Productive externalities and endogenous growth
Makroøkonomi 2: Vækstteori og konjukturpolitik (MakØk2) Michael Bergman Københavns Universitet
Productive externalities and endogenous growth • The Solow model assumes that the rate of technological progress is exogenous, i.e., given outside the model. Technological progress is unexplained! • This is a potential problem since we know that in the Solow model, growth in output per worker is equal to the growth rate of technology. The Solow model can therefore not explain economic growth, it explains how technological progress explains growth. • What is the source of technological progress? • We will now study models where we assume that the growth rate of technological progress is endogenous (explained by factors within the model). These models are called endogenous growth models. • Why is this extension of the Solow model important? • There is no role for economic policy to aect economic growth in the Solow model! We know that changes in the savings rate only aects economic growth in the shortrun not in the longrun. What we would like to have is a model of economic growth where economic policy can be used to aect the longrun growth rate. • We will now relax the assumption that g is exogenous and therefore we will not assume that At+1 = (1 + g) At anymore. • We will still denote technology by A but there is no explicit process for technological progress.
1
• In our model we will assume that on the individual rm level there is constant returns to scale in capital and labor as before. But at the aggregate level there is increasing returns to scale. • Let us assume that the production function for a representative rm is given by ¡ ¢α Yt = Ktd (At Lt )1−α . (1) This is the same production function we have used earlier. The production function exhibits constant returns to scale in capital and labor since α + (1 − α) = 1.
• Assume now that the level of technology used by these rms is dependent on the aggregate stock of capital in the economy Kt . In particular, we assume that At = Ktφ (2) where φ > 0.
• What is the motivation for this expression? • In short, learning by doing! If one rm is learning to use new capital in production, all other rms may be looking over their shoulders and therefore also learn how to use this new capital good in their production. It may also be the case that workers move between jobs in dierent rms and they also bring new capabilities and knowledge with them to the new rm. There is a spillover eect from the use of new capital in one rm to the other rms. • Note that we should de ne the level of technology in equation (2) in terms of investment instead of the capital stock. However, we implicitly ignore depreciation and assume that the level of technology depends on the capital stock. • The representative rm takes the level of technology as given (there is no way the individual rm can in uence the aggregate capital stock if there are an in nite number of rms). 2
• The aggregate production function can be written as (insert equation (2) into (1) and use the market clearing conditions Kt = Kt and Ldt = Lt ) ³ ´1−α α+φ(1−α) 1−α φ α Lt . (3) Yt = Kt Kt Lt = Kt This production function exhibits increasing returns to scale since α + φ (1 − α) + 1 − α > 1. If we double both capital and labor, aggregate output is multiplied by the factor 21+φ(1−α) > 2.
• We can easily verify that the rental rate and the wage are given by their marginal products. Take the partial derivatives of the production function in (1), use the market clearing conditions above and use equation (2) to obtain rt = αYt
and
wt = (1 − α) Yt .
(5)
• The share of capital in output (and the share of income in production) is constant and equal to α. • All other aspects of the general Solow model are intact. • We can now write down the full model in the following way Yt = (Kt )α (At Lt )1−α ,
(6)
At = Ktφ ,
(7)
Kt+1 = sYt + (1 − δ) Kt ,
(8)
Lt+1 = (1 + n) Lt .
(9)
and
• Note that if φ = 0, then we have the basic Solow model. • We have stated above that φ > 0. In this case, we have increasing returns to scale in capital and labor. If φ < 1, then there is diminishing returns to scale in capital (α + φ (1 − α) < 1) and if φ = 1 then there will be constant returns to scale in capital. We, thus have two versions of the model to analyze! Are these two cases dierent? 3
• Yes! • Let us rst consider the model when φ > 1. This model is called semiendogenous growth model.
Semiendogenous growth • To solve the model we once again de ne the `∼' variables, i.e., technologyadjusted output per worker and capital per worker. It is then straightforward to rewrite the production function as y˜t = k˜tα .
(10)
• Use the relationship between technology and capital given in (7) such that µ ¶φ Kt+1 At+1 . (11) = At Kt • Use this relationship, the de nition of k˜, equations (9), (8) and (10) to get ´1−φ k˜t+1 1 ³ ˜α−1 = . skt + (1 − δ) 1+n k˜t • Multiply both sides of this equation by k˜t ´1−φ 1 ˜ ³ ˜α−1 ˜ . kt+1 = k skt + (1 − δ) 1+n t
(13)
This is the transition equation!
