ECON 101B Final Review

Created by Yunhao Cao (Github@ToiletCommander) in Fall 2022 for UC Berkeley ECON 101B (Benjamin Schoefer).

Reference Notice: Material highly and mostly derived from Prof Schoefer's lecture slides, some ideas were borrowed from wikipedia.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Textbook: Modern Macroeconomics by Sanjay Chugh (1st edition, MIT Press)

Generic Terms

Stocks vs. Flows

Stock variables (accumulation variables)

Quantity variables whose natural measurement occurs at a particular moment in time

e.g.
1. checking account balance
1. credit card indebtedness
1. Mortgage loan payoff

Flow variables

Quantity variables whose natural measurement occurs over the course of a given interval of time

e.g.
1. Income
1. Consumption
1. Savings

Fluctuations vs. Trend

fluctuation_{t,trend}^x = \frac{x_t - x_t^{trend}}{x_t^{trend}}=\frac{x_t}{x_t^{trend}}-1

1+ fluctuation_{t,trend}^x=\frac{x_t}{x_t^{trend}} \\

Using trick $\ln(1+x) \approx x, \forall x \ll 1$

\text{fluctuation}_{t,trend}^x\approx \ln(x_t) - \ln(x_t^{trend})

Present Discounted Value (PDV)

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How to value variables in the future (suppose there’s no inflation)

The present value of a payout $x$ that is paid $d$ periods in the future is amount of $PDV(x,i,d)$ that would yield exactly $x$ if continuously invested for $d$ periods starting today at interest rate $i$

(1+i)^d \cdot PDV(x,i,d) = x

Fisher equation

1+r = \frac{1+i}{1 + \pi}

$i$ ⇒ nominal interest rate between periods

$r$ ⇒ real interest rate between periods

$\pi_t$ ⇒ net inflation rate between period $t-1$ and period $t$

“Approximate fisher equation”

1+r+\pi+\underbrace{r\pi}_{\mathclap{\text{In advanced economics, $r$ and $\pi$ are both generally small}}} = 1+i \\ \vdots \\ r = i - \pi

Can also be derived from:

1+r = \frac{1+i}{1 + \pi} \\ \ln(1+r) = \ln(1+i) - \ln(1+\pi) \\ \downarrow \text{approximately} \\ r \approx i - \pi

Fisher Effect

The effect of inflation on the nominal interest rate, for a given real interest rate.

If inflation increases, the nominal interest rate increases roughly 1-to-1

Capital and Investment

Capital
- A stock variable
- Takes time to build
- Analogous to consumers’ wealth/asset
- Except $k$ cannot be negative
- Firms must decide in period $t$ how much capital they want to use in the production process $t+1$

Investment
- A flow variable
- Change in a firm’s capital stock between two consecutive periods
- One of the components of GDP
  - Investment compreses around 15% of GDP in US
    - Around 40% in China ⇒ High investment drives rapid growth

Capital investment - the change in a firm’s capital stock between two consecutive periods

GDP Gross Domestic Product

Three views:

Production

Expenditure

Income

Counting GDP using production

We want to measure only the price of the final good, not intermediate goods to avoid double-counting.

Either

Survey very last producer in the production chain

Use value added approach
1. $GDP = \sum VA_i = p_n q_n$

Nominal GDP 名义/货币GDP

Nominal Production of good $i$ :

y_{i,t}^{nom} = p_{i,t} \times q_{i,t}

Then we calculate nominal GDP by summing over all product categories

NGDP_{t} = \sum_{i \in I} p_{i,t} \times q_{i,t}

Growth for Good $y$

growth_{t}^y=\frac{y_t-y_{t-1}}{y_{t-1}}=\frac{y_t}{y_{t-1}}-1

Real GDP(实际GDP)

GDP Deflator(平均物价指数)

Conencted with Real GDP, see that section as well

GDPDef_t^b = \frac{NGDP_t}{RGDP_t^b} = \frac{p_t}{p_b}

Note $GDPDef$ captures inflation:

growth_t^{GDPDef}=growth_t^p

We can set a aseline year $b$ and the GDP deflator converts nominal GDP in any year into a real GDP measure

\underbrace{RGDP_t^b}_{\mathclap{\text{Real GDP in year $t$ provided baseline year $b$}}}=\sum_{i \in I} p_{t=b} \times q_{t=t} = \frac{NGDP_t}{GDPDef_{t}^b}

Also note that for production of product $i$ :

y_{i,t}^b = p_{i,b} \times q_{i,t}=y_{i,t}^t \frac{p_{i,b}}{p_{i,t}}

Growth of RGDP

growth_t^{RGDP}=growth_t^{NGDP}-growth_t^{GDPDef}

GDPpc(GDP Per Capita) 人均GDP

GDPpc_t = \frac{GDP_t}{Pop_t}

Counting GDP using expenditure

Y=C+I+G+NX

$Y$ - output

$C$ - private consumption

$I$ - private investment

$G$ - government expenditure and investment

$NX$ - net export

$X/Y$ - share of component $X$ in $Y$

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Important Fact:

\sum_{X \in \{C, I, G, NX\}} dX/dY = 0

Prince Indices 物价索引

GDP Deflator

GDPDef_t^b = \frac{NGDP_t}{RGDP_t^b} = \frac{p_t}{p_b}

Other Indexes

Consumer Price Index (CPI)

Personal Consumption Expenditure (PEC)

Expenditure

Consumer / Household Theory

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Utility Maximization!

