ECON 211 MACROECONOMICS
Key Question: Can we raise national income by increasing aggregate demand alone?
Some economists refer to the Keynesian Model to advocate various expansionary policies that raises the model’s Aggregate Demand (AD) to achieve growth of national income. Ironically, the model implies that those policies can increase national income only in the short run. In the long run, those expansionary policies fail to raise the national income beyond its potential level determined by the supply side. In addition, the model also implies that expansionary policies would have the effect of raising the price level above the expected price level causing a decline in the real wage rate because the nominal wages are fixed in the short run, i.e., for the contract period. The real wage declines leading to an increase in the hours of labour employed by firms and consequently an increase in national output Y. However, in the long run, workers re-negotiate their wages upward to compensate them for the reduced standard of living due to the increase in the price level. This will have the opposite effect of reducing the hours of employment by the firms leading to a decline in output. Thus, an expansionary policy raises the national output Y only temporarily and typically when nominal wages are contractually fixed.
AD e AD 1 SRAS LRAS P e P e 1 P 1 SRAS 1
The policy of increasing AD, therefore, acts in the model to increase growth only in the short run. For any long run growth the value of the maximum potential income Yf itself has to increase and this can only be achieved by policies that address the supply-side of the economy. In the next few lectures we will develop a model of the long run national income (Yf) and examine how it grows in the model as various factors of production that determine national output and income grow. The Solow’s model relies heavily on the notion of an aggregate production function determined by economy’s stock of capital (K), labour force (L) and technology (A) as follows:
Yf = A Kα Lf 1-α
The technology parameter A can also be viewed as a measure of output per input or total factor productivity (TFP) such that TFP = Y / KαL1-α, or simply, technology, which represents the stock of knowledge about available methods (techniques) of production.
This knowledge can come from various non-economic sources such as religious, cultural or political factors. In addition, the growth of this productive knowledge depends on economic as well as various political, institutional, religious, sociological and other non-economic factors. ------------------------------------------------------------------------------------------------------
The Neoclassical Theory of Investment and the Foundation of Economic Growth Theories:
Q. How would the investment demand Id change, if the expected marginal product of capital (MPKe) increases?
1 + t K • • • Step 4: A new equilibrium is reached at * ′ t K where * * ′ ′ −= t t r MPK δ . Step 1: At * * , t e t t r K MPK −> ′ δ . (Initial Shock) Step 3: As ↑ t K by the law of diminishing returns ↓ t MPK . Step 2: . to from increases stock capital of level desired * * * ′ ′ −> ⇒ t t t e t K K r MPK δ 1 0 0 * * ′ t K 2 δ − = e t e t r MPK K t δ − ′ e t MPK * t r δ − ′ e t MPK δ − e t MPK
↑ Itd =↑ Kt* − −(1 δ)Kt−1
§ An increase in Kt* to Kt*′ implies an increase in Itd for the given real interest rate rt*in the goods market.
§ Therefore, ↑ MPKte ⇒↑ Itd given rt*.
Theories of Economic Growth and Evidence
© Debasis Bandyopadhyay
Key Question: Why do countries grow and why do they grow at different rates? What can the policymakers do to promote growth?
1. Introduction: The Facts of Growth
The national income and consumption per capita in today’s rich countries have grown steadily in the last fifty years. However, the growth rates appear to be slowing down. Among the top five OECD countries, the relatively poor countries appear to grow faster than others. Consequently, some economists argue that the standard of living in those countries is converging to the same level. Many economists believe that the Asian countries are also converging to the same level. However, there is no evidence of convergence for the African countries. To make sense of those observations we will first study a model of economic growth known as the Solow (1956) model.
In Solow (1956) model, growth is driven by the accumulation of physical capital. An increase in the capital stock will increase the national income of an economy. Therefore, if we can describe the process of accumulation of physical capital, we can describe growth of national income.
2. Saving, Capital Accumulation and Output
The Solow (1956) model assumes that the saving is a constant fraction of output, i.e.
St = sYt , 0 < s < 1 (1)
and that the production technology exhibits diminishing returns to capital and labour. For example, we can use the Cobb-Douglas production function:
Yt = A K Lt α αt 1t− , 0 < <α 1, At > 0. (2)
• We interpret alpha as the output elasticity of the capital stock. It measures the percentage change of aggregate output for a given percentage change in the aggregate capital input. If firms maximise profit in a competitive environment, then the value of alpha turns out to be equal to the fraction of national income distributed to the owners of capital.
