The Solow Growth Model (SGM) system of equations:
$ \frac{d\left(L_t\right)}{dt} = nL_t $ :: labor force growth equation
$ \frac{d\left(E_t\right)}{dt} = gE_t $ :: efficiency of labor growth equation
$ \frac{d\left(K_t\right)}{dt} = sY_t - \delta{K_t} $ :: capital stock growth equation
$ Y_t = \left(K_t\right)^{\alpha}\left(L_tE_t\right)^{1-\alpha} $ :: production function
$ \lim\limits_{t\to\infty}\left(\frac{K_t}{Y_t}\right) = \frac{s}{n+g+\delta} $ :: steady-state balanced-growth path capital-output ratio
$ \lim\limits_{t\to\infty}\left(\frac{Y_t}{L_t}\right) = \left(\frac{s}{n+g+\delta}\right)^{\frac{\alpha}{1-\alpha}} \left(E_0{e^{gt}}\right) $ :: steady-state balanced-growth path output-per-worker ratio
$ \lim\limits_{t\to\infty}\left(\frac{K_t}{L_t}\right) = \left(\frac{s}{n+g+\delta}\right)^{\frac{1}{1-\alpha}} \left(E_0{e^{gt}}\right) $ :: steady-state balanced-growth path capital-worker ratio
convergence rate $ = -(1-\alpha)(n+g+\delta) $
$ \frac{K_t}{Y_t} = \left(1- e^{-(1-\alpha)(n+g+\delta)t}\right)\left(\frac{K}{Y}\right)^* + \left(e^{-(1-\alpha)(n+g+\delta)t}\right)\left(\frac{K_o}{Y_o}\right) $
:: convergence to the steady-state balanced-growth capital-output ratio
H: ideas—non-rival: growth rate h
E: efficiency of labor: growth rate g
L: labor force: growth rate n
N: natural resources—rival: growth rate 0
$ g = \left(\frac{\gamma}{1+\gamma}\right)h - \left(\frac{1}{1+\gamma}\right)n $
$ n = {\gamma}h $ :: steady-state balanced-growth path with g = 0
$ n = {\phi}\ln\left(\frac{Y/L}{y^s}\right) $ :: Malthusian population growth
$ g = \left(\frac{\gamma}{1+\gamma}\right)h - {\phi}\left(\frac{1}{1+\gamma}\right)\ln\left(\frac{Y/L}{y^{s}}\right) $
Two sets of theories for escape:
Or:
Plus:
(not on exam)
Kaldor facts:
*C onstant wL/Y (= 1-α)
Piketty facts:
Plutocracy and its fear of creative destruction
Divergence, 1800-1975
Convergence 1975-present?
How to understand?
We need a high capital share α:
We need n to be inversely and s strongly correlated with E
And we need education to be a key link:
Full employment (because of flexible wages and prices and debt)
Shifts of production and spending across categories
$ Y^* = Y = \mu\left(c_o + I_o + G\right) + \mu\left(x_f{Y^f} + x_{\epsilon}{\epsilon}_o + x_{\epsilon}{\epsilon}_r{r^f}\right) - \mu\left(I_r + x_{\epsilon}{\epsilon}_r\right)r $
$ Y = E = \mu\left(c_o + I_o + G\right) + \mu\left(x_f{Y^f} + x_{\epsilon}{\epsilon}_o + x_{\epsilon}{\epsilon}_r{r^f}\right) - \mu\left(I_r + x_{\epsilon}{\epsilon}_r\right)r $
Causation from left to right:
Influences on spending from:
$ Y = C + I + G + (GX - IM) $
$ Y = (c_o + c_y(1-t)Y) + I + G + (GX - im_y{Y}) $
$ (1 - c_y(1-t) + im_y)Y = c_o + I + G + GX $
$ Y = \frac{c_o + I + G + GX}{(1 - c_y(1-t) + im_y)} $
$ Y = {\mu}(c_o + I + G + GX) $
$ \mu = \frac{1}{(1 - c_y(1-t) + im_y)} $
The interest rate in the IS Curve is the long-term risky real interest rate: r
The interest rate the central bank controls is the short-term safe nominal interest rate: i
$ {\pi_t} = {\pi_t}^e - \beta\left(u_t - u^*\right) + SS_t$
Expectations:
$ r_t = r^{**} + r_{\pi}(\pi_t - \pi^T) - r_u(u_t - u^{**}) $
$ r_t = r^{**} + r_{\pi}(\pi_t - \pi^T) $
$ u_t - u^* = \phi(\pi_{t-1} - \pi^T) + \psi(r^{**} - r^*) + \delta_t $
Combine the MPRF with the "inflation dynamics" version of the Phillips Curve...
Boost government purchases by ΔG—if no Federal Reserve offset because at ZLB
“Hysteresis” parameter η
r - g greater or less than 2η?