Utility Function

Describes how much “happiness” or “satisfaction” an individual experiences from “consuming”

Marginal Utility $\frac{\partial u}{\partial c}$
1. extra total utility resulting from consumption of an extra unit of good

Diminishing Marginal Utility
1. $\frac{\partial u}{\partial c} > 0, \frac{\partial^2 u}{\partial c^2} < 0$

Usual Forms:

Log Utility
1. $u(c_1, \dots, c_n) = \sum_{i=1}^n \ln(c_i)$

More general utility
1. $u(c_1, \dots, c_n) = \sum_{i=1}^n \frac{c_i^{1-\sigma}-1}{1-\sigma}$
1. Equals to log utility when $\sigma = 1$

Subjective Discount Factor $\beta$

Impatience is an potential issue when we think about time

So we develop a simple model of consumer impatience

\beta \in (0,1)

The lower $\beta$ , the less does individual value future utility

Marginal Rate of Substitution

the quantity of one good( $c_1$ ) that a consumer can forego for additional units of another good( $c_2$ ) at the same utility level.

dc_1\frac{\partial u(c_1, c_2)}{\partial c_1}+dc_2 \frac{\partial u(c_1, c_2)}{\partial c_2} = 0 \\ MRS = \bigg| \frac{dc_1}{dc_2} \bigg|=\bigg| -\frac{\nabla_{c_2} u(c_1,c_2)}{\nabla_{c_1}u(c_1,c_2)}\bigg| = \frac{u_2(c_1,c_2)}{u_1(c_1,c_2)}

Income Types

Two broad categories:

Labor income

Asset income

Two-period Consumption-savings framework

Notations:

$c_t$ ⇒ consumption in period $t$

$P_t$ ⇒ nominal price of consumption in period $t$

$Y_t$ ⇒ nominal income in period $t$

$A_t$ ⇒ nominal wealth at the beginning of period $t+1$ (end of period $t$ )

$i$ ⇒ nominal interest rate between periods

$r$ ⇒ real interest rate between periods

$\pi_t$ ⇒ net inflation rate between period $t-1$ and period $t$

$y_t$ ⇒ real income in period $t$ ( $=Y_t / P_t$ )

(Lifetime) Budget Constraint

We will assume that $A_2$ is 0 (spend all in period 2 and leave no inheritances)

We will also assume $A_0 = 0$ for graphical simplicity

Assume strictly increasing utility with increase in consumption, but diminishing marginal utility in $c_1, c_2, \dots, c_n$ .

P_tc_t + A_t = Y_t + (1+i)A_t

List all of them

P_1c_1+A_1 = Y_1 + (1+i)A_0 \\ P_2c_2 + A_2 = Y_2 +(1+i)A_1

LBC:

\underbrace{P_1c_1+\frac{P_2 c_2}{1+i}}_{\text{PDV of all lifetime expenditure}}=\underbrace{Y_1+\frac{Y_2}{1+i}+(1+i)A_0}_{\text{PDV of all lifetime income}}

\begin{split} c_2&=-(\frac{P_1(1+i)}{P_2})c1+(\frac{1+i}{P_2})Y_1+\frac{Y_2}{P_2} \\ &=-(\frac{1+i}{1+\pi_2})c_1+(\frac{1+i}{P_2})Y_1+\frac{Y_2}{P_2} \end{split}

We can also write LBC in real terms

c_1 + \frac{c_2}{1+r} = y_1 + \frac{y_2}{1+r}

Optimum

At the optimal choice (max lifetime utility s.t. LBC)

\underbrace{\frac{u_1(c_1^*,c_2^*)}{u_2(c_1^*,c_2^*)}}_{\text{MRS}}=\underbrace{\frac{1+i}{1+\pi_2}}_{\text{price ratio}} = 1+r

Consumption Smoothing

Concave utility ( $u’>0, u’’<0$ ) implies that households will prefer to “spread out” consumption over time, rather than concentrate it in a single period.

Factors that can affect consumption smoothing:

Discount rate

Shape of utility function

Interest rate

Borrowing/saving constraints

Consumption Smoothing by LBC

The lifecycle budget constraint (LBC) allows consumers to consume the present value of their wealth and income and therefore allows consumers to smooth consumption.

Consumption smoothing by Propensity to Consume

Propensity to consume: The fraction of income that the household decides to consume when income changes

Transitory income change $dY$
1. When income suddenly falls for a single period, consumers will reduce consumption less than one-to-one with income. Income drop is smoothed out between both periods.
1. When income in both periods changes, consumption will fall by the full amount in both periods.

Consumption smoothing in a person’s entire life

Preference imply that consumption should be smoothed

Over an individual’s lifecycle, income flows vary greatly, and often predictably
1. schooling, first job (high wage growth), retirement

Consumption smoothing by wealth effects

When “period 0” assets increase or fall, consumption (in both periods) will change by the same amount.

Interest Rate Changes

Private savings function

s_1^{priv}(r,y_1,y_2)=y_1-c_1(r,y_1,y_2)

Although some households are borrowers while some may be savers, both borrowers and savers increase saving/decrease debt when the interest rate increases. Therefore, the aggregate response to interest rate changes goes into the same direction as we predict from our microeconomic model of the household. Going from “micro” to “macro” is seamless here!

Credit Crunch

financial sector has restricted the quantity of loans it is willing to extend to consumers in the “short run”

Consequence of “credit crunch”:

A larger fraction of consumers unable to borrow to pay for their desired early-period consumption ⇒ their consumption falls
1. Then GDP falls

Consumption in next period actually rises
1. More saving / less debt taken out by credit-constrained households

Infinite Period Consumption-savings Framework

Notations

$c_t$ ⇒ consumption in period $t$

$P_t$ ⇒ nominal price of consumption in period $t$

$Y_t$ ⇒ nominal income in period $t$ (assume it falls from the sky)

$a_{t-1}$ ⇒ real wealth (stock) holdings at beginning of period $t$ / end of period $t-1$

$\pi_{t+1}$ ⇒ net inflation rate between period $t$ and $t+1$

$y_t$ ⇒ real income in period $t$

Utility Function

Assume

u(c_t, c_{t+1}, \dots) = u(c_t) + \beta u(c_{t+1})+\beta^2 u(c_{t+2}) + \cdots

Budget Constraint

c_t+a_t=y_t+(1+r_{t-1})a_{t-1}

Consumer Optimum

Construct the lagrangian, we get

\begin{split} L(c_t, c_{t+1}, \dots,\lambda_t, \lambda_{t+1},\dots) &= u(c_t)+\beta u(c_{t+1}) + \cdots \\ & \quad + \lambda_t[y_t+(1+r_{t-1})a_{t-1}-c_t-a_t] \\ &\quad + \beta \lambda_{t+1}[y_{t+1} + (1+r_t)a_t - c_{t+1}-a{t+1}] + \cdots \end{split}

Compute FOCs

w.r.t $c_t$
1. $u’(c_t)-\lambda_t = 0$

w.r.t $a_t$
1. $-\lambda_t + \beta \lambda_{t+1}(1+r_t) = 0$

w.r.t. $c_{t+1}$
1. $\beta u’(c_{t+1}) - \beta \lambda_{t+1} = 0$

Combine them, we get euler’s equation

\lambda_t = \beta(1+r_t)\lambda_{t+1} \Longleftrightarrow u'(c_t)=\beta(1+r_t)u'(c_{t+1})