• We interpret At as an exogenous technological parameter. No household, or firm, or government can change the value of At in the model even though the latter influences the values of MPL and MPK and hence the behaviour of households of the economy. This can also be seen as total factor productivity and is sometimes called technology. Technology here is the knowledge about available methods (techniques) of production. This knowledge can come from anywhere; for example, it may include religious, cultural or political factors.
Given the goods market clearing condition, Yt = Ct + It, and the definition of saving, St = Yt – Ct, it is clear that in this model, investment must be equal to saving, i.e. St = It. This implies that
St = sYt = It = sAt K Ltα α1t− (3)
Recall that the capital stock in the next period is equal to the capital stock left over after depreciation from the current period, plus investment from the current period, i.e.
Kt+1 = + −It (1 δ)Kt (4)
Capital accumulation is the change in the capital stock between periods. Rearranging the above equation, the change in the capital stock between date t and t + 1 is given by:
Kt+1 − Kt = It −δKt (5)
For the time being to simplify notation let us assume a constant level of population and technology level. In other words, Lt = L and At = A. Next by (3) and (5) we find that
Kt+1 − Kt = St −δKt α α1− ) −δKt (6)
= s A K L( t t t
In other words, the change in the capital stock equals saving minus depreciation. Note that the diminishing returns to capital due to decreasing MPK (or, equivalently, due to α<1) implies that as capital stock increases, national income and hence saving increases only less than proportionately while the maintenance costs (i.e., depreciation) increases in constant proportion to the existing capital stock. Consequently, the net growth in the capital stock (the L.H.S. of equation 6) declines as the capital stock increases. It follows then that a nation cannot continue to become richer indefinitely by saving a constant proportion of income.
3. The Steady State
In the steady state saving would be just enough to meet investment required for maintenance of the existing capital stock such that capital stock does not change, i.e., Kt + 1=Kt =K*, where
sA(K*)α αL1− =δK * (7)
Note that as the capital stock increases, aggregate saving increases, but at a decreasing rate (due to diminishing returns to capital). However, depreciation increases in direct proportion to the capital stock. We can illustrate this on the following graph:
δ K t sY t K* K t ∆ K t sY = t - δ K t sY t = δ K t ⇒ ∆ K t = 0 ∆ K t ,>0 Figure 1
When the level of the capital stock is below K*, aggregate savings exceeds depreciation. Thus the change in the capital stock is positive, and Kt increases. However, at K* (the steady state), growth of the capital stock ceases because aggregate saving is just enough to make up for depreciation.
Since growth of output depends on growth of the capital stock, once the capital stock reaches K* growth of output ceases.
Proposition 1: There is a unique steady state K* > 0, Y* > 0 for the above economy and their values as functions of the parameters of the model are given by
K* = L (8)
Y* = α L (9)
⇒ Y * =δ . (10)
K * s
Derivation of K* and Y*: We consider the case when Lt=L and At=A. In other words, the population and technology do not grow. To find K* we set Kt=K* in equation (7) and then solve for K* to get (8). Then we plug the value of K* into (2) to get Y*.
Proposition 2: If the capital stock of an economy is different from its steady state value, then the capital stock and output approach their steady state values.
• Proposition 2 implies that growth rate of capital or output may be positive in the short run but it disappears in the long run. As the capital stock converges to the steady state, the growth rate converges to zero. This can be illustrated as follows:
t=0,1,2,... K* K 0 K t ∆ K K t t Growth Rate Figure 2
Recall that the no-arbitrage condition for investment implies that
rt = MPKt – δ
Proposition 3: As the capital stock increases towards the steady state, the real interest rate falls because of diminishing MPK. The steady-state level of the real interest rate is given by:
r* = MPK(K*) − =δ αAK *α α−1 1L − −δ
=α(Y */K*) −δ
=αδ−δ (11) s
Q: How does the model’s prediction of the real interest rate compare with the data? First, we define real interest rate as follows: r=R-πe, where R denotes the nominal interest rate on government bonds and πe denotes expected inflation rate. If we have data on R and π we can calculate r from the data such that rdata=Rgovt. bonds - πe. Suppose that R=……. , πe=…….. Then rdata=……. Now compare this with rmodel: α=……. , δ=……, s=……, rmodel=……. If the estimated values of the parameters of the model as well as the model are reasonable then the gap between the data on the real interest rate and its counterpart in the model should be very small.
