ARE/ECN 115A Summer Session 1, 2015 PROBLEM SET 2 - GROWTH MODELS - Harrod-Domar Growth Model

ARE/ECN 115A	Summer Session 1, 2015
PROBLEM SET 2: GROWTH MODELS (Due: Thursday, July 9 in sections)

In this problem set we become familiar with the mechanics and the implications of the Harrod-Domar and Solow growth models. You will be asked to present several graphs. Please title your graphs and make sure you clearly label the axes of each graph.

1. 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= 1.5);

· Capital depreciates at a rate of 4% per year (d= 0.04);

· The population grows at 3% per year (n = 0.03);

· The savings rate is 15% (s= 0.15).

For South, in the initial year (t=0) the capital stock equals 300 and the population is 1

A. 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 to 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 1. 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	300.00	300.00
1
2							NA	NA

B. Use the Harrod-Domar growth 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 an approximation.

For parts C and D you will need to create graphs in Excel. You might find it useful to set up a spreadsheet with the same structure as Table 1 above.

C. Using the same parameter values, compute and graph South’s income per capita and growth rate of per capita income over the next 100 years.

· Title the graph of South’s income per capita “Figure 1A”

· Title the graph of South’s growth rate of per capita income “Figure 1B”

· What is South’s income per capita after 50 years? _____

· What is South’s income per capita after 100 years? _______

D. 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 800. On the same graphsthat you made in part C, graph North’s income per capita and growth rate of income per capita over the next 100 years.

· Include North’s income per capita in “Figure 1A”

· Include North’s growth rate of per capita income in “Figure 1B”

· 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.

2. Solow Model with No Technological Progress

Assume we have the same parameter values and initial conditions (L(0)=1 for both South and North, and that K(0) =300 in Southand K(0) = 800 in North) as in question 1. But, let’s now modify the technology assumption used 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:

A. Express the production function in per capita terms (i.e., derive an expression for y as a function of k).

B. What are the steady state levels of capital per worker and income per worker?

C. On one graph plot the income per capita levels for the two countries over 100 years under exogenous technological change.

· Title this graph “Figure 2A”.

· What is South’s income per capita after 50 years? _____

· What is North’s income per capita after 50 years? _______

D. On a separate graph, plot the income per-capita growth rates over the same time period for both North and South.

· Title this graph “Figure 2B”.

· What is South’s growth rate of income per capita after 100 years? _____

· What is North’s growth rate of income per capita after 100 years? _______

E. In the long-run, is there a steady-state level of income per capita?

F. In the long-run, do the levels of income per capita for South converge to that for North?

G. Suppose now that the savings rate in Southincreases to 20%.

· Include South’s income per capita if savings rate is 20% in “Figure 1A”

· Do the levels of income per capita for South with s=20% and North with s=15% converge? Why or why not?

3. Solow Model with Exogenous Technological Progress

Assume we have the same parameter values and initial conditions (L(0)=1 for both South and North, and that K(0) =300 in Southand K(0) = 800 in North) as in question 1. But, now assume that there is an exogenous rate of technological progress of 1.5% per year. To capture this, we need to slightly modify the production function as follows:

In the equation above,, is the state of technology and measures how “effective” our workers are. To capture exogenous technological progress, assume that in our initial period, and that grows exogenously by 1.5% every year for every country.

A. Express the production function in (i) per capita terms and the production function in (ii) per effective workerterms.

B. What are the steady state levels of capital per effective worker and income per effective worker?

C. On one graph plot the income per capita levels for the two countries over 100 years under exogenous technological change.

· Title this graph “Figure 3A”.

· What is South’s income per capita after 50 years? _______

· What is North’s income per capita after 50 years? _______

D. On a separate graph, plot the income per-capita growth rates over the same time period for both North and South.

· Title this graph “Figure 3B”.

· What is South’s growth rate of income per capita after 100 years? _______

· What is North’s growth rate of income per capita after 100 years? _______

E. In the long-run, is there a steady-state level of income per capita? Why or why not?

F. In the long-run, do the levels of income per capita for South converge to that for North? Why or why not?