So, from St = It = sYt . By making everything into the per effective worker terms, we divide by AtLt:
We have, Kt+1 /At+1Lt+1 = s F (Kt, Lt)/AtLt AtLt/At+1Lt+1 + (1- α) Kt/AtL AtLt/Bt+1Lt+1
Yt+1 = αK αt + (1- α) Kt/ {(1 + g) (1 + n) = φ (Kt)
At steady state, dkt+1 = 0,
Now solving for the value of K gives:
To effectively check for the overall stability of the steady sate, we need to check its limit if it is less than unity. That is to say; lim K as Kt tends to infinity should be less than one (Acemoglu 2009)
For output per worker
From the growth rate of output per worker, yt – Yt/ Lt in steady state is gives the following expression (Acemoglu 2009)
Yt/Lt = [Kαt (AtLt) 1-α]/Lt = (Kt/Lt) α At1-α = (Kt/AtLt) α At = KtαAt
Therefore, as the economy reaches the steady state then;
Ytss = KαBt
Now, from above we get;
yt+1ss/ytss – 1 = At+1/At – 1 =g. Similarly, by taking natural logs on both terms gives the following;
Logyt+1ss – log ytss – log At+1 – log At ≡ g
The consumption per effective worker
Under the steady state, the consumption per effective worker is obtained from the general equation as (Acemoglu 2009); yt = ct + it, where it = syt and also ct = (1-s) yt. Therefore, in the steady state;
Capital per worker at steady state is obtained from; Kt+1ss = Ktss = K. it implies that at steady state, yk = 0. Now getting the K from the expression of ykt gives (Noel and Mark 2017)
D)Therefore, the growth rate of output per worker is given as;
Also from the capital per effective worker, Taking natural logs where Kt/Lt = AtKt , where the Kt = Kt/ (AtLt).Now defining the per capita stock of capital as kt =Kt / Lt:
It then gives;
Kt/Lt = kt = ktAt. By taking logarithms to get the growth rate of capital gives;
Log kt+1 – log kt = log At+1 – log At ≡ g(Noel and Mark 2017
The value of at the golden rule in the steady state is the capital stock per worker which maximizes the consumption at the steady state (Haine etal 2006). Therefore, from;
Ct = (1-s) yt = f (kt) –sf(kt).
It is believed that there is now way one can maximize consumption in all the states (Haine etal 2006). This is because consumption is a function of f (kt) that is not bounded (Haine et.al, 2006). Therefore, with such a reason, a corner solution is only obtained as kt = 0 that is true when stationary points are obtained first. But the steady state condition is given as;
Sf (k) =k. Hence it is true for all steady states because ct = f(kt) – sf(Kt) in steady states (Halsmayer etal 2016).
In simple terms; c = f (k) – k. The maximization problem now becomes;
∂c/ ∂k = f’ (k) –
Where y = Akα a closed form of solution for K’ can be obtained as’
f’ (k’) = αA(k’)α-1 = n +
k’ =1/(1-α)
And now consumption becomes;
Unemployment in the Solow Model
Therefore from; yt ≡ f (kt) = Ktα [(1-u) Lt] 1-α/(Lt)
This gives, kαt (1-u) 1-α
as the growth rate of output per effective worker. This unemployment will affect his economy in the way that the job separation rate s in this case will increase and there are high chances of losing the job (Haine etal 2006).
Part B: Computational Work
?* = {0.2/ (0.02 + 0.025 + 0.05)} 1/ (1-0.33) = (2.11)1.5 = 3.064952
? = ?α = 3.0649521/3 = 1.452584
?t = s ?t = 0.2 *1.122497 = 0.224499
Ct = 1-0.2 (1.122497)
= 0.897992
C old = (1-sold) yt = 1-0.12 (1.122497) = 0.988
Kold ={0.12/(/(0.02 + 0.025 + 0.05)} 1/(1-0.3333) = 1.414346. therefore, yold becomes
Yold= ?α = 1.4143461/3 = 1.122497
The line graph shows that there is no significant relationship between the three variables of kt, yt and ct. It is observed that from the data, there deviations from when time is o to time are 7. From that point there is no correlation as all the variables are kept constant (Halsmayer etal 2016)
?* = {0.2/(0.02 + 0.025 + 0.05)}1/(1-0.33) = (2.11)1.5 = 3.064952
? = ?α = 3.0649521/3 = 1.452584
However, in the short run, consumption reduces with time as saving increases. This is explained since saving increases, this would imply that, consumption in the short run will also reduce and finally no consumption at all in the long run (Halsmayer etal 2016).
From the graph above, there is not relationship between the variables as it is seen in the diagram above. According to most scholars, they believed that a change in saving rate only affects the consumption in the short run and long run(Halsmayer et al 2016). They stated that, all other variables are not heavily affected as they are independent from saving. Except only investment has a direct relationship with the change in the saving(Halsmayer et al 2016).