4. Implications for Growth Rates
Solow (1956) implies the following propositions regarding the dynamics of growth rates that offer possible answers to the question why countries grow at different rates:
Proposition 4: A poorer country with relatively lower aggregate capital stock will experience a relatively higher rate of growth.
Proposition 5: In the development process of each country, growth rates will be higher at earlier stages of development than at later stages.
5. Policy Implications
• Conventional policies for growth such as those advocated by the World Bank encourage poorer countries to save a higher fraction of their income.
• In the context of the Solow model, an increase in the saving rate will mean that saving will be higher at every level of the capital stock. The saving function will shift upwards and the steady state capital stock will increase, as shown:
K* δ K t s 2 Y t K*’ K t An Increase in the Rate of Saving s 1 Y t s 2 > s 1 Figure 3
• Such a policy will temporarily increase the growth rate. However the long run growth rate is still zero, i.e. the policy has no effect in sustaining long run growth.
• Policies such as capital gains tax cuts or investment tax credits may encourage accumulation of physical capital in the short run but would fail to secure long run growth of a country.
6. Capital Accumulation with Population Growth
So far we have assumed zero population growth, i.e. a constant Lt. A once-and-for-all increase in the population will have the same effect as an increase in the saving rate as more people are saving. The saving function will shift upwards and the steady state capital stock will increase. The growth rate will increase temporarily, but growth still disappears in the long run.
What about population growth at a constant rate? Suppose that the labour force grows at a constant rate n, i.e. Lt+1 = (1+ n L) t .
What this will mean is that the saving function will keep shifting upwards forever. Thus the value of the capital stock (and output) will never reach a steady state.
Increasing national income by increasing population is not a good strategy according to the model, however. Suppose we redefine all our variables in per capita terms. A lowercase letter stands for a per capita value, i.e. y = Y/L and k = K/L. Note that in absence of depreciation, to keep the value of k constant K must grow at the same rate as population, i.e., at a rate n. If, however, the per capita capital stock k depreciates at a rate δ, then to keep its value constant, the minimum required investment would be equal to (δ + n)k. In other words, population growth effectively requires additional capital to maintain a constant stock of capital per capita. Given a fixed amount of capital stock, if the population grows at a rate 1% per year, then 1% of the existing capital stock per capita needs to be budgeted from any additional saving for only maintenance of the status quo before any part of saving can be used to supply additional capital or output or consumption goods for someone.
Q. How to get a production function in per capita terms:
Yt = A Kt α αt L1t− production function
Yt AKtα L1t−α divide through by the population , but Lt = L Lα αt 1t− .
Y K Lα α1−
⇔ t = A t 1−tα the L1t−α on the top and bottom cancel out.
Lt L Lt t
⇔ t = A Kt α where Yt is ouptut per capita and Kt is capital per capita.
Lt Lt Lt Lt
⇔ yt = Aktα production function in per capita terms.
An Empirical Note: The term per capita may be misleading if we wish to do empirical work using this model. In the model, the variable L refers to labour input and not the total population. Consequently, in the data, the variable L corresponds to total labour hours employed in productive activities. However, given the availability of the data, economists often measure L by using data on output per hour or output per worker or output per labour force.
Figure 4 below plots the saving and minimum required investment per capita as a function of per capita capital stock.
( δ + n)k t sy t k* t k t ∆ k t sy = t - ( δ + n)k t sy t = ( δ + n)k t ⇒ ∆ k t = 0 ∆ k t ,>0 Figu re 4
In the model, the per capita capital stock and per capita output cease to grow and converge to their steady state levels denoted by k* and y* respectively.
• Steady state values k* > 0 and y* > 0 when population grows at a rate n are:
sy* sAk *α sA
k* = (δ+ n) = (δ+ n) ⇔ k* = δ+ n
y*= A k( *)α
Proposition 7: Population growth at a constant rate fails to produce a sustained growth of per capita output. Moreover, an increase in the population growth rate causes a decrease in the long run standard of living, measured by per capita income.
Note that all remaining propositions of the model without population growth remain true but they hold only in per capita terms. If two countries starts economic growth with the same size of initial population and with the same saving rate then the country with a higher growth rate of population will end up with a lower capital stock, K* in the steady state.
7. The Golden Rule of Saving (Or Investment) – How to Maximise Consumption Per Capita So far we are encouraged to think that increasing our saving rate may be good, since it does increase long run per capita income. However, a country should not be obsessed in raising its per capita income by any means. In fact, a more important measure of the standard of living is consumption, which is income minus saving. Therefore, the real question is if we can increase consumption and not just income by increasing our saving. Or, even better, we can ask for a “golden rule”: What is the optimal rate of saving that will maximise the consumption per capita in the long run?
