Unicas. Master program in Global Economy and Business. Prof. Nadia Cuffaro. Development Economics 2012-13. Lecture notes
Endogenous growth
The revival of interest for growth since the eighties is essentially linked to two disappointments with the Solow growth model. The first is the essential shortcoming that the long-run per-capita growth rate is determined entirely by an element –the rate of technological progress –which is exogenous to the model. Secondly, the empirical evidence does not support the model’s prediction that countries with lower starting values of the capital/labour ratio have higher per capita growth rates and tend thereby to catch up (‘converge’) to those with higher capital labour ratios[i].
It is easy to show that endogenous growth can be obtained by eliminating the hypothesis of diminishing returns to capital, i.e. by specifying the production function of the Solow model as
1.
where A is a positive constant reflecting the level of technology. Since equation 1 implies that
2.
the growth rate of the capital labour ratio can be expressed as
3.
The relationship between k and its growth rate is represented graphically in figure 3.3, where
Figure 3.3 The AK model
the downward sloping sf (k) function is replaced by the horizontal line sA: all per capita variables in the model grow at the same rate, given by the difference between sA and (n+) (Barro and Sala-i-Martin, 1995).
Technical progress
The tendency for diminishing returns could be eliminated –while maintaining the assumption of perfect competition – if there were knowledge creation as a side effect of investments. Arrow had made the accumulation of knowledge endogenous already in a 1962 article, by assuming that the growth of A was proportional to the accumulation of physical capital K as a result of learning-by-doing, i.e. by interpreting the accumulation of knowledge as a by product of mechanisation.
Romer used a similar framework in a 1986 seminal paper where the condition of decreasing marginal productivity of capital is maintained at the level of the firm, and a positive technological externality, linked to the total amount of capital is introduced at the aggregate level [ii]. The production function for firm i is
4. Yi=F(Ki,AiLi)
The additional assumptions are that an increase in a firm’s capital stock leads to a parallel increase in its stock of knowledge Ai; and that once discovered knowledge spills over instantly and all firms can use it at zero cost (i.e. a firms knowledge is a public good that any other firm can access at zero cost). Therefore
5. For all i.
Hence the production function for firm i is
6. Yi= KiβLiαKγ with (α+β)=1
With N identical firms Y=NYi; K=NKi; L=Nli and since (α+β)=1 N=NαNβ
Substituing in 6
7. Y=NYi=ANβKiβNαLiαKγ= A(NKi)β(NLi)αKγ= AKβLαKγ
With (β +γ)= 1 there are constant returns to scale in capital at the aggregate level
As a way of further illustration of this point we may refer to a simplified version (as formulated by D. Romer, 1996) of models where –like in the Solow and the AK model– knowledge accumulation is the driving force of growth, but knowledge production is explicitly modelled. The model is described by equations 7 to 10
7.
8.
9.
10.
Where aK andaL are the fractions of capital and labour used in the research and development sector (R&D), (1- aK ) and (1-aL) are the fractions used in the goods producing sector, and 0<<1[iii] and a dot over a variable denotes its derivative with respect to time.
If for simplicity we start by eliminate capital from the model
11.
12.
Since equation 11 implies that output per worker y is proportional to At, the growth rate of y is the same as the growth rate of At, and the equilibrium growth rate depends positively on the parameters γ, η (population growth) , and θ
Human capital
1. Yt=KαtH1-αt
Where K isphisical capital and H human capital.Familiessave in twoways, oneisbyallocatingtheirincometotheirchildreneducation.
2. Kt+1-Kt=sYt
3. Ht+1-Ht=qYt
Wheres and q are the marginalpropensitytosave and the marginalpropensitytosaveforeducation.
4.
Output growth is a function of inputs growth and specifically depends on the propensity to save and invest in physical capital per the average product of physical capital, and the propensity to save and invest in human capital per the average product of human capital
Microeconomic analysis of families’ choices
The relation between economic and demographic change is explored at a different level by the literature on the economics of the family, which mostly uses a neoclassical framework to provide microeconomic analysis of families’ choices and especially of the motives for limiting family size. Non-market activities of household are viewed as optimising decisions representing economic choices and changes in the family are viewed as a result of external shifts in prices and incomes caused by technological change[iv]. In the ‘household demand’ model of fertility families maximise an utility function depending on children quantity (N), children quality (Q), and other consumption goods (C ),
U= f(N,Q,C)
subject to a production function constraint in which the production of children and of C requires inputs not only of conventional goods but also of time.
Although children are normal goods –i.e. their demand increases with income– high and rising income may be associated with low and falling fertility. Since each child imposes a time cost of rearing, as the value of time (its opportunity cost in terms of wage)-and especially the value of women’s time– increases, the quantity of children becomes more expensive. As Schultz (1973) points out ‘Whereas Malthus assumed that the price of children would remain constant... the cost of children increases with the rise in price of human time’. An increase in wages induces households to choose small families and larger capital (physical and human) investments on each child (Birdsall, 1988; Tamura, 1995).
