MP





Steps for growth accounting 8

◮ TFP residual, At , for K only production function

◮ TFP residual, At , across countries: K only

◮ TFP residual, At , including human capital

◮ TFP residual, At , across countries: K and H

◮ Decomposing growth in GDP per worker: K only

◮ Decomposing growth in GDP per worker: K and H

◮ Summary of results

◮ Critique





TFP, At , as residual for K only production function 9

◮ From the production function in year t

Yt = AtK αt L1−α

t

◮ Denoting “per worker” variables in lower case letters,i.e., output per w. yt = Yt

Ltand capital per w. kt = Kt

Lt

◮ After dividing by Lt , we rewrote production functionYtLt

= At(KtLt

)α(LtLt

)1−α as yt = Atkαt

◮ Hence, by rearranging we got

At = ytkα

t





CRS production function with physical and humancapital 10

◮ Now include human capital (e.g., years of education) intoproduction function (Make sure production function is CRSand positive and diminishing MP)

◮ We will use Yt = AtF (Kt , Ht , Lt) = AtK αt Hβ

t L1−α−βt ,

with α, β ∈ [0, 1] (parameters)

◮ Note: AtF (λKt , λHt , λLt) = λAtF (Kt , Ht , Lt) → CRS





TFP as residual for production function with K and H11

◮ From the production function in year t

Yt = AtK αt Hβ

t L1−α−βt

◮ Denoting “per worker” variables in lower case letters,i.e., output per w. yt = Yt

Lt, phys. capital per w. kt = Kt

Ltand

human capital per w. ht = htLt

◮ After dividing by Lt , we can rewrite the production functionYtLt

= At(KtLt

)α(HtLt

)β(LtLt

)1−α−β as yt = Atkαt hβ

t◮ Hence, by rearranging we get

At = yt

kαt hβ

t





Compare At for the 2 production functions 12

◮ Residual for production function with physical capital onlyAt = yt

kαt

◮ Residual for production function with human capitalAt = yt

kαt hβ

t

◮ Difficult to measure β





Decomposing GDP per worker growth: K only 13

◮ Now that we have a series for At , we want to decomposegrowth in GDP per worker into growth in the capital stockversus growth in productivity.

◮ Last time, we derived

log yt+1 − log yt = log At+1 − log At + α(

log kt+1 − log kt

)

log yt+1−log ytlog yt+1−log yt

=log At+1−log Atlog yt+1−log yt

+ αlog kt+1−log ktlog yt+1−log yt

1 = 100% =log At+1−log Atlog yt+1−log yt






Decomposing GDP per worker growth: K and H 14

◮ Now that we have a series for At , we want to decomposegrowth in GDP per worker into growth in the capital stockversus growth in human capital versus growth inproductivity (TFP).

◮ Growth in output per worker is

yt+1yt

=At+1(kt+1)α(ht+1)β

At(kt )α(ht+1)β

yt+1yt

=At+1At

(kt+1kt

)α(ht+1ht

)β

◮ Next, as before we go to logs so we have a sum.





Decomposing GDP per worker growth: K and H 15

log yt+1 − log yt =

log At+1 − log At + α(

log kt+1 − log kt

)

+ β(

log ht+1 − log ht

)

log yt+1−log ytlog yt+1−log yt

=log At+1−log Atlog yt+1−log yt


+ βlog ht+1−log htlog yt+1−log yt

1 = 100% =log At+1−log Atlog yt+1−log yt


+ βlog ht+1−log htlog yt+1−log yt





Summary of Results 16

On my webpage, you can find an example of growth accountingcomparing the UK, India and South Korea. The main result isthat

◮ GDP growth accounting:

increase in human capital (average years of education)accounts for a major part of growth in India

◮ When accounting for physical capital only, a lot of thegrowth is assigned to the residual. Keep in mind that thisresidual may have a different meaning than justtechnological progress.