• From equation (13) we note that
it passes through (0,0), it is increasing in k˜, the slope is diminishing, above 1 for small k˜ and then diminishing ˜ implying that it crosses the 450 degree line. when increasing k
• This implies that there is a steady state where k˜ is constant! 4
• Insert k˜t+1 = k˜t = k˜ in equation (13) and solve for k˜ Ã k˜∗ =
1 ! 1−α
s (1 + n)
1 1−φ
− (1 − δ)
.
(15) 1
Note that there is a unique positive solution if (1 + n) 1−φ > (1 − δ).
• The convergence to steady state is illustrated in Figure 8.1. Figure 8.1: The transition diagram of the model of semiendogenous growth.
• Insert this solution into the production function to nd the steady state solution of technologyadjusted output per worker as α Ã ! 1−α s y˜∗ = . (16) 1 (1 + n) 1−φ − (1 − δ) • What is the growth rate of output per worker, capital per worker and technology in this model?
5
• First, we have found a steady state where k˜ and y˜ are constants. This implies that output per worker and capital per worker must grow at the ˜ = k/A implying that if same rate. (Remember that y˜ = y/A and k these two variables are constant, then capital per worker and output per worker must grow at the same rate in the steady state.) • What is the growth rate of technology? The de nition of k˜ implies that µ ¶1−φ 1 Kt+1 k˜t+1 . (12) = 1+n Kt k˜t In the steady state the LHS is equal to one
1 1= 1+n
µ
Kt+1 Kt
¶1−φ
which implies that
1 Kt+1 = (1 + n) 1−φ . Kt Use equation (11) such that 1 At+1 = (1 + n) 1−φ At
or
1 At+1 − At = (1 + n) 1−φ − 1 ≡ gse . (17) At We have now found the growth rate of technology in the steady state. This growth rate is endogenous since it is dependent on the parameters of the model.
• Take logs and time derivatives of the production function in equation (6) (in per worker terms) we nd that gty = αgtk + (1 − α) gtA . In the steady state, the growth rate of output per worker is equal to the growth rate of capital per worker implying that
g y = g k = gse . 6
• Here we note that if n = 0, then gse = 0 and g y = 0. This is the same prediction as in the basic Solow model. • Note that capital K and labor L are not growing at the same rate in the steady state, k = K/L grows at the rate gse implying that the growth rate of K is gse + n.
Structural policy for steady state • Can structural policy aect the growth rate of output per worker? • We know that yt∗ = y˜∗ At where At = Ktφ . • This implies that yt∗ = y˜∗ Ktφ . • We also have de ned k˜ = K/(AL) = K 1−φ /L. In the steady state, the capital stock is 1 ´ 1−φ ³ ∗ ˜ Kt = k L t . • Insert this relation into the steady state growth path for output per φ worker (yt∗ = y˜∗ Kt ) and use equation (9) such that yt∗
φ ³ ´α+ 1−φ φ φ 1−φ ∗ ˜ L0 (1 + n) 1−φ t . = k
7
• Insert the steady state solution in (15) and use (17) such that φ
à yt∗ =
1−φ ! α+1−α
s (1 + n)
1 1−φ
φ
L01−φ (1 + gse )t .
− (1 − δ)
(18)
This is the steady state growth path for output per worker in this economy.
• The growth rate of output per worker is given by φ
gty = gtc = gse = (1 + n) 1−φ − 1. • Now we can answer the question whether structural policies can aect the steady state growth path of output per worker. • From equation (18) we nd that
a higher investment rate s shifts the growth path up, and higher population growth also shifts the steady state growth path y
up, gt increases. The rst prediction implies that it is optimal to increase the savings rate unless it is equal to the golden rule savings rate whereas the second prediction implies that we should promote population growth. This latter prediction is questionable for other reasons!
• Why?
8
• In Figure 8.2 we show the relationship between the average annual growth rate of real GDP and population growth. The relationship is negative, higher population growth is associated with lower growth! There is a negative relationship, not a positive one as our theory predicts! Figure 8.2: Average annual growth rate of real GDP per worker against average annual growth rate in population, 55 countries.
