Investigate the speed of convergence in the Solow growth model.

1. This question has you investigate the speed of convergence in the Solow growth model. Specifically, assume that the functional form for the production function is given by: Y t = A K tθ L 1 − θ t Assume the following values for all of the model’s parameters: A=1, θ=.3, s=.2, δ=.10, n=.02, l=1000

(a)  Find the value of the steady state capital stock per capita, k∗, for this economy.

(b)  Assume that the economy starts quite far away from its steady state, in particular, assume that k0 = .20k∗, i.e., the economy starts with only 20 percent of its steady state level of capital. One measure of speed of convergence that many people use is what is referred to as the half- life. Specifically, how long does it take for the economy to get halfway to the steady state relative to where it is today. So, for example, if the economy were to start with k0 = .20k∗, the half life would reflect how long it takes the economy to reach a level of capital equal to k0 = .60k∗. Compute this value. Then compute the half life starting from k0 = .60k∗, i.e., how long it takes to go from .60k∗ to .80k∗. Do the same starting from .80k∗, .90k∗, .95k∗, .975k∗. Summarize your findings and explain why they suggest that the half life is a meaningful way to summarize the speed of convergence.

(c)  We can ask how various factors might influence the speed of convergence to the steady state. For example, consider another economy that is identical to the economy above except that in this economy individuals work much harder. Specifically, assume that l = 1500. Compute the value of k∗ and compute again how long it takes to go from .20k∗ to .60k∗, from .60k∗ to .80k∗, etc… How do the answers compare with what you found in (b)? Consider someone who makes the following claim: “If people work harder then the economy will get to the steady state faster.” Respond to this statement in light of your findings. Specifically, is this consistent with your findings? If not, briefly explain why is this statement is not true?

(d)  The poorest countries in the world have per capita income that is only about 3% as large as that in the world’s richest economies. Using the values in part (a), if two economies were identical except for the fact that one of them is in steady state and the other is at a lower level of capital, find out what the value of k0 would have to be in order that the value of 1 y0 is only 3% of y∗. Assume that the economy is in this position as of 1950. If we let this economy evolve, what will the value of y be in 2010 relative to y∗? If the only difference between the rich and poor countries in 1950 were differences in initial capital stocks, what would have happened to the ratio of income per capita of the rich countries relative to the poor countries? This question has you consider the consequences of a one time increase in immigration using the Solow growth model. Specifically, consider for now the version of the model in which population is assumed to be constant and equal to N . Assume that the economy has reached its steady state level of capital stock per capita.

Now assume that the economy experiences a one time influx of immigrants that raises the population from N to N′.

(a) Describe the short and long run consequences of this for kt, yt, ct. In doing this calculation assume that the influx of immigrants does not affect the saving rate or the amount of work performed per person. That is, implicitly we are assuming that the decisions of immigrants regarding consumption and labor supply are the same as for the individuals who were initially living in this economy.

(b) Assume the same values as in question 1 part (a), and assume that N = 100, 000, 000 and N′ = 110,000,000, i.e., that the influx of immigrants corresponds to 10% of the original population. In the initial period of the immigration, what happens to per capita consumption relative to the previous period? How many years will it take before consumption returns to within 1% of its previous value.

(c) Consider the following statement: “Allowing immigration will increase the population and therefore lead to permanently less output per person in the long run, thereby lowering the living standards of all current residents.” Does the Solow model support this claim? If not, why not? This question has you examine some data on the evolution of income per capita. The attached file “TEDI.xls” gives you access to different economic series for various countries.1 In the file, there are two different series that provide information on real per capita income across countries: GDP- capita GK and GDP-capita-EKS. These represent two different ways to compute price indices that are used to measure real output. We will not get into the details of the differences, and for this question please use the GDP-capita GK series.
Clicking on this tab will bring up an excel spread sheet with the data for a large set of countries.
(a)  (A miracle story) In 1950 South Korea and the Philippines were at similar levels of develop- ment. Produce a figure that shows how these two countries evolved since then, in each case plotting the value of GDP per capita in each of these countries relative to the US (i.e., as a percentage of the US GDP per capita).
(b)  (A disaster story) In 1950, Belgium and Argentina are at similar levels of development. Pro- duce a figure like the one in part (a) for those two countries. 1The source of the data is the Groningen Growth and Development Center website.
(c) (Falling Behind) Do the same for New Zealand and Australia.
(d)  (Sub-Saharan Africa) Do the same for Uganda and Zimbabwe.
(e)  This question asks you to assess how the recent behavior of China and India compares with that of earlier miracle growth experiences in East Asia. Let’s date China’s high growth period as starting in 1980, South Korea’s as starting in 1965, Japan’s as starting in 1950, and India in 1987. Plot the level of GDP per capita for each country on the same graph, with time 0 reflecting the appropriate starting point for each country. This allows you to visually contrast the pace of the growth experiences across countries. What patterns do you find?

This question asks you to evaluate the magnitude of the golden rule value of the saving rate. Specifically, for the economy in question 1 part a, consider values of the saving rate equal to “.05, .1, .15, .20,…”, etc… For each value, compute the value of steady state consumption per capita, and determine at what point the value of c∗ starts to decline.

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