If you are interested in research in economic growth, this kindof accounting exercise (including variables suspected tocontribute to growth) is a useful start.




Model SetupBalanced Growth (with popul. and prod. growth)

From Growth Accounting to the Solow Model 17

◮ In growth accounting

→ link of inputs in period t to output in period t→ no link of inputs or output across periods (t versus t + 1)

◮ Solow model links

→ population/labor force, productivity and, in particular,capital stock in year t

to→ labor force, productivity and capital stock in year t + 1

◮ Solow (1956), Solow (1957) and Solow (1960)





From Growth Accounting to the Solow Model 18

◮ Solow’s story about how the capital stock evolves over time

◮ Households save → investment◮ Households save a (constant) fraction s ∈ [0, 1] of their

income every period/year◮ Households consume the rest, i.e., fraction (1 − s) of

income◮ Aggregate income : Yt◮ Aggregate investment = It = sYt

◮ Law of motion of aggregate capital (δ ∈ [0, 1])

Kt+1 = (1 − δ)Kt + It





Kaldor facts: Stylized facts of economic growth 19

1. The labor share and the capital share are almost constantover time.

2. The ratio of aggregate capital to output is almost constantover time.

3. The return to capital is almost constant over time.

4. Output per capita and capital per worker grow at a roughlyconstant and positive rate.

5. Different countries and regions within a country that startout with a different level of income per capita tend toconverge over time.





Understanding growth differences across time andacross countries 20

◮ Why do (developed) countries grow?

◮ Will developing countries catch up to developed countries?

◮ Solow model:a first attempt to explain the mechanics of growth

◮ Implications of Solow’s theory:differences in initial condition, effectiveness of labor andpopulation growth matter





Assumptions of the Solow model 21

◮ Assumptions

◮ Inputs: capital, Kt and labor Lt

◮ Production function: neo-classical production function

◮ Depreciation:capital depreciates at rate δ ∈ [0, 1] from t to t + 1

◮ Evolution of technology:At+1 = (1 + g)At ,

◮ Evolution of population (labor force*):Lt+1 = (1 + n)Lt

◮ where δ, g and n are exogenously given parameters





Assumptions of the Solow model 22

◮ Last Assumption

◮ Consumption and savings:

consumers save a constant fraction s of their income, yt ,consume fraction (1 − s) (s parameter)

◮ Per person income is: yt = rtkt + wtℓt

◮ Labor is supplied inelastically & normalized to ℓt = 1

◮ Savings per person are: syt





Aggregating consumers 23

◮ Savings per person are: syt = rtkt + wt

◮ Multiplying by the number of people in period t

Aggregate Savings/Investment

= It = Ltsyt = Lts(rtkt + wt) = s(rt Kt + wtLt)





Profit maximizing firm(s) 24

Π(K , AL) = F (K , AL)− rK − wL◮ Firms take prices as given and choose inputs K and L

◮ First order conditions

◮∂Π∂K = FK (K , AL) − r = 0

◮∂Π∂L = FL(K , AL) − w = 0

◮ Firm picks K and L such that

◮ FK (K , AL) = r

◮ FL(K , AL) = w





Profit maximizing firms with CD production function 25

◮ With CD production function, FOCs become

FK (K , AL) = αK α−1(AL)1−α = r

FL(K , AL) = (1 − α)K αA1−αL−α = w

◮ Or, rearranging

FK (K , AL) = α(AL

K

)1−α= r

FL(K , AL) = (1 − α)A( K

AL

)α= w





Euler’s Theorem with CD production function 26

rK + wL = F (K , AL)

Factor payments exhaust production.

