Augmented Solow Model Essay, 1300 Words

Lecture 3: The Augmented Solow Model

March 5th, 2018

Recap

I In the last lecture, we studied the basic Solow model without population growth and without technological progress (chapter 5 Garin et al 2018)

I We studied the key equation: the transitional dynamics [TD] equation

kt+1 = sAf (kt) + (1 − δ)kt

I The properties of f (k) (diminishing returns + INADA) ensure this equation converges to a long-run equilibrium (k∗)

I We discussed that the model predicts conditional convergence

I Today, we will finish working with the model and extend it to have technological progress and population growth

Graphical analysis

Similar if the economy starts at kt > k∗

Transition dynamics

What about (short-term) economic growth? (Assume f(k) = kα)

Assume for now that At = A for all t (no technological progress)

kt+1 = sAk α t + (1 − δ)kt,

divide it by kt 1 + gk = sAk

α−1 t + (1 − δ)

gk = sAk α−1 t −δ

We know that yt = Yt Nt

= Akαt . Take logs and subtract the lag:

ln yt − ln yt−1 = α(ln kt − ln kt−1)

use the fact that ln ( yt

yt−1

) = ln(1 + gy) ≈ gy, for gy ≈ 0, to get

gy = αgk = α(sAk α−1 t −δ),

which depends on the level of technology, the capital share (α), the investment rate (s), the current level of capital (kt), and the depreciation rate (δ)

Is there long run growth when At = A for all t? No!

When the economy reaches its steady state k∗ = kt = kt+1, the rate at which capital is created (total investment per-capita) equals the rate at which capital is “’destructed’

kt+1 = sAk α t + (1 −δ)kt [TD]

∆kt+1 = sAk α t −δkt [TD

′]

At steady state, ∆kt+1 = 0, implying

sAk∗α = δk∗,

implying that in the long run

gk = 0 ; gy = 0

Capital accumulation only leads to per-capita economic growth along the transition dynamics

Long-run GDP Although this simple model is not consistent with the data (gy ≈ 2%), it is useful to understand the transition dynamic of alike countries

Use the [TD] equation to find an expression for k∗ and y∗

kt+1 = sAk α t + (1 − δ)kt,

k∗ = sAk∗α + (1 − δ)k∗,

δk∗ = sAk∗α,

k∗ = ( sA δ

) 1 1−α ,

y∗ = ( sA δ

) α 1−α ,

A key prediction from the model is that economies with similar characteristics (s, A, δ, α) converge to similar levels of income y∗ = k∗α. Is this true in the data?

Conditional Convergence (more empirics in lecture 5)

Figure: Source: Garin et al (2017)

Experiments: increased saving rate

Figure: Source: Garin et al (2017)

Experiments: increased saving rate

Figure: Source: Garin et al (2018)

Experiments: increased saving rate

Note that c∗ = (1 − s)y∗. So, it is unclear whether the new c∗ is larger or smaller

Figure: Source: Garin et al (2018

Experiments: increased saving rate

Figure: Source: Garin et al (2018)

Experiments: increased saving rate

Figure: Source: Garin et al (2018)

Initial burst of higher growth but eventually come back to 0 growth.

Experiments: increased productivity (A)

Figure: Source: Garin et al (2018)

Experiments: increased productivity (A)

What is the main difference wrt a change in s? a direct level effect on y

Figure: Source: Garin et al (2018)

Experiments: increased productivity (A)

Figure: Source: Garin et al (2018)

Experiments: increased productivity (A)

Figure: Source: Garin et al (2018)

Discussion

How do we know we achieved our long-run equilibrium? Is the observed positive growth (gy > 0) due to transitional growth?

Or, is it due to frequent shocks that move the economy from one long-run equilibrium to another?

Use the model to shed lights on this. Calibrate our [TD] equation. In the data, α ≈ 0.3, δ ≈ 0.1 (annual), s ∈ (0.2, 0.4), A? k0? Use

kt+1 = sAk α t + (1 −δ)kt,

and

k∗ = (sA δ

) 1 1−α ,

How should the growth path look like when the economy is in the transition?

