Economic Growth: Solow Model

-References: There are many good presentations of the Solow Model. There are two on the website.

G. McCandless The ABCs of RBCs Ch. 1 ; Jones and Vollrath Introduction to Economic Growth Ch. 2

- Standards of living, size of capital stock are typically growing in the long-run.

- it is desirable to have models with long-run growth as an outcome.

- long-run growth: is it more important to well-being than cycles?

- ideally: want a macroeconomic model allowing for both long-run growth and cyclical deviations from the long-run growth path.

(this is one aim of DSGE Macroeconomics)

- Starting point: Aggregate production function Yt = F(Kt, Nt, t)

Y = output, N=labour, K = capital, t - time

(Notational point : lower case will mean per unit labour or per worker e.g. y ≡ Y/N, k≡K/N)

- Why might growth occur?

- accumulation: more capital and other non-labour inputs.

- labour force growth (skills too? or are they like capital?)

- technological and organizational progress (“t” in the production function might capture this): raises Total Factor Productivity.

- Growth accounting exercises take this as a starting point: choose a specific functional form for F(), obtain measures of Y, K, N; infer the effect of ‘t’.

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Solow Model

- A widely used growth model.

- Robert Solow (1956) "A Contribution to the Theory of Economic Growth". Quarterly Journal of Economics 70 (1): 65–94.

“The Solow model is a mixture of an old-style Keynesian model and a modern dynamic macroeconomic model” D. Acemoglu.

- Let’s start with a basic version of the Solow model:

- Basic? no technological change.

- Often presented this way.

- Can allow for a quite general production function.

- Model has an interesting equilibrium outcome.

- Parts of the basic model:

- Aggregate Production function.

- Labour force growth and capital accumulation assumptions.

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- Aggregate Production Function:

Yt = F(Kt, Nt)

- Economy-wide production function (earlier notes: firm level production)

- Form of F(): reflects technology, organization, incentives. - These affect productivity of a given quantity of inputs.

- Other inputs? e.g. natural resources; assume fixed in this version: built into F; another possibility: part of K.

- Make our usual technical assumptions about the production function:

Positive marginal products: ∂F/∂N≡FN>0, ∂F/∂K≡FK>0

Diminishing returns: ∂2F/∂K2≡FKK and ∂2F/∂N2≡FNN<0

- Also assume “constant returns to scale”i.e. raise all inputs by the same proportion and output rises by that

proportion.

So where z is some constant: z∙Yt = F(z∙Kt, z∙Nt)

Let: z=1/Nt

Then the production function can be written in per worker terms:

Yt/Nt = F(Kt/Nt ,1)

or: yt = F(kt,1) (y≡Y/N, k≡K/N)

notice : ∂y/∂k = FK

- Lastly assume the “Inada conditions” hold:

lim FK = 0 lim FK = ∞ k→∞ k→0

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- Consequences of technical assumptions:

Inada conditions will push outcomes away from extremes (0 or infinite k): convenient! Realistic?

Are they true? No empirical evidence! (don’t see these extremes)

Diminishing returns: return to k investment declines in level of k.

- Plausibility? Two stories: - high ‘k’ then each unit of K has little N to work with

and is therefore less productive.

-aggregate production: initial K goes to highest return activities, later investment in K to remaining highest return activity, etc.

- Diminishing returns places a limit on growth in this model.(See Appendix ‘AK Model’ to see what happens

without diminishing returns)

- R. Allen Global Economic History : is diminishing returns empirically sensible at the aggregate level?

an aggregate production function of this form might be a good approximation across countries and time (see his Figures 8-11).

(see next page)

- A production function that satisfies the technical assumption? Cobb-Douglas with 0<a<1, A >0

Yt= AKtaNt

1-a

per worker form (divide by N):

yt= Akta FK=aAkt

a-1 >0FKK=(a-1)aAkt

a-2<0(Inada conditions hold too)

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Other pieces of the Solow model:

- Labour force growth: labour grows at a constant per period rate of “n” (n>0).

so: Nt+1 = Nt (1+n)

- model assumes full employment: so labour force = employment

- Alternative ideas: endogenous ‘n’-- Malthus; Demographic transition (n depends on living standards?)

