Discrete-Time Solow Growth Model: Analyzing Dynamics And Main Results

1. It is divided between labor and capital income according to their marginal productivity. That is to say, from; y_t= F (K_t, A_tL_t) = K^α_t (A_tL_t) ^1-α

Develop understanding of discrete-time Solow Growth Model

So, from S_t = I_t = sY_t . By making everything into the per effective worker terms, we divide by A_tL_t:

We have, K_t+1 /A_t+1L_t+1 = s F (K_t, L_t)/A_tL_t AtLt/A_t+1L_t+1 + (1- α) K_t/A_tL   A_tL_t/B_t+1L_t+1

Y_t+1 = αK ^α_t + (1- α) Kt/ {(1 + g) (1 + n) = φ (K_t)

1.  

At steady state, dk_t+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 K_t tends to infinity should be less than one (Acemoglu  2009)

For output per worker

From the growth rate of output per worker, y_t – Y_t/ L_t in steady state is gives the following expression (Acemoglu 2009)

Y_t/L_t = [K^α_t (A_tL_t) ^1-α]/L_t = (K_t/L_t) ^α A_t^1-α = (K_t/A_tL_t) ^α A_t = K_t^αA_t

Therefore, as the economy reaches the steady state then;

Y_t^ss = K^αB_t

Now, from above we get;

y_t+1^ss/y_t^ss – 1 = A_t+1/A_t – 1 =g. Similarly, by taking natural logs on both terms gives the following;

Logy_t+1^ss – log y_t^ss – log A_t+1 – log A_t ≡ g 

The consumption per effective worker

Under the steady state, the consumption per effective worker is obtained from the general equation as (Acemoglu 2009); y_t = c_t + i_t, where i_t = sy_t and also c_t = (1-s) y_t. Therefore, in the steady state; 

Capital per worker at steady state is obtained from; K_t+1^ss = K_t^ss = K. it implies that at steady state, yk = 0. Now getting the K from the expression of yk_t 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 K_t/L_t = A_tK_t , where the K_t = K_t/ (A_tL_t).Now defining the per capita stock of capital as k_t =K_t / L_t:

It then gives;

K_t/L_t = k_t = k_tA_t. By taking logarithms to get the growth rate of capital gives;

Log k_t+1 – log k_t = log A_t+1 – log A_t ≡ g(Noel and Mark 2017

1. E) The Golden –Rule Steady State

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;

C_t = (1-s) yt = f (k_t) –sf(k_t).

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 (k_t) that is not bounded (Haine et.al, 2006). Therefore, with such a reason, a corner solution is only obtained as k_t = 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 c_t = f(k_t) – sf(K_t) in steady states (Halsmayer etal 2016).

In simple terms; c = f (k) – k. The maximization problem now becomes;

∂c/ ∂k = f’ (k) – 

Where y = Ak^α 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

1. The output per effective worker y_t= Y_t/L_t as the function of capital per effective worker K_t+1 = K_t/L_t and the natural rate of the unemployment u is obtained by dividing the total output by the number of workers (Haine etal 2006).

Therefore from; y_t ≡ f (k_t) = K_t^α [(1-u) L_t] ^1-α/(L_t)

This gives, k^α_t (1-u) ^1-α

Solve the model analytically for steady-state and Golden-Rule steady-state levels

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

1. The value of y

?* = {0.2/ (0.02 + 0.025 + 0.05)} ^{1/ (1-0.33)} = (2.11)^1.5 = 3.064952

? = ?^α = 3.064952^1/3 = 1.452584

?_t = s ?_t = 0.2 *1.122497 = 0.224499

C_t = 1-0.2 (1.122497)

                   = 0.897992

1. K_old^*= {0.12/(0.02 + 0.025 + 0.05)}^1/(1-0.3333) = (1.26)⁵ = 1.414346 

C _old = (1-s_old) y_t = 1-0.12 (1.122497) = 0.988

K_old ={0.12/(/(0.02 + 0.025 + 0.05)} ^1/(1-0.3333) = 1.414346. therefore, y_old becomes

Y_old= ?^α = 1.414346^1/3 = 1.122497 

The line graph shows that there is no significant relationship between the three variables of k_t, y_t and c_t. 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)  

1. The new value are;

?* = {0.2/(0.02 + 0.025 + 0.05)}^1/(1-0.33) = (2.11)^1.5 = 3.064952

? = ?^α = 3.064952^1/3 = 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).

1. The impact of this policy in the short run is that, consumption is not affected but saving increases and capital ratio slightly falls. While in the long run, most of the variables are seen to change like the consumption which declines heavily and also capital ratio reduces as investment increases as a result of saving(Halsmayer et al 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

1. J) It is believed that the Solow Model is built upon the neoclassical dynamic model with the aggregate production function(Halsmayer et al 2016).

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;

1. Y(t) = K(t)^α(A(t)L(t))^1-α…………………………………..3. Where A (t) L (t) is the labor effective. Then when A is expressed as the factor which increases labor productivity, we obtain income per unit of effective labor as a function of the capital per unit of effective labor as given by; (Noel and Mark 2017)
2. (Y(t)/A(t)L(t)={k/(A(t)L(t))}^α*{A(t)L(t)/(A(t)L(t)}^1α?y=K^α…4. where y is the output per unit of the effective labor while k is the capital per unit of effective labor. However, any changes in capital will lead to changes in the level of total income as determined by;

 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;

1. K…………………………………………………………….5.This marks the point at which capital no longer accumulates but the capital ratio becomes constant. Thus the growth of income is seen to stabilize with the steady stae being reached with the output per effective worker. Inserting the expression into the steady sate equation gives the production function ( Yi Man and Yi Lut 2013). 

………………………………………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).

1. The predicted elasticity’s of income per capita is obtained as follows

Y_t = K_t^α (A_tH_t) ^1-α

Y_t = C_t + I_t

C_t = (1-s) Yt

?_t/A_t = g

L^’_t/L_t = n

Therefore, growth becomes .But also, S= sY_t( Robert and  Xavier. 2004)

 (Wall and Griffiths2008)

1. The authors believe that these coefficients supports the Solow model predictions because they knew that once the specification turns out of a large aspect of the problem due to systematic differences would not affect the Solow model predictions (Field 2011).Since the versions of the Solow model studied was abstract from achieving the predictions. However, the model as estimated is not completely successful because it needs improvements in its empirical performance (Field 2011). The empirical performance was not attained as expected thus improvement is needed (Agénor 2004).
2. The authors propose to improve the empirical performance of the Solow model as its performance was not so good to the standard. They also proposed to improve its significance by modifying it version and introducing in more variables like the human labor and capital (Breton 2013).
3. Defining x = X/L as per worker term;
4. Then, y = Y/L = AK^αL^1-α/L = A(K/L)^α (L/L)^1-α = AK^α but y =Y/AL, K =K/AL and H =H/AL

y = Y/AL/L = {A (K/AL) ^αL1-α}/L

From the production function given as; Y = F (K, L) = K^α L^1-α but in per capita terms it becomes, Y/L = = K^α L^1-α/L (Romer 2011).

Which implies that, y = K^α L^1-α and similarly, (K/L) ^α = K^α so no substituting gives

1. O)The author usedin his equation simply because he wanted to account for the changes in share of capital. The authors realized that there are variations in the share of capital and so to account for them was introduced (Romer 2011)

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