Infinite Period Consumption-savings Framework with Asset Pricing

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Not in scope for the exam

Notations

$c_t$ ⇒ consumption in period $t$

$P_t$ ⇒ nominal price of consumption in period $t$

$Y_t$ ⇒ nominal income in period $t$

$y_t$ ⇒ real income in period t ( $=Y_t/P_t$ )

$A_t$ ⇒ nominal wealth at the beginning of period $t+1$ (at the end of period $t$ )

$a_t$ ⇒ real wealth at beginning of period $t+1$ (at the end of period $t$ )

$i$ ⇒ nominal interest rate between periods

$r_t$ ⇒ real interest rate between periods $t-1$ and $t$

$\pi_t$ ⇒ net inflation rate between period $t-1$ and period $t$
1. $= \frac{P_{t}}{P_{t-1}} -1$

$S_t$ ⇒ nominal price of a unit of stock in period $t$

$D_t$ ⇒ nominal dividend paid in period $t$ by each unit of stock held at the start of $t$

Budget Constraints

We need infinite budget constraints to describe economic opportunities and possibilities

\forall t, P_tc_t+S_ta_t=Y_t+S_ta_{t-1}+D_ta_{t-1}

Solving for for lagrangian optimal,

With respect to $c_t$ : $u’(c_t)-\lambda_tP_t = 0$

With respect to $a_t$ : $-\lambda_tS_t+\beta \lambda_{t+1} (S_{t+1} + D_{t+1}) = 0$

With respect to $c_{t+1}$ : $u’(c_{t+1})-\lambda_{t+1}P_{t+1} = 0$

Combine

With this we can derive asset pricing

Asset Pricing

S_t = (\frac{\beta \lambda_{t+1}}{\lambda_t}) (S_{t+1}+D_{t+1})

Why buy an asset?

Pay a dividend in the future

Market value may rise in the future

Microeconomic events affect asset prices

Solve for $\lambda_{t+1}$ , $\lambda_{t}$ and substitute into equation,

S_t =(\frac{\beta u'(c_{t+1})}{u'(c_t)})(S_{t+1}+D_{t+1})(\frac{P_t}{P_{t+1}})

Use the definition of inflation, $1+\pi_{t+1}=P_{t+1}/P_t$

S_t =(\frac{\beta u'(c_{t+1})}{u'(c_t)})(S_{t+1}+D_{t+1})(\frac{1}{1+\pi_{t+1}})

We see that

Consumption across time ( $c_t$ and $c_{t+1}$ ) affects stock prices

Inflation affects stock prices

Any factor (monetary policy, fiscal policy, globalization) that affects inflation and GDP/consumption in principle impacts stock/asset markets

Consumer Optimization

S_t =(\frac{\beta u'(c_{t+1})}{u'(c_t)})(S_{t+1}+D_{t+1})(\frac{P_t}{P_{t+1}})

\frac{u'(c_t)}{\beta u'(c_{t+1})} = \underbrace{(\frac{S_{t+1} + D_{t+1}}{S_t})(\frac{1}{1+\pi_{t+1}})}_{\text{Analogy with Ch 3 \& 4, must be $(1+r_t)$}}

Production

Modeling Production

Production Function

A_tf(k_t,n_t)

Takes in production inputs (in real terms)

Fundamental production factors
1. labor ( $n_t$ / $l$ / $L$ )
1. capital $k_t$

Intermediate inputs
1. raw materials
1. patents

Land

Three Properties:

$\frac{\partial A_t f(k_t, n_t)}{\partial i} > 0$ ⇒ the function is always increasing in each factor

$\frac{\partial^2 A_t f(k_t, n_t)}{\partial i^2} < 0$ ⇒ the function has decreasing marginal product in the same factor

$\frac{\partial^2 A_t f(k_t,n_t)}{\partial i \partial j} > 0$ ⇒ the function has increasing marginal product in factors that are not the same

When allow time-varying $A_t$ , changes in $A$ cause shifts in production function

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Almost always ⇒ we use Cobb-Douglas Production Function

A_t f(k_t, n_t) = A_t k_t^{\alpha} n_t^{1-\alpha}

$\alpha \in (0,1)$ measures capital’s share of output
1. $(1-\alpha)$ measures labor’s share of output
1. US economy: $\alpha \approx 0.3$
1. Chinese economy: $\alpha \approx 0.15$

Productivity

3 Concepts

Marginal Product of Labor
1. $mp_{n_t} = \frac{\partial A_t f_n(k_t,n_t)}{\partial n_t} = (1-\alpha) A_t k_t^\alpha n_t^{-\alpha}$

Average product of labor
1. $ap_{n_t} = \frac{A_t f(k_t, n_t)}{n_t} = \frac{A_t k_t^\alpha n_t^{1-\alpha}}{n_t}$

Efficiency/total factor of productivity $A_t$

Profits

revenue of sold goods/services minus costs

Ptofit Maximization

\max_{k,n} A_t f(n_t, k_t) - Cost(n_t, k_t)

Notation:

$n_t$ - labor used for production

$k_t$ - capital (”machines and equipment”) used for production

$A_t$ total factor productivity
- Easy to assume $A_t = 1$

Labor

Production input provided by workers.

Unit in hours, workers, etc.

