1. This question deals with the Malthusian model of economic growth. Provide a brief outline of the model and its equilibrium.
問題涉及經濟增長的馬爾薩斯模型。提供模型的簡要概述和它的平衡論介紹。
Here is the brief introduction of the model of the Economic Theory of Malthus: at the very first let us suppose an economy with a given area of land; if the labor force keeps growing in this economy, the area of the spare land appropriated to the labor force will surely decrease by that time. In this processing, the Law of Diminishing Marginal Returns begins to take its effect(David N. Weil, 2005, p547). The continuing increase of the ratio of labor against land leads the margin output of labor to go down accompanied with a decrease in the productivity. If this process arrives to the extreme circumstance, as a result, the economy can barely provide the products which are supposed to sustain the people's basic survival owing to the excessive labor.（Weil, 2005 and Williamson, 2005）
The model of the Economic Theory of Malthus does not give a determinative formula; the following formula will help to its understanding.
簡要介紹馬爾薩斯的經濟理論模型：在第一個讓我們假設一個經濟體的土地給定的區域，如果勞動力市場一直在這個經濟成長，備用土地面積劃撥給勞動力一定會到那個時候減少。在這種處理，邊際報酬遞減法開始采取其效果（大衛·N.韋爾，2005，P547）。對土地勞動的比例持續增加導致勞動的余量輸出下井伴隨的生產率下降。

This theory applies to the economic model as below:
The total income is the function of all the production factors; Production equation is Y= F(K,L). Y represents the output; K stands for the capital and L means the labor force. is the revenue per capital.( Dornbusch, Fischer and Si Tanzi 2007)
這一理論適用于如下的經濟模式： 總收入是所有生產要素的功能;生產方程為Y= F（K，L）。 Y代表輸出，K代表資本和L表示勞動力。是人均收入（多恩布什，費希爾和姒貪呲2007）

(a) For each of the following, illustrate what happens to a country's population size and per capita income in the short run and in the long run. Explain your results.
i. Scientists discover a new strain of wheat that can produce twice as much grain per acre.
Population will expand, the revenue per capita declines in the short term while increases in the long run..
The output of the wheat increases by 200%, in that case, the income per capita in the short term - (which is the result of the total income divided by the total population)--will rise because the number of the total population can't increase sharply in the short run. While in the perspective of a long term, more food will contribute to the growth of population, and the population size will expand at the speed of a somewhat geometric series, which is always higher than the growth rate of the food. Therefore, the income per capita in the long term will decrease. If the expansion of the population reaches at the extreme point at which people can only sustain a mere survival, it will come to a halt.

ii. A war kills half of the population
Population declines sharply, revenue per capita in the short ascends but declines in the long run.
Suppose that half of the population is killed by a war, then the population of the whole nation will descend by half while the total production will remain the same in the short term, resulting, in the short-term ,the income per capita's increase. Conditions in the long term just functions in the same way as what we have mentioned above in Example 1. "More foods and few people" makes the population grow by leaps and bounds while the income per capita in the long term declines. Economy finally may fall into a poverty trap.
iii. A volcanic eruption kills half the people and destroys half the land
Population declines sharply, the revenue keeps steady in the short term but declines in the long run.
Suppose half of the population is killed and half of the total lands are destroyed by the eruption of a volcano. In this situation, labor and lands are reduced by 50% at the same time, while the short-term income per capita will not change according to the income formula. Nevertheless, in the long run, the total population will increase and the ruined lands will be reclaimed, that is to say, in a broader sense, the velocity of population growth is always higher than that of the lands' reclamation, which accounts for the decrease of the long-term income.

