CLASS #18: ENDOGENOUS GROWTH

• Have poor countries really caught up with the rich ones?
• Some empirical facts seem to be starkly at odds with the standard neoclassical growth model (which we called the Solow growth model).
• Twin Peaks growth
• There is some evidence (Quah, 1994) that countries tend to cluster in one of the following two groups: a `rich country' club, and a `poor country' club.
• Historically, for a country that is in the lower end of the income per capita distribution the chances of substantially improving its rank (i.e. to end up in the upper end of the distribution) are slim. Likewise, for a country that is in the upper end of the distribution the chances to end up in the lower end of the distribution are also small.
• In simple words, poor countries seem to stay poor (where poor is defined in relative terms, i.e., relative to the word average income), and rich countries tend to stay reach: this leads to a Twin Peaks distribution of income.
• Of course there are exceptions - countries that move substantially in the ranking: historically, Japan moved from being poor to being the second richest, and Argentina moved in the opposite direction.
• This evidence made some economists disenchanted with the neoclassical growth model.
• Moreover, growth in the neoclassical model is due to exogenous technological progress, and what is exogenous is, by definition, unexplained.
• A model that does not fully explain why India never took off, and why Thailand did, is not very appealing:
  Is the government in India doing something wrong, and in Thailand something right? The neoclassical model does not really address this questions: all the differences in growth rate -after reaching the steady state- to exogenous technological shocks.
• We will study two models of endogenous growth, which display the following characteristics:
  • growth is endogenous to the model (whence the name) - that is, explained within the model- and persistent
  • there is no convergence, if anything, countries diverge
  • the government can affect the growth rates by its policies

A first model of Endogenous Growth: Growth and Spillovers (the AK model)

• Recall the production function -in per capita terms:
  
  $$y_t=A_t k_t^\alpha$$
  
  

• Let us make the assumption that the level of technology A_t is proportional to the per capita amount of capital in the economy:
  
  $$A_t=A k_t^{1-\alpha}$$
  
  

• I will give two examples to justify this assumption
• First example: ``Capital and Knowledge Spillovers"
• Think at the following situation: a firm (ITAM) renews its current set of computers (capital investment). This implies, for instance, that secretaries not only can do what they did before faster, but also that they can do something new, like exploring the Internet. As a consequence, they can be more efficient (for instance, book flights, buy books, search information, by internet- instead of spending time on the phone). Furthermore, they may teach their children, or their co-workers, how to do the same things: information spreads to the rest of the society.
• Bottom line: investment may have benefits that can go beyond simply having more `input': it may also increase efficiency.
• Therefore, the higher the level of the capital stock, the higher the amount of spillovers to the rest of the economy, and the higher A<sub>t</sub>.
• Second example: ``Learning-by-Doing"
• Suppose you buy a machine that makes espresso. you give it to the person of the coffee shop here at ITAM, and suppose instead that you give it to a coffee shop in Italy. I can guarantee you that with the same machine, and a similar quality coffee, the espresso. is much better in Italy.
• Why? Learning by doing!
  The Italian waiter has been using a similar espresso. machine for the last twenty years, or has been instructed by someone who has being knowing how to make espresso. for twenty years.
• Having more capital also implies that you have a better knowledge of how to use it. In places where there is already a lot of capital an additional unit of capital is more productive, because people know how to use it. And that additional unit of capital improve skills even further, as people get more and more experienced at using it.
• These two example justify the assumption:
  
  $$A_t=A k_t^{1-\alpha}$$
  
  

• If we plug this into the production function we get:
  
  $$y_t=A k_t^{1-\alpha} k_t^\alpha=A k_t$$
  
  

• Remember the standard law of motion for per capita capital (assuming that savings is a constant fraction s of output, as in the Solow growth model):
  
  $$(1+\lambda)k_{t+1}=(1-\delta)k_t+sy_t.$$
  
  
  Plugging in the expression for y_t we obtain:
  
  $$(1+\lambda)k_{t+1}=(1-\delta)k_t+sAk_t$$
  
  