- National income constraint (economy-wide budget constraint):

Yt = Ct + It C= consumption, I=investment

(no government spending; closed economy so no exports of imports)

- Investment and saving:

- Closed economy: investment is financed with domestic savings (S):

It=St

- Savings function: St = s∙Yt (then Ct=(1-s)Yt )

- total savings (S) in time t are a fixed proportion (s) of Y.

- “s” is the savings rate (a constant in Solow)

- No “microfoundations” for this: ‘s’ is assumed constant. ‘s’ is not derived from intertemporal optimization.

(an old-style macroeconomic assumption)

- Alternative ideas: endogenous ‘s’ (depends on income; rate of return)

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- Capital growth: (Kt+1 = Kt+1-Kt)

- Capital accumulation: Kt+1 = It - Kt,

= depreciation rate (share of K that wears out each period)

use St = s∙Yt and St=It :

Kt+1 = s∙Yt - Kt

divide by K to express it as the growth rate in K:

Kt+1/Kt = s∙Yt/Kt -

- Is k=K/N growing, shrinking or staying the same?

Growing if: Kt+1/Kt > t+1/Nt (= n)

or: Kt+1/Kt - t+1/Nt > 0

s∙Yt/Kt - - n >0

multiply last condition through by Kt/Nt:

s∙Yt/Nt - ( n)∙Kt/Nt >0

or (using definitions of y and k):

s∙yt - ( n)∙kt >0

- So: K/N is growing if: s∙yt - ( n)∙kt >0

K/N is shrinking if: s∙yt - ( n)∙kt <0

K/N is constant if: s∙yt - ( n)∙kt =0

- Look more closely at these conditions:

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syt = amount of savings and investment (expressed per N)i.e. amount of new capital being created (per N).

( n)∙kt = amount of new capital needed to hold K/N constant (expressed per N)

i.e. must replace depreciation: ∙kt

must equip new workers with same level of K as old workers: n∙kt

So:

k shrinks if new capital is insufficient to replace depreciation and equip new workers.

k grows if new capital is greater than what is needed to replace depreciation and equip new workers.

k constant if new capital and needed capital are the same.

- Steady state equilibrium:

K/N (k) is constant if: s∙yt - ( n)∙kt =0

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Steady State Equilibrium in the Solow Model:

- In diagrams: remember y=F(k,1)

- If: k<kss then s∙yt - ( n)∙kt >0 ↑k, ↑y

k>kss then s∙yt - ( n)∙kt <0 ↓k, ↓y

k=kss then s∙yt - ( n)∙kt =0 k, y unchanging.

- Long-run outcome: kss, yss

- k and y are constant in this “steady state” equilibrium.(consumption per worker: c=(1-s)y is also constant)

- Notice: Y, K, C and N are all growing at rate “n”.

- So economy is growing in this model. ( but the standard of living measures y, c are not!)

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- Another way to solve the model? Use the underlying difference equation in k

- Take: Kt+1 = It - Kt, with Kt+1=Kt+1-Kt

Kt+1= It + (1-Kt,

- Divide by Nt:K t+1

N t=

It

N t+(1−δ )

K t

N t

- Use Nt+1=Nt(1+n) to write:

K t+1

N t+1=( I t

N t+(1−δ )

K t

N t)/(1+n)

- Write in per N terms:k t+1=

it+(1−δ )k t

1+n

- Substitute: it=syt (with yt=F(kt,1) )

k t+1=sF (k¿¿ t ,1)+(1−δ )k t

1+n¿

- This is a first order non-linear difference equation in k.

- Steady state equilibrium? kt=kt+1=k* (k* is same as kss above)

k ¿=sF (k¿ , 1)+(1−δ )k ¿

1+n

rearrange: s F(k*,1) - (+nk* = 0

i.e. same equilibrium condition as above.

