Intermediate Macroeconomics - Lecture 3 - The …econ.sciences-po.fr/sites/default/files/file/barany/int_macro/...Intermediate Macroeconomics Lecture 3 - The Solow model ... sustained

Intermediate MacroeconomicsLecture 3 - The Solow model

Zsofia L. Barany

Sciences Po

2014 February

Recap of last week

I two distinct phases of growth:

1. pre-industrial revolution - stagnation: growth in total output isoffset by population growth, constant per capita consumption

2. post-industrial revolution - sustained growth: around 1.5% to2% growth per year sustained over 150 years

I different models can explain the different stages:

1. Malthusian model - main assumptions: population growthdepends on consumption per capita & land is available inlimited supply

2. Solow model - the topic of today’s lecture

The Solow model

I key: the description of the saving - investment - capitalaccumulation - growth process

I the model has stark predictions relating to the growth facts

I main prediction: technological progress is necessary forsustained growth in living standards

I note: not a micro-founded model, i.e. the goals andconstraints of the agents are not explicit

I the Solow model is the basis for the modern theory ofeconomic growth

Assumptions on Consumers/Workers

Labor supply

I there are N consumers

I who do not make a labor-leisure choice

I ⇒ each provides his one unit of labor

I ⇒ N is also the size of the labor force

I the population is growing at exogenous constant rate nN ′ = (1 + n) · N

Consumers/Workers

Income and consumption

I consumers are assumed to save an exogenous, constantfraction s of their income, and consume the rest

I the total income of consumers is Y , which is equal to totaloutput, as there is no government

I note: we do not need to distinguish between labor income andcapital income or profits (since households own the firms anddo not make a labor supply decision)

I the consumer’s budget constraint is Y = C + S

I ⇒ total saving in a period is s · YI ⇒ total consumption in a period is (1− s) · Y

Production

Production functionY = z · F (K ,N)

I Y total output

I z total factor productivity (TFP)

I K stock of capital

I N labor input

Important features:

I output depends on the quantity of labor and capital, and TFP

I no other inputs, such as land or natural resources, matter

Production function is assumed to be neoclassical

I positive, but diminishing marginal products

I Inada conditions

I constant returns to scale (CRS)

Characteristics of the neoclassical production function I.positive, but diminishing marginal products:

I z · F (K ,N) is monotone increasing in K and N

→ positive marginal product of each inputs

I MPK = ∂z·F∂K = z · FK > 0: if K increases by 1, z · F (K ,N)

increases by MPK

I MPN = ∂z·F∂N = z · FN > 0: if N increases by 1, z · F (K ,N)

increases by MPN

I z · F (K ,N) is concave - decreasing returns to K and N

→ decreasing marginal product of both inputs

I FKK = ∂2F∂K 2 = FK

∂K < 0 and FNN = ∂2F∂N2 = FN

∂N < 0

I as K increases more and more, F (K ,N) increases by less andless

I as N increases more and more, F (K ,N) increases by less andless

Characteristics of the neoclassical production function II.

Inada conditions:

I the marginal product of an input goes to zero as that inputgoes to infinity (holding all other inputs constant):

limK→∞ FK (K ,N) = 0 and limN→∞ FN(K ,N) = 0

I the marginal product of an input goes to infinity as that inputgoes to zero (holding all other inputs constant):

limK→0 FK (K ,N) =∞ and limN→0 FN(K ,N) =∞

Characteristics of the neoclassical production function III.

constant returns to scale:

F (λK , λN) = λF (K ,N) for any λ > 0

I e.g. λ = 2: if K and N double, then F (K ,N) and Y doubles

consequence of CRS:

I since we can pick λ = 1/N, output per worker can be writtenas:

y = YN = z · F (KN ,

NN ) = z · F (

K

N, 1)︸︷︷︸

≡f (k)

= z · f (k)

→ allows us to work with per capita quantities

I denote: k = KN , capital per capita; y = Y

N output per capita

I f inherits the properties from F : fk(k) > 0, fkk(k) < 0

Shape of the production function

K

Y z · F (K ,N)

K

YN

z · F (KN , 1)

k = KN

y = YN

z · f (k)

For a given labor supply (N) Y as a function of capital (K ) lookssomething like this.


K

Y z · F (K ,N)

K

YN

z · F (KN , 1)

k = KN

y = YN

z · f (k)

In per capita terms, Y /N looks something like this.


K

Y z · F (K ,N)

K

YN

z · F (KN , 1)

k = KN

y = YN

z · f (k)

The per capita production function, f (k) = F (KN , 1) lookstherefore something like this.