Assumption: “spot markets” for labor: in each period a worker is either on job or not

Labor Demand

Firm-level demand for labor defined by the relation

\underbrace{w_t}_{\mathclap{\text{wage at time $t$}}} = (1-\alpha) k_t^\alpha n_t^{-\alpha} = mpn_t = (1-\alpha)(\frac{k_t}{n_t})^\alpha

Capital Demand

Firm-level demand for capital defined by the relation

r_t = \alpha k_t^{\alpha - 1}n_t^{1-\alpha} = mpk_t = \alpha(\frac{n_t}{k_t})^{1-\alpha}

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But capital is a stock variable, we want to actually study a flow variable - investment

Investment is a change in capital stock between consecutive periods

$inv_t = k_{t+1} - k_t$ ⇒ capital demand and investment demand functions have the same shape

Two-period production schedule

Notation:

$k_1$ capital use for production in period 1 (decided upon in “period 0”)

$n_1$ labor used for production in period 1

$W_1$ nominal wage rate for labor in period 1

$w_1$ real wage rate for labor in period 1 ( $w_1 = W_1 / P_1$ )

$i$ nominal interest rate

$P_1$ nominal price of output produced and sold by firm in period 1
- Also nominal price of one unit of capital good bought by the firm in period 1 for use in period 2
  - Underlying assumption: capital goods are not necessarily “distinct” from consumption goods

So profit maximization with dynamic profit function:

P_1 f(k_1,n_1) + P_1k_1 - W_1n_1-P_1k_2 + \frac{P_2 f(k_2, n_2)}{1+i} + \frac{P_2 k_2}{1+i} - \frac{W_2 n_2}{1+i} - \frac{P_2 k_3}{1+i}

With a two-period model assumption,

$k_3 = 0$
- No more capitals needed in “period 3”

Solve with first order conditions

With respect to $n_1$ :

P_1 \underbrace{f_n(k_1,n_1)}_{\mathclap{\text{derivative of $f$ with respect to $n$}}} - P_1w_1 = 0 \\ \equiv f_n(k_1,n_1) - w_1 = 0

With respect to $n_2$ :

\frac{P_2 f_n(k_2, n_2)}{1+i} - \frac{P_2 w_2}{1+i} = 0

With respect to $k_2$ :

\begin{split} -P_1 + \frac{P_2 f_k(k_2, n_2)}{1+i} + \frac{P_2}{1+i} = 0 \\ \frac{P_2 f_k(k_2, n_2)}{P_1 (1+i)} + \frac{P_2}{P_1(1+i)} = 1 \\ (\frac{P_2}{P_1})(\frac{1}{1+i})f_k(k_2, n_2)+(\frac{P_2}{P_1})(\frac{1}{1+i}) = 1 \\ (\frac{1+\pi_2}{1+i})f_k(k_2, n_2) + (\frac{1+ \pi_2}{1+i}) = 1 \\ \text{apply fisher equation} \\ \frac{f_k(k_2,n_2)}{1+r} + \frac{1}{1+r} = 1 \\ f_k(k_2,n_2) = r \end{split}

Summary:

Profit-maximizing labor hiring implies
- $f_n(k_1,n_1) = w_1$
- $f_n(k_2, n_2) = w_2$

Profit-maximizing capital purcahses (for the future) implies about return (real interest rate)
- $f_k(k_2, n_2) = r$

Real Interest Rate

Previous lectures:

$r$ measures the price of period-1 consumption in terms of period-2 consumption

$r$ reflects degree of impatience (in the long run)

$r$ often reflects rate of consumption growth between periods

📢

Now,

r

measures the price of capital purcahses by firms Reflects real opportunity cost of sinking funds into capital today that won’t bear fruit until the future

See it mathematically:

-P_1 + \frac{P_2 f_k(k_2, n_2)}{1+i} + \frac{P_2}{1+i} = 0 \quad \text{(FOC on $k_2$)} \\ f_k(k_2,n_2) = r

Many interest rates

Mortgage loans
- 3.5% for now (30 year fixed rate)

Savings account
- ~0% for now

Credit card debt
- ~14%

Income distribution to $L$ and $K$

(Assume with CRS)

Y = MPL \times L + MPK \times K

Long Term Economic Growth

Solow Framework

Central Idea:

🔥

Accumulation of

k

(capital stock per capita) leads to ever-improving economic standard of living (proxied by

y

, GDP per capita)

Assumptions

CRS(Constant Return to Scale)

Exogenous(外源的) TFP(Total Factor Productivity) Growth

Exogenous Savings Rule

Closed Economy ⇒ Investment = Saving

no “consumption-savings optimality condition”

Question

How much of aggregate production is devoted to future economic growth?

Notation

$K$ - aggregate physical capital stock of economy

$N$ - aggregate labor stock of economy

$X$ - “Labor-augmenting” productivity
- “Solow residual” ⇒ developed by Solow and Swan (1956)
- measures other (difficult-to-measure) sources of
  - Rule of law, clean water, institutional structures, safe roads, access to coputers, etc.

$gr_X$ is growth rate (% change) of $X$ in every period
- $gr_X = \frac{X_{t+1} - X_t}{X_t}$
- Exogenous (not determined by model)
- e.g. Increasing education over the generations

$gr_N$ is growth rate (% change) of $N$ in every period
- $gr_N = \frac{N_{t+1} - N_t}{N_t}$
- Exogenous (not determined by model)
- e.g. birth rates or immigration

Aggregate Production Function

Also Total Factor Productivity (given Cobb-Douglas production funciton)

Y_t = K_t^{\alpha} \cdot {\underbrace{(X_t \cdot N_t)}_{\mathclap{\text{Efficiency units of labor}}}}^{1-\alpha}

Where $\alpha \in (0,1)$ measures capital’s share of output, $(1-\alpha) \in (0,1)$ measures (efficiency units of) labor’s share of output

If we define $A_t = X_t^{1-\alpha}$ (total factor productivity), by rearranging equation, we get

Y_t = A_t \cdot K_t^{\alpha} \cdot N_t^{1-\alpha}

Define $k = \frac{K}{XN}$ per-capita capital and $y = \frac{Y}{XN}$ per-capita GDP

\begin{split} \frac{Y_t}{X_t \cdot N_t} &= (\frac{K_t}{X_t \cdot N_t})^{\alpha} \cdot (\frac{X_t \cdot N_t}{X_t \cdot N_t})^{1-\alpha} \\ y_t &= k_t^{\alpha} \end{split}

Law of motion of aggregate capital stock

K_{t+1} = K_t + I_t - Depreciation_t = K_t + I_t - \delta K_t

Where $\delta$ is the fraction of capital that depreciates each period

$I$ - gross investment

$(I - Depreciation_t)$ - net investment

In per-capita terms:

\begin{split} \frac{K_{t+1}}{N_{t+1}} \frac{N_{t+1}}{N_t} &= \frac{I_t}{K_t}\frac{K_t}{N_t} + (1-\delta)\frac{K_t}{N_t} \\ \frac{K_{t+1}}{N_{t+1}}(1 + g_{t+1}^N) &= \frac{I_t}{K_t}\frac{K_t}{N_t} + (1-\delta)\frac{K_t}{N_t} \\ \frac{k_{t+1}}{k_t}(1+g_{t+1}^N) &=\frac{I_t}{K_t}+(1-\delta) \\ g_{t+1}^k \approx \ln (1 + [\frac{I_t}{K_t}-\delta]) - g_{t+1}^N &\approx \frac{I_t}{K_t}-\delta - g_{t+1}^N \end{split}