◮ We have

r = FK (K , AL) = α(AL

K

)1−α

w = FL(K , AL) = (1 − α)A( K

AL

)α

◮ Therefore,

rK + wL = FK (K , AL)K + FL(K , AL)L





Euler’s Theorem with CD production function 27

◮ We had

rK + wL = FK (K , AL)K + FL(K , AL)L

◮ Therefore, substituting in functional forms for FK (K , AL)and FL(K , AL) from the previous slide, we get:

rK + wL =[

α(AL

K

)1−α]

K +[

(1 − α)A( K

AL

)α]

L

= αK α(AL)1−α + (1 − α)K α(AL)1−α

= K α(AL)1−α

= F (K , AL)





One large firm or many small firms 28

◮ Since firms take prices as given,◮ and assuming A and α are the same for all firms◮ from FOCs, we get

r = FK (Ki , ALi) = αA1−α( Li

Ki

)1−α

w = FL(Ki , ALi) = (1 − α)A1−α(Ki

Li

)α

◮ Capital-labor ratio, k∗ = LiKi

, chosen is the same for all firms(indexed by i).

◮ Using the CRS assumption, total output by many firms canbe represented by output of one firm





Law of motion of aggregate capital stock 29

◮ Using the solution to the firm’s problem, we showed that

rtKt + wtLt = F (Kt , AtLt) = Yt (lecture 2)

◮ Using the aggregation over consumers, we saw earlier

It = s(rtKt + wtLt)

◮ Therefore, It = sYt = sF (Kt , AtLt)

◮ Law of motion of aggregate capital

Kt+1 = (1 − δ)Kt + It

◮ Consider Kt+1 as a function of Kt





Balanced growth: n 6= 0 and g 6= 0 30


◮ Evolution of population (labour force*):Lt+1 = (1 + n)Lt

◮ Law of motion of aggregate capitalKt+1 = (1 − δ)Kt + sF (Kt , AtLt)

◮ Want to find growth rate of capital per worker, kt = KtLt

and

GDP per capita yt = YtLt





Balanced Growth: Steady state in units of effectivelabour 31

◮ Letyt = Yt

At Ltoutput per unit of effective labour

kt = KtAt Lt

capital per unit of effective labour

◮ Then we can writeytAtLt = Yt

ktAtLt = Kt

◮ Law of motion becomesKt+1 = (1 − δ)Kt + sYt or,kt+1At+1Lt+1 = (1 − δ)kt AtLt + sytAtLt





→ Law of Motion 32

◮ Law of motion becomes

Kt+1 = (1 − δ)Kt + sYt or,

kt+1At+1Lt+1 = (1 − δ)kt AtLt + sytAtLt or,

kt+1(1 + g)At(1 + n)Lt = (1 − δ)kt AtLt + sytAtLt

kt+1(1 + g)(1 + n) = (1 − δ)kt + syt





→ Law of Motion 33

◮ Law of motion for capital per unit of effective labour

kt+1 = 1(1+g)(1+n)

[

(1 − δ)kt + syt

]

◮ Note that yt = YtAt Lt

= F (Kt ,At Lt )At Lt

= kαt





Steady state in detrended variables(i.e. per unit of effective labour) 34

◮ Using yt = kαt , law of motion for kt

kt+1 = 1(1+g)(1+n)

[

(1 − δ)kt + skαt

]

◮ Show that kt+1 is an increasing and concave function of kt

if α, δ ∈ [0, 1], g, n ∈ [−1, 1]





Solow’s law of motion (capital per u. of eff. labour) 35

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50K_t

K_

t+1

Kt+1 = Kt (45 degree line)

Kt+1 = (1-delta) Kt + s F(Kt,AL)





Solow’s law of motion (capital per u. of eff. labour) 36

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50K_t

K_t+

1

Kt+1 = Kt (45 degree line)

Kt+1 = (1-delta) Kt + s F(Kt,AL)





Balanced Growth: Per capita/worker var’s 37

◮ Steady state ito capital per u. of eff. labour

kt+1 = 1(1+g)(1+n)

[

(1 − δ)kt + skαt

]