In the steady state, the capital per capita k* depends on the rate of saving s. Now our goal is to find the level of k* and s* which maximizes the steady state consumption per capita. This is not the greatest level of capital per capita that we can sustain, as we shall see. However, maximising consumption per capita should be our goal if we gain pleasure only from consuming rather than saving, which is after all only a vehicle for increasing our future consumption!!
The Golden Rule: The optimal rate of saving that maximizes the value of steady state consumption per capita is called the golden rule saving rate (sGR) and the corresponding capital stock is called the golden rule level of capital stock (kGR).
Question: What level of k* maximizes the steady state consumption per capita c(k*)? What rate of saving would yield such k*?
By definition, consumption equals income minus saving also in per capita terms. In the Solow (1956) model, the steady state saving equals the minimum required investment in per capita terms to make up for depreciation and population growth. Consequently, as a function of the steady state per capita capital k*, per capita consumption, c*=c(k*), is given by: c*= y*−s*, where y*= f k( *) and s*= (n +δ) *k . In other words,
c(k*) = f(k*) – (δ +n)k*.
A Graphical Illustration:
The Golden Rule
f(k t national income pc )= ( δ + n)k t = investment for maintenance pc k t sy 1 t k* Figure 5 s GR y 1 t k GR c(k* 1 ) c max Slope = ( δ + n ) c(k t f ( )= k t ) - ( δ + n)k t consumption pc =
From looking at the graph we can see that the optimal level of capital occurs when the slope of the production function and the capital maintenance ( δ+n) line are the same. This occurs when the gap between production and capital maintenance (ie consumption) is at it’s greatest.
Algebraic Derivation: f k( t ) = Aktα; c k( *) = f k( *)−(δ+ n k) *.
To maximize consumption per capita we simply set the derivative of c(k*) to zero. Or, equivalently, set: c k′( *) = 0 .
⇔ f ′(kGR) =δ+ n. (Graphically, this refers to the tangency condition.)
⇔ αAkGRα−1 =δ+ n .
⇔ kαGR−1 =δα+An .
⇔ kGR = δα+An = δα+An−1−α
⇔ kGR =δα+An . [Note: x−a = 1xa ].
Q. What is the optimal rate of saving?
[Hint: Compare the specific expression for kGR and the general formula for k* and note that sGR= ].
8. Growth with Technological Change
So far we have been assuming that the state of technology has remained constant. All of the Solow (1956) model’s conclusions about convergence to a steady state arise from the assumption of diminishing returns to capital.
The per-capita growth rate will also increase temporarily, as shown below:
δ k t sy 2 t k* 2 k t A Once - and - for - all Technological Ch ange sy 1 t k* 1 y 1 = A 1 k α t y 2 = A 2 k α t A 2 > A 1
Suppose that we have continuous technological progress at a rate γ such that At+1 = (1+γ)At , γ > 0, as well as continuous population growth. In this case, the per-capita savings function will continuously shift upwards. Thus in the Solow (1956) model, a sustained technological progress generates a sustained rate of growth of per capita output in the long run. The model implies that all countries should grow at the same rate (the rate of technological progress) in the long run, provided that technology is a public good so that everyone can take a free ride.
Problems with Solow (1956) Model:
1. Growth based on the accumulation of physical capital alone disappears in the long run because ofthe diminishing marginal productivity of capital.
2. The long-term growth rate depends entirely on factors like the population growth rate andtechnological progress that are beyond the scope of the theory.
3. The model cannot explain why countries experience sustained technological growth and whylong run growth may vary across countries.
Checklist of key concepts
q How the capital accumulation process and convergence to the steady state.
q The effect of an increase in the savings rate.
q The effect of population growth at a constant rate.
q The effect of technological change.
q Problems of the Solow (1956) growth model.
Endogenous Growth and Equity: A Human Capital Approach
© Debasis Bandyopadhyay
Key Question: Why might the growth rates between two countries vary in the long run?
In the Solow (1956) model that we have studied in the previous lesson, the only source of sustainable long run growth of per-capita output is continuous technological progress. However, technological progress is exogenous to the model, that is, it is not explained by the Solow (1956) model.
In this lesson we will first look at some “new growth theories”. These theories or models are of endogenous growth. This means that the growth is an outcome of the model or system itself and as such is fully described by the model. Some theories of endogenous growth attempt to explain technological progress itself, while others model how continuous growth can result even in the absence of technological change.