Savings Supply

We assumed that there’s no “consumption-savings optimality condition”

⚠️

Assumption 1: Constant savings rate

s \in (0,1)

of output in every period

\text{total savings of economy} = s \cdot y_t = s \cdot k_t^\alpha

⚠️

Assumption 2: Fraction

\delta

of physical

k

depreciates every period

Physical wear out of equipments

Chips and wires frying

So general case:

\text{total break-even investment} = (gr_X + gr_N + \delta) \cdot k_t

⚠️

Assumption 3: savings = (gross) investment in every period (G = 0, NX = 0)

Y_t = C_t + I_t \\ Y_t -C_t = I_t \\ s \cdot Y_t = S_t^{net} = I_t = K_{t+1} - (1-\delta)K_t

Therefore, we have $K_{t+1}$ in normalized form

\begin{split} \frac{K_{t+1}}{X_t \cdot N_t} &= s \cdot k_t^\alpha + (1-\delta)k_t \\ &=\frac{K_{t+1}}{X_{t+1} \cdot N_{t+1}} \cdot \frac{X_{t+1} \cdot N_{t+1}}{X_t \cdot N_t} = k_{t+1} \cdot (1 + gr_X) \cdot (1+gr_N) \end{split}

Equilibrium Law of Motion of k (describes how $k_t$ transitions over time)

k_{t+1} = \frac{s \cdot k_t^\alpha}{(1 + gr_X) \cdot (1+ gr_N)} + \frac{(1-\delta)k_t}{(1+gr_X) \cdot (1+gr_N)}

Long-run equilibrium $k = k_{t+1} = k^*$ :

\begin{split} k^* &= \frac{s \cdot (k^*)^{\alpha}}{(1+gr_X)(1+gr_N)} + \frac{(1-\delta) k^*}{(1+gr_X)(1+gr_N)} \\ &= [\frac{s}{(1+gr_X)(1+gr_N)-(1-\delta)}]^{\alpha - 1} \end{split}

y^* = k^{* \alpha} = [\frac{s}{(1+gr_X)(1+gr_N)-(1-\delta)}]^{\frac{\alpha}{1-\alpha}}

\begin{split} \ln(y^*) &= \ln(s) \cdot \frac{\alpha}{1-\alpha} - \ln((1+gr_X)(1+gr_N)-(1-\delta))\frac{\alpha}{1-\alpha} \\ & \approx \ln(s) \cdot \frac{\alpha}{1-\alpha} - \ln(gr_X + gr_N+\delta)\frac{\alpha}{1-\alpha} \end{split}

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Population growth lowers GDPpc, and the savings rate increases it.

Transitional Dynamics

How does economy converge to steady state?

Transitional Dynamic Equilibrium

If $k_t < k^*$ , does economy converge towards $k^*$ ?

To answer this question, we need to analyze the transition dynamics

k_{t+1} = \frac{s \cdot k_t^\alpha}{(1 + gr_X) \cdot (1+ gr_N)} + \frac{(1-\delta)k_t}{(1+gr_X) \cdot (1+gr_N)}

Neoclassical Framework

Growth model with optimization
- Yields consumption-savings optimality condition that endogenizes(内生化) “saving rate”

\max_{\{c_t,k_{t+1}\}_{t=0}^\infin} \sum_{t=0}^\infin \beta^t u(c_t) \\ \text{s.t.} \\ \sum_{t=0}^\infin \beta^t \lambda_t [k_t^\alpha - (c_t + k_{t+1} - (1-\delta)k_t)] = 0

First Order Conditions
- with respect to $c_t$ : $u’(c_t) - \lambda_t = 0$
- with respect to $k_{t+1}$ : $-\lambda_t + \beta \lambda_{t+1} [f’(k_{t+1}) + 1 - \delta] = 0$

Solve FOC
- $u’(c_t)=\beta u’(c_{t+1})[f’(k_{t+1}) + 1 - \delta]$
  - Euler Equation (consumption-savings optimality condition)
  - $f(k_{t}) = k_t^{\alpha}$

Solve endogenous ss equilibrium using Euler Equation
- $\frac{1}{\beta} = 1+f’(k^*)-\delta$
- $\frac{1}{\beta} = 1+\alpha(k^*)^{\alpha-1}-\delta$
- $k^* = [\frac{1}{\alpha} \cdot (\frac{1}{\beta} - (1-\delta))]^{\frac{1}{\alpha-1}}$

And endogenous ss equilibrium $c^*$ using RC:
- $(k^*)^\alpha = c^* + k^* - (1-\delta)k^*$
  - Using the constraint applied to $k^*$
- $(k^*)^\alpha = c^* + \delta k^*$
- $c^* = (k^*)^\alpha - \delta k^*$
  - savings = investment
- $c^* = [\frac{1}{\alpha} \cdot (\frac{1}{\beta} - (1-\delta))]^{\frac{\alpha}{\alpha-1}} - \delta \cdot [\frac{1}{\alpha} \cdot (\frac{1}{\beta} - (1-\delta))]^{\frac{1}{\alpha-1}}$

Long run “savings rate”
- $s = \frac{\delta k^*}{y} = \frac{\delta k^*}{(k^*)^\alpha}=\delta \cdot (k^*)^{1-\alpha} = \delta \cdot [\frac{1}{\alpha} \cdot (\frac{1}{\beta} - (1-\delta))]^{-1}$
- If $\beta = 1$ ⇒ $s_{neo} = \alpha$

🧙🏽‍♂️

Which savings rate maximizes steady-state consumption?