◮ This again can be solved for k∗, the value for which capitalper unit of effective labour does not change anymore, i.e.kt = kt+1 = k∗

* k∗ =(

sg+n+ng+δ

)1

1−α

◮ Higher population growth implies lower level of capitalstock per unit of effective labor in the long run, but growthrate of per capita variables unaffected





Capital per worker and GDP per capita 38

◮ When the capital stock per unit of effective labour, kt ,reaches its steady state level k∗, we get:

◮ Growth rate of capital per worker:

kt+1

kt=

Kt+1Lt+1

KtLt

=

At+1Kt+1At+1Lt+1

At KtAt Lt

=At+1k∗

At k∗

= (1 + g)

◮ Growth rate of output per capita:

yt+1

yt=

kα

t+1A1−α

t+1

kαt A1−α

t

=(kt+1

kt

)α(At+1

At

)1−α

= (1 + g)α(1 + g)1−α

= (1 + g)





Balanced Growth: Wage and rental rate of capital 39

◮ Growth rate of wages

wt+1

wt=

FK (t + 1)

FK (t)=

(1 − α)K αt+1(At+1Lt+1)

−αAt+1

(1 − α)K αt (AtLt)−αAt

=( k∗

k∗

)α(At+1

At

)

= (1 + g)

◮ Show that the rental rate on capital, rt ,is constant along the BGP




Preferences, Budget constraint, Optimal choiceConsumption smoothingIntertemporal-substitution and wealth effects

Solow model and Savings Behaviour 40

Recall that in the Solow model

◮ the savings rate was an exogenous constant (parameter)

◮ therefore aggregate investment was a constant fraction ofoutput/aggregate income

Suppose you know that whatever you save, the government willtax at 100% next year. How much would you save versusconsume this year? About nothing – unless you can hide itreally well...

Hence questions such as: What is the effect of capital gainstaxes? cannot seriously be addressed in the Solow model.

However, the Solow model DID teach us that the savings rate isimportant for growth.





A two period model of saving 41

Income

◮ Consider a household that receives an exogenous flow ofincome in each period of time

◮ We restrict the number of periods to be 2: t and t + 1

◮ Denote income in each period by yt and yt+1

◮ Assume there are perfect financial markets where thehousehold can freely borrow and lend by holding assets ordebt, at+1, at an interest rate r






Preferences

◮ Preferences of the household are defined over sequencesof consumption {ct , ct+1}

◮ We assume that instantaneous utility can be representedby a standard utility function: u(c) [i.e. u(.) is increasing,twice differentiable, concave and satisfies Inadaconditions*]

◮ Life-time utility is the discounted sum of instantaneousutilities → The agent has a subjective rate of timepreference ρ so that the discount factor is 1/(1 + ρ) < 1→ high ρ means impatient→ low ρ means patient






Preferences

◮ Life-time utility is

V (ct , ct+1) = u(ct) +1

1 + ρu(ct+1)

◮ Sometimes we will define β ≡ 11+ρ and write utility as

V (ct , ct+1) = u(ct) + βu(ct + 1)

β is the discount factor






Budget constraint

◮ The agent faces two period-by-period constraints

ct + at+1 = yt

ct+1 = yt+1 + (1 + r)at+1

◮ The assumption of perfect financial markets means thatconsumption is not restricted to equal income

◮ Agent can allocate consumption in many different ways

◮ In fact, he faces a single constraint:

the intertemporal budget constraint

◮ It follows from aggregating over time as follows:

ct + 11+r ct+1 = yt + 1

1+r yt+1Lecture 1.1, Exogenous Growth 44/76 Topics in Growth, Part 2





Budget constraint

◮ It follows from aggregating over time that:

ct + 11+r ct+1 = yt + 1

1+r yt+1

◮ In other words, the present value of consumption cannotexceed the present value of income (or wealth)

◮ This can be represented graphically*

◮ Only at the point corresponding to the endowment, saving(borrowing (-) or lending (+)) is zero