Having introduced some of the new growth theories, we will then look at the issue of growth and income inequality.
2. Romer’s (1986)2 Model of Endogenous Growth
• Romer’s vision was to re-define capital to include human (H) as well as physical capital (K).
• Human capital consists of the abilities, skills and knowledge of particular workers. It is distinct from “knowledge” in that it is rival and excludable. For example, if an architect is using their full effort to design a building, it cannot be used for any other purpose simultaneously.
• Physical and human capital work as complimentary inputs in production. Output depends on the stock of human capital (H), as well as the stock of physical capital (K) and unskilled labour (L).
• An increase in the stock of human capital can offset diminishing returns to physical capital. For example, if factory workers learn how to use their machines better and improve the factory processes, they can increase the productivity of existing machines.
• Similarly, an increase in the stock of physical capital can offset diminishing returns to human capital.
• Therefore, we may have constant returns to both human and physical capital together. This is illustrated below:
2 Romer, P. M. (1986). Increasing returns and long run growth. Journal of Political Economy, 94.
In Romer’s model national output Y is a function of H and K and is given by:
Y = AKα β αβH L1− − now let us transform everything into per capita terms such that k =
H and h = , by dividing through both sides by the population L. L
Y K H Lα β αβ1− −
⇔ = A α β αβ1− − where
L L L L
⇔ y = Akα βh
For constant returns to both human and physical capital we assume α β+ =1and K and H are perfect substitutes such that we can replace h by k and get y=Ak.
0 0 K H MPK MPK (H 1 ) MPK (H 2 ) H 1 H 2 MPH (K 1 ) r K 1 K 2 MPH (K 2 )
Figure 7 k t sy t sAk = t sAk t ( δ n)k + t ∆ k t
As the economy’s physical capital stock increases, the marginal productivity of human capital in the economy increases. Consequently, households increase their investment in human capital for a given interest rate. As the human capital stock in the economy increases, the marginal productivity of physical capital stock increases. It follows that for any given interest rate, the desired capital stock and hence investment in physical capital increases. The resulting increase in the physical capital stock increases the marginal productivity of human capital and the economy grows forever endogenously.
There are two important points to note. (1) Endogenous growth requires constant returns to capital as opposed to diminishing returns that Solow (1956) assumes. (2) In this model, an increase in the savings rate will increase the growth rate of per-capita output permanently unlike Solow (1956) where the effect is only temporary.
3. Lucas’s (1988) Model of Endogenous Growth
This model is similar to Romer (1986) in that it expands the definition of capital to include human capital. Lucas’s (1988) vision was to assume that the average productivity (or technology parameter A) of labour in a country depends on the average stock of human capital due to external effects. This would measure not only the productivity of an individual, but also the productivity of all the people in the economy.
The average human capital stock is endogenously determined in the model. In particular, the rate of growth of human capital is determined by the fraction of time that the average household devotes
dH for skill formation instead of working or enjoying leisure. In other words, = h v( ), where v is the dt fraction of a year that the households in the economy devote, on average, to skill formation, h(.) is a increasing function of function of v. A country can grow faster than others if it allocates a higher fraction v of year to skill formation.
The process of allocation of labour between production of human capital and the production of other goods effectively determines the average human capital of a country. In particular, the rate of technological change and hence the rate of growth is proportional to the average fraction of time that households of a country devote to the accumulation of human capital.
sy t 1 sy t 2 k t k 1 * k 2 * H y Hk Hk y H ExternalityParameter H AverageHumanCapitalStock t t t t 1 1 2 2 2 1 0 = = > > → → θα θα θ , , ; ) ( δ + nk t Figure 8 g H H hv h t t = > = ∆ ()0 ∆ hvH H t t = > () 0 ⇒∆ > y t 0 for all t
The graph above shows the effect of a once-and-for-all increase in the stock of human capital (from H1 to H2). The steady state physical capital stock increases permanently while the growth rate increases temporarily. However, in Lucas (1988), the human capital stock increases continuously, at a rate that depends on the amount of time households invest in developing it, as discussed above.
Consequently, a sustainable long run growth in physical capital and national income will result at a rate determined by the rate of investment in human capital.
4. Disparity in Growth Rates and Income Inequality Across Countries
• The World Development Report (1991) points out a quite different problem related to economic growth. It reports that a higher income inequality is associated with a lower rate of growth.
