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In this problem set we become familiar with the mechanics and the implications of the Harrod-Domar and Solow growth models. You may (but do not have to) work in pairs. If you choose to work in pairs, hand in a single copy of the answer sheet and make sure you put write in both your names above. In this problem set, you will be asked to present several graphs. Place all graphs in an appendix at the end of your homework. Please title each of your graphs with the suggested titles I have provided and make sure you clearly label the axes of each graph.
1. Empirical Trends in Growth Rates of the Ex-Colonies
This problem helps us warm up to the use of real data in Excel. As we will see later in the quarter, one hypothesis (that is outside the realm of classical growth models) to explain differences in economic performance across developing countries is that the different European colonizers established institutions of different quality. A colonial “legacy” of high quality institutions would then be expected to lead to higher income and growth compared to a legacy of low quality institutions.
In this problem we take a brief look at this hypothesis, by examining trends in growth between 1960 and 2010 for different groups of developing countries based on the identity of the European colonizer. Download the Excel data file “Penn Table” from Smartsite. The worksheet “World” gives per-capita income levels for the 77 countries in Asia, Africa and Latin America that were European colonies (or mandates) and for which we have data for both 1960 and 2010.
A. Fill in Table 1 below. The first two columns ask you to calculate the population weighted average per-capita income levels for 1960 and 2010 for eight groups of countries; each group contains the ex-colonies of the eight European colonial powers. The weight for each country is just the fraction of population among all countries in that group that comes from that country. The third column asks for the annual growth rate of per-capita income, g*, over these 50 years for each group. To calculate g*, just use the numbers in the first two columns and assume that g* was constant over the entire period. (Hint: the equation you need is: y(t)=y(0) 〖(1+g^*)〗^t.)
Table 1. Trends in Per-Capita Income by European Colonizer: 1960 - 2010
Colonial Power y(1960)
($000) y(2010)
($000) Annual Growth rate of per capita income
g*
(%)
Belgium
England
France
Germany
Italy
Netherlands
Portugal
Spain
ALL COUNTRIES
B. Which colonial power is associated with the highest growth rate among ex-colonies in the 1960-2010 period? Which colonial power is associated with the lower growth rate?
C. This question asks you to speculate a bit (since we haven’t yet discussed institutions!), so do your best speculation! In order to explore the hypothesis of institutional quality and colonization, we would need to identify those institutions that are critical for economic performance. Which three institutions would you suggest looking at?
D. The file “Penn Table” contains a second worksheet named “1960-2010", that contains complete time series data on per-capita GDP for the full 1960 to 2010 period for a select groups of countries: China, Senegal, Denmark and Argentina.
(i) Create a graph that plots per-capita GDP (vertical axis) against time (horizontal axis) for these countries. Put the plots for each country in a single graph. Title this graph “Figure 1B”.
(ii) Based on your graph, would you conclude that we observe unconditional convergence? Why or why not? (“Unconditional” simply means that we haven’t conditioned on the exogenous parameters: s, n, d and v.)
2. Mechanics of the Harrod-Domar Growth Model
Now let’s walk through the mechanics of the H-D model for an imaginary country, which we’ll call South. Let’s assume the following parameter values for South: The incremental capital-output ratio, v, is 1.25; capital depreciates at a rate of 2% per year (d = 0.02); the population grows at 2% per year (n = 0.02); and the savings rate is 10% (s = 0.1).
A. In the initial year (t=0) the per-capita capital stock equals 200 and the population is 1. Continuing under the assumption that the economy evolves according to the Harrod-Domar model, fill in Table 2 below. (NOTE: Do not use the H-D growth equations yet. Instead, first completely fill in columns A – G using the production function from the H-D model and the equation for capital accumulation, then calculate the growth rates using the definition of growth rate: g(t) = [Y(t+1)-Y(t)]/Y(t))
Table 2. Evolution of a Harrod-Domar Economy
(A)
Year
t (B)
Popu-
lation
L(t) (C)
Total
Capital Stock
K(t) (D)
Per Capita
Capital
Stock
k(t) (E)
Total
Income
Y(t) (F)
Per Capita
Income
y(t) (G)
Total Savings
S(t) (H)
Growth Rate of Total Income
g(t) (I)
Growth Rate of Per Capita Income
g*(t)
0 1.00 200.00 200.00
1
2 NA NA
B. Now let’s check your work. Write down the Harrod-Domar growth equations (one for aggregate income and one for per-capita income). Use these equations to find the growth rate of aggregate income, g, and the growth rate of per-capita income, g* for South. Recall that the equation for the growth rate of per-capita income is approximate.
C. Use the approximate doubling time formula from lecture to calculate approximately how many years it will take South to double:
-Aggregate Income: _______
-Per Capita Income: _______
For parts D - F you will need to create graphs in Excel. You might find it useful to set up a spreadsheet with the same structure as Table 2 above.
D. Using the same parameter values, compute and graph South’s income per capita over the next 100 years. Title this graph “Figure 2D/E”.