Household’s optimization problem 46

Given yt , yt+1 and r

maxct ,ct+1

u(ct) + βu(ct+1)

s.t. ct +1

1 + rct+1 = yt +

11 + r

yt+1

or, equivalently,

maxct ,ct+1

u(ct) + βu(ct+1)

s.t. ct + at+1 = yt

ct+1 = (1 + r)at+1 + yt+1





Solution to the optimization problem 47

Graphical solution*

◮ Using the first formulation, the problem and the solutioncan be represented in the typical indifference-curvediagram on the (ct , ct+1) - space

◮ The optimal choice is characterized by the allocation wherethe intertemporal budget constraint (with slope −(1 + r)) istangent to an indifference curve





Solution to the optimization problem 48

Analytical solution

◮ This solution is characterized by a FOC and the budgetconstraint (2 equations for 2 unknowns (ct , ct+1))

◮ The FOC reads,

u′(ct) =1 + r1 + ρ

u′(ct+1)

= β(1 + r)u′(ct+1)

◮ This is called the Euler equation





Consumption smoothing 49

◮ The FOC implies that the change in consumption over timedepends entirely on the form of the utility function, u(.), ρand r

◮ The time-profile of income does not matter for thetime-profile of consumption (holding present value oflife-time income fixed)

◮ The present value of income is only important indetermining the level consumption in the two periods, butnot the steepness of the consumption path





Consumption smoothing 50

◮ Consider in particular the situation where interest rateequals the rate of time preference: r = ρ

◮ In this case, consumption is the same in the two periodseven if income is not

◮ This captures the implication of concave utility functions forconsumption: agents tend to prefer smooth consumptionpaths

◮ They can do that because they can borrow and lend

◮ To see more specifically how the interest rate can alter theoptimal path of consumption, it proves convenient to use aspecific yet fairly general form for the utility function...





Constant-elasticity-of-substitution utility 51

◮ We use the following,

u(c) =c1−σ

1 − σif σ 6= 1

= log c if σ = 1

◮ It turns out that σ determines the household’s willingnessto shift consumption across periods:→ the smaller is σ,→ the more slowly marginal utility ↓ as consumption ↑→ the more willing is the household to allow itsconsumption to vary over time (if r differs from ρ)






◮ This can be seen from the FOC,

u′(ct ) =1 + r1 + ρ

u′(ct+1)

◮ Using the CES utility function*,

ct+1

ct=

(1 + r1 + ρ

)1σ

◮ If σ is close to zero, then utility is close to linear and thehousehold is willing to accept large swings in consumptionto take advantage of small differences between ρ and r






◮ In fact the intertemporal elasticity of substitution, IES, isclosely related to σ

◮ The IES is defined as

θ(c) = − u′(c)u′′(c)c

◮ This is essentially a measure of the curvature of the utilityfunctions and, therefore, of the willingness to acceptswings in consumption over time

◮ With the CES utility function, the IES becomes*

θ(c) = − u′(c)u′′(c)c = 1/σ





Intertemporal-substitution and wealth effects 54

Intertemporal substitution and r

◮ θ = 1/σ determines the responsiveness of the slope of theconsumption path to changes in the interest rate

◮ Higher r implies that optimal consumption grows fasterover time

◮ This does not depend on the time path of income

◮ This is the intertemporal-substitution effect of a change inthe interest rate → (1 + r) is just the relative price of ct interms of ct+1






Intertemporal substitution and r

◮ Thus intertemporal substitution is the standard substitutioneffect when the relative price of two commodities changes

◮ This effect of an increase in r tends to increase savinga = yt − ct






Wealth effect and r

◮ But, as usual, there is also a wealth effect

◮ Here, it is useful to draw the indifference-curve diagram

◮ If initially saving is zero, then the wealth effect is nil and thesubstitution effect dictates an increase in saving

◮ If initially the household is borrowing, both the wealth andsubstitution effects go in the direction of increasing saving(or reducing borrowing)

◮ If the household is initially saving, then the wealth effecttends to reduce saving and the net effect is ambiguous

◮ Follow graphical analysis and discussion in D.Romer(1996,p325-327)].






