Economic Growth * Aggregate production function relates the aggregate output (Y) to:
(a) Physical capital
(b) Labor
(c) Index of aggregate productive efficiency

Y = F(AL, K)

* Y is a flow variable because it measures GDP in a given period * K and L are stock variables, can measure at a given point in time * Y measures the services produced by K and L in period or instant t (e.g. hours of computation and hour of labors affects output, not just number of computers) * Y = 100K + 40L * One computer works 100 hours per week; each labor works 40 hours a week. If there are 2 worker then L=2; computer = 4 then K=4 * Assume economy is working at full potential * Capital is depreciated at constant proportionate rate, * L is assumed to be homogenous (everyone has same skill, education, productivity) and it grows at rate n. * If n>0 growing exponentially, n<0 declining exponentially, n=0 stagnant * Growth of labour is due to growth of population * A is defined as costless labour-augmenting technological progress * A multiplies the labour input, while leaving capital unchanged (e.g.: new organization routines, rearrangement of the flow of material in a factory, better management of inventory, other changes that do not require knowledge to be embodied in new equipment) * A is also known as index of productive efficiency or index of productivity or level of technology * A grows at an exponential rate of g; g is normally constrained to be non-negative * If the production function is Y = AF(K,L) then A in this case is the total factor productivity, it measures how much output is produced with a given combination of K and L.
Assumption about production function 1. CRS: F(Kb, ALb) = bF(K, AL) = bY 2. Positive marginal products of each factor but diminishing in corresponding factor: Y>0, Y<0 3. Inada conditions: MP are extremely large at low levels of the factor and extremely small at very high levels of the factor

* y = f(k) is the production function in intensive form that depends only one input k called capital per effective worker * Assume there is no adjustment cost in hiring K and L * r + is real rental price of capital * w is real wage * Assume that the supplies of factor services are inelastic. There is full employment and all desired savings are channeled towards investment * In a perfectly competitive firm, the profit max behavior is to hire until MPK = r+; MPL = w * Perfect competition guarantees that total payment to factor inputs must equal to total aggregate output * 3 ways to measure GDP: expenditure (CIGXM), income (rent, dividend, wage & profit) and final output approach * I = sY; s = S/Y; s is the constant fraction of income that is saved k= sf(k) + (n+g+)k * sf(k) = investment per effective worker * (n+g+)k = break even investment = the investment per effective worker that must take place per period just to keep up with capital depreciation and the fact that the number of effective worker is growing over time and each effective worker needs to be equipped with the same amount of capital as the existing ones * When sf(k) is greater than (n+g+)k, it means that more capital can be supplied to each effective worker, hence k>0 * When sf(k) is less than (n+g+)k, it means that each worker will have less capital to work with, hence k<0 * Steady state can be found when k=0
At steady state
(a) Variables per effective worker is constant
(b) variables per worker grow at rate g, the growth rate of productive efficiency or the ‘index of technology’
(c) aggregate variables (those multiplied by AL) will grow at rate of n+g
(d) since w = MPL, the steady rate is given by w = A[f(k) – kf’(k)], which means w is proportionate to A, it grows at rate g * Steady state = balanced growth path * A higher level of A will increase productivity and income per capita, but this would be a one-off effect. This will be followed by a faster population growth, and due to diminishing returns, income per capita would revert to its pre-shock value. To sum up, the growth rate of productive efficiency was zero (g=0); income per capita could not grow continuously over long periods of time. * Y/L = Af(k) output per worker can only grow from a higher level of A, a higher level of capital per effective worker or both * A higher k can be obtained from a high capital-output ratio and this is a result from a high saving rate and a low population growth * When an economy switched from a low s and high n to a high s and low n, steady state will increase, this is known as capital deepening. During the transition, much of the new capital that was invested in each period is used to equip existing workers with even more capital. * When the opposite occurs, it is a capital widening effect. The new capital that is invested per period is used to equip the new workers with the same amount of capital as the existing ones. * Conditional convergence is due to diminishing returns to capital. It is because MPK decreases that the APK also decreases (can show from the graph) * When a country has a low k, investment returns will be very high and this allows output and capital to grow very quickly * When country has high k, investment returns will be much lower because the obvious and high return investment projects will have been undertaken in the past. * The lower the starting point of a country’s initial income per worker (which is a function of k), the faster its growth rate over the period should be. There is a negatively correlation

