Assignment: INTERMEDIATE MACROECONOMICS

Question 1. The Solow Model of Economic Growth

Consider the following Solow growth model with technological change and population growth:

Y_t = K_t^0.5 (A_tN_t)^0.5                       (1)
S_t = sY_t, 0 < s < 1                        (2)
K_t+1 = (1 - δ)K_t + l_t                      (3)
(N_t+1 / N_t) = 1 + g_n, g_n = 0.01      (4)
(A_t+1 / A_t) = 1 + g_A, g_A = 0.02.     (5)

a) Explain in words what each of these equations means or describes.

b) Write down the goods market equilibrium condition for the model.

c) Combine the goods market equilibrium condition with equations (1) through (3) to find an equation that describes the change in the capital stock between dates t and t+1 in terms of the levels of inputs to production at date t. Explain in words what determines this change over time; whether it is positive or negative or zero.

d) Now take each variable in the model and divide it by A_tN_t. Use these transformed variables to re-express the equation you derived in c) as an equation that describes the change in the capital stock per effective worker between dates t and t+1. Explain in words what determines this change over time; whether it is positive or negative or zero.

e) Define and describe in words a long-run, steady state equilibrium of this economy. Depict a long-run, steady state equilibrium in a diagram and label the diagram carefully. What condition on the equation that you derived in d) would measure or captured this steady state?

f) In the steady state equilibrium, what will be the numerical values of the growth rates of aggregate output, the aggregate capital stock, aggregate investment, and aggregate savings? What will be the numerical values of the growth rate of output per worker, and capital per  worker? What will be the numerical values of the growth rate of output per effective labor unit and capital per effective labor unit?

g) What would be the qualitative impact of an increase in s for the steady state level of capital per effective worker and output per effective worker? Show this in a diagram.

h) What would be the qualitative impact of an increase in s for the steady state growth rates of output, capital, savings and investment? What is the impact of an increase in s for output per worker?

i) What factors will cause a change the steady state growth rate of this economy? What types of policies would a government have to enact to increase the steady state growth rate of the economy? Would an increase in the steady state growth rate of the economy increase living standards in the steady state? Explain carefully.

Question 2. More on the Solow Growth Model

a) Re-do Question 1, parts b) through d), assuming that the production function is given by

Y_t = K_t^0.3 (A_tN_t)^0.7                        (1')

b) If the production function is given by (1') how does this affect your answers to Question 1 parts e) through i), if at all?

c) Derive expressions for the steady state level of capital per effective worker and output per effective worker for the case where the production function is given by (1) and the case where it is given by (1').

d) Assume that depreciation is 10 percent per year, and the savings rate is twenty percent. What are the steady state levels of capital and output per effective worker for the case where the production function is given by (1) and the case where it is given by (1'). Compare them and explain in words how and why they are different, if they are.

e) Now assume that the savings rate increases from twenty percent to thirty percent. How does this affect the steady state levels of capital and output per effective worker when the production function is given by (1) and when it is given by (1'). Compare them and explain in words how and why they are different, if they are.

f) Now assume that the growth rate of technological change increases from 2 percent to 3 percent per annum. How does this affect the steady state levels of capital and output per effective worker when the production function is given by (1) and when it is given by (1'). Compare them and explain in words how and why they are different, if they are.

Question 3. Growth Accounting

Assume the Cobb-Douglas production function (1') determines aggregate real GDP.

a) Derive a "growth rate" version of (1') which shows how the growth rate of real GDP is determined by the growth rates of technology, employment, and capital.

b) Now collect at least ten years of annual data on:

Real GDP
Employment
Capital

c) Compute the annual growth rate of each of the data series you downloaded or otherwise collected in b). Compute the implied annual growth rate of technology at each date. Now compute the sample average annual growth rates of all four variables.

d) Using the equation you derived in a), and your results in c), what fraction of real GDP annual average growth over your sample period is accounted for by i) capital growth, ii) employment growth, and iii) technological change.