- Is it stable? depends on slope of the difference equation at k*.

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d k t+1

d k t=

s FK+(1−δ )1+n

<1 so it is stable.

(Inada conditions tell you slope is infinite at k=0, as k→∞ slope approaches (1-)/(1+n) <1, and you have FKK<0 so the

difference equation will intersect the 45-degree line with slope<1)

- Diagram: start at klow and the economy moves to k*(same is true if start with k>k*)

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Comparative statics in the Solow model: How does the steady state change?

- Exogenous parameters: s, n, and any parameters of F(k,1).

(1) Rise (fall) in s: k and y both rise (fall).

(2) Rise (fall) in n or : k and y both fall (rise).

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(3) Shift up (down) in F: k and y both rise (fall).

picture: rise in F is like a rise in s except y=F(k,1) shifts up as well)

Why might one country have a high level of y and another a low y?

- Steady-state differences:

- If both are in a steady state higher y could be the result of high s, low n and/or high F (high productivity).

- Disequilibrium differences:

- Another possibility? same s, n and F but maybe low y country is further from the steady state.

(is this case convergence to the same steady state occurs over time)

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From: Jones and Vollrath Introduction to Economic Growth (y vs. s, y vs. n)

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Open economy and Solow’s Predictions:

- Solow model is a closed economy model.

- Differences in ‘y’ are rooted in differences in s, n and F(K,N).

- If economies are open will differences in s, n and F be smaller?

- will savings flow from high to low k countries to take advantage of higher FK?

- will workers migrate from poor (high n) countries to rich countries (low n)?

- will diffusion of technology reduce inter-country differences in F?

- Is convergence between rich and poor countries more likely if economies are open?

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Consumption per Worker and the Golden Rule Outcome

- Consumption per worker (c=C/N): ct= yt-syt with yt=F(kt,1)

- Remember that y is not the same as c.

- What is the socially best level for “s”?

- Golden rule: choose s to maximize steady state c.

c = F(kt,1) – s y for a steady state: sy=(+n)k = F(kt,1)-(+n)kt

so in picture maximize distance between F(k,1) and (+n)k.

f.o.c. (max c with respect to k):

FK-(+n)=0 choose s that gives a steady state at this point (see diagram).

- Is the Golden rule savings rate what policy makers should aim for? (see later discussion in Ramsey model)

- Policy tools to influence ‘s’?

- Tax system design: can effect incentives to save.

- Financial system: encourage development, regulation.

- Economic planning: can ‘the plan’ determine the savings rate, e.g. via government savings.

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Growth in Solow without Technological Change

- Steady state: N, K, Y and C are all growing at rate ‘n’. k, y and c don’t grow.

- Away from the steady state?

- Growth is driven by changes in k (=K/N).

- Distance from the steady state determines the growth rate.

- Higher is ‘k’ the lower is the growth rate in ‘k’.

- More formally: what determines the growth rate in k? (see McCandless)

From p. 10:

k t+1=sF (k¿¿ t ,1)+(1−δ)k t

1+n¿

Divide by kt to get 1+growth rate in k:

k t+1

k t=

sF (k¿¿t ,1)+(1−δ )k t

(1+n)k t¿

So the growth rate in k is higher when s is high, n or is low.

How is the growth rate in k affected by the level of k:

d ( k t+1

k t)

d k t=s (1+n )

[F ¿¿ K (k¿¿ t , 1) ∙ k t−F (k t , 1)](1+n)2 k t

2 ¿¿<0

(this uses: Fk k – F(k,1) <0 )

- so k growth rate (and so y and c growth) is higher lower is k (other things equal)

i.e. when k is low extra k increases y, savings and investment more.

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Solow Model with Exogneous Technological Change:

- With exogenous technological progress.

- Production function shifts up over time in some specified way.