An increase in saving rate, s will increase the fraction of output that is devoted to capital accumulation and so over time the economy will move to a new higher steady state. * From time 0 onwards, the old investment curve is no longer relevant, it is now represented as a higher curve. From t=0 onwards, the relevant fundamental equation is k= s’f(k) + (n+g+d)k and it’s corresponding steady state value is k* * At time t=0, the economy has capital stock, k(0) where it is lower than the steady state value, the rate of change of capital at time 0 must be positive. Intuitively, once a society begins to save a larger fraction of its income per period, its investment in the early stages must outweigh the investment needed to keep k constant. So capital per effective worker will gradually increase over time and eventually converge to new steady state.

A decrease in population growth rate, n will decrease the slope of the break-even investment line from the fundamental equation. * From time 0 onwards, the original line (n+g+)k is no longer irrelevant, as the new dynamics of k are given by k= sf(k) + (n’+g+d)k. since at time 0, capital per effective worker is below the new steady state value k*, k(0) must be positive. * The economic reason is that investment is now greater than what is needed to keep k constant

An increase in rate of technological progress, g reduces k*, but from the time of the shock onwards, all variables per worker will be growing at a faster rate than before the shock. Eventually, these per worker variables will be growing at a higher steady state rate than before. * An increase in s and decrease in n is known as level effects, they do not have a permanent increase in growth rate of output per worker, only a temporary one.

* in Cobb-Douglas case captures how quickly diminishing returns set in * Graphically, influences the shape of the production function, how concave it is * When is higher, k is more productive. The slope is greater, diminishing returns sets in slowly * When diminishing returns set in very quickly, a higher saving rate is not very effective in raising standards of living
Key extensions of neoclassical growth model 1. Natural resources: a fixed stock of land – Z
Y = F(K, Z, AL) * Main advantage of working with per effective worker variables is that we reduce the no of factors of production by one * Income per capita = income per worker = Y/L * (According to equation in notes) When n=0, g is a necessary and sufficient factor for sustained rising income per worker. There is a growth drag effect from fixed resources Z. The drag growth increase with the importance of the fixed resources Z in production. Captures the share of output/income that is paid to the factor Z * When n>0, there is perpetual rise in standard of living (Y/L) when there is high technological progress, low importance of fixed resources in production and a low rate of population growth.

2. Human capital, H is just an index of the stock of human capital and it captures impact of activities such as training, learning on the job and health have on a worker’s productivity * The higher the average stock of human capital per worker, the greater a worker’s productivity
Y = F(K, H, AL) – augmented neoclassical model * L – raw labour * Assume that H is determined by s(h) which is the fraction of output devoted to human capital accumulation (it’s the part of savings that is devoted to human capital) * The fraction of output is split between physical capital accumulation, human capital accumulation and consumption * Assume that K and H are depreciated at the same rate, * Countries with high level of income per worker are societies that benefit from a high level of productive efficiency, high saving rates in both physical and human capital and to a lesser extent, low population growth rate