Example: the Cobb-Douglas production function

Y = z · KαN1−α

I has constant returns to scale as the coefficients sum to 1:

zF (λK , λN)

= z(λK )α(λN)1−α = λα+1−αzKαN1−α =λzKαN1−α

=λF (K ,N)

Per capita output is

y = Y /N = zkα

I positive, but diminishing marginal products, which are givenby:

MPK = αzKα−1N1−α

MPN = (1− α)zKαN−α

I Inada conditions hold

The capital accumulation equation

Capital stock

I increases - due to investment

I decreases - due to depreciation, d

Capital evolves according to:

K ′ = (1− d)K + I

How do we determine I?

This is part of the equilibrium condition of the model.

Competitive equilibriumTwo options:

1. Income - expenditure identity holds as an equilibrium conditioni.e. the goods market is in equilibrium

Y = C + I

combine this with the consumer’s budget constraint: Y = C + S⇒ gives us market clearing on the capital market:

S = I

2. Capital market is in equilibrium

total saving = total investment

S = I

combine this with the consumer’s budget constraint: Y = C + S⇒ gives us the income-expenditure identity:

Y = C + I

Equilibrium dynamics of capital per worker/per capita

The equilibrium dynamics of capital is then:

K ′ = I+(1−d)K = S+(1−d)K = s·Y+(1−d)K = s·z ·F (K ,N)+(1−d)K

Transforming this into per capita terms:

K ′

N= s · z · F (K ,N)

N+ (1− d)

K

N

N ′

N ′︸︷︷︸=1

K ′

N= s · z · F (K ,N)

N+ (1− d)

K

N

(1 + n)N

N

K ′

N ′=s · z · F (K ,N)

N+ (1− d)

K

N

(1 + n)k ′=s · z · f (k)+(1− d)k





K ′

N= s · z · F (K ,N)

N+ (1− d)

K

N

N ′

N ′︸︷︷︸=1

K ′

N= s · z · F (K ,N)

N+ (1− d)

K

N

(1 + n)N

N

K ′

N ′=s · z · F (K ,N)

N+ (1− d)

K

N

(1 + n)k ′=s · z · f (k)+(1− d)k





K ′

N= s · z · F (K ,N)

N+ (1− d)

K

N

N ′

N ′︸︷︷︸=1

K ′

N= s · z · F (K ,N)

N+ (1− d)

K

N

(1 + n)N

N

K ′

N ′=s · z · F (K ,N)

N+ (1− d)

K

N

(1 + n)k ′=s · z · f (k)+(1− d)k





K ′

N= s · z · F (K ,N)

N+ (1− d)

K

N

N ′

N ′︸︷︷︸=1

K ′

N= s · z · F (K ,N)

N+ (1− d)

K

N

(1 + n)N

N

K ′

N ′=s · z · F (K ,N)

N+ (1− d)

K

N

(1 + n)k ′=s · z · f (k)+(1− d)k

the key equation of the Solow model

k ′ =s · z · f (k)

1 + n+

(1− d)k

1 + n

→ determines capital per worker in the next period as a functionof capital per worker today

capital per capita in the next period is

I investment in the current period

I plus whatever is left of capital after depreciation

I adjusting for the fact that the population is higher in the nextperiod

Finding the steady state

the steady state k∗ is where k = k ′

I if k < k∗ ⇒ k ′ > k

I if k > k∗ ⇒ k ′ < k

The steady state

Equation determining the steady state of the economy:

k∗ = k ′ = k

in the steady state capital per worker is constant

k∗ =s · z · f (k∗)

1 + n+

(1− d)k∗

1 + n

(1 + n)k∗ = s · z · f (k∗) + (1− d)k∗

(n + d)k∗︸︷︷︸break-even investment

= s · z · f (k∗)︸︷︷︸actual investment

In the steady state

I the amount of actual investment is exactly such that

I it compensates for depreciation and the growth in population,i.e. for the break-even investment

⇒ the level of capital per worker stays constant

A useful graphical representation:

The ‘savings curve’ (actual investment) and the ‘effectivedepreciation line’ (break-even investment)

break-even investment

actual investment

steady state: where thebreak-even investment == actual investment

This graph is very useful for studying the effects of n, s and z

Three important questions about the Solow model:

1. does the steady state exist?

YES, if the two curves cross

2. is the steady state unique?

YES, if the two curves only cross once

3. is the steady state stable, i.e. does the economy converge tok∗ from any initial k0?

YES, if the actual investment curve is above the break-eveninvestment curve for k < k∗, and if the actual investmentcurve is below the break-even investment curve when k > k∗

This is due to the neoclassical production function.

































Example 1.

k

i

(d + n)k

szf (k)

szf (k)

Which of the assumptions on a neoclassical production function isviolated?