Savings is a = yt − ct

Suppose r increases

◮ Substitution effect ⇒ a ↑

◮ Income effect ⇒ a ?

◮ If initially a = 0, no wealth effect ⇒ a ↑

◮ If initially a > 0, positive wealth effect ⇒ a ?

◮ If initially a < 0, negative wealth effect ⇒ a ↑




Main Ingredients of the ModelDefinition of EquilibriumSteady state and Dynamics

The One Sector Growth Model 58

Main Ingredients

◮ Neoclassical model of the firm

◮ Consumption-savings choice for consumers

◮ “Solow model + incentives to save”

(recall example with taxes)





The 1SGM 59

Markets and ownership

Agents◮ Firms produce goods, hire labor and rent capital◮ Households own labor and assets (capital),

receive wages and rental payments, consume and save

Markets◮ Inputs: competitive wage rates, w , and rental rate, R◮ Assets: free borrowing and lending at interest rate, r◮ Output: competitive market for consumption good





The 1SGM 60

Firms / Representative Firm

Seeks to maximize profits

Profit = F (K , L) − RK − wL

The FOCs for this problem deliver

∂F∂K

= R∂F∂L

= w

In per unit of labor terms, let f (k) ≡ F (k , 1)

f ′(k) = R f (k) − kf ′(k) = w

Recall Euler’s Theorem: factor payments exhaust outputLecture 1.1, Exogenous Growth 60/76 Topics in Growth, Part 2




The 1SGM 61

Households / Representative household

Preferences

U0 =

∞∑

t=0

βtu(ct)

Budget constraint

ct + at+1 = wt + (1 + r)at ,

for all t = 0, 1, 2, ...

a0 given

Note: labor supplied inelastically, lt = 1





The 1SGM 62

Households / Representative household

Intertemporal version of budget constraint

∞∑

t=0

t∏

s=0

(

11 + rs

)

ct = a0 +

∞∑

t=0

t∏

s=0

(

11 + rs

)

wt

We rule out that debt explodes (no Ponzi games)

at+1 ≥ −B for some B big, but finite

More compactly, PDV (c) = a(0) + PDV (w)





The 1SGM 63

Household’s problem

max(at+1,ct )

∞

t=0

∞∑

t=0

βtu(ct)

s.t.

ct + at+1 = wt + (1 + r)at , for all t = 0, 1, 2, ...

at+1 = −B for some B big, but finite

a0 given





The 1SGM 64

Euler equationIn general,

u′(ct) = β(1 + rt+1)u′(ct+1)

From here on, CES utility, u(c) = c1−σ

1−σ , Euler eqn. becomes,

(

ct+1

ct

)σ

= β(1 + rt+1)





The 1SGM 65

Transversality conditionHH do not want to “end up” with positive values of assets

limt→∞

βtu′(ct)at ≥ 0

HH cannot think they can borrow at the “end of their life”

limt→∞

βtu′(ct)at ≤ 0

Hence,

limt→∞

βtu′(ct)at = 0





Definition of Equilibrium 66

A competitive equilibrium is defined by sequences of quantitiesof consumption, {ct}, capital, {kt}, and output, {yt}, andsequences of prices, {wt} and {rt}, such that

◮ Firms maximize profits

◮ Households maximize U0 subject to their constraints

◮ Goods, labour and asset markets clear





Benevolent planner’s problem* 67

What is the allocation of resources that an economy shouldfeature in order to attain the highest feasible level of utility?

