Solow Model: Normalization, Steady State, and Savings Rate

Normalization and Steady State

In the Solow model, consider an economy with an aggregate production function !" = $" %('"(")*+%, where !", $", (", and '" are output, capital stock, number of workers and efficiency of workers at time t. The number of workers grows over time at rate n, while the efficiency of these workers grows at rate g. Assume capital depreciates at a constant rate d and 0 < 3 < 1. Last, assume the households’ savings rate is s%.

(a) Which normalization of the endogenous variables allows us to specifiy a steady state in the model (i.e. the normalized variables are constant in the long-run)? Explain why.

(b) Write the law of motion of the normalized capital. 

(c) Illustrate the steady state level of normalized capital on a graph. 

(d) Solve for the steady state level of normalized capital

(e) Solve for the steady state levels of normalized output and normalized consumption. At which rate does output per worker grow in the steady state?

(f) Assume the households’ savings rate increases to s’ %> s%. Show graphically what effects this has on the long-run (steady state) normalized capital.

(g) Continue assuming the households’ savings rate increases to s’ %> s%. Show graphically how the economy converges to the long-run level of normalized capital (that is, the transition path).

(h) Continue assuming the households’ savings rate increases to s’ %> s%. Show graphically how the economy converges to the long-run level of output per worker.