• It defines income inequality by the ratio of the income shares of the richest 20% of the population to the poorest 20%.
• Using cross-country data it plots the growth rate of per capita income against this measure of income inequality and suggests a negative relationship between income inequality and the rate of growth (Ref: World Development Report 1991, Page 12, Page 137).
• In this context we can also contrast the growth experience of Latin America to that of East Asia and Japan. The Latin American countries are typically characterised by high degrees of income inequality and low rates of growth while the rapid growth with reduction of income inequality generally characterises the experiences of the East Asia and Japan.
• The existing growth theories have yet to explain these phenomena. Establishing a theoretical link between income inequality and the rate of growth thus offers a challenge to the new growth theories of the 1990s.
5. Growth with Equity with Investment in Human capital, Solow (1992)
Traditional policies to promote economic growth encourage saving and investment in physical capital. Such policies will increase per-capita income (i.e. raise the standard of living) and increase the growth rate (at least temporarily). However, it is usually the rich who are owners of physical capital. Thus the gains from the growth accrue to the rich, which widens income inequality.
Solow (1992) argues that policies that encourage investment in human capital will have similar implications for economic growth as policies that encourage investment in physical capital. Solow (1992) proposed that higher growth with reduced income inequality could result from policies that encourage investment in human capital. Growth spurred by additional human capital generates more income among newly skilled individuals who can be a member of any income group, including the poor.
Solow (1992) Key Points:
• Given that human capital cannot be easily defined or measured, Solow (1992) considers two possible ways to account for the effect of human capital using the Solow (1956) model that we discussed in the previous lesson. In first case (1) human capital is a perfect substitute for physical capital. In the second case (2) it is a perfect substitute for labour. However he mentions that in the real world it is probably the intermediate case that prevails.
• Given the value of alpha is only 1/3 (the returns to physical capital), the contribution of human capital to growth will be larger in the second case than the first case.
• Growth with equity is possible through investment in human capital.
Two Alternative Model Formulations:
Model 1: Human capital is a substitute for physical capital:
Yt = (K Ht t )α αL1t−
By taking logarithms on both sides, differentiating w.r.t. time and setting dt=1 we get,
∆YYtt =α∆KKtt + ∆HHtt + −(1 α) ∆LLtt
Note: For the US, econometric estimates show that α = 0.33. According to Model 1, if population growth remains constant, a 10% increase in the stock of either form of capital would lead to a 3.3% increase in output.
Model 2: Human capital is a substitute for labour:
Yt = (Kt ) (α H Lt t )1−α
By taking logarithms on both sides, differentiating w.r.t. time and setting dt=1 we get,
∆YYt =α∆Kt + −(1 α)∆LLtt + ∆HHtt t Kt
According to Model 2, population growth remaining constant, a 10% increase in the stock of physical capital would lead to a 3.3% increase in output,while a 10% increase in human capital would increase output by 6.7%.
How to get growth rates for model 1 and model 2:
We start off with the original equation of the production function:
Yf = A Kα Lf 1-α
Next we take the natural log of both sides: lnYf = ln AKα αL1f−
Now we differentiate both sides with respect to t (use the chain rule) and set dt=1
1 dYt 0 α 1 dKt + −(1 α) 1 dLt
Yt dt Kt dt Lt dt
which simplifies to:
∆Yt ∆Kt (1 α)∆Lt where ∆Yt is the growth rate of output and ∆Kt is the
Yt =α Kt + − Lt Yt Kt
growth rate of capital and ∆Lt is the growth rate of
Lt labour force
Q: How does an investment tax credit or a capital gains tax cut compare to a tuition or training subsidy in influencing a country’s long run growth rate and its underlying income distribution?
Checklist of key concepts:
q The concept of human capital.
q How endogenous growth results from Romer (1986).
q How the rate of growth is determined in Lucas (1988).
How growth with equity is possible and the key points of Solow
 When comparing standard of living across countries, we should not use the official exchange rates but the purchasing power parity (PPP) adjusted exchange rate to construct comparable index of income or consumption per capita.  Lucas, R. E. (1988). On the mechanics of economic development. Journal of Monetary Economics, 22.  Here we will briefly introduce the basic model that Solow introduced in his George Seltzer Distinguished Lecture delivered in 1992 at the University of Minnesota. The lecture title was “Growth with Equity through Human Capital Investment.” You can watch the videotaped version of that lecture in the Audio-Visual Library. There are five copies made for short loan. The tape runs for 30 minutes.