M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory Toulouse School of Economics Notes written by Ernesto Pasten ([email protected] ) Slightly re-edited by Frank Portier ([email protected] )

Macroeconomics I Chapter 2. New Growth Theory November 10, 2010

1

Introduction

The Solow growth model tells us that capital accumulation cannot account for a large part of long run growth and cross-country income differences. Instead, it points out to technology. This chapter focuses on: • Understanding the accumulation of knowledge as the main determinant of the labor productivity factor At . The result will be that knowledge is central for growth, but not for cross-country differences. • Regarding cross-country differences, we will then study the effect of human capital accumulation.

2

Endogenous production of knowledge

We now build on the Solow model in continuous time to introduce a production function for knowledge, which will use capital, labor and knowledge as inputs.

2.1

Model set-up

Assume that fixed proportions of capital (1 − aK ) and labor (1 − aL ) are used for producing a output according to a Cobb-Douglas production function: Y (t) = [(1 − aK ) K (t)]a [(1 − aL ) A (t) L (t)]1−a where 0 < α < 1. This production function preserves our assumption of CRS. 1

M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory

In addition, let interpret At as the stock of productive knowledge in the labor force, which may be increased by new research. In particular, knowledge will increase according to dA (t) = B [aK K (t)]β [aL L (t)]γ A (t)θ dt where B > 0, β ≥ 0 and γ ≥ 0. The parameter B is an exogenous ”shift” that could capture unmodelled determinants of the production of knowledge. For simplicity, these other determinants are assumed constant. Parameters β and γ are not forced to sum 1, so there could be either decreasing (β + γ < 1), constant (β + γ = 1) or increasing returns to scale (β + γ > 1). Certainly we cannot think in producing technology as producing potatoes, so the idea that doubling inputs will double production may or may not work. The parameter θ is also set free, so we allow for existing technology to either make easier (θ > 0) or more difficult (θ < 0) the production of new technologies. We will keep the assumption of the Solow Growth model that labor grows at a exogenous rate n : dL (t) = nL (t) dt and that savings is a fixed proportion of total income and that savings must be equal to investment, so I (t) = sY (t) .

2.2

The model without capital

Without capital, the production function for output becomes Y (t) = (1 − aL ) A (t) L (t) , which keeps the CRS assumption. In addition, the accumulation of knowledge becomes dA (t) = B [aL L (t)]γ A (t)θ . dt The production function implies that per-capital income is proportional to A (t), so it might grow at a rate gA (t) =

dA (t) 1 = B [aL L (t)]γ A (t)θ−1 . dt A (t)

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M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory

The first observation we could make, in comparison to the Solow model, is that this growth rate is not constant any more: dgA (t) 1 d log gA (t) = = γn + (θ − 1) gA (t) . dt gA (t) dt Thus, dgA (t) = γngA (t) + (θ − 1) [gA (t)]2 . dt The equation for gA (t) determines initial growth of per-capita income as a function of the initial level of labor and technology, and the equation for

dgA (t) dt

determines its subsequent evolution. As

we can see in the last expression above, the parameter θ determines whether per-capital income growth is constant or increasing along a steady growth path (by steady growth path, we mean a path along which the growth rate is constant).

Case 1: θ < 1. This assumption means that technology has decreasing returns in producting new technology, which implies that a constant growth rate is obtained along a steady growth path: gA∗ =

γ n. 1−θ

In other words, regardless of initial conditions, per-capita income of all countries grow at the same rate gA∗ along the steady growth path. This growth rate depends on the exogenous growth of the only input for the production of output and technology, labor L (t). This case also implies that growth in technology will stop if population stop growing. Note that the share of labor used in technological research has no impact on g ∗ , but it has an impact on the initial growth rate. If the initial growth rate gA (0) is below gA∗ , then gA (t) will be higher than gA∗ for some time, until gA (t) = gA∗ . The converse dynamics occurs if gA (0) > gA∗ .

Case 2: θ = 1. This assumption means that technology has constant returns in producing new technology, so gA (t) = B [aL L (t)]γ , dgA (t) = γngA (t) . dt In this case there is no level of gA∗ at which growth constant for positive n and γ. Instead, there is an explosive growth. Note that in this case the growth of technology is also linked to population 3

M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory

growth. In the case in which population is constant, whatever growth gA (0) comes up from the initial conditions of the economy, that growth will prevail along the steady growth path.

Case 3: θ > 1. For completeness, we briefly study this case, which, as case 2, also implies an explosive growth rate, even for n = 0. If n < 0, then steady state growth is zero.