Confront the model with the data

How can the U.S and Australia achieve higher levels of GDP per-capita?

More investment (s)? Higher productivity (A)?

Confront the model with the data

How can the U.S and Australia grow persistently more than Chile and Argentina?

Did they do it by persistently increasing the saving rate (s)? https://fred.stlouisfed.org/series/RTFPNAAUA632NRUG

https://fred.stlouisfed.org/series/KIPPPGAUA156NUPN

Confront the model with the data How do we account for Singapore’s and Venezuela’s case?

https://fred.stlouisfed.org/series/RTFPNASGA632NRUG

https://fred.stlouisfed.org/series/RTFPNAVEA632NRUG

Golden Rule When is long-run consumption maximized?

c∗ = (1 − s)A∗f (k∗)

Figure: Source: Garin et al (2017)

Golden Rule When is long-run consumption maximized?

Figure: Source: Garin et al (2017)

Golden Rule

When is “good” to save more? (think of c as a measure of welfare)

Figure: Source: Garin et al (2017)

Policy Implications

I Is the model’s equilibrium Pareto efficient? I Does the model provide insights on how to improve

productivity? I In the next lectures, we will extend the model to

incorporate mechanisms through which the long-run growth rate is endogenously determined.

Take aways

I A simple version of the Solow model explains the observed conditional convergence in GDP per-capita levels

I Physical capital accumulation is key in the generation of economic growth along the transition dynamics

I Changes in the saving rate, the capital share, or the depreciation of capital can only have a transitory effect on growth

I Long-term growth requires technological progress

The Augmented Solow model

Learning resource: chapter 6 Garin et al (2018)

I So far, the model misses the observed trend in GDP per-capita observed in the data (gy ≈ 1.7% in Australia)

I Thus, we will augment the Solow growth model to allow for: population growth and technological progress

I We will, for now, stick to the assumption of exogenous growth

I In lecture 4, we will study endogenous growth models

Intro to the augmented Solow model

Assume there is labor-augmenting productivity Zt (At is the factor neutral productivity), this is

Yt = AtF(Kt, ZtNt),

where ZtNt is called efficiency units of labor. Assume that

Zt = (1 + z)Zt+1, z > 0

Nt = (1 + n)Nt+1, n > 0

We assume that At controls for the level of technology and that Zt is the one growing overtime.

In reality, A and Z grow and are important drivers of economic growth. See Doraszelski and Jaumandreu (2018) “Measuring the Bias of Technological Change”, Journal of Political Economy, (Forthcoming).

Equations that characterize the augmented model

Production function : Yt = AtF(Kt, ZtNt) Goods’ market clearing : Yt = Ct + It Evolution of capital : Kt+1 = It + (1 −δ)Kt Investment “decision” : It = sYt

Equilibrium wage rate : wt = AtZt ∂F(Kt,ZtNt) ∂ZtNt

Equilibrium rental rate : rt = At ∂F(Kt,Nt) ∂Kt

Population growth : Nt = (1 + n)t

Labor productivity growth : Zt = (1 + z)t

The key difference equation of the model

Kt+1 = sAtF(Kt, ZtNt) + (1 −δ)Kt,

divide it by ZtNt (efficiency units of labor) to get

Kt+1 ZtNt

= Kt+1

Zt+1Nt+1 Zt+1Nt+1

Nt = sAtF(

Kt ZtNt

, 1) + (1 − δ) Kt

ZtNt

k̂t+1(1 + z)(1 + n) = sAtf (k̂t) + (1 − δ)k̂t,

k̂t+1 = 1

(1 + z)(1 + n) [ sAtf (k̂t) + (1 −δ)k̂t

] [TD2]

Today’s level of capital per-efficiency units (predetermined) determines tomorrow’s capital stock per-efficiency units k̂t+1. Note that we still care about per-capita variables. However, the system converges to a steady state in efficiency units variables.

