Suppose that you have the standard Solow model with both labor augmenting productivity growth and population growth. The production function is Cobb-Douglas. The central equation of the Solow model, expressed in per efficiency units of labor, is given by:
k̂t+1 = 1/(1 + z)(1 + n) [sAk̂αt + (1 − δ)k̂t].
The other variables of the model are governed by Equations:
ît = sŷt (6.23)
ŷt = Af(k̂t) (6.24)
ŷt = ĉt +ît (6.25)
Rt = Af′ (k̂t) (6.26)
wt = Zt[Af(k̂t) − Af′ (k̂t)k̂t]. (6.27)
(a) Create an Excel file. Suppose that the level of productivity is fixed at A = 1. Suppose that s = 0.2 and δ = 0.1. Suppose that α = 1/3. Let z = 0.02 and n = 0.01. Solve for a numeric value of the steady state capital stock per efficiency unit of labor.
(b) Suppose that the capital stock per worker initially sits in period 1 in steady state. Create a column of periods, ranging from period 1 to period 100. Use the central equation of the model to get the value of k̂ in period 2, given that k̂ is equal to its steady state in period 1. Continue to iterate on this, finding values of k̂ in successive periods up through period 9. What is true about the capital stock per efficiency unit of labor in periods 2 through 9?
(c) In period 10, suppose that there is an increase in the population growth rate, from n = 0.01 to n = 0.02. Note that the capital stock per efficiency unit of labor in period 10 depends on variables from period 9 (i.e. the old, smaller value of n), though it will depend on the new value of n in period 11 and on. Use this new value of n, the existing value of the capital stock per efficiency unit of labor you found for period 9, and the central equation of the model to compute values of the capital stock per efficiency unit of labor in periods 10 through 100. Produce a plot showing the path of the capital stock per efficiency unit of labor from period 1 to period 100.
(d) Assume that the initial levels of N and Z in period 1 are both 1. This means that subsequent levels of Z and N are governed by Equations:
Nt = (1 + n)^t. (6.7)
Zt = (1 + z)Zt−1, z ≥ 0. (6.8)
Zt = (1 + z)^t. (6.9)
Create columns in your Excel sheet to measure the levels of N and Z in periods 1 through 100.
(e) Use these levels of Z and N, and the series for k̂ you created above, to create a series of the capital stock per worker, i.e. kt = k̂tZt. Take the natural log of the resulting series, and plot it across time.
(f) How does the increase in the population growth rate affect the dynamic path of the capital stock per worker?