Part C: A Contribution to the Empirics of Economic Growth. Solow Model with Human Capital
From the general equation as, Y (t) = [A (t), K (t), L (t)]………………………….1
Where the Y (t) is the aggregate output at time t, output is then presented as the function of the capital inputs (Halsmayer etal2016). From the same equation, the households have got capital that is available for renting to other individuals and firms which
Where ? presents the time derivatives. Firms under Solow aim at profit maximization. Also from the Solow model, the neoclassical production is obtained from the Cobb Douglas functions as (Halsmayer et al 2016);
Y(t) = A(t)K(t)αL(t)1-α, that is to say; 0<α<1 where α is the output share paid to the capital while 1- α is the output share that is paid to the labor. When labor productivity is increased by the presence of technology, the function becomes;
K, where sf (k) is the income fraction saved. But similarly, capital change per unit effective worker finally will become zero, that is to say;
………………………………………6. However, the Solow model was further modified by introducing in the concept of human capital (Field & Alexander, 2011). Then the equation becomes;
Y(t) = K(t)α H(t)β(A(t)L(t))1-α-β ………………………………7. Where H represents the human capital stock and β represents the income fraction paid to the human capital stock (Field 2011).
Yt = Ktα (AtHt) 1-α
Yt = Ct + It
Ct = (1-s) Yt
?t/At = g
L’t/Lt = n
Therefore, growth becomes .But also, S= sYt( Robert and Xavier. 2004)
(Wall and Griffiths2008)
y = Y/AL/L = {A (K/AL) αL1-α}/L
From the production function given as; Y = F (K, L) = Kα L1-α but in per capita terms it becomes, Y/L = = Kα L1-α/L (Romer 2011).
Which implies that, y = Kα L1-α and similarly, (K/L) α = Kα so no substituting gives
p). The results of estimating eq. (11) show that there is an increase in the share of capital was introduced which caused a fall in the human capital. The Authors are sure that this equation is best to fit for the data because the biasness was removed by introducing the into the equation. Therefore, the equation would produce more realistic results once it is fitted for the data (Breton 2013).
References
Acemoglu, Daron. 2009. “The Solow Growth Model”. Introduction to Modern Economic Growth. Princeton: Princeton University Press. pp. 26–76. Accessed 12 Oct.2018.
Haines, Joel D.; Sharif, Nawaz M. 2006. “A framework for managing the sophistication of the components of technology for global competition”. Competitiveness Review: An International Business Journal. Emerald. Vol 16 (2). Pp 106–121. doi:10.1108/cr.2006.16.2.106.
Halsmayer, Verena; Hoover, Kevin D. 2016. “Solow’s Harrod: Transforming macroeconomic dynamics into a model of long-run growth”. The European Journal of the History of Economic Thought. 23 (4): 561–596. doi:10.1080/09672567.2014.1001763. Accessed on 12 October 2018
Romer, D. 2011. “The Solow Growth Model”. Advanced Macroeconomics (Fourth ed.). New York: McGraw-Hill. pp. 6–48. Accessed on 12 October 2018.
Agénor, Pierre-Richard. 2004. “Growth and Technological Progress: The Solow–Swan Model”. The Economics of Adjustment and Growth (Second ed.). Cambridge: Harvard University Press. pp. 439–462. Accessed on 12 October 2018
Barro, Robert J.; Sala-i-Martin, Xavier. 2004. “Growth Models with Exogenous Saving Rates”. Economic Growth (Second ed.). New York: McGraw-Hill. pp. 23–84. Accessed on 12 October 2018
Field, A. J. 2011. A Great Leap Forward: 1930s Depression and U.S. Economic Growth. New Haven, London: Yale University Press. Accessed on 12 October 2018
Li, Rita Yi Man; Li, Yi Lut. 2013. “Is There a Positive Relationship between Law and Economic Growth? A Paradox in China”. Asian Social Science. 9 (9): 19–30. Accessed on 12 October 2018
Johnson, Noel D.; Koyama, Mark (2017). “States and Economic Growth: Capacity and Constraints”. Explorations in Economic History. Vol 64. Pp 1–20. doi:10.1016/j.eeh.2016.11.002.
Berg, Andrew G.; Ostry, Jonathan D. 2011. “Equality and Efficiency”. Finance and Development. International Monetary Fund. Vol 48 (3). Retrieved on 12 October 2018.
Field, A. J. 2011. A Great Leap Forward: 1930s Depression and U.S. Economic Growth. New Haven, London: Yale University Press. Accessed on 12 October 2018.
Johnson, Noel D.; Koyama, Mark. 2017. “States and Economic Growth: Capacity and Constraints”. Explorations in Economic History. 64: 1–20. doi:10.1016/j.eeh.2016.11.002
Breton, T. R. 2013. “The role of education in economic growth: Theory, history and current returns”. Educational Research. 55 (2): 121. doi:10.1080/00131881.2013.801241.
Wall, S.; Griffiths, A. 2008. Economics for Business and Management. Financial Times Prentice Hall. ISBN 978-0-273-71367-8. Retrieved 6 March 2010.https://books.google.com/books? Id=TrRtUr_Wn2IC
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