Sep 7
Phase Portraits
dx/dt = rx(1 − x/K)
dx/dt = rx(1 − x/K) − H
Domar Growth Model
Evsey Domar, Capital Expansion, Rate of Growth, and Employment, Econometrica,
Vol. 14, No. 2. (Apr., 1946), pp. 137-147.
Y (t) income flow per year
I(t) investment flow per year
Assume dY /dt = (1/s)dI/dt, where s is constant marginal propensity to save
κ is the potential output of the economy
Assume κ = ρK, where ρ is a constant capacity-capital ratio
Then dκ/dt = ρdK/dt = ρI
Domar assumes Y = κ
To maintain this equilibrium, we need dY /dt = dκ/dt
Substitute to get (1/s)dI/dt = dκ/dt = ρI. Need exponential growth in investment at a rate ρs.
Solow Growth Model
Robert M. Solow, A Contribution to the Theory of Economic Growth, The Quarterly
Journal of Economics, Vol. 70, No. 1. (Feb., 1956), pp. 65-94.
Production function Q = f (K, L)
Q is output (net of depreciation)
K is capital, K > 0
L is labor, L > 0
Assume fK , fL > 0 (positive marginal products) and fKK , fLL < 0 (diminishing
returns)
Assume Q is homogeneous, i.e. Q = Lf (K/L, 1) = Lφ(k), where k = K/L
Show φ0 (k) > 0 and φ00 (k) < 0. Use fK > 0, fKK < 0 and L > 0.
Solow assumes dK/dt = sQ (constant proportion of Q invested, s is the marginal
propensity to save) and dL/dt = λL, λ > 0 (labor force grows exponentially, λ is the
rate of growth of labor)
Note that assumptions are about how the rates of change are determined.
Substitute for Q, dK/dt = sLφ(k)
Differntiate K = kL, dK/dt = Ldk/dt + kdL/dt = Ldk/dt + λkL
Obtain dk/dt = sφ(k) − λk.
Find the phase portrait for the Solow model. Show that any equilibrium is stable.
Assume Q = Lk α .
Assignment:
Find the phase portraits for 3.25, 3.37, 3.40, 3.45