2.3

The model with capital

For simplicity, we assume that depreciation δ = 0. We also take an approach slightly different from the one we took in Chapter 1 for the Solow model. The reason for this departure is that the function for knowledge accumulation does not necessary have CRS, so we cannot get a simple intensity version of it. The production function now is Y (t) = [(1 − aK ) K (t)]a [(1 − aL ) A (t) L (t)]1−a where 0 < α < 1. Given the assumption that I (t) = sY (t) and that δ = 0, capital accumulation follows the rule dK (t) = s [(1 − aK ) K (t)]a [(1 − aL ) A (t) L (t)]1−a dt gK = cK K (t)α−1 [A (t) L (t)]1−α where cK = s (1 − aK )α (1 − aL )1−α . Applying logs and taking derivatives w.r.t. time, we get the motion rule of gK : dgK 1 = (1 − α) [gA (t) + n − gK (t)]. dt gK where gK (t) changes with time according to the growth gA (t) of the technological factor A (t), which is also time-varying, the population growth n and the level of growth of capital itself gK (t) . In the model with capital, the accumulation of knowledge is given by gA (t) =

dA (t) 1 = B [aK K (t)]β [aL L (t)]γ A (t)θ−1 , dt A (t)

which has a motion rule dgA 1 = βgK (t) + γn + (θ − 1) gA (t) . dt gA

4

M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory

Whether the growth of gA (t) increases, decreases or remains constant depend on the RHS of this last expression above, which in turn depends on the growth of capital gK (t), population growth n, and the growth of the knowledge stock gA (t). Parameters β, γ and θ also play a role. To ∗ find the steady growth path, we need to find a combination of growth rates {gK , gA∗ } such that dgA 1 dt gA

=

dgK 1 dt gK

= 0. In other words, we need to solve the system: dgK ∗ = 0 : gA∗ + n − gK = 0, dt dgA ∗ + γn + (θ − 1) gA∗ = 0. = 0 : βgK dt

which implies γ+β n, 1 − (θ + β) = gA∗ + n.

gA∗ = ∗ gK

To study the characteristics of this economy, in terms of transitional dynamics and steady growth path, we need to study different cases. For simplicity, we just focus on the most interesting cases: when β + θ < 1 and when β + θ = 1 and n = 0 [suggestion: try and discuss other cases]. 2.3.1

Case 1: β + θ < 1

This is a case in which there is decreasing returns to scale of joint changes in capital and knowledge. The following figure helps us to represent this economy: The arrows in the figure represent the transitional dynamics in this economy. For instance, assume that the pair {˜ gK , g˜A } is such that dgA 1 = β˜ gK + γn + (θ − 1) g˜A = 0 dt gA In words, if the growth rates are {˜ gK , g˜A }, then

dgA dt

= 0. Note that there is no reason why

dgK dt

=0

∗ unless {˜ gK , g˜A } = {gK , gA∗ }. Also note the pair {˜ gK , g˜A } is not the unique pair that satisfies dgA dt

= 0. Actually, any point along the curve

dgA dt

= 0 in the figure above satisfy this condition.

Suppose now that we choose a point {˜ gK , gA } , with gA < g˜A , in the figure above. Then we ask which sign

dgA dt

should have. Since we are studying the case in which β + θ < 1, θ < 1. Thus,

according to the last equation above,

dgA dt

> 0 at the point {˜ gK , gA }. Therefore, gA should increase,

which justifies the upward arrow below the curve 5

dgA dt

= 0.

2.3.1

Case 1:

+ 0. Let keep the simplifying assumptions of no depreciation δ = 0 and constant saving rate I (t) = sY (t). Hence, the production function becomes Y (t) = B 1−α K (t)a K (t)φ(1−α) L (t)1−α , 7

M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory

and capital accumulation follows dK (t) = sB 1−α K (t)a K (t)φ(1−α) L (t)1−α . dt In this economy capital has α + φ (1 − α) return to scale. Thus, we can use our analysis above for cases when α + φ (1 − α) is either smaller, equal or greater than 1. If φ < 1, then capital has decreasing returns to scale, so long-run growth is stable and is governed by population growth n. And if φ = 1, then there is explosive growth if n > 0 and a continuum of possible steady growth rates if n = 0, where initial conditions matter for which growth rate the economy converges to.

3

Human Capital

This Section augments the Solow Growth Model to introduce human capital. The main motivation for this investigation is to study the determinants of cross-country growth differences.