Example 1.

k

i

(d + n)k

szf (k)

szf (k)


Example 1.

k

i

(d + n)k

szf (k)

szf (k)


Example 2.

k

i

(d + n)k

szf (k)


Example 3.

k

i

(d + n)k

szf (k)


With a neoclassical production function

A unique and stable steady state exists.

break-even investment

actual investment

steady state:break-even investment=actualinvestmentk∗ = k ′ = k

Implications 1.

In the steady state

I k∗ is constant

I y∗ = f (k∗) is constant as well

I c∗ = (1− s)y∗ is constant as well

I K ∗ = N · k∗ grows at rate n

I Y ∗ = N · y∗ grows at rate n

I C ∗ = N · c∗ grows at rate n

in the long-run no growth in output per worker

Implications 2.

I if two countries have the same n, s, and d , as well as the sameproduction function⇒ they have the same steady state capitalper worker, k∗ ⇒ they converge to the same steady state

⇒ conditional convergence

along this convergence path poorer countries grow faster(why?)

Implications 3.

I what happens if two countries have different saving rates, i.e.sA < sB?

I ⇒ their steady state per capita capital is different, i.e.k∗A 6= k∗B

I both countries converge to their own steady state, but theseare different

⇒ no absolute convergence

when saving rates are different poorer countries don’tnecessarily grow faster

Saving rate and the steady state per capita capital

k

i

(d + n)k

sAzf (k)

k∗A

sBzf (k)

k∗B

The steady state capital per worker is higher, when the savingsrate is higher.

sA < sB ⇒ k∗A < k∗B


k

i

(d + n)k

sAzf (k)

k∗A

sBzf (k)

k∗B




k

i

(d + n)k

sAzf (k)

k∗A

sBzf (k)

k∗B




k

i

(d + n)k

sAzf (k)

k∗A

sBzf (k)

k∗B




k

i

(d + n)k

sAzf (k)

k∗A

sBzf (k)

k∗B




k

i

(d + n)k

sAzf (k)

k∗A

sBzf (k)

k∗B



Data: Real income per capita and the investment rate

→ per capita GDP and the investment rate are positively correlated

Income per capita and saving rate in the modelWhat is the implication of a higher saving rate in the Solowmodel?

I intuitively, a higher saving rate implies that the saving curve,the actual investment curve is higher

I ⇒ higher steady state k∗ ⇒ higher y∗

What is the effect of an increase in the saving rate?

I long-run increase in the level of per capita capital

I long-run increase in the level of per capita output

I temporary increase in the growth rate of per capita capitaland per capita output

I no effect on the long-run growth rate of capital per workerand output per worker, which are equal to zero

A change in the savings rate has a long-run level effect, but doesnot have a long-run growth effect.


I so a higher saving rate, s, leads to higher per capital capital inthe steady state, k∗, and thus to a higher steady state percapita income, y∗

I ⇒ should people increase their saving rate in order to increaseoutput? would this be good for them?

I put another way: Does it mean that a higher saving rate isalways better?

I what is the ’best’ steady state?

I define the ’best’ steady state as the one that maximizesconsumption per capita, c∗


I so a higher saving rate, s, leads to higher per capital capital inthe steady state, k∗, and thus to a higher steady state percapita income, y∗

I ⇒ should people increase their saving rate in order to increaseoutput? would this be good for them?

I put another way: Does it mean that a higher saving rate isalways better?

I what is the ’best’ steady state?

I define the ’best’ steady state as the one that maximizesconsumption per capita, c∗

Consumption and steady state capital

c∗ = (1− s)zf (k∗) = zf (k∗)− szf (k∗)

= zf (k∗)−(n + d)k∗

k

i

szf (k)

zf (k)

(d + n)k

k∗

c∗

To maximize c∗, need to find the biggest distance between zf (k)and (d + n)k .


c∗ = (1− s)zf (k∗) = zf (k∗)− szf (k∗)= zf (k∗)−(n + d)k∗

k

i

szf (k)

zf (k)

(d + n)k

k∗

c∗




k

i

szf (k)

zf (k)

(d + n)k

k∗

c∗


The golden ruleI what are the effects of a higher s on c∗?

I higher s ⇒ higher k∗ ⇒ higher y∗

I higher s ⇒ smaller consumption share in output

c∗︸︷︷︸?

= (1− s)︸︷︷︸↓

y∗︸︷︷︸↑

I net effect in general is ambiguousI due to the assumptions made on the production functionI c∗ first increases, then decreases in sI golden rule of saving: s such that c∗ is the highestI how to find it? by choosing s maximize


note: the last equality makes use of the steady state condition(szf (k∗) = (n + d)k∗)




c∗︸︷︷︸?

