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Problem Set 1 (Solow-Swan and Ramsey Models)

ECON 6002

1. Suppose the aggregate production function in the Solow-Swan model is Cobb-Douglas, y = k α, with α = 1/3 . Assume population growth n = 1%, technology growth g = 2%, and depreciation δ = 5%.

(a) Derive k ∗ , y ∗ , and c ∗ as functions of the model parameters and determine their values when the saving rate s = 24%.

(b) Assume both labour and capital are paid their marginal products and the economy is on a balanced growth path at time t = 0:

i) What is the real wage w(0) if A(0) = 1?

ii) What is the growth rate of wages ˙w/w?

iii) What is the return to “working” capital r (net of depreciation)?

iv) What are the shares of income going to (both “working” and “dead/depreciated”) capital and to labour?

(c) How would your answers to part (b) change if the savings rate were s ′ = 30% economy were on the balanced growth path?

(d) Draw the transition given a change in the saving rate from s = 24% to s ′ = 30% in the basic diagram for the Solow model.

(e) What are the growth rates of real wages, ˙w/w, and the return on working capital, ˙r/r at the beginning of the transition when k = 1? What do these results predict about real wage growth and the return on working capital as an economy with a high saving rate approaches its new new steady state?

(f) Can the economy achieve a higher c ∗ than for s = 30%? Why or why not?

2. Now consider the Ramsey model with a Cobb-Douglas aggregate production function, y = k α and α = 1/3. Assume the discount rate ρ = 4%, population growth n = 1%, technology growth g = 2%, and there is no capital depreciation.

(a) Derive k ∗ , y ∗ , and c ∗ as functions of the model parameters and determine their values when the coefficient of relative risk aversion θ = 5.

(b) How do your answers to part (a) change if the coefficient of relative risk aversion changes to θ = 2 (i.e., the intertemporal elasticity of substitution rises from 0.2 to 0.5)?

(c) Draw the transition given the change in the coefficient of relative risk aversion in the phase diagram for the Ramsey model. Consider the change to be unexpected.

(d) What happens to the interest rate r ∗ = f ′ (k ∗ ) given the change in θ?

(e) For this economy, what is the impact of a permanent fall in the growth rate of technology on the saving rate along the balanced growth path? How does your answer depend on the intertemporal elasticity of substitution? (hint: compute ∂s∗/∂g, where s ∗ = 1 − c ∗/y∗ )(f) What happens to the steady-state return on capital r ∗ given a permanent fall in g to 1% if θ = 5?

3. Essay Question: Drawing arguments from the models seen in class so far, make an argument about the level of global interest rates in the future as the population ages and the popula-tion growth rate falls. You may comment on whether you think the models make realistic predictions, and ponder the result one of one against the other. You may also consider the role of policy. To have an idea of the magnitudes involving the decline in population growth, you can use the World Population Prospects, available in this link. The essay should be no more than one page in length. (Note: with the exception of the population growth n, the base parameters provided in exercise 1 are approximately correct, considering the world economy).