https://github.com/braddelong/LSF18E101B/blob/master/Intermediate_Macroeonomics_Review.ipynb
The Solow Growth Model (SGM) system of equations:
$ g_L = \frac{d\left(L_t\right)}{dt} = nL_t $ :: labor-force growth equation
$ g_E = \frac{d\left(E_t\right)}{dt} = gE_t $ :: efficiency-of-labor growth equation
$ g_K = \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
U.S. today:
Afghans back at where U.S. was 200 years ago
$ \left(\frac{K}{Y}\right)^* = \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
$ \left(\frac{Y_t}{L_tE_t}\right)^* = \lim\limits_{t\to\infty}\left(\frac{Y_t}{L_tE_t}\right) = \left(\frac{s}{n+g+\delta}\right)^{\frac{\alpha}{1-\alpha}} $ :: steady-state balanced-growth path output-per-worker ratio
$ \left(\frac{K_t}{L_tE_t}\right)^* = \lim\limits_{t\to\infty}\left(\frac{K_t}{L_tE_t}\right) = \left(\frac{s}{n+g+\delta}\right)^{\frac{1}{1-\alpha}} $ :: steady-state balanced-growth path capital-worker ratio
convergence rate $ = -(1-\alpha)(n+g+\delta) $
convergence to the steady-state balanced-growth capital-output ratio:
$ \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) $
$ \frac{K_t}{Y_t} = \left(1 - \frac{1}{[(1+(1-\alpha)(n+g+\delta)]^t}\right)\left(\frac{K}{Y}\right)^* + \left(\frac{1}{[(1+(1-\alpha)(n+g+\delta)]^t}\right)\left(\frac{K_o}{Y_o}\right) $
Start with an economy on its balanced-growth path, with:
And let's change something: $ s_{alt} = s_{ini} + {\Delta}s $...
What is $ {\Delta}\left(\frac{K}{Y}\right)^* $ going to be?
A. $ {\Delta}\left(\frac{K}{Y}\right)^* = \left(\frac{{\Delta}s}{n+g+\delta}\right)^{\alpha} $
B. $ {\Delta}\left(\frac{K}{Y}\right)^* = \left(\frac{{\Delta}s}{n+g+\delta}\right)^\left({\frac{\alpha}{1-\alpha}}\right) $
C. $ {\Delta}\left(\frac{K}{Y}\right)^* = \frac{{\Delta}s}{n+g+\delta} $
D. $ {\Delta}\left(\frac{K}{Y}\right)^* = \frac{(1-\alpha){\Delta}s}{n+g+\delta} $
E. None of the above
If $ \alpha = 2/3, \delta = 0.025, n_{ini} = 0.005, $
$ s_{ini} = 0.16, g_{ini} = 0.01,$ and $ {\Delta}s = 0.04, $
then $ {\Delta}\left(\frac{K}{Y}\right)^* $ is going to be?
A. 9
B. 0.404
C. 1
D. 1/3
E. None of the above
3 years after a jump $ {\Delta}s $ in the savings rate, an economy that started out on its initial balanced-growth path would have seen its capital-output ratio reduce approximately what fraction of the gap between the initial and the alternative balanced-growth path values?
A. $ (1-\alpha)(n+g+\delta) $
B. $ e^{(1-\alpha)(n+g+\delta)} $
C. $ e^{3(1-\alpha)(n+g+\delta)} $
D. $ 3(1-\alpha)(n+g+\delta) $
E. None of the above
H: ideas: non-rival factors: growth rate h
E: efficiency of labor: growth rate g
L: labor force: growth rate n
N: natural resources: rival factors: growth rate 0 (in most applications)
$ g = \left(\frac{\gamma}{1+\gamma}\right)h - \left(\frac{1}{1+\gamma}\right)n $
$ 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) $
In a Malthusian equilibrium, what is
the relationship between the rate of
growth $ h $ of the useful ideas stock
and the rate of population growth $ n $?
A. $ n = h $
B. $ n = {\gamma}h $
C. $ n = \frac{h}{\gamma} $
D. None of the above
Between the year 1 and the year 1800,
$ \gamma = 3, n = 0.075% $ per year, and
the economy was in Malthusian equilibrium.
What, roughly, was the rate of growth of h?
A. 0.025% per year
B. 0.075% per year
C. 0.225% per year
D. None of the above
In Malthusian equilibrium, what is the
Malthusian balanced-growth path level
of output per worker (Y/L)*?