iv. There is a technological advance (or perhaps a public health programme) which reduces
death rates
Population's expansion leads to the stable condition of the revenue per capita in the short term , however, in the long run, the revenue per capital will decline.
The development in terms of the technology won't do a great change to the size of the total population and the income per capita in the short term. However, in a long term, the decrease of death rate will result in the increase of the population. The Improvement here can be interpreted as something similar to a public health care plan which will not add the income. Hence, the income per capita in the long term decreases due to the increase of population.
v. A change in cultural attitudes towards parenthood means that for a given income, people now want to have more children than they formerly had.
Population increases, which attributes to the steady condition of the revenue per capita in the short run and to the declining of it in the long term.
Since the income is fixed and the fluctuations in population will be small in a short run, the increase in the growth rate of people will result for the growth of the population in the long term. In that case of a fixed income and a growing population, the long-term income per capita declines.
(b) The equation that relates the population growth rate, n, to income per capita, y = Y /L, is
n(y)=
Let T be the total quantity of land in the economy, which is fixed. The function that relates
land per capita and income per capita is
y=A( )
where A is a measure of productivity.
i. Suppose A is constant. What will the steady-state level of income per capita be?
Hypothesize a fixed A, the only two factors that have an influence on the income per capita are the proportion of lands and the population. At the very beginning, the area of the lands seems to be unlimited compared with the number of the population. With this result, along with the growth of the population, people are prone to be more scattered according to the increase of the lands, which in turn results in the income's rising in this stage. However, with more and more lands reclaimed, there are no longer adequate spare lands for the growing population and the production per capita decreases, so does the income per capita owing to the growth resulting from the more input of labor force at the very beginning begins to decline. To conclude, if the population is allowed to continues to escalate, the growing population will leads to starvation and starvation will in turn cause the reduction of the population in that way. We call it Steady-state equilibrium in population when the foods supplies can only sustain people's survival.

ii. Now suppose A grows at 10% per year. What will be the steady-state level of income per capita? Explain what is going on. the steady-state level of income per capita will increase
Assume that A increases by 10% each year. One year later, it is 1.01y, two years later, 1.01*1.01y, etc. Consequently, the equation breaks away from the Law of Diminishing Marginal Returns and will be growing as time goes on. The improvement in terms of the area of the technology adds to the income per capita and contributes to the Steady-state equilibrium with a higher income per capita and a higher ratio of capital against labor.

2. This question deals with the endogenous growth model with human capital due to Lucas. (We'll
use the notation due to Williamson). Provide a brief outline of the model and its equilibrium.
The Full Essential Factor economic Growth model is: △Y/Y=△K/K+△L/L+△A/A =MPK/K+MPL/L+△A/A （1）(Dornbusch, Fischer, Si Tanzi, 2007)
The core hypothesis of the Lucas model is:

1. The ratio of the growth in the area of the human resource is somewhat like a linear function of the time which people need to accumulate the human resource, that's why comes out of the branch of the office of the human resource.

2. The level of each individual human capital not only has an effect on the productivity of the individual himself, but also produces a deep influence on the productivity of the whole society (however, every individual economy doesn't take this into consideration), which is the basis of the model resulting in the growing scale revenue (seeing from the large picture of the whole society) and the increasing effects of the government policy.
H(t)=H(t)g[1-u(t)] (2)
Q(t)=AK(t)u(t)H(t) （3）
Q represents the output quantity（-ies）,k stands for the material capital, while h means the human resource.( Williamson, 2005)
(a) Suppose that z, the marginal product of efficiency units of labour, increases. What effect does this have on the rates of growth and the levels of human capital, consumption, and output? Explain your results
The function ascending indicates the increase of the assumption goods.
If MPL-the marginal output of the labor force increases, according to the model 1, there will appear an increase of the economical productivity ratio △Y/Y. It plus with the fact that the growth of MPL equals to the increase of the level of the human resource capital, as a result, through the analysis of the third pattern of the Lucas Model, brings an increase of the output. In accordance with formula Y=C+I+G+NX(Dornbusch, Fischer, Si Tanzi, 2007), the curve of the assumption ascends, which accounts for the growth of the assumption products.

(b) Introduce government activity as follows. In addition to working u units of time in producing
goods, the representative consumer works v units of time for the government and produces gvH
goods for government use in the current period, where g > 0. The consumer now spends 1 - u - v units of time each period accumulating human capital
i. suppose that v increases with u decreasing by an equal amount. Determine the effects on the
level and the rate of growth of consumption. Draw a diagram showing the initial path followed
by the natural logarithm of consumption and the corresponding path after these changes.
△u=△v, The curve will slow down.
.According to the hypothesis of the Lucas Model, .the ratio of the increasement in the area of the human resource is somewhat like a linear function of the time which people need to accumulate the human resource. Moreover, from the implied meaning of the above subject, l-u-v is steady while the human capital increases at a constant speed, on another hand, u is decreasing while v is developing on the opposite direction, therefore, the assumption goods reduce by u while the community products escalate by gvH which is bigger than u. To sum up, what has been mentioned above together with the equation formula Y=C+I+G+NX(Dornbusch, Fischer, Si Tanzi, 2007.) leads to the reduction of C as well as the growth of G ( nevertheless, all the other elements remain the same).