• This implies that the growth rate of per capita capital is equal, in every period, to:
  
  $$\frac{k_{t+1}-k_t}{k_t}=sA-\frac{\delta+\lambda}{1+\lambda}$$
  
  

• This model implies that:
  • There is no steady state: the economy keeps on growing
  • There is no convergence: economies with higher savings rate, lower depreciation and lower population growth will keep on growing faster.
    This is not surprising, as there are no diminishing returns to capital
    countries diverge.
  • Growth is endogenous to the model: it is due to technology, but technology is `explained' by capital accumulation.
  • The government can affect the growth rates by its policies:
    there are externalities in the model, because my decision to invest favors everybody -through A_t: if the government puts investment tax credits so that I internalize the externality, I am going to invest more, and the economy is going to grow faster.

A second model of Endogenous Growth: Human Capital Accumulation (the AK model)

• This model emphasizes growth through human capital accumulation
• assume that the production function is of the kind:
  
  $$Y_t=A K_t ^{\alpha} (u h_t N_t) ^{1-\alpha}$$
  
  
  This production function says that the effective input of workers depends not only on the amount of hours worked, N_t, but also on the `efficiency' of these hours worked.
  Such efficiency in turn depends on the amount of human capital you have, and on the fraction of human capital u that you put in work.
  A is constant, but you can think of $A_t=A (u h_t) ^{1-\alpha}$: efficiency increases with human capital.
• The law of motion for per capita capital is as usual:
  
  $$(1+\lambda)k_{t+1}=(1-\delta)k_t+sy_t$$
  
  

• While the law of motion for per capita human capital is assumed to be as follows:
  
  $$h_{t+1}=(1-\delta_h)h_t+B(1-u)h_t.$$
  
  
  This means that there are constant returns to scale in the accumulation of human capital.
• What is u? Say that I spend all my time preparing classes: then u=1: my input to classes is very high, but my knowledge of economics is not progressing.
  At some point, because of $\delta_h$, my knowledge will become obsolete and I will not be able to teach anything interesting.
• Other example: 1-u can be the time I spend teaching my daughter, or in general the amount of human capital a society spends in education.
  It is clear that if a society does not teach to the young generations, and puts all human capital into current production, at some point the overall level of human capital will decrease.
• Note that the law of motion for human capital implies that human capital grows at the rate $\gamma=B(1-u)-\delta_h$:
  
  $$h_{t+1}=(1+B(1-u)-\delta_h)h_t$$
  
  

• The production function per capita is:
  
  $$y_t=A k_t ^{\alpha} (u h_t) ^{1-\alpha}$$
  
  
  and the law of motion of capital is:
  
  $$k_{t+1}=\frac{(1-\delta)k_t+sA k_t ^{\alpha} (u h_t) ^{1-\alpha}}{1+\lambda}$$
  
  

• Define $\bar{k}_t=\frac{K_t}{h_tN_t}$, the "capital per effective labor" ratio, and recall that $h_{t+1}=(1+\gamma)h_t$.
• Using these definitions we obtain the law of motion in terms of the "capital per effective labor" ratio:
  
  $$\bar{k}_{t+1}=\frac{(1-\delta)\bar{k}_t+sA u ^{1-\alpha} \bar{k}_t^\alpha} {(1+\lambda)(1+\gamma)}$$
  
  

• But these is the same thing as the model with exogenous technological progress!
• Conclusion: capital per capita will grow -in steady state- at the same rate of growth of human capital: $\gamma=B(1-u)-\delta_h$
• Human capital, in this model, is the real engine of growth, as its production function has constant returns to scale.
• This model has the following features:
  • As in the previous model, the economy keeps on growing
  • As in the previous model, there is no convergence: economies with higher 1-u (put more time in education and in developing its skills), and with lower depreciation of human capital will keep on growing faster.
    Countries diverge
  • Growth is endogenous to the model: it is due to human capital accumulation.
  • Unlike the previous model, all that matter here is human capital: increasing s has no effect in the long run, since decreasing marginal returns to physical capital will kick in soon or later.
  • The government can affect the growth rates by its policies:
    in this model, the thing to do is to favor education -increase 1-u. investment tax breaks do nothing to long run growth in this model.