• The steady state growth path for consumption per worker is given by (1 − s) times this expression, i.e., φ
à c∗t
= (1 − s)
1−φ ! α+1−α
s (1 + n)
1 1−φ
− (1 − δ)
φ 1−φ
L0
(1 + gse )t .
Note that the golden rule value for s is given by (take logs and the partial derivative with respect to s)
s∗∗ = α + φ (1 − α) which is greater than the golden rule savings rate in the general Solow model. 9
• Is our model wrong? • There are, at least, two reasons why our model may be correct even if our test suggest the opposite.
First, it is not clear if our model covers the whole world or a region or a country. For that reason it may be the case that crosscountry evidence is not relevant. In other words, we cannot draw conclusions about the relevance of our model using crosscountry evidence.
Second, our model has a steady state solution and it converges to the steady state in the longrun. During this convergence, there is a negative relationship between population growth and output y growth. Why? Higher population growth tends to increase gt but at the same time reduces the slope of the growth path of yt∗ as can be seen in equations (17) and (18) above. Assume that the economy is converging towards the steady state. If population growth falls, then the economy is farther away from its steady state and the growth rate of output per worker is higher than initially. This is illustrated in Figure 1 (within the basic Solow model). It could be the case that the countries we include in our crosscountry comparison in Figure 8.2 all are converging to steady state. If so, higher population growth will reduce output per worker growth in the shortrun (all countries are closer to the steady state) but increase output per worker growth in the longrun.
10
Figure 1: Modi ed Solow diagram. 6 6
³
kt+1 −kt kt
´0
kt+1 −kt kt
?
n+δ
?
n0 + δ
-
kt
11
n0 < n
• Looking at longer time periods and certain countries does not change the conclusion that output per worker growth is negatively related to population growth, see Table 8.1. Table 8.1: Average annual growth rates in population and real GDP per capita in 17 industrialized countries, 18701990.
• Are all countries converging to steady state even over this very long time period? If so, then the model may be correct but the steady state solution may not be very informative. The main prediction from the model must then be the transition dynamics.
12
Endogenous growth • We will now study our model under the assumption that φ = 1. • We will also abstract from population growth, we are interested in a model where population growth does not contribute to output per worker growth. Assume therefore that Lt+1 = Lt = L, i.e., population is constant. • Letting φ = 1 we can write the production function (using equation (3)) as Yt = Kt L1−α = AKt (20) where A ≡ L1−α . Note that A is not equal to technology At . The capital accumulation equation is
Kt+1 = sYt + (1 − δ) Kt .
(21)
• The real factor prices are now (inserting φ = 1 in the relevant equations above) K rt = r = αL1−α and wt = (1 − α) αt . (19) L • This model is called the AK model! That was the reason why we used the notation A ≡ L1−α above.
Growth according to the AK model • Proceeding as we have done before (transforming our relationships into per worker terms) we nd that the transition equation is kt+1 = (sA + 1 − δ) kt .
(22)
• Subtract kt from both sides such that the Solow equation is kt+1 − kt = sAkt − δkt . 13
(23)
• Finally, divide both sides by kt such that we obtain the modi ed Solow equation kt+1 − kt = sA − δ ≡ ge . (24) kt • Equation (24) implies that the growth rate of capital per worker is constant and equal to ge ! • This implies that the growth rate of capital Kt is also equal to ge ! • Since yt = Akt this also implies that the growth rate of output per worker is equal to ge ! • Finally, since ct = (1 − s) yt , the growth rate of consumption per worker is equal to ge ! • What about technology? Since At = Kt when φ = 1 it implies that the growth rate of technology is also equal to ge ! • We have found that all variables in the AK model grow at the same rate ge = sA − δ which is endogenous since it depends on the parameters of the model. There is growth even in the absence of technology growth! • Let us now illustrate the model in the transition diagram, the Solow diagram and the modi ed Solow diagram, see Figure 8.3. • Note that there is no steady state in the model, k is not converging to a steady state, see the top graph in Figure 8.3.
14
Figure 8.3: The transition diagram (top), the Solow diagram (middle), and the modi ed Solow diagram (bottom) of the model of endogenous growth.
15
Implications for structural policy and the scale eect • What are the implications of the model? • Equation (24) gives the common growth rate of the variables in the model restated here for convenience kt+1 − kt = sA − δ ≡ ge . kt • This equation implies that
A higher savings rate s tends to increase growth in output per worker permanently.