3.1

Model set-up

Consider the production function Y (t) = K (t)α [A (t) H (t)]1−α which has constant returns to scale. The only departure from the Solow Mode is that total working hours L (t) have been replaced by H (t) in the production function. This variable H (t) represents the total amount of productive services supplied by workers. The key difference between ”working hours” and ”productive services” is that the latter is related to workers’ skills, which in turn depend on the level of education: H (t) = L (t) G (E) were L (t) is the total number of working hours, and G (E) represents human capital as a function of the years of education per worker E. Microeconomic evidence suggests that one extra year of education increases an individual’s wage by the same percentage amount. Wages and productivity should have a one-to-one relationship. Thus, let us assume that G (E) = eφE , φ > 0, 8

M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory

with G (0) = 1. To complete the model, we keep the assumption that A (t) is a productivity factor that affects all workers and that is driven by an exogenous process, such that dA (t) = gA (t) . dt Population is also an exogenous variable growing at rate n, such that dL (t) = nL (t) . dt Capital accumulation is given by dK (t) = sY (t) − δK (t) dt after we impose the assumption of a constant saving rate, so I (t) = S (t) = sY (t).

3.2

Solving the model

The solution of this model follows closely that of the Solow model. To see this, note that we can also exploit the CRS property in the production function, so we can define k (t) =

K (t) A (t) H (t)

as the stock of capital per effective units of labor services (i.e., effective working hours adjusted by skill level). Therefore, the accumulation of k (t) is given by dk (t) = sf (k (t)) − (n + g + δ) k (t) dt assuming that E will be constant along the steady growth path. This assumption will be confirmed later. It also kind of makes sense: We are augmenting the Solow model to include human capital to explain cross-country differences in growth in the long run. Coming back to the solution of the model the latter expression above implies a constant steady state k ∗ , which is defined by 

s k = n+g+δ ∗

9

1  1−α

M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory after we check that the intensive form of the production function now is still f (k (t)) = k (t)α . This analysis implies that the quantitative and the qualitative effects of the change in the saving rate s are also the same than in the Solow model: no effect on growth rates, but an effect on the scale. The introduction of human capital does make a difference in levels, though. Per-capital income in this economy along the steady growth path is  ∗ Y = AG (E) y ∗ L where y ∗ is output per-effective working servive, y ∗ = f (k ∗ ). The path of AG (E) is not affected by changes in the saving rate, but y ∗ is affected. Therefore, growth rates are not affected by
∗ education. Nevertheless, in level terms, how much is the change in the saving rate s affects YL does depend of the level of education of the country, E.

3.3

Students and workers

We cannot study and work at the same time. Until now we have assumed that education does not require an investment in terms of time. In this Section we relax this assumption. Suppose that individuals have a finite lifespan T years and spend the first E years of their lives in getting educated. To keep our assumption that the population growth is n, we assume now that every year a new generation of people is born which is n times larger than the generation born in the previous year. Thus, we can denote N (t) as total population at t, and B (t) as the total births at t, such that Z T N (t) = B (t − τ ) dτ . τ =0

In words, the equation above indicates that total population is the sum of births of all generations that are still alive, i.e., those who have been born at most T years ago. Given the assumtion that dB (t) = nB (t) , dt we can get an explicit solution for N (t): Z

T

B (t) e−nτ dτ

N (t) = τ =0

1 − e−nT = B (t) . n 10

M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory

Similarly, the number of workers at time t, L (t), equals the number of individuals who are alive and no longer in school, i.e., those generations born between T years ago and T − E years ago. Thus Z

T

B (t − τ ) dτ

L (t) = τ =E Z T

=

B (t) e−nτ dτ

τ =E

e−nE − e−nT = B (t) . n after we have also used the assumption that the births growth is n. Combining our expressions for N (t) and L (t), we get L (t) e−nE − e−nT = . N (t) 1 − e−nT Now we can use our results in the previous section (which remain invariant), to compute output per-capita, as opposed to output-per-worker, which is what strictly speaking we obtained in the previous section. 

Y N

∗

 ∗   Y L e−nE − e−nT . = = A (t) G (E) y ∗ L N 1 − e−nT

where y ∗ = f (k ∗ ), the steady state level of output per effective units of labor services. Therefore, an increase in the number of years of education E of the population has two effects in the living standards in this economy in the steady state. On one hand, it increases the productivity of workers because they have more skills (i.e., higher G (E)). But, on the other hand, it also decreases the proportion of people who actually work. Therefore, the overall effect of an increase
Y ∗ in E could either increase or decrease N . To study transitional dynamics, assume that everybody born after t0 has E1 years of education, while everybody born before t0 only has E0 years of education, with E1 > E0 . This change decreases output per capita for the first E1 − E0 years since the working force still has E0 years of education, therefore the function G (E0 ) remains constant. This is because the new generations that should go to work stay for longer in the school. But at the same time, and because of the same reason, the proportion of workers

L(t) N (t)

decreases, which justifies the decrease in output per

capita. 11

M1-TSE. Macro I. 2010-2011. Chapter 2: New Growth Theory

However, for t > t0 + (E1 − E0 ), the more educated new generations enter to work, which both slowly increases G (E) and

L(t) . N (t)

12