A. $ \left(Y/L\right)^* = e^{\left(\frac{{\gamma}h}{\phi}\right)}y^s $
B. $ \left(Y/L\right)^* = {\left(\frac{{\gamma}h}{\phi}\right)}y^s $
C. $ \left(Y/L\right)^* = {\phi}{\gamma}hy^s $
D. None of the above
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 $
Alternatively:
$ 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)(i - \pi + \rho) $
$ Y^* = Y = AD = \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 $
Start from one equilibrium "ini" and consider a shift to another "alt"...
What things change?
$ Y^* = Y = AD = \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 $
$ 0 = \mu\left(x_{\epsilon}{\epsilon}_r{{\Delta}r^f}\right) - \mu\left(I_r + x_{\epsilon}{\epsilon}_r\right){\Delta}r $
What is $ {\Delta}r $?
A. $ {\Delta}r = {\Delta}r^f $
B. $ {\Delta}r = \frac{x_{\epsilon}{\epsilon}_r}{I_r + x_{\epsilon}{\epsilon}_r}{\Delta}r^f $
C. $ {\Delta}r = \frac{I_r}{I_r + x_{\epsilon}{\epsilon}_r}{\Delta}r^f $
D. None of the above
If $ I_r = 20 , \mu = 2, x_{\epsilon} = 1, $ and
$ {\epsilon_r = 10 } $, then $ {\Delta}r^f = + 0.03 $ will
have what effect on domestic investment I?
A. $ {\Delta}I = + 0.6 $
B. $ {\Delta}I = - 0.6 $
C. $ {\Delta}I = - 0.2 $
D. None of the above
If $ I_r = 20 , \mu = 2, x_{\epsilon} = 1, $ and
$ {\epsilon_r = 10 } $, then $ {\Delta}G = + 0.3 $ will have
what effect on domestic investment I?
A. $ {\Delta}I = \left(x_{\epsilon}{\epsilon_r}{I_r + x_{\epsilon}{\epsilon}_r}\right){\Delta}G $
B. $ {\Delta}I = \left(\frac{I_r}{I_r + x_{\epsilon}{\epsilon}_r}\right){\Delta}G $
C. $ {\Delta}I = \left(\frac{1}{I_r + x_{\epsilon}{\epsilon}_r}\right){\Delta}G $
D. None of the above
$ Y = AD = \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 $
Alternatively:
$ 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)(i - \pi + \rho) $
Causation from right to left:
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 Simple Multiplier: $ \frac{{\Delta}Y}{{\Delta}G} = \mu $ :: holding r, t fixed
The Balanced Budget Multiplier:
Allowing t to vary with G, so that: $ {\Delta}t = \frac{{\Delta}G}{Y} $ so that $ {\Delta}(G - T) = 0 $...
Then: $ {\Delta}Y = \frac{{\Delta}G - c_y(1-t){\Delta}G}{1-c_y(1-t)+im_y} = \frac{1-c_y(1-t)}{1-c_y(1-t)+im_y}{\Delta}G $
The Monetary-Offset Multiplier:
Allowing r to vary with G, so that: $ {\Delta}r = \frac{{\Delta}G}{I_r + x_{\epsilon}{\epsilon}_r} $...
Then: $ {\Delta}Y = 0{\Delta}G $
$ \mu = \frac{1}{(1 - c_y(1-t) + im_y)} $
Suppose $ c_y = 5/6, t = 1/4, im_y = 1/8 $,
then if $ {\Delta}G = 0.4 $ and r and t are fixed:
A. $ {\Delta}Y = 0.4 $
B. $ {\Delta}Y = 0.5 $
C. $ {\Delta}Y = 0.8 $
D. $ {\Delta}Y = 0.6 $
E. None of the above
Suppose $ c_y = 2/3, t = 1/4, im_y = 1/4 $,
then if $ {\Delta}G = 0.3 $ and r and t are fixed:
A. $ {\Delta}Y = 0.4 $
B. $ {\Delta}Y = 0.5 $
C. $ {\Delta}Y = 0.8 $
D. $ {\Delta}Y = 0.6 $
E. None of the above
Suppose $ c_y = 1/3, t = 1/4, im_y = 1/4 $,
then if $ {\Delta}G = 0.6 $ and r and t are fixed:
A. $ {\Delta}Y = 0.4 $
B. $ {\Delta}Y = 0.5 $
C. $ {\Delta}Y = 0.8 $
D. $ {\Delta}Y = 0.6 $
E. None of the above
$ Y = AD = \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 right to left...