Economic policy should promote savings! Can structural policy aect δ ? No, this is di cult! What about A? Since A = L1−α , then population growth will increase growth in output per worker permanently. This scale eect is controversial! We have earlier looked at the data which suggest a negative relationship between population growth and growth in output per worker. We still have the same problem with the prediction in this model.
• Is it possible to get rid of the scale eect? • Yes! Assume that technology is given by µ ¶φ Kt At = . Lt If φ = 1 then the production function can be written as
Yt = Ktα (At Lt )1−α = Kt .
16
• This implies that the modi ed Solow equation is kt+1 − kt = s − δ ≡ ge kt implying that there is no scale eect from population. This follows from our assumption that production is independent on labor input. In other words, labor input is unproductive.
• This is a potential problem with endogenous growth models, either there is a scale eect or labor is unproductive.
Empirics for endogenous growth • The main prediction from the endogenous growth models studied here is that there is a positive relationship between the savings rate s and the growth rate of output per worker.
17
• Can this prediction be rejected by the data? See Figure 8.4. Figure 8.4: Average annual growth rate of real GDP per worker against average investment rate in physical capital, 90 countries.
• The empirical evidence cannot reject endogenous growth models. But, the data is consistent with the exogenous growth models also! Higher savings rate lead to higher transitory growth in output per worker in the Solow models!
Exogenous versus endogenous growth • Both versions of endogenous growth explained growth in output per worker using structural model parameters. Growth is endogenous! • In the Solow models we assume that there is exogenous growth in technology. These models do not explain why technological progress occurs, they take technological progress as given. • However, the endogenous growth models presented here do not model technological progress, i.e., they do not explain why there is technological progress. (Such models are discussed in Chapter 9 which is not included in the curriculum!) 18
• Note that the endogenous growth model suggests that there is a positive relationship between the savings rate and technological progress. The growth rate of technology At is equal to ge which is, in turn, equal to sA − δ . Empirical evidence seems to suggest that this prediction is not rejected, see Figure 8.5. Figure 8.5: Average annual rate of laboraugmenting technological progress against average investment rate in physical capital, 84 countries.
• It is often argued that this is the strongest empirical evidence for endogenous growth models. Research is ongoing! The future will tell if this is a fruitful approach to economic growth.
19
Productive externalities and endogenous growth • The Solow model assumes that the rate of technological progress is exogenous, i.e., given outside the model. Technological progress is unexplained! • This is a potential problem since we know that in the Solow model, growth in output per worker is equal to the growth rate of technology. The Solow model can therefore not explain economic growth, it explains how technological progress explains growth. • What is the source of technological progress? • We will now study models where we assume that the growth rate of technological progress is endogenous (explained by factors within the model). These models are called endogenous growth models. • Why is this extension of the Solow model important? • There is no role for economic policy to aect economic growth in the Solow model! We know that changes in the savings rate only aects economic growth in the shortrun not in the longrun. What we would like to have is a model of economic growth where economic policy can be used to aect the longrun growth rate. • We will now relax the assumption that g is exogenous and therefore we will not assume that At+1 = (1 + g) At anymore. • We will still denote technology by A but there is no explicit process for technological progress.
1
• In our model we will assume that on the individual rm level there is constant returns to scale in capital and labor as before. But at the aggregate level there is increasing returns to scale. • Let us assume that the production function for a representative rm is given by ¡ ¢α Yt = Ktd (At Lt )1−α . (1) This is the same production function we have used earlier. The production function exhibits constant returns to scale in capital and labor since α + (1 − α) = 1.
• Assume now that the level of technology used by these rms is dependent on the aggregate stock of capital in the economy Kt . In particular, we assume that At = Ktφ (2) where φ > 0.
• What is the motivation for this expression? • In short, learning by doing! If one rm is learning to use new capital in production, all other rms may be looking over their shoulders and therefore also learn how to use this new capital good in their production. It may also be the case that workers move between jobs in dierent rms and they also bring new capabilities and knowledge with them to the new rm. There is a spillover eect from the use of new capital in one rm to the other rms. • Note that we should de ne the level of technology in equation (2) in terms of investment instead of the capital stock. However, we implicitly ignore depreciation and assume that the level of technology depends on the capital stock. • The representative rm takes the level of technology as given (there is no way the individual rm can in uence the aggregate capital stock if there are an in nite number of rms). 2
• The aggregate production function can be written as (insert equation (2) into (1) and use the market clearing conditions Kt = Kt and Ldt = Lt ) ³ ´1−α α+φ(1−α) 1−α φ α Lt . (3) Yt = Kt Kt Lt = Kt This production function exhibits increasing returns to scale since α + φ (1 − α) + 1 − α > 1. If we double both capital and labor, aggregate output is multiplied by the factor 21+φ(1−α) > 2.