- Exogenous? rate of technological progress is independent of what is happening in the economy.

- A desirable addition since technological change happens!

- Also desirable as a way of making steady state y and c grow over time. (living standards seem to grow with time!)

- Introducing technological change: a common approach

- define ek and en as “efficiency” units of K and N respectively.

- efficiency units: adjust for productivity of K and N.

- so the effective amount of each input is: ekK and enN

- inputs grow because there are more units of input or because a given unit of input is more “efficient”.

- technological change is assumed to drive changes in ek, en. in this presentation.

(note: en could reflect skill levels of workers)

- production function is :

Y = F(ekK,enN)

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- Common to assume technology works through en (let ek=1): “labour-augmenting” technological progress.

- advantage? has a steady state much like that of the basic model (see the note below)

- Another common assumption: “neutral” technological progress: as if ek=en and both rise by same proportion as technology improves.

So let: ek=en=e

Then: Y = F(ekK,enN)

Y = e F(K,N) due to constant returns

How Fast does the Economy Grow when Technology is Improving?

- Assume neutral technological progress.

- Assume that “e” rises at an exogenous rate: and e=1 in period t=0

- So: Yt = (1+)t F(Kt,Nt) so et = (1+)t

- In per worker form:

yt = (1+)t F(kt,1)

- Growth rate in y reflects both the growth rate in kt (as above) and the rate of technological progress ().

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Growth with a Cobb-Douglas Production Function:

- Assume a Cobb-Douglas production function (F):

- So: Yt = (1+)t KtNt

1- where 0<<1

- In per N form: yt = (1+)t kt

- Growth rate in output per worker (y), call it gy ? (note c grows at the same rate) Output per N: yt = (1+)t kt

then: yt+1 /yt = (1+) (kt+1kt)

(1+gy) = (1+) (1+(kt))(k) is the growth rate of k)

take logs: ln(1+gy) = ln(1+) + ln(1+(kt))

or, using the approximation, ln(1+x) x for small x:

gy = + (kt)

- Growth rate in y is higher the higher is:

- the rate of technological progress

- the growth rate of k(kt) : see p. 19)

- and the growth rate of k (and so y as well) is higher:- higher is ‘s’- lower is ‘n’ or ‘’- lower is k (see above)

- Note implications for differences in country growth rates: rooted in these parameters.

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Steady State Outcome with Technological Progress:

- The growth rate conditions describe adjustment toward a possible equilibrium (focus on disequilibrium adjustment)

- What about a steady state?

- Does it exist?

- How might it differ from the model with no technological progress?

- Take an easy case: labour-augmenting technological progress.

( it is possible to find a similar steady state with neutral technological progress with certain productions functions, e.g. Cobb-Douglas)

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A note on labour-augmenting technological progress in the Solow model:

- Start with: Yt = F(ekKt,enNt)

- Assume ek=1: Yt = F(Kt,enNt) so technological progress is labour augmenting.

- Call: enNt=Nt’ (this is efficiency-adjusted or effective labour)

- Much as in the basic Solow model divide through by Nt’ to get (possible due to our assumption of constant returns to scale):

Yt /Nt’= F(Kt,/Nt’,1)

or with y’=Y/N’ and k’=K/N’

yt’ = F(kt’,1)

- Capital accumulation is the same as in the basic Solow model, so:Kt+1/Kt = s∙Yt/Kt -

- Effective labour: N’=enN

- if en grows at the rate (so en in time t is (1+)t ) and N grows at rate n then N’ approximately grows at the rate: n+

’t+1/Nt’ = n+

- This model has a steady state when:

Kt+1/Kt - ’t+1/Nt’ =0 (k’ and y’ will be unchanging)So:

s∙Yt/Kt - - n - = 0

or s∙yt’ - ( + n + )kt’= 0 (multiply above by Kt/Nt’)

- In this steady state y’, k’ are not changing (Y, K and N’ all grow at the rate n+).

- But y=Y/N and k=K/N are growing at the rate .