= (1− s)︸︷︷︸↓

y∗︸︷︷︸↑

I net effect in general is ambiguous

I due to the assumptions made on the production functionI c∗ first increases, then decreases in sI golden rule of saving: s such that c∗ is the highestI how to find it? by choosing s maximize






c∗︸︷︷︸?

= (1− s)︸︷︷︸↓

y∗︸︷︷︸↑

I net effect in general is ambiguousI due to the assumptions made on the production function

I c∗ first increases, then decreases in sI golden rule of saving: s such that c∗ is the highestI how to find it? by choosing s maximize






c∗︸︷︷︸?

= (1− s)︸︷︷︸↓

y∗︸︷︷︸↑

I net effect in general is ambiguousI due to the assumptions made on the production functionI c∗ first increases, then decreases in s

I golden rule of saving: s such that c∗ is the highestI how to find it? by choosing s maximize






c∗︸︷︷︸?

= (1− s)︸︷︷︸↓

y∗︸︷︷︸↑

I net effect in general is ambiguousI due to the assumptions made on the production functionI c∗ first increases, then decreases in sI golden rule of saving: s such that c∗ is the highestI how to find it? by choosing s maximize



The golden rule

c∗ = (1− s)zf (k∗) = zf (k∗)− szf (k∗) = zf (k∗)− (n + d)k∗

→ take the derivative with respect to s, and find s where ∂c∗

∂s = 0

where does s enter the right hand side?

only through k∗, as thesteady state level of per capita capital increases in s

∂c∗

∂s=z

∂f (k∗)

∂s− (n + d)

∂k∗

∂s= zf ′(k∗)

∂k∗

∂s− (n + d)

∂k∗

∂s

=(zf ′(k∗)− (n + d)

) ∂k∗∂s

= 0

⇒ zf ′(k∗)︸︷︷︸MPk

= n + d︸︷︷︸effective depreciation rate

→ First find the k∗ that maximizes c∗ (where ∂c∗

∂k∗ = 0).→ Then find the s that achieves this k∗ using the steady statecondition szf (k∗) = (n + d)k∗.

The golden rule



∂s = 0

where does s enter the right hand side? only through k∗, as thesteady state level of per capita capital increases in s

∂c∗

∂s=z

∂f (k∗)

∂s− (n + d)

∂k∗

∂s= zf ′(k∗)

∂k∗

∂s− (n + d)

∂k∗

∂s

=(zf ′(k∗)− (n + d)

) ∂k∗∂s

= 0

⇒ zf ′(k∗)︸︷︷︸MPk




The golden rule



∂s = 0

where does s enter the right hand side? only through k∗, as thesteady state level of per capita capital increases in s

∂c∗

∂s=z

∂f (k∗)

∂s− (n + d)

∂k∗

∂s= zf ′(k∗)

∂k∗

∂s− (n + d)

∂k∗

∂s

=(zf ′(k∗)− (n + d)

) ∂k∗∂s

= 0

⇒ zf ′(k∗)︸︷︷︸MPk




The golden rule

I so the marginal product of capital should equal n + d in thesteady state to achieve the highest per capita consumption

I → we can find the saving rate that results in this specificsteady state: sgr

I if the government, central planner would prescribe this savingrate, the economy would reach the steady state and themaximum possible consumption

I in practice: we can estimate MPK , we know n, we cancalculate d ⇒ we can actually calculate sgr

I should the government do something, try to impose this?

1. redistribution across generations2. consumers save optimally (given their preferences)⇒ is there a market failure that prevents them from achievingthe correct trade-off between current consumption andsavings?

Data: Income per capita and the population growth rate

→ GDP per capita and pop growth rate negatively correlated

Income per capita and the population growth rate in themodel

What is the effect of a higher population growth rate in the Solowmodel?

I a higher n increases the break-even investment, rotates theeffective depreciation line up

I lower steady state per capita capital and output: k∗, y∗ ↓

I temporary decrease in the growth rate of output per worker

I long-run decrease in the level of output per worker


I of course the growth rate of aggregate variables increases

Income per capita and the population growth rate in themodel

Income per capita and technology in the model

What is the effect of higher productivity in the Solow model?

I a higher productivity, z increases actual investment in themodel→ the savings curve pivots up

I higher steady state per capita capital and output: k∗, y∗ ↑

I temporary increase in the growth rate of output per worker

I long-run increase in the level of output per worker



but what is the effect of a sustained productivity growth?


but what is the effect of a sustained productivity growth?

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Intermediate Macroeconomics - Lecture 3 - The …econ.sciences-po.fr/sites/default/files/file/barany/int_macro/...Intermediate Macroeconomics Lecture 3 - The Solow model ... sustained