• We can easily verify that the rental rate and the wage are given by their marginal products. Take the partial derivatives of the production function in (1), use the market clearing conditions above and use equation (2) to obtain rt = αYt
and
wt = (1 − α) Yt .
(5)
• The share of capital in output (and the share of income in production) is constant and equal to α. • All other aspects of the general Solow model are intact. • We can now write down the full model in the following way Yt = (Kt )α (At Lt )1−α ,
(6)
At = Ktφ ,
(7)
Kt+1 = sYt + (1 − δ) Kt ,
(8)
Lt+1 = (1 + n) Lt .
(9)
and
• Note that if φ = 0, then we have the basic Solow model. • We have stated above that φ > 0. In this case, we have increasing returns to scale in capital and labor. If φ < 1, then there is diminishing returns to scale in capital (α + φ (1 − α) < 1) and if φ = 1 then there will be constant returns to scale in capital. We, thus have two versions of the model to analyze! Are these two cases dierent? 3
• Yes! • Let us rst consider the model when φ > 1. This model is called semiendogenous growth model.
Semiendogenous growth • To solve the model we once again de ne the `∼' variables, i.e., technologyadjusted output per worker and capital per worker. It is then straightforward to rewrite the production function as y˜t = k˜tα .
(10)
• Use the relationship between technology and capital given in (7) such that µ ¶φ Kt+1 At+1 . (11) = At Kt • Use this relationship, the de nition of k˜, equations (9), (8) and (10) to get ´1−φ k˜t+1 1 ³ ˜α−1 = . skt + (1 − δ) 1+n k˜t • Multiply both sides of this equation by k˜t ´1−φ 1 ˜ ³ ˜α−1 ˜ . kt+1 = k skt + (1 − δ) 1+n t
(13)
This is the transition equation!
• From equation (13) we note that
it passes through (0,0), it is increasing in k˜, the slope is diminishing, above 1 for small k˜ and then diminishing ˜ implying that it crosses the 450 degree line. when increasing k
• This implies that there is a steady state where k˜ is constant! 4
• Insert k˜t+1 = k˜t = k˜ in equation (13) and solve for k˜ Ã k˜∗ =
1 ! 1−α
s (1 + n)
1 1−φ
− (1 − δ)
.
(15) 1
Note that there is a unique positive solution if (1 + n) 1−φ > (1 − δ).
• The convergence to steady state is illustrated in Figure 8.1. Figure 8.1: The transition diagram of the model of semiendogenous growth.
• Insert this solution into the production function to nd the steady state solution of technologyadjusted output per worker as α Ã ! 1−α s y˜∗ = . (16) 1 (1 + n) 1−φ − (1 − δ) • What is the growth rate of output per worker, capital per worker and technology in this model?
5
• First, we have found a steady state where k˜ and y˜ are constants. This implies that output per worker and capital per worker must grow at the ˜ = k/A implying that if same rate. (Remember that y˜ = y/A and k these two variables are constant, then capital per worker and output per worker must grow at the same rate in the steady state.) • What is the growth rate of technology? The de nition of k˜ implies that µ ¶1−φ 1 Kt+1 k˜t+1 . (12) = 1+n Kt k˜t In the steady state the LHS is equal to one
1 1= 1+n
µ
Kt+1 Kt
¶1−φ
which implies that
1 Kt+1 = (1 + n) 1−φ . Kt Use equation (11) such that 1 At+1 = (1 + n) 1−φ At
or
1 At+1 − At = (1 + n) 1−φ − 1 ≡ gse . (17) At We have now found the growth rate of technology in the steady state. This growth rate is endogenous since it is dependent on the parameters of the model.
