# 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

(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… • 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
• "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?