Central Planner’s optimal choice problem

max(kt+1,ct)

∞

t=0

∞∑

t=0

βtu(ct)

s.t.

ct + kt+1 = f (kt) + (1 − δ)kt , for all t = 0, 1, 2, ...

k0 > 0 given





Benevolent planner’s problem 68

Welfare

Socially optimal allocation coincides with the equilibriumallocation.

The competitive equilibrium leads to the social optimum.

Not surprising: no distortions or externalities→ Welfare Theorems hold





Steady state* 69

Definition

A balanced growth path (BGP) is a situation in which output,capital and consumption grow at a constant rate.

If this constant rate is zero, it is called a steady state.

We can usually redefine the state variable so that the latter isconstant (i.e. the growth rate is zero)

Recall from the Solow model:

capital per capita for n = 0, g = 0 capital per unit of effectivelabor for n > 0, g > 0





Steady state 70

From the Euler equation,

ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/σ , for all t

If consumption grows at a constant rate (BGP), say γ

1 + γ = [β(1 + f ′(kt+1) − δ)]1/σ , for all t

Thus RHS must be constant→ kt+1 = kt = k∗ must be constant along the BGP





Steady state 71

But then, from the resource constraint with kt = kt+1 = k∗:

ct + kt+1 = f (kt) + (1 − δ)kt , for all t

i.e.,ct = f (k∗) − δk∗

ct+1 = f (k∗) − δk∗

We find that consumption must be constant along the BGP,→ ct+1 = ct = c∗ or γ = 0

Hence we have a steady state in per capita variables.





Steady state* 72

Hence from the Euler equation

1 + γ = 1 = [β(1 + f ′(k∗) − δ)]1/σ

or, simplified

f ′(k∗) =1β− (1 − δ) = ρ + δ

we can solve for k∗and from the (simplified) resource constraint

c∗ = f (k∗) − δk∗

we can solve for c∗





Off steady state dynamics* 73

Off the steady state, consumption and capital adjust to reachthe steady state eventually.

To analyze these dynamics, consider the movements of c and kseparately.

Let ∆c = ct+1 − ct and ∆k = kt+1 − kt . See graphical analysis.





Off steady state dynamics* 74

◮ Consider the set of points such that ∆c = 0, then from theEuler eqn, the optimal k satisfies f ′(k) = ρ + δ

→ draw vertical line at k∗(< kGR)

To the left: kt < k∗ ⇒ f ′(kt) > f ′(k∗) ⇒ ∆c > 0 ⇒ c ↑To the right: kt > k∗ ⇒ f ′(kt) < f ′(k∗) ⇒ ∆c < 0 ⇒ c ↓

◮ Consider the set of points such that ∆k = 0, then from theResource cstrt, the optimal c satisfies c = f (k) − δk

→ draw hump-shaped line from origin, maximized at kGR

cross 0 again for k such that f (k) = δk

Above: ct > f (kt) − δkt ⇒ ∆k = f (kt ) − δkt − ct < 0 ⇒ k ↓Below: ct < f (kt) − δkt ⇒ ∆k = f (kt) − δkt − ct > 0 ⇒ k ↑





Adding population and productivity growth 75


◮ Evolution of population (labour force*):Lt+1 = (1 + n)Lt

◮ Resource constraint in units of effective labourct + (1 + n)(1 + g)kt+1 = (1 − δ)kt + f (kt)

◮ Euler equation in units of effective labour becomes

ct+1ct

=[β(1+f ′(kt+1)−δ)]

1σ

(1+n)(1+g)





Balanced Growth: St. state in units of effective labour76

◮ In steady state:ct+1ct

= 1 and kt+1

kt= 1

◮ Solving the Euler Equation for k∗ gives

k∗ =

[

βα

((1 + n)(1 + g))σ − (1 − δ)β

]

decreasing in population growth rate. But long run growthrates of per capita variables are independent of n

◮ Thus, in terms of population growth rates, One SectorGrowth Model gives same conclusions as Solow model.


growth theory: review - university of southampton

Documents