• Take logs and time derivatives of the production function in equation (6) (in per worker terms) we nd that gty = αgtk + (1 − α) gtA . In the steady state, the growth rate of output per worker is equal to the growth rate of capital per worker implying that
g y = g k = gse . 6
• Here we note that if n = 0, then gse = 0 and g y = 0. This is the same prediction as in the basic Solow model. • Note that capital K and labor L are not growing at the same rate in the steady state, k = K/L grows at the rate gse implying that the growth rate of K is gse + n.
Structural policy for steady state • Can structural policy aect the growth rate of output per worker? • We know that yt∗ = y˜∗ At where At = Ktφ . • This implies that yt∗ = y˜∗ Ktφ . • We also have de ned k˜ = K/(AL) = K 1−φ /L. In the steady state, the capital stock is 1 ´ 1−φ ³ ∗ ˜ Kt = k L t . • Insert this relation into the steady state growth path for output per φ worker (yt∗ = y˜∗ Kt ) and use equation (9) such that yt∗
φ ³ ´α+ 1−φ φ φ 1−φ ∗ ˜ L0 (1 + n) 1−φ t . = k
7
• Insert the steady state solution in (15) and use (17) such that φ
à yt∗ =
1−φ ! α+1−α
s (1 + n)
1 1−φ
φ
L01−φ (1 + gse )t .
− (1 − δ)
(18)
This is the steady state growth path for output per worker in this economy.
• The growth rate of output per worker is given by φ
gty = gtc = gse = (1 + n) 1−φ − 1. • Now we can answer the question whether structural policies can aect the steady state growth path of output per worker. • From equation (18) we nd that
a higher investment rate s shifts the growth path up, and higher population growth also shifts the steady state growth path y
up, gt increases. The rst prediction implies that it is optimal to increase the savings rate unless it is equal to the golden rule savings rate whereas the second prediction implies that we should promote population growth. This latter prediction is questionable for other reasons!
• Why?
8
• In Figure 8.2 we show the relationship between the average annual growth rate of real GDP and population growth. The relationship is negative, higher population growth is associated with lower growth! There is a negative relationship, not a positive one as our theory predicts! Figure 8.2: Average annual growth rate of real GDP per worker against average annual growth rate in population, 55 countries.
• The steady state growth path for consumption per worker is given by (1 − s) times this expression, i.e., φ
à c∗t
= (1 − s)
1−φ ! α+1−α
s (1 + n)
1 1−φ
− (1 − δ)
φ 1−φ
L0
(1 + gse )t .
Note that the golden rule value for s is given by (take logs and the partial derivative with respect to s)
s∗∗ = α + φ (1 − α) which is greater than the golden rule savings rate in the general Solow model. 9
• Is our model wrong? • There are, at least, two reasons why our model may be correct even if our test suggest the opposite.
First, it is not clear if our model covers the whole world or a region or a country. For that reason it may be the case that crosscountry evidence is not relevant. In other words, we cannot draw conclusions about the relevance of our model using crosscountry evidence.
Second, our model has a steady state solution and it converges to the steady state in the longrun. During this convergence, there is a negative relationship between population growth and output y growth. Why? Higher population growth tends to increase gt but at the same time reduces the slope of the growth path of yt∗ as can be seen in equations (17) and (18) above. Assume that the economy is converging towards the steady state. If population growth falls, then the economy is farther away from its steady state and the growth rate of output per worker is higher than initially. This is illustrated in Figure 1 (within the basic Solow model). It could be the case that the countries we include in our crosscountry comparison in Figure 8.2 all are converging to steady state. If so, higher population growth will reduce output per worker growth in the shortrun (all countries are closer to the steady state) but increase output per worker growth in the longrun.
10
Figure 1: Modi ed Solow diagram. 6 6
³
kt+1 −kt kt
´0
kt+1 −kt kt
?
n+δ
?
n0 + δ
-
kt
11
n0 < n
• Looking at longer time periods and certain countries does not change the conclusion that output per worker growth is negatively related to population growth, see Table 8.1. Table 8.1: Average annual growth rates in population and real GDP per capita in 17 industrialized countries, 18701990.
• Are all countries converging to steady state even over this very long time period? If so, then the model may be correct but the steady state solution may not be very informative. The main prediction from the model must then be the transition dynamics.
12
Endogenous growth • We will now study our model under the assumption that φ = 1. • We will also abstract from population growth, we are interested in a model where population growth does not contribute to output per worker growth. Assume therefore that Lt+1 = Lt = L, i.e., population is constant. • Letting φ = 1 we can write the production function (using equation (3)) as Yt = Kt L1−α = AKt (20) where A ≡ L1−α . Note that A is not equal to technology At . The capital accumulation equation is
Kt+1 = sYt + (1 − δ) Kt .