ii. suppose that v increases with u held constant. Determine the effects on the level and the rate  http://www.ttc-switch.com/Thesis_Writing/Marketing/
of growth of consumption. Draw a diagram showing the initial path followed by the natural
logarithm of consumption and the corresponding path after v increases
The curve will at first descends vertically and ascends slowly, ending up in go beyond the former result.
.△v>0，△u=0, hence l-u-v decreases, the speed of the increase of the human resource capital decreases at the same time, contributing to the result that the assumption goods remain the same while the community products keeps increasing, which finally leads to the growth of the output and the more and more decreasing speed of its growth ratio. When the speed's increasing is reduced to zero and arrives at a balanced state, the assumption goods increase at the very first but remain stable at the very last.

iii. Explain your results and any difference between parts 2(b)i and 2(b)ii.
In the comparison i with ii, we can find out that the major difference is that △u and △v are a pair of counterpart in their relationship in the model of i, while ii does not work in that way, hence comes out the diversified consequences.
(c) Suppose the government makes a one-time investment in new state school buildings, which results in a one-time reduction in consumption. The new buildings increase the efficiency with which human capital is accumulated. Determine the effects of this on the paths of aggregate consumption and aggregate output over time.
The final result is the increase of the whole assumption goods.

From the Endogenous Economy Model of Lucas, we can arrive at the conclusion that, the human capital, situating at a centered position, is a very important element in stimulating the social economy. Government has increased investment in education, aiming in improving the stock of the human capital in terms of knowledge and skills, which, in another word, lays a solid foundation for the development of the human capital as well as its full use(Jagdish Handa, 2000; Eckhard Siggel, 2005). By this way, although the popularity of the construction of school buildings will reduce the assumption output in the short term, it will increase the accumulation productivity and speed ration of the human resource capital and will enlarge the accumulative quantity of the human resource quickly in the long run, which will attribute to the high-speed development of our economy as well as the substantial increase of the assumption level in general and the output in total, that is to say, i increases while c goes down, with the implied meaning of the subject, H（t）'s increase will lead to the growth of the output at the very last, to go further, the assumption level c in general will work in the same way (Philippe Aghion, Peter Howitt, Maxine Brant-Collett,1998).

(d) Reinterpret the endogenous growth model as follows. Suppose there are two groups of people in a country, the low skilled and the high skilled. The low skilled have less human capITal per person initially than do the high skilled. In the economy as a whole, output is produced using efficiency units of labour, and total factor productivity is z, just as in the ‘standard' model. Each individual in this economy accumulates human capital on his/her own, and each has one unit of time to split between human capital accumulation and work. However, now b = bh for the high skilled and b = bl for the low skilled. In the United States, there has been an increase in the gap between wages of high-skilled workers and low-skilled workers that has occurred over the last 30 years or so. Determine how this model can explain this observation and discuss(John C. H. Fei, John Fei, Gustav Ranis, 1999).

Give the explanation to the following question: why the income gap between the high-skilled labor and the low-skilled labor has been widened in the last 30 years in United States.

The time belonging to each individual has been divided into two parts: one part is used for the work, another for the accumulation of the human capital himself, which ends in the high efficiency of the high-skilled labor. Since they demand less time when doing the same job, they enjoy the priority of more time to accept the professional training and the vocational education, which in turn, helps to increase their H(t) at a relative much higher speed(Gordon W. Schmidt, 2003). In terms of the salary, if the policy of allotting the wages according to one's performance were adopted, because of their higher efficiency, the salary of those high-skilled labor would be much higher than that of those low-skilled labor at the very outset, in addition, the gap will expand along with the deepening of the performance gap in the above-mentioned situation.

References:
Acocella, Nicola and Brendan , Jones. ( 2005). economic policy in the age of globalisation , Cambridge University Press.
Aghion, Philippe., Howitt, Peter., Brant-Collett and Maxine (1998) Endogenous growth theory. The MIT Press
Fei, John C. H., Fei, John., and Ranis, Gustav. (1999). Growth and development from an evolutionary perspective. Wiley-Blackwell
Fischer, Dornbusch., Tanzi, Si. (2007), Macroeconomics. McGraw-Hill/Irwin;
Handa, Jagdish. (2000) Monetary economics. Routledge
Schmidt, Gordon W. (2003) .Dynamics of endogenous economic growth: a case study of the "Romer Model' (Contributions to Economic Analysis). North Holland
Siggel, Eckhard. (2005) Development economics: a policy analysis approach. Ashgate Publishing
Weil, David N. (2005).Economic growth. Addison Wesley
Williamson, Stephen D (2005).Macroeconomics. Addison Wesley.