Rewrite the whole system of equations in efficiency units

Production function : ŷt = Atf (k̂t) Goods’ market clearing : ŷt = ĉt + ît Evolution of capital : k̂t+1 = ît + (1 −δ)k̂t Investment “decision” : ît = sŷt

Equilibrium wage rate : wt = Zt[Atf (k̂t) − Atf ′(k̂t)k̂t]

Equilibrium rental rate : rt = At ∂f(k̂t) ∂k̂t

Graphical analysis

Figure: Source: Garin et al (2018)

Steady state and long-run growth of K and Y We know that in the long-run

k̂t+1 = k̂t = k̂ ∗

Kt+1 Zt+1Nt+1

= Kt

ZtNt

Kt+1 Kt

= Zt+1Nt+1

ZtNt

1 + gK = (1 + z)(1 + n)

gK ≈ z + n

In the long-run, the level of capital stock grows at the product (approx. the sum) of the growth rates of Zt and Nt

Steady state and long-run growth of k and y

What about per-capita variables?

Kt+1 Zt+1Nt+1

= Kt

ZtNt

Kt+1 Nt+1

= Zt+1Kt+1

ZtNt

kt+1 kt

= 1 + gk = 1 + z

gk ≈ z

Using the same approach, in the long-run, gy ≈ z. Thus, technological progress accounts for the trend in GDP per-capita in the data.

Steady state and long-run growth of w and r

What about the rental rate and the wage rate?

r = A∗f ′(k̂∗)

wt = Zt[A ∗f (k̂∗ − A∗f ′(k̂∗k̂∗].

We know that f (k̂) is an increasing function of k̂, while f ′(k̂) is a decreasing function of k̂. In the long-run equilibrium, k̂ stays constant.

Thus, r is constant and equal to r∗ in the long run, while w grows at the same rate at which Z grows (z).

Consistent with Kaldor facts?

1. Labor productivity has grown at a sustained rate

2. Capital per worker has also grown at a sustained rate

3. The real interest rate, or return on capital, has been stable

4. The ratio of capital to output has also been stable

5. Capital and labor have captured stable shares of national income

Consistent with Kaldor facts?

Source: Garin et al (2018)

Consistent with Kaldor facts?

Source: Garin et al (2018)

An increase in the saving rate

Source: Garin et al (2018)

An increase in the saving rate

Effect on per-efficiency units variables:

Source: Garin et al (2018)

An increase in the saving rate

Source: Garin et al (2018)

An increase in the saving rate

Source: Garin et al (2018)

An increase in the saving rate

Effect on per-capita variables:

Source: Garin et al (2018)

An increase in the saving rate

Source: Garin et al (2018)

An increase in the saving rate

Source: Garin et al (2018)

An increase in the saving rate

Source: Garin et al (2018)

An increase in the level of productivity A

The effect on per-efficiency units variables

Source: Garin et al (2018)

An increase in the level of productivity A

The effect on per-efficiency units variables

Source: Garin et al (2018)

An increase in the level of productivity A

Source: Garin et al (2018)

An increase in the level of productivity A

The effect on per-capita variables

Source: Garin et al (2018)

An increase in the level of productivity A

The effect on per-capita variables

Source: Garin et al (2018)

An increase in the level of productivity A

Source: Garin et al (2018)

Take aways

1. The Augmented Solow model (with population growth and labor-augmenting technological progress) converges to a steady state in per-efficiency units

2. The model predicts long-run steady growth in per-capita variables and the wage rate (growth rate z)

3. These predictions are partially consistent with the Kaldor facts

4. Blog article for discussion (at the end of Lecture 4) https:

//www.bloomberg.com/view/articles/2017-11-16/

the-robot-revolution-is-coming-just-be-patient

5. Additional readings (not for discussion): 1) https://pubs. aeaweb.org/doi/pdfplus/10.1257/jep.31.2.145 and 2) https://www.aeaweb.org/articles?id=10.1257/mac.2.1.224