Assume only depreciation (and no growth in TFP/pop),

y(k(s)) = c+i = c+sy(k(s)) \\ y(k(s)) = c+\delta(s) \\ y(k(s)) = y(k(s)) - \delta k(s)

\frac{\partial c}{\partial s} = \frac{\partial y(k(s))}{\partial k} \frac{\partial k(s)}{\partial s}-\delta \frac{\partial k(s)}{\partial s} = (\frac{\partial y(k(s))}{\partial k} - \delta) \frac{\partial k(s)}{\partial s} = (f_k - \delta) \frac{\partial k(s)}{\partial s}

Now solve for explicit savings rate (”Golden Rule”)

\frac{\partial c(s_{max})}{\partial s} = (f_k(k(s_{max})) - \delta) \frac{\partial k(s_{max})}{\partial s} = 0

Transitional Dynamics

Long-run Theory of Macro

We have

\frac{u'(c_t)}{\beta u'(c_{t+1})} = 1+ r_t

Simplify (with long-run assumptions):

\frac{1}{\beta} = 1+r

Now a second interpretation of $r$

long-run (i.e., steady state) real interest rate
- simply a reflection of degree of impatience of individuals in an economy
- The lower is $\beta$ , the higher is $r$

The more impatient a populace is, the higher are interest rates

Which came first, $\beta$ or $r$ ?

Modern macro view: $\beta < 1$ causes $r > 0$

Labor Market

Income $Y$

C_t + A_t - A_{t-1} = iA_{t-1} + Y_t

Two perspectives on income types:

The “mean” prespective household cares about both sources of income quite a bit

The “median” perspective household may have a very different view in income split
- Capital assets are distributed much more unequally than labor income

How important is financial wealth?

Median wealth is small and capital income will be small too

Most households don’t actually receive large inheritances

Asset ownership is very cocentrated

Consumption smoothing: hosueholds borrow/dissave early on, pay back debt over course of life

Price of an asset is the present value of the income it yields, so cannot on average gai by buying assets

Income Flows

One third of national income is capital income
- $CapitalShare_t = \frac{i_t A_{t-1}}{GDP_t}$

The other two thirds: labor income
- $LaborShare_t = \frac{Y_t}{GDP_t}$

These shares are very stable over time

Measuring financial wealth

C_t + A_{t+1} = A_t + iA_t + Y

\sum_{t= YoB}^{YoD} \frac{c_t}{(1+r)^{t - YoB}} = a_0 - \frac{a_{YoD}}{(1+r)^{YoD - YoB}} + \sum_{t=YoB}^{YoD} \frac{y_t}{(1+r)^{t-YoB}}

Intial assets are your financial wealth
- Above lifetime budget constraint (LBC) combines period BCs, eliminating interim assets

Asset pricing
- The price of an asset = present value of its income streams
- On average will not yield positive net PV

Stock prices
- $PV(\text{stock paying dividend $d$}) = d + \frac{1}{1+r}d + \dots \approx \frac{(1+r)}{r}d$

Assets are vehicles for intertemporal shifting of funds

Measuring “human” wealth (from labor income)

Apply our present value(PV) calculation by just summing up discounted salaries over lifecyle

\begin{split} PV_{Birth}(\text{All $Y$}) &= \sum_{age = 0}^{\text{Age at death}} Y_{age} (\frac{1+ \pi}{1+i})^{age} = \sum_{age = 0}^{AaD} \frac{y_{age}}{(1+r)^{age}} \\ &=y \cdot \frac{1 - (\frac{1}{1-r})^{AaD + 1}}{1 - \frac{1}{1+r}} = y \cdot \frac{1+r-(\frac{1}{1+r})^{AaD}}{r} \approx y \cdot \frac{1+r}{r} \end{split}

Labor Supply

Labor income and its PV can be affected by certain factors

others are at least partially “chosen”
- education, occupation, location, industry

Those choices are part of the household’s labor supply

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Education is the key source of GDPpc growth and differences between countries

Mincer model of education and wages

W_t = p_t^{E} \cdot S_t

$p_t^E$ price of skill - the salary the worker receives per skill unit

$S_t$ skill unit variable

Skill is a stock, variable
- $S_0$ - Your skill endowment “inherited” in period 0
  - Inherent ability; IQ; dexterity
- $S_{t} = S_{t-1}$ if not in school, then skills remain fixed

$T$ total lifetime

$n$ years in workforec

$s$ years in schooling (years not worked)

Households(individuals) can invest in skill
- $S_t = (1+a) S_{t-1}$
  - If in school, skills grow at rate $a$
- $S_{t>T_{grad}} = S_0 (1+a)^s$
  - After graduation, this is your skill level
- During work life, earn wage based on skill
  - $W_t = p_t^E S_t = p_t^E \cdot S_0 (1+a)^s$

\begin{split} \ln(W_t) &= \ln (p^E) + \ln(S_t) = \ln(p^E) + \ln(S_t) \\ &= \ln(p^E) + s \cdot \ln(1+a) + \ln(S_0) \\ &\approx \ln(p^E) + a \cdot s + \ln (S_0) \end{split}

Trade-off between schooling time and work time (with time constraint)

T = s+n \equiv n = T-s

If we assume no discount

\max_s[(T-s) \cdot W(s) + s \cdot 0]

FOC reveals

-W(s^*) + (T-s^*) \cdot W'(s^*) = 0 \\ W(s^*) = (T-s^*) \cdot W'(s^*)

Stay in school until marginal effect on lifetime income equals the opportunity cost of staying in school

Suppose $W_t(s) = p_t^E \cdot S_t = p_t^E \cdot S_0(1+a)^s$

p_t^E \cdot S_0(1+a)^s = (T-s^*) \cdot S_0(1+a)^s \ln(1+a) \\ 1/\ln(1+a) =T-s^* \\ 1/\ln(1+a) \approx 1/a \rightarrow T-s^* = 1/a

Tradeoff between Labor and Leisure (Consumption-Leisure Framework)

Tight link between labor and leisure through time constraint of total available time $T$ :
- $T = Labor + Leisure$
- $Leisure = T - Labor$

Notation
- $c$ consumption
- $n$ percentage of time working per unit period
- $l$ percentage of leisure per unit period
- $P$ dollar price of one unit of consumption (nominal)
- $W$ hourly wage rate in terms of dollars (nominal)
- $t$ tax rate on labor income

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Consumer’s decision problem: maximize utility subject to budget constraint - bring together both cost side and benefit side

\max u(s,l) \\ \text{s.t.} \\ Pc+Wl = W

At the optimal choice

\frac{u_l(c^*, l^*)}{u_c(c^*, l^*)} = \frac{W}{P}

How do micro-level consumption/leisure choices change as real wage changes

For low-levels of real wages, a rise in the real wage causes optimal leisure to decrease

