Economic Growth: Solow Model
-References: There are many good presentations of the Solow Model. I have
posted three on the website.
G. McCandless The ABCs of RBCs Ch. 1;
C. Jones and D. Vollrath Introduction to Economic Growth Ch. 2;
M. Wickens Macroeconomic Theory Chapter 3
- 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 Real Business Cycle models)
- here we introduce a basic growth model.
- 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 organization progress (“t” in the production
function might capture this).
- Growth accounting exercises take this as a starting point: choose a specific functional form for F(), obtain measures of Y, K, N; infer effect of ‘t’.
Solow Model
- Sometimes called the Solow-Swan model.
- Most widely used growth model in economics.
- 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.
- Aggregate Production Function:
Yt = F(Kt, Nt)
- Economy-wide production function (earlier notes: firm level production)
- Form of F(): reflects technology, organization, incentives: all determine how productive 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/∂L≡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
- Consequences of technical assumptions:
Inada conditions will push outcomes away from extremes
(0 or infinite k): convenient!
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: ‘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= AKtaNt1-a
per worker form (divide by N):
yt= Akta FK=aAkta-1 >0
FKK=(a-1)aAkta-2<0
(Inada conditions hold too)
- Labour force growth: Nt=N0(1+n)t (N0 - initial value of N in
period 0)
i.e. labour grows at a constant per period rate of “n”.
if you prefer: Nt+1 = Nt (1+n)
(model assumes full employment: labour force and number
employed are the same)
- National income constraint (economy-wide budget) :
Yt = Ct + It C= consumption, I=investment
- Investment and saving:
- Closed economy: investment is financed with domestic savings (S):
It=St
- Savings function: St = s∙Yt ( 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)
- Capital growth: (DKt+1 = Kt+1-Kt)
- Capital accumulation: DKt+1 = It - dKt,
d = depreciation rate (share of K that wears out each period)
use St = s∙Yt and St=It :
DKt+1 = s∙Yt - dKt
divide by K to express it as the growth rate in K:
DKt+1/Kt = s∙Yt/Kt - d
- Is k=K/N growing, shrinking or staying the same?
Growing if: DKt+1/Kt > DNt+1/Nt (= n)
or: DKt+1/Kt - DNt+1/Nt > 0
s∙Yt/Kt - d - n >0
multiply last condition through by Kt/Nt:
s∙Yt/Nt - (d + n)∙Kt/Nt >0
or (using definitions of y and k):
s∙yt - (d + n)∙kt >0
- So: K/N is growing if: s∙yt - (d + n)∙kt >0
K/N is shrinking if: s∙yt - (d + n)∙kt <0
K/N is constant if: s∙yt - (d + n)∙kt =0
- Look more closely at these conditions:
syt = amount of savings and investment (expressed per N)
i.e. amount of new capital being created (per N).
(d + n)∙kt = amount of new capital needed to hold K/N constant (expressed
per N)
i.e. must replace depreciation: d∙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.
- In diagrams: remember y=F(k,1)
- If:
k<kss then s∙yt - (d + n)∙kt >0 ↑k, ↑y
k>kss then s∙yt - (d + n)∙kt <0 ↓k, ↓y
k=kss then s∙yt - (d + 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!)
- Another way to solve the model? Use the underlying difference equation in k
- Take: DKt+1 = It - dKt, with DKt+1=Kt+1-Kt
Kt+1= It + (1-d)Kt,
- Divide by Nt:
Kt+1Nt=ItNt+(1-δ)KtNt
- Use Nt+1=Nt(1+n) to write:
Kt+1Nt+1=ItNt+1-δKtNt/(1+n)
- Write in per N terms:
kt+1= it+(1-δ)kt1+n
- Substitute: it=syt (with yt=F(kt,1) )
kt+1= sF(kt,1)+(1-δ)kt1+n
- This is a first order non-linear difference equation in k.
- Steady state equilibrium? kt=kt+1=k*
k*= sF(k*,1)+(1-δ)k*1+n
rearrange: s F(k*,1) - (d+n)k* = 0
i.e. same equilibrium condition as above.
- Is it stable? depends on slope of the difference equation at k*.
dkt+1dkt=sFK+(1-δ)1+n<1 so it is stable.
(Inada conditions tell you slope is infinite at k=0, as k→∞ slope
approaches (1-d)/(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*)
- Another way of looking at dynamics:
- 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:
kt+1= sF(kt,1)+(1-δ)kt1+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:
kt+1≅ sFk*,1+FKk*,1kt-k*+1-δkt1+n
or:
kt+1≅sFk*,1-sFKk*,1k*1+n+sFKk*,1+1-δ1+n kt
This linear difference equation is stable if:
sFKk*,1+1-δ1+n<1
or if: sFK(k*,1) < n+d (as in the diagram above)
- Comparative statics in the Solow model? How does the steady state change?
- Exogenous parameters: s, n, d and any parameters of F(k,1).
- Rise (fall) in s: k and y both rise (fall).
- Rise (fall) in n or d: k and y both fall (rise).
(picture: (n+d)k pivots up for a rise or down for a fall)
- 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 will occur over time)
From: Jones and Vollrath Introduction to Economic Growth (y vs. s, y vs. n)
- 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=(d+n)k
= F(kt,1)-(d+n)kt
so in picture maximize distance between F(k,1) and (d+n)k.
f.o.c. (max c with respect to k):
FK-(d+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)
- Open economy and Solow:
- 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 reduces inter-country differences in F?
- Is convergence between rich and poor countries more likely if economies are open?
Solow Model with Exogneous Technological Change:
- Add exogenous technological change.
- Production function shifts over time in some specified way.
- Exogenous: rate of technological progress is independent of what is
happening in the economy.
- Desirable addition since technological change happens!
- also in basic Solow model steady state y does not rise over time.
(living standards seem to grow with time)
- Possible ways of 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)
- Most common approach assumes 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)
- Wickens assumes “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
- Wickens and McCandless have “e” rise at an exogenous rate: m
- So: Yt = (1+m)t F(Kt,Nt) so et = (1+m)t
- If they also assume a Cobb-Douglas production function for F:
- So: Yt = (1+m)t KtaNt1-a
- In per N form: yt = (1+m)t kta
- The Cobb-Douglas does have a steady-state with this type of
technological change
- can treat neutral progress like the labour-augmenting case.
(not all constant returns production functions do).
- Growth rate in output per worker (y), call it gy ?
Output per N: yt = (1+m)t kta
then: yt+1 /yt = (1+m) (kt+1/kt)a
(1+gy) = (1+m) (1+g(kt))a (g(k) is the growth rate of k)
take logs: ln(1+gy) = ln(1+m) + a ln(1+g(kt))
or using the approximation ln(1+x) @ x for small x:
gy = m + a g(kt)
- What determines the growth rate in k (i.e. g(k)) ?
- Savings, capital accumulation and labour force growth are the same as
earlier.
- So as before: DKt+1/Kt = s∙Yt/Kt - d = s∙yt/kt - d
and: DNt+1/Nt = n
so the growth rate in K/N=k is approximately:
g(kt)= Dkt+1/kt = s∙ yt/kt - d - n