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- Do economies settle into a steady-state of this last sort?

- Living standards grow at the rate of technological progress.

- Barro and Sala-i-Martin (2004) : argue that long-term US growth is well approximated by something like this.

- Can the model explain differences across countries or across time?

- For reasonable production function estimates: differences in ‘k’ do not appear to be the critical factor.

- Is it technology (or its growth rate)? or something the model misses?

(see Romer Ch. 1 discussion; Solow’s growth accounting exercise; example?)

- ‘Technology’ is often used as a catch-all term for form of F().

- Total factor productivity (TFP).

- Reflects the state of technological knowledge in the economy.

- Also reflects institutions, state of economic organization.

- Modern macroeconomics (RBC-style models): Solow growth accounting exercises suggest that TFP changes account for most growth. Are productivity shocks a key driver of the economy?

- Aziariadis ‘Riddles and Models’ Journal of Economic Literature (2018):

‘we have learned that the Solow residual is more often than not the most conspicuous proxy force behind output movements… Plausible sources of productivity growth at the moment appear to be credit and politics in the short run, technology and institutions in the long run’

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Appendix: the AK Model

- Drops the diminishing returns to capital assumption from Solow.

- Production: Yt =AKt A = a constant (reflects state of technology)

yt=Akt if expressed per N.

- no diminishing returns (marginal product: FK = A is constant)

- Diminishing returns justifications:

- an economy has fixed input(s) e.g. land, natural resources.

Add more capital then each unit of capital has less fixed input to work with: its productivity falls.

- Ricardo-style diminishing returns: extra capital goes to the highest productivity uses first, then next highest etc.

- How might these forces be offset?

- define capital more broadly:

- physical K (as before) plus skills, knowledge.

- all three increasing perhaps value of K in new uses can stay the same or even rise.

- new K and new technology may go hand-in-hand (knowledge as K above fits this too)

- Consequences?

- as if in Solow model a rise in K means a simultaneous shift up in the production function.

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- Capital accumulation:

- K is determined as in the Solow model.

- The growth rate in k is:

For K: Kt+1/Kt =sYt/Kt- = syt/kt -

- Growth rate for k=K/N:

= sy/k – -n = sA - (+n) (using y=Ak )

- k-growth rate is constant (s, A, and n are all constants).

- this is also the growth rate of y and c

i.e. since yt=Akt and ct=(1-s)yt and s is constant.

- k, y and c all grow if: sA - (+n) >0

- k, y and c all shrink if: sA - (+n) <0

(no growth if =0)

- Say the economy is growing (sA - (+n) >0 ):

- growth rate rises with higher s and/or higher A,

- growth rate lower with higher depreciation () or higher labour force growth (n).

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- With no diminishing returns the growth rate does not depend on the value of k.

- if two countries have the same A, s, n and they grow at the same rate.

- the one with lower initial k and y will not catch up.

- so convergence type stories gone.

- AK model has long-run growth without technological progress.

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Appendix: Linear approximations and dynamics in the Solow Model

- a common technique is to use a linear approximation to assess stability near the equilibrium.

- Here the production function is the source of non-linearity in:

k t+1=sF (k¿¿ t ,1)+(1−δ)k t

1+n¿

- Replace the production function with a first-order Taylor series approximation evaluated at k*:

F(kt,1) ≅ F(k*,1) + FK(k*,1) (kt-k*)

this gives:

k t+1≅ s ( F (k¿ ,1 )+FK ( k¿ , 1 ) ( k t−k¿ ))+(1−δ ) k t

1+n

or:

k t+1≅ [ sF (k ¿ ,1 )−sF K (k¿ , 1 ) k¿

1+n ]+[ sFK (k ¿ ,1 )+ (1−δ )1+n ]k t

This linear difference equation is stable if:

[ sFK ( k¿ , 1 )+ (1−δ )1+n ]<1

or if: sFK(k*,1) < n+(as in the diagram above)

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