For intermediate levels of real wages,a rise in the real wage causes optimal leisure to remain unchanged

For high levels of real wages, a rise in the real wage causes optimal leisure to increase

Price changes

Substitution effect
- General two-good case
  - If relative price of good 1 increases, purchase of good 1 decrease
- Application to consumption-leisure framework
  - If real wage $W/P$ increases, choose less leisure = choose to work more

Income effect
- General two-good case
  - if total income increases (Y increases), purchases of all goods increases
- Application to consumption-leisure framework
  - If real wage $W/P$ increases ⇒ Total income increases ⇒ Choose more leisure ⇒ choose to work less

Government

Government Spending

Household income goes into savings, taxes, consumption

Y = C+S+T = C+I+G+(E-M)

Therefore

(T-G) = (I-S)+(E-M)

Bonds

Bonds are financial assets that pay out a specific interest in each period, and usually the principal later on)

Interest payment: nominal interest times face value of debt

Market price of a bond is the discounted present value of the interest payment and the principal repayment

Coupon bonds
- pay something back(”coupon payments”) every so often until the final date of maturity

Zero-coupon bonds
- Only pay back at final date of maturity

Government Bonds

Simplify by supposing that all bonds are one-period government bonds (short-term bonds)
- a “riskless” debt instrument
- U.S. gov never defaulted on bond payment
- Excess inflation is a backdoor way of “defaulting”

“Debt”: $B_t$ quantity of short-term bonds with face value 1

Interest due to tomorrow: $i_t \cdot B_t$

Principal due: $B_t$

Today: think of debt as negative bonds

Gov. Budget Constraint

G_t + B_t = T+ i_{t-1}B_{t-1} + B_{t-1}

Government expenditure $G$

Tax revenue $T_t$

Government debt $B_t$ (in negative number)

Government budget balance
- Flow variable
- Deficit or surplus
- Primary Deficit $D_t^P = G_t - T_t$
- Total Deficit $D_t^S = G_t - T_t - i_{t-1} \cdot B_{t-1} = -(B_t - B_{t-1})$

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Useful way to look at “debt burden” in data:

-B/Y

⇒ debt to GDP ratio

Fiscal gap: formal debt plus debt-like liabilities, minus expected government revenue (from tax or other sources)

Ricardian Equivalence

“Benchmark result” of fiscal policy effects

Households anticipate that a debt-financed tax cut today entails an increase in future taxes that is equal - in present value - to the tax cut
- As a consequence, households save the full tax cut in order to repay thefuture tax liability

Private saving rises the amount public saving falls, leaving national saving unchanged

Dynamic Model of the Gov.

Notation

$g_1$ real government spending i period 1

$g_2$ real government spending in period 2

$b_0$ government asset position at beginning of period 1 / end of period 0

$b_1$ government asset position at beginning of period 2 / end of period 1

$b_2$ government asset position at beginning of period 3/ end of period 2

$r$ real interest rate between periods

Period-t government budget constraint

g_t+b_t = (1+r) b_{t-1} + t_t \rightarrow g_t + \underbrace{b_t - b_{t-1}}_{\mathclap{\text{savings during period $t$}}} = t_t + \overbrace{rb_{t-1}}^{\mathclap{\text{asset income during period $t$}}}

Combine period-1 and period-2 government budget constraint, and combine into Lifetime Budget Constraint (LBC):

g_1 + \frac{g_2}{1+r} = t_1 + \frac{t_2}{1+r} + (1+r)b_0

Consumer LBC with tax introduced

c_1 + \frac{c_2}{1+r} = y_1 - t_1 + \frac{y_2 - t_2}{1+r} + (1+r)a_0

Economy-wide Resource Frontier (PPF)

Also called “Production Possibilities Frontier” (PPF)

Conbining Government LBC with consumer LBC, we have

c_1 + \frac{c_2}{1+r} = y_1 - g_1 + \frac{y_2 - g_2}{1+r} + (1+r)(a_0+b_0)

intermediate micro theorem: If taxes are lump-sum(fixed amount), then consumer optimal choices can be analyzed using either the consumer LBC or the economy-wide resource frontier (superimpose indifference map), and either approach will yield the same predictions.

National Savings

National Savings = Savings by Consumers + Savings by Government + Savings by Firms

For now, we have not added firms in our model,

s_1^{nat} = s_1^{priv} + s_1^{gov} = y_1 - t_1 -c_1 + t_1 -g_1 = y_1-c_1-g_1

Richardian Equivalence Theorem: For a given PDV(Present Discounted Value) of government spending, neither consumption nor national savings is affected by the precise timing of lump-sum taxes.

Rational consumers understand that a tax cut today means a tax increase in the future (because total government spending is unchanged)
- entire tax cut is saved by consumers in order to pay higher taxes in the future
- Private savings and government savings move in exactly offsetting ways

At aggregate level, total tax collections sometime “seem” lump-sum (independent of aggregate macroeconomic activity)

Ricardian Equivalence

Ricardian Equivalence Theorem: For a given PDV of government spending, neither consumption nor national savings is affected by the precise timing of lump-sum(fixed amount) taxes.

Economic Interpretation: Rational consumers understand that a tax cut today means a tax increase in the future (because total government spending is unchanged)

Entire tax cut is saved by consumers in order to pay higher taxes in the future

Priviate savings and government savings move in exactly offsetting ways

Taxation

Lump-sum tax
- A tax whose total incidence (total amount paid) does not depend in any way on any decisions/choices an individual makes
- We used this in our two-period framework

Proportional Tax / distortionary tax
- Total incidence depends on decisions/choices an individual makes
- Affects prices differentially
- Changes LBC of two-period consumption
  - $(1+\tau_1) c_1 + \frac{(1+\tau_2)c_2}{1+r} = y_1 + \frac{y_2}{1+r} + (1+r)a_0$
  - Change in tax rates distort optimal consumption choices because they change slope of consumer LBC

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The higher the tax rates, the lower the tax base (amount of money to which proportional tax applies) Because tax rates affect consumption and labor supply!