A growth model in which economic development influences family decisions about the number of children and, hence, the fertility rate, is presented by Barro and Sala-i-Martin(1995). The model uses (in a continuous time framework) the results of Becker and Barro (1988) that relate parents’ utility to the utility of their children through the function
1. U0=u(c0,n0)+a(n0)n0U1
where U0is the utilityof an adult, U1is the utility of each child, n0is the number of children per adult, anda(n0) represents the degree of altruism towards each child. Becker and Barro assume that a()>0, a’()<0, i.e. the utility of an adult is increasing in the utility of each child (one-sided altruism), but the utility of an additional child is decreasing in the number of children. Hence the utility of an adult is increasing and linear in the utility of each child and increasing and concave in the number of children[v].
As for the family’s budget constraint, a key assumption is that a child rearing costs is =bo+bk where bo represents the goods cost of child rearing and the bk part represents a part of the cost that increases with the capital intensity k, which in the model represents the per capita quantities of human and physical capital. This formulation reflects the notion that, because child rearing is intensive in parental time, its cost tends to raise with parents’ wage rates or with other measures of the opportunity costs of such time. Hence, the family’s budget constraint in the model is expressed as
2.
where is the growth rate of the household’s own per capita assets, w and r are respectively the wage rate and therate of return on assets, n and d are the family’s birth and mortality rate, c and k are the family’s per capita consumption and assets respectively.
The transitional dynamics of this model imply that if the economy begins with a capital/labour ratio lower than its steady state values, k increases and c/k – which expresses the ratio of the income effect on the demand for children, represented by c, to the cost of children, which depends linearly on k –declines monotonically towards its steady state value. Since during the transition to the steady state the fertility rate moves in the same direction as c/k, n also declines –. That is (with a given mortality rate) fertility declines steadily as the economy develops, a result that accords with empirical evidence. The positive association between fertility and per capita GDP which the data show at extremely low levels of income appear when considering the goods cost -b0 of child rearing, as this introduces a positive income effect in the relation between fertility and per capita product. Moreover, since the goods cost is relatively more important in poor countries, the net positive relation between fertility and per capita product tends to appear only at low levels of per capita product.
Becker, Murphy and Tamura (1990) obtain more complex results in terms of the population-development linkages in a model with endogenous fertility. The model uses the Barro and Becker (1988) dynastic utility function and two additional assumptions. An increase in the stock of human capital raises per capita income and hence has a positive income effect as well as a negative substitution effect –the cost of the time input must rise as H increases– on the demand for children. The income effect dominates in economies with little human capital but the substitution effect eventually becomes dominant when H is large enough. The additional, key assumption is that the rate of return on human capital H, unlike for physical capital, does not monotonically decline as the stock of human capital increases. Rates of return are low when there is little human capital and they grow at least for a while as human capital increases.
The dynamics of the model are described by figure 3.4
Figure 3.4 Human capital and growth
The function hh gives human capital in period t+1 as a function of the amount in t. Low levels of Htgenerate Ht+1<Ht and high levels of Ht generate Ht+1Ht .
The shape of the hh function depends on the behaviour of the rates of return on investments in human capital versus the discount rate on future consumption. At the origin and at low levels of H the economy does not invest in human capital or the amount invested is less than the capital that wears out (because low H implies high fertility which in turn rises the discount rate on the future while the rates of return on H are low). Thus there is a locally stable steady state at H=0. Countries may be trapped in a Malthusian, underdeveloped steady state, with high birth rates and low level of human capital. As the amount invested in H continues to rise, fertility declines, the discount rate on future consumption decreases while the rates of return on H increases and a new steady state emerges (when H is sufficiently large that the rate of return equals the discount rate).
[i] Countries with same production function and the same values of the parameters s, n and d would have the same steady-state values k* and y*. Since the relation between the level of k and its growth rate is negative in the model, if the initial k values differed among countries –because of past specific events– poor economies should grow faster per capita than rich ones, converging to the common steady-state values. Although the debate on this issue is far from being conclusive, we may say that the empirical evidence accords better with another notion of convergence, the idea that an economy grows faster the further it is from its own steady state, as determined by differing values of the key parameters –i.e. “conditional”, rather than “absolute” convergence (Barro and Sala-i-Martin, 1995).
[ii] The presence of a positive externality in the production function has also the effect of generating a divergence between the private and social marginal productivity of capital and therefore generates a sub-optimal rate of growth of the decentralised economy.
[iii] For ,, aL and aKequal to 0 and =1 the model simplifies to the Solow model with a constant, exogenous rate of growth of technology and Cobb-Douglas production function.
[iv] A contrasting institutionalist interpretation of fertility decline is given by John Caldwell (1996).
[v] An additional restriction on a is that a<1, i.e. the weight attributed to each child’s utility is never larger than the weight attributed to the marginal utility of own consumption.