Question

Piketty (2014) argues that a fall in the growth rate of the economy is likely to lead to an increase in the difference between the real interest rate and the growth rate. This problem asks you to investigate this issue in the context of the Ramsey-CassKoopmans model. Specifically, consider a Ramsey-Cass-Koopmans economy that is on its balanced growth path, and the two equations describing the dynamics of capital and consumption per unit of effective labor are given by k˙(t)=f(k(t))−c(t)−(n+g)k(t),c(t)c˙(t)=θf′(k(t))−rho−θg. Now suppose there is a permanent fall in g.

(A) How, if at all, does this affect the k=0 curve?
(B) How, if at all, does this affect the c˙=0 curve?
(C) At the time of the change, does c rise, fall, or stay the same, or is it not possible to tell?
(D) At the time of the change, does r−g rise, fall, or stay the same, or is it not possible to tell?
(E) In the long run, does r−g rise, fall, or stay the same, or is it not possible to tell?
(F) Find an expression for the impact of a marginal change in g on the fraction of output that is saved on the balanced growth path. Can one tell whether this expression is positive or negative?

Answer 1

In the Ramsey-Cass-Koopmans model, a fall in the growth rate does not affect the k=0 curve but does shift the c˙=0 curve. Initially, the impact on consumption cannot be definitively determined, and the difference between the real interest rate and the growth rate, r−g, is likely to rise both in the short term and in the long run.

Step-by-step explanation:

The question is regarding the Ramsey-Cass-Koopmans model and how a permanent fall in the growth rate of the economy, denoted by g, affects different curves and variables within the model. The dynamics of capital and consumption per unit of effective labor are governed by equations involving k(t) and c(t), where k˙(t) represents the change in capital stock over time, and c(t)c˙(t) represents the change in consumption over time.

(A) A fall in g would not affect the k=0 curve, which determines the steady-state level of capital because this curve is determined by the economy's production function and depreciation rate, which do not directly depend on g.

(B) The c˙=0 curve, which defines the consumption growth rate, is directly affected by a change in g because it includes (θf′(k(t)) − rho − θg), and a change in g would shift this curve.

(C) It's not possible to definitively say what happens to consumption c immediately after the change without additional information.

(D) The real interest rate minus growth rate (r−g), would likely increase at the time of the change since g has decreased but it's not certain without knowing the behavior of the real interest rate (r).

(E) In the long run, r−g is expected to rise, because a decrease in g without a corresponding decrease in r will cause the difference to become larger.

(F) The expression for the impact of a change in g on the fraction of output that is saved on the balanced growth path would depend on the particular model specifics. It is difficult to make a clear determination on whether this expression is positive or negative without additional information.