3. Poverty trap – countries with output per worker stagnating at a very low level for the most part of the last three or more decades * Assume growth rate of population, n is endogenous. For low level of income per effective worker and therefore capital per effective worker, the population may not be able to sustain itself, so L decreases, and this results in population decline n<0, not enough money for healthcare * Once income and capital per effective worker reaches a threshold, the relation between k and L turns positive, n>0. This is because income reduces the vulnerability of the population to certain diseases, or a higher income allows people to pay for better health care. * Higher incomes may at some point impact fertility decisions. E.g., the opportunity cost for educated women staying at home looking after children will now be greater, parents might choose to have less but more educated children, or there may be no easy opportunities to use children as productive assets. So they choose to have less children hence making n approach to 0 as capital and income become very large * Assume rate of technological progress, g is endogenous; A does not change until income and capital reach a certain level. This might be that most modern or relatively modern technologies are simply not appropriate for very poor economy. The country may lack the basic infrastructure to use them. It may also be that those technologies require a certain amount of capital expenditure by individuals and as people cannot get funding to invest (since they lack collateral and/or financial markets are not well developed), they cannot adapt more sophisticated technologies than the ones in use. Until the economy reaches a certain threshold, g will be 0. But beyond that threshold, g will turn positive and then it stagnates at a positive level. * Therefore, g and n are a function of k = g(k) and n(k) * This causes the break even investment line to be curved, it intersects sf(k) three times (plus the origin, but that is not a plausible steady-state) * The middle point is known as an unstable steady state; even a tiny shock to income would force it to move away from it. * The third intersection is a good steady state. In this equilibrium, g is positive but constant and the rate of population is also constant (and possibly equal to 0) * K1 is a bad steady state, because g=0, there is no technological progress, because the rate of population is very high, this is the poverty trap. It is a trap in the sense that it is a self-reinforcing mechanism that causes poverty to persist. At this point, even though there are some people using modern cell phones, there is a vast majority of the population will be engaged in activities such as subsistence farming that makes no use of modern technology. * People at these dire circumstances still choose to have many children and also not to adopt modern production methods because of the “vicious circle”. * High fertility rates may be the result of the combined importance in a traditional society of social norms, lack of opportunities and poor information. * Eg: a poor women bears many children, they are malnourished and/or end up poorly educated. As a result, these children will not be very productive and so they will not be able to use sophisticated technologies when they grow up. Also, because the poor lack collateral to be able to borrow to buy ‘modern tools’ they can easily be stuck with traditional relatively unproductive technologies. As a result, they produce little output and income and the circle begins again. * Club convergence is when there is 2 separate groups of countries that will converge to different steady states. If it start below k* then it will converge to the bad steady state, if start above k*, it will converge to good steady state * Today there is a group of countries that have benefitted from sustained income per capita growth for more than a century due to invention and/or adoption of even better productive methods and the accumulation of various types of capital. By contrast, today’s poorest countries have never been able to sustain growth for long enough to lift most of their people out of poverty and many seem to have fallen into a poverty trap which is proving difficult to escape from.

Technology gaps and catching up hypothesis * Poor, backward countries could industrialize much faster than rich countries because they can rely on technology transfer. It should be much easier to imitate or implement existing technologies than to develop entirely new ones. * The technology growth rate is given by g=s(t)A^/A * S(t) is the fraction of GDP (resources) devoted to “adapting” foreign technologies * is the diminishing return to technology adoption. The greater the value, the slower the diminishing return sets in * A^ is the world technology frontier * A is the technology level of the country in question * A^/A is the measure of “relative productivity” or the ‘technology gap” which is greater than one * The greater s(t) the greater g * The quality of firms and people adopting foreign technologies is not the same across economies, some might be more effective than others, making larger * g also depends positively upon the distance between the technology frontier and the country’s current productivity level. A larger gap suggest it should not be difficult for the relatively poor country to adopt technologies that are a bit more sophisticated, so should result in big g * A small current gap suggest imitating almost state of the art technologies has been difficult * The graph is downward slopping cause the larger the gap, the greater g is * At time 0, if the less developed country s(t) and if it starts with a relative productivity equal to A(0)/ A(0)^, A will grow at g(0). Since A grows faster than A^, the gap between two productivity levels must narrow, g must decline over the catching up process. * Once steady state is reached, income per capita in the economy will be growing at a constant rate g^, might be still at a lower productivity level, A than developed country. * However, this catching up process must be supported by other conditions such as human capital, education, and it should also be ensure that the returns to the investment in modern technology be properly protected by property rights.

Definitions: 1. Y is a quantity index that measures the real value of all the final good and services produced in the economy in a given period in physical units 2. There is conditional convergence when an economy’s growth rate is greater the further away it is from its steady state or BGP.