What is South’s income per capita after 50 years? _____
What is South’s income per capita after 100 years? _______
E. Our second country is called North. North is exactly the same as South except that it starts with a greater capital stock. In the initial year (t=0), North has a capital stock of 600. On the same graph that you made in part D, graph North’s income per capita over the next 100 years.
What is North’s income per capita after 50 years? ______
What is North’s income per capita after 100 years? ______
Do North and South converge over time given their common savings rate according to this model? Explain the reasons for convergence, or lack thereof, in this model.
F. Finally, suppose a newly elected, populist leader of South wants her country to catch up with North. She is a brilliant orator and is confident that she can convince her people to consume less and save more in order to fulfill her economic catch-up mission. She has hired you to tell her what savings rate is necessary to catch up with North over a 50-year time horizon.
What is this savings rate? ________ (You can answer this either by trying different savings rate in the spreadsheet you have been using for parts D and E or by figuring it out with pencil and paper.)
Are you surprised by your answer? Do you think that Zimbabwe could really catch up with the USA by following this savings plan? (BRIEFLY explain the rationale for your answer to this question.)
3. Solow Model with No Technological Change
Let’s now modify the technology assumption used in the prior problem and assume that output in all countries (South and North) is produced according to the following constant-returns-to-scale production function that lies at the heart of the standard Solow growth model:
Y(t)=〖A[K(t)]〗^α [L(t)]^(1-α)
A is just an exogenous, constant, parameter of the production function. Assume that A = 10 and α = 0.50.
A. Express the production function in per capita terms (using the values of A and α given above).
B. Assume we have the same values as in problem two for the following exogenous parameters: s = 0.10; n = 0.02; d = .02. Assume also that our two countries start with the same initial conditions as in problem two; namely, that L(0)=1 for both South and North and that K(0) = 200 in South and K(0) = 600 in North.
On one graph plot the income per capita levels for the two countries over 200 years under exogenous technological change. Title this graph “Figure 3A”.
On a separate graph, plot the income per-capita growth rates over the same time period for both countries. Title this graph “Figure 3B”.
Does the model reach a steady state?
Does South converge to North according to this model when both countries have the same savings rate?
How and why are the implications of this model for South different from those in the Harrod--Domar model? Why can growth not be sustained?
C. Suppose now that the populist leader raises savings rate in South to the levels that you identified in 2F above.
What happens in both short and long terms to per-capita income levels and growth rates after the savings rate increases in South?
Do South and North still converge? Why or why not?
4. Solow Model with Exogenous Technological Progress
Now assume that there is an exogenous rate of technological progress of 2% per year. To capture this, we need to slightly modify the production function as follows:
Y(t)=A[K(t)]^α [T(t)L(t)]^(1-α)
In the equation above, T(t), is the state of technology and measures how “effective” our workers are. To capture exogenous technological progress, assume that in our initial period, T(0)=10 and that T(t) grows exogenously by 2% every year for every country.
A. Assume that both North and South have a 10% savings rate and that initial capital stocks and all other exogenous parameters are the same as in problem 3.
On one graph plot the per capita income levels for the two countries over 200 years under exogenous technological change. Title this graph “Figure 4A”.
On a separate graph, plot the per-capita income growth rates over the same time period for both countries. Title this graph “Figure 4B”.
B. Compare and contrast your findings in 4A with the results you obtained under the Harrod-Domar model. Even though long run growth rates converge between South and North in both the Harrod--Domar and this “Solow-with-exogenous-technological-change” model, why do per-capita income levels converge in one model, but not in the other?
Appendix: Weighted vs. Unweighted Average
Let’s look at a simple example of test scores. Say we have two sections of class with the following scores:
Section 1: 70, 80, 90
Section 2: 80, 90, 90, 90, 100
The average test score in Section 1 is (70+80+90)/3 = 80. The average test score in Section 2 is: (80+90+90+90+100)/5 = 90.
The unweighted average across the two sections is just the simple (arithmetic) average of the averages from the two sections:
Unweighted average = (80+90)/2 = 85.
In this unweighted case, the average of each section was given the same weight (1/2) when calculating the overall average. Notice, however, that if we want to know the average test scores of all students in class, this unweighted average will give us an incorrect average. Why? Because there are more students in Section 2 than in Section 1. To calculate the mean test score across all students, we must take the different class sizes (i.e., population sizes in the problem set) into account. Specifically, the average of Section 1 should receive a lower weight than the average of Section 2 because Section 1 has fewer students.
The weights we assign to each Section are just the fraction of the total enrollment in class that are in each section. Since total enrollment is 8, Section 1 will have a weight of 3/8 and Section 2 will have a weight of 5/8. The class-size weighted average is then:
Weighted average = (3/8)*80 + (5/8)*90 = 86.25.
Notice that the weighted average is higher than the unweighted average because we’ve given more weight to Section 2 which had a higher average.

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