Solow Growth Model

Framework

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

 

Balanced-Growth Path

  • $ \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

  • 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

 

Malthusian Efficiency of Labor Growth

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) $

 

How Did We Escape?

Two sets of theories for escape:

  • Eye of the needle
    • Cultural-scientific
    • Resource-technology
    • Plunder-exploitation
    • Variants: "We almost got there many times" and "we never got close before" variants
    • Variants: Commercial Revolution, Industrial Revolution, or Modern Economic Growth?

Or:

  • Two heads are better than one...
    • $ h = \left(h_1\right)L^{\lambda} $ :: idea generation

Plus:

  • Demographic transition...
    • $ n = \min\left({\phi}\ln\left(\frac{Y/L}{y^s}\right), \frac{n_1}{Y/L}\right) $

 

Escape: Industrial Revolution and Modern Economic Growth

  • Elasticity of Demand as a Key (not on final)
  • Productivity Trends in the North Atlantic
    • Britain the First Industrial Nation
    • Britain richer—but with low real wages
    • British growth acceleration
    • But America growing faster from 1800
    • And American growth acceleration—modern economic growth and the industrial research lab
    • Until the productivity growth slowdon of the 1970s
      • And then the speed up of the new-economy 1990s
      • And then the growth collapse of the Great Recession

 

Income and Wealth Inequality

(not on exam)

Kaldor facts:

  • Constant r (=αK/Y) *C onstant wL/Y (= 1-α)
  • Constant K/Y
  • Constant g
  • d(ln(w))/dt = g

Piketty facts:

  • Increase in W/K
  • Increase in market-to-book ratio for K
  • Divergence between marginal product of capital and average return
  • Substantial decrease in real interest rates in financial markets

Plutocracy and its fear of creative destruction

 

Measuring Economic Growth Truly

(not on exam)

 

Global Patterns

Divergence, 1800-1975

  • Britain and U.S. growing together
  • OECD convergence 1945-present
  • Behind Iron Curtain divergence
  • General divergence 1800-1975
  • From a fivefold to a fifty-fold divergence

Convergence 1975-present?

  • East Asia
  • Japan
  • China

How to understand?

  • $ \alpha = 3/5 $
  • Schooling very important for the efficiency of labor

 

Modeling Global Patterns

We need a high capital share α:

  • To make “convergence” take a long time
  • To amplify the effects of differences in (K/Y)* on prosperity

We need n to be inversely and s strongly correlated with E

  • Demographic transition
  • Favorable relative price structure

And we need education to be a key link:

  • We need technology transfer to a poorly educated population to be nearly impossible…

 

Business Cycles

2018 03 08 Macroeconomics Flexprice MRE key

  • Okun's Law

 

Flexible-Price Models

Full employment (because of flexible wages and prices and debt)

  • Unemployment rate equal to NAIRU
  • Production equal to potential output

Shifts of production and spending across categories

  • In response to changes in the economic environment
  • And in response to changes in economic policy
  • As a result of shifts in the long-term real risky interest rate r

 

The Business Cycle NIPA Framework

  • $ Y = C + I + G + (GX - IM) $ :: national income and product
  • $ C = c_o + c_y(1-t)Y $ :: consumption function—consumer confidence; marginal propensity to consume; net taxes-less-transfers rate
  • $ I = I_o - I_r{r} $ :: investment spending; "animal spirits"
  • $ G $
  • $ IM = im_y{Y} $ :: imports
  • $ \epsilon = \epsilon_o + \epsilon_r(r^f - r) $ :: exchange rate; foreign exchange speculators; "gnomes of Zurich"
  • $ GX = x_f{Y^f} + x_\epsilon{\epsilon} $ :: gross exports

 

The Flexible-Price Model IS Curve Equation

$ 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 $

 

Sticky-Price Models

The Sticky-Price Model IS Curve Equation

$ 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:

  • Spending determines aggregate demand
  • Aggregage demand via the inventory adjustment channel determines national income and product

 

Influences on spending from:

  • Policy variables: G, t, $ r = i - \pi +\rho $
  • Expectations: $c_o, I_o, \epsilon_o $
  • Foreign economic conditions: $ Y^f, r^f $

 

The Keynesian Multiplier

$ 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)} $

 

Monetary Policy and the Zero Lower Bound

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

  • $ r = i - \pi + \rho $ subject to $ i ≥ 0 $
  • $ \rho = \rho^R + \rho^T $
    • $ \rho^R $ :: the risk premium for lending to privates rather than to the government
      • Moral hazard
      • Adverse selection
      • "Skin in the game" from borrowers
      • Financial crises
    • $ \rho^T $ :: lack of confidence that the central bank will keep i where it currently is

 

Phillips Curve

$ {\pi_t} = {\pi_t}^e - \beta\left(u_t - u^*\right) + SS_t$

Expectations:

  • Static: $ {\pi_t}^e = \pi^{*} $
  • Adaptive: $ {\pi_t}^e = \pi_{t-1} $
  • Rational: $ {\pi_t}^e = \pi_{t} $
  • Hybrids: $ {\pi_t}^e = \lambda(\pi_{t}) + (1-\lambda)(\pi_{t-1}) $ or $ {\pi_t}^e = (1-\lambda)(\pi^*) + \lambda(\pi_{t-1}) $

 

Inflation Dynamics

  • 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} $
  • Hybrids:
    • $ {\pi_t} = {\pi_{t-1}} - \frac{\beta\left(u_t - u^*\right) + SS_t}{1-\lambda} $
    • $ {\pi_t} - \pi^* = \lambda({\pi_{t-1}}-\pi^*) - \beta\left(u_t - u^*\right) + SS_t $

 

Monetary Policy Reaction Function

$ 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...

 

Hysteresis and Budget Arithmetic in a Depression

Boost government purchases by ΔG—if no Federal Reserve offset because at ZLB

  • Get boost to real GDP by μΔG
  • Get boost to taxes by tμΔG
  • Increase in debt of (1 - tμ)ΔG = ΔD
  • Financing cost of this debt: (r-g)ΔD = (r-g)(1 - tμ)ΔG

“Hysteresis” parameter η

  • Gain tημΔG in tax revenue from heading off “hysteresis”
  • (r-g)(1 - tμ)ΔG greater or less than ηtμΔG?
    • t = 0.33
    • μ = 2
    • 0.33(r - g) greater or less than 0.66η?

r - g greater or less than 2η?