#!/usr/bin/env python # coding: utf-8 # # Econ 101b Review: May 1, 2018 # # | # ## 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?