Government policy determines G, t via fiscal policy and (influences) r via monetary policy that determines i in $ r = i + \rho - \pi $
The economic environment:
Add these up (and apply the multiplier $ \mu $) to calculate aggregate demand AD
Aggregate demand AD determines production Y through the inventory-adjustment process
And potential output $ Y^* $ is nowheresville here...
$ Y = AD = \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 $
Suppose $ \mu = 2 $ and $ {\Delta}G = 0.3, $ then:
A. $ {\Delta}Y = 0.2 $
B. $ {\Delta}Y = 0.3 $
C. $ {\Delta}Y = 0.5 $
D. $ {\Delta}Y = 0.6 $
E. None of the above
Suppose $ \mu = 2, I_r = 20, x_{\epsilon} = 4, $ and $ {\epsilon}_r = 5 $.
Then if $ {\Delta}r = -0.015 $ :
A. $ {\Delta}Y = 0.2 $
B. $ {\Delta}Y = 0.3 $
C. $ {\Delta}Y = 0.5 $
D. $ {\Delta}Y = 0.6 $
E. None of the above
Suppose $ \mu = 2, {\Delta}G = 0.3, I_r = 20, x_{\epsilon} = 4, $ and $ {\epsilon}_r = 5 $.
Then if $ {\Delta}r = -0.015 $ :
A. $ {\Delta}Y = 0.2 $
B. $ {\Delta}Y = 0.3 $
C. $ {\Delta}Y = 0.5 $
D. $ {\Delta}Y = 0.6 $
E. None of the above
$ {\pi_t} = {\pi_t}^e - \beta\left(u_t - u^*\right) + SS_t$
Expectations:
Static: $ {\pi_t} = \pi^* - \beta\left(u_t - u^*\right) + SS_t$
Adaptive: $ {\pi_t} = {\pi_{t-1}} - \beta\left(u_t - u^*\right) + SS_t $
Rational: $ {\pi_t} = {\pi_t}^e $ and $ u_t = u^* - \frac{SS_t}{\beta} $
Hybrid: Adaptive and Rational:
Hybrid: Adaptive and Static:
$ r_t = r^{n} + r_{\pi}(\pi_t - \pi^T) $
$ u_t - u^* = {u_r}({r_t}^n - r^*) + \delta_t $
$ \pi_t = {\pi_t}^e - \beta(u_t - u^*) + SS_t $
Combine the MPRF with the "inflation dynamics" version of the Phillips Curve...
$ u_t - u^* = {u_r}r_{\pi}(\pi_t - \pi^T) $
$ \pi_t = {\pi_t}^e - \beta{u_r}r_{\pi}(\pi_t - \pi^T) + SS_t $
$ \pi_t = \left(\frac{1}{1 + \beta{u_r}r_{\pi}}{\pi_t}^e + \frac{\beta{u_r}r_{\pi}}{1 + \beta{u_r}r_{\pi}}\pi^T\right) + \frac{1}{1 + \beta{u_r}r_{\pi}}SS_t $
Expectations: Static:
\frac{1}{1 + \beta{u_r}r_{\pi}}SS_t $
Expectations: Adaptive:
\frac{1}{1 + \beta{u_r}r_{\pi}}SS_t\right) $
$ Y^* = Y = AD = \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 $ :: Flexprice IS
$ r^* = \frac{\left[\frac{Y^*}{\mu} - \left(c_o + I_o + G\right) + \left(x_f{Y^f} + x_{\epsilon}{\epsilon}_o + x_{\epsilon}{\epsilon}_r{r^f}\right)\right]}{I_r + x_\epsilon\epsilon_r} $ :: r-star
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
Boost government purchases by ΔG—if no Federal Reserve offset because at ZLB
“Hysteresis” parameter η
r - g greater or less than 2η?
Keynote: https://www.icloud.com/keynote/0HA_6uYorQfMYi_hRNLo23p5Q
Lecture Support:http://nbviewer.jupyter.org/github/braddelong/LSF18E101B/blob/master/Intermediate_Macroeonomics_Review.ipynb https://github.com/braddelong/LSF18E101B/blob/master/Intermediate_Macroeonomics_Review.ipynb