Solow Growth Model with Cobb-Douglas Production Function

Deriving the Law of Motion of Capital per Capita

Consider the Solow growth model with a Cobb-Douglas production function with capital share a, a constant savings rate s, a constant depreciation rate 8, and a constant population growth rate Suppose a equals 1/3, and TFP initially is A₁ > 0. 

1. Derive the law of motion of capital per capita.

2. Derive the steady state levels of capital and output per capita.

3. Explain intuitively why there is a steady state.

Now suppose that productivity increases by 10% to A₂.

4. Derive the new steady state levels of capital and output per capita. Compare them to the old ones. 

5. In three separate figures, draw the evolution through time of productivity A, capital per capita K_it, and output per capita Y_it. Make sure to include several periods of the pre-shock steady state, the time of the shock, the transition to the final steady state, and several periods of the final steady state. Discuss the differences between the two figures.

6. Suppose that the economy has converged to its new steady state. Do a growth accounting exercise on output per capita. How much of the change in output per capita was due to changes in productivity, how much due to changes in capital per capita? 

7. Does this result reflect well what actually occurred? 8. In the last decades, East Asian economies have grown strongly. Some studies attribute this mainly to capital accumulation. Discuss. 

Assume that the Solow model provides an accurate representation of the two Germanies around the late 80s and early 90s. Although in 1990 both countries re-unified, for this problem we will treat them as two separate economies, and we will model the effects of the reunification as a one-time flow of resources (capital and/or labor) across the common border. Aside from being in steady state initially, assume that before the re-unification both economies had the same saving rate, the same rates of population growth and depreciation, and that capital was equally important for production in both countries, with a common value of = 1/2.

Let's focus first on the period before the re-unification.

1. In this initial steady state, output per capita in the West was twice the level in the East, due to its higher productivity A. Use this information and equations for the steady state to calculate the ratio of productivity between the two countries. Explain your derivations.

2. Calculate the ratio of capital per capita between the two countries. Explain your deriva-tions carefully.

3. Did the owners of capital in West Germany have any incentives to reallocate their capital from the capital abundant Western economy to the capital scarce Eastern one? Explain your answer (think about the payment per unit of capital in both countries).

4. Would East German workers have any economic incentives to migrate to the West? Explain your answer (think about wages in both countries). Now consider reunification. Let's think of it as a one-time migration of workers from East to West Germany. In the remainder of the problem, we will take the perspective of East Germany.

5. Use the phase diagram of the Solow model to trace the effects of the reunification, i.e. of a one-time decrease in the labor force of East Germany. Discuss the forces at work along the transition. 

6. In three figures, sketch the evolution through time of capital per capita, investment per capita and (the logarithm of) aggregate output. Make sure your figures include the initial steady state (that is, some periods before the re-unification), the shock, several periods after the shock, and the final steady state.