(21)
• The real factor prices are now (inserting φ = 1 in the relevant equations above) K rt = r = αL1−α and wt = (1 − α) αt . (19) L • This model is called the AK model! That was the reason why we used the notation A ≡ L1−α above.
Growth according to the AK model • Proceeding as we have done before (transforming our relationships into per worker terms) we nd that the transition equation is kt+1 = (sA + 1 − δ) kt .
(22)
• Subtract kt from both sides such that the Solow equation is kt+1 − kt = sAkt − δkt . 13
(23)
• Finally, divide both sides by kt such that we obtain the modi ed Solow equation kt+1 − kt = sA − δ ≡ ge . (24) kt • Equation (24) implies that the growth rate of capital per worker is constant and equal to ge ! • This implies that the growth rate of capital Kt is also equal to ge ! • Since yt = Akt this also implies that the growth rate of output per worker is equal to ge ! • Finally, since ct = (1 − s) yt , the growth rate of consumption per worker is equal to ge ! • What about technology? Since At = Kt when φ = 1 it implies that the growth rate of technology is also equal to ge ! • We have found that all variables in the AK model grow at the same rate ge = sA − δ which is endogenous since it depends on the parameters of the model. There is growth even in the absence of technology growth! • Let us now illustrate the model in the transition diagram, the Solow diagram and the modi ed Solow diagram, see Figure 8.3. • Note that there is no steady state in the model, k is not converging to a steady state, see the top graph in Figure 8.3.
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Figure 8.3: The transition diagram (top), the Solow diagram (middle), and the modi ed Solow diagram (bottom) of the model of endogenous growth.
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Implications for structural policy and the scale eect • What are the implications of the model? • Equation (24) gives the common growth rate of the variables in the model restated here for convenience kt+1 − kt = sA − δ ≡ ge . kt • This equation implies that
A higher savings rate s tends to increase growth in output per worker permanently.
Economic policy should promote savings! Can structural policy aect δ ? No, this is di cult! What about A? Since A = L1−α , then population growth will increase growth in output per worker permanently. This scale eect is controversial! We have earlier looked at the data which suggest a negative relationship between population growth and growth in output per worker. We still have the same problem with the prediction in this model.
• Is it possible to get rid of the scale eect? • Yes! Assume that technology is given by µ ¶φ Kt At = . Lt If φ = 1 then the production function can be written as
Yt = Ktα (At Lt )1−α = Kt .
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• This implies that the modi ed Solow equation is kt+1 − kt = s − δ ≡ ge kt implying that there is no scale eect from population. This follows from our assumption that production is independent on labor input. In other words, labor input is unproductive.
• This is a potential problem with endogenous growth models, either there is a scale eect or labor is unproductive.
Empirics for endogenous growth • The main prediction from the endogenous growth models studied here is that there is a positive relationship between the savings rate s and the growth rate of output per worker.
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• Can this prediction be rejected by the data? See Figure 8.4. Figure 8.4: Average annual growth rate of real GDP per worker against average investment rate in physical capital, 90 countries.
• The empirical evidence cannot reject endogenous growth models. But, the data is consistent with the exogenous growth models also! Higher savings rate lead to higher transitory growth in output per worker in the Solow models!
Exogenous versus endogenous growth • Both versions of endogenous growth explained growth in output per worker using structural model parameters. Growth is endogenous! • In the Solow models we assume that there is exogenous growth in technology. These models do not explain why technological progress occurs, they take technological progress as given. • However, the endogenous growth models presented here do not model technological progress, i.e., they do not explain why there is technological progress. (Such models are discussed in Chapter 9 which is not included in the curriculum!) 18
• Note that the endogenous growth model suggests that there is a positive relationship between the savings rate and technological progress. The growth rate of technology At is equal to ge which is, in turn, equal to sA − δ . Empirical evidence seems to suggest that this prediction is not rejected, see Figure 8.5. Figure 8.5: Average annual rate of laboraugmenting technological progress against average investment rate in physical capital, 84 countries.
• It is often argued that this is the strongest empirical evidence for endogenous growth models. Research is ongoing! The future will tell if this is a fruitful approach to economic growth.
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