T = t \cdot W \cdot (1-l^*(t)) \\ \frac{\partial T}{\partial t} = W \cdot (1 - l^*(t)) + t \cdot W \cdot \frac{\partial (1-l^*(t))}{\partial t}

Many shapes of the Laffer curve is possible betwene 0 and 0 revenue for 0/100% taxes

Monetary Policy

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Is money neutral? (if changes in the money supply such as monetary policy have no effect on the real economy)

“Keynesian” view:
- money is non-neutral
- because prices are rigid/sticky, often for long periods of time

“Classical” view:
- money is neutral
- because prices are not rigid/sticky in any important way

Roles of money

Medium of exchange
- Liquidity is an object’s ability to be excahnged for purchases

Unit of account

Store of value
- Vegetables will perish in short amount of time, money will not.

🔥

It’s hard to conceptually model “money”

Shortcut: suppose money directly yields utility

Period-t utility function
- $u(c_t, \frac{M^D_t}{P_t})$
- Money-in-the-utility-function(MIU) formulation
- $M_t / P_t$ is a key variable for macroeconomic analysis
  - Unit: $M_t$ in $, $P_t$ in $/Unit of consumption, therefore $M_t/P_t$ in unit of consumption
- $M_t/P_t$ is the real money, while $M$ is the nominal money

Bonds

Bond is a debt obligation

We simplify our annalysis by supposing that all bonds are one-period zero-coupon government bonds
- Traditional simplification for analysis of monetary policy
- Conventional channel through which Federal Reserve conducts monetary policy

Key relationship between price of a bond and nominal interest rate

P_t^b = \frac{FV_{t+1}}{1+i_t}

$P^b_t$ : nominal price of a one-period bond

$i_t$ : nominal interest rate between period t and period t+1

$FV_{t+1}$ : Face-value of bond(how much will be repaid in t+1)

Inverse relationship between $P^b$ and $i$ :

i_t = \frac{1}{P_t^b} - 1

$i$ can be thought of in two (mirror-image) ways

Interest payoff of a bond

Opportunity cost of holding money
- Each unit of wealth held as a dollar is a unit of wealth not held as a bond
  - entails loss of chance to earn interest
- $i$ interpreted as the “price of money”

Conventional monetary policy
- Fed open-market operations conducted via short-term bond markets, so Fed operations do affect bond supply
- Fed buys short bonds from financial sector, reducing open-market supply
  - by printing new money, increasing supply in money market
  - which causes short-term $i$ to decrease

Types of money

Unconventional monetary policy

Conventional monetary policy:
- interest-rate targeting via open-market operations

Unconventional monetary policy
- Fed to purchase other assets, not just short-term US treasuries
- fed to issue its own bonds
- Bail out/lend to banks and firms in times of distress
  - “lender of last resort”
- “Communicate” with the public and markets about “how the economy is doing”
  - regular press conferences
  - expectations-management / confidence-managment role for monetary policy

Infinite Period Monetary Policy

Now three types of asets (stocks, short-term bonds, money) for representative consumer to use for savings purposes

Allows us to think further about what the “pricing kernel” is

Allows us to understand connection between bond prices and stock prices

Allows us to consider monetary neutrality

Sequential Lagrangian Analysis
- With money in the utility function

MIU(Money-in-the-utility) function

Individual obtain utility from purchasing power of money
- medium of exchange
- unit of account
- store of value

DEMAND holdings of nominal money

u(c_t,\frac{M_t^D}{P_t})

In consumer optimization it’s $M_D$ everywhere, where $M^S$ is determined by the central bank.

In money market equilibrium,

\frac{M_t}{P_t} = \frac{M_t^D}{P_t}=\frac{M_t^S}{P_t}

Notation

$c_t$ - consumption in period $t$

$P_t$ - nominal price of consumption in period $t$

$Y_t$ - nominal income in period $t$ (assume it “falss from the sky”)

$a_{t-1}$ - real stock holdings at beginning of period $t$ /end of period $t-1$

$M_{t-1}$ - nominal money holdings at the beginning of period $t$ / end of period $t-1$

$B_{t-1}$ - nominal bond holdings at the beginning of period $t$ / end of period $t-1$

$S_t$ - nominal price of a bond in period $t$

$D_t$ - nominal dividend paid in period $t$ by each unit of stock held at the start of $t$

$P^b_t$ - nominal price of a bond in period $t$

$i_t$ - nominal interest rate on a bond purchased in $t$ and which pays off in $t+1$

$\pi_{t+1}$ - net inflation rate between period $t$ and period $t+1$

$y_t$ - real income in period $t$ ( = $Y_t / P_t$ )

We need infinite budget constraints to describe economic opportunities and possibilities, one for each period

P_t c_t + P_t^b B_t + M_t + S_ta_t = Y_t+M_{t-1}+B_{t-1}+S_ta_{t-1}+D_ta_{t-1}

LHS: Total outlays in period $t$
- period-t consumption + stocks to carry into into period t+1 + money to carry into period t+1 + bond purchases

RHS: Total income in period $t$
- period-t $Y$ + income from stock holdings carried into period $t$ + bond-holdings carried into period $t$ (each unit repays $FV = 1$ )

Lagrangian function:

\begin{split} L &= \sum_{t'=t}^\infin \beta^{t'-t}u(c_{t'},M_{t'}/P_{t'}) \\ &\quad +\sum_{t'=t}^\infin \beta^{t'-t}\lambda_{t'}[Y_t + (S_t+D_t)a_{t-1}+M_{t-1}+B_{t-1}-P_tc_t-S_ta_t-M_t-P_t^bB_t]\end{split}

Compute FOC with respect to $c_t, a_t, B_t, \dots$

Stock-pricing equation (from equation 2)

\underbrace{S_t}_{\text{Period-$t$ stock price}} = \underbrace{(\frac{\beta \lambda_{t+1}}{\lambda_t})}_{\text{pricing kernel}} \underbrace{(S_{t+1}+D_{t+1})}_{\text{Future return}}

This is where macro theory and finance theory intersect

Bond-pricing equation (from equation 3)

P_t^b = \frac{\beta \lambda_{t+1}}{\lambda_t}

📌

Price of short-term bond is the pricing kernel

Stock prices and bond prices are conected

Most asset prices are fundamentally connected to bond prices

Finance: pricing kernel reflects the price/return of the least risky asset / “riskless” asset in the economy (US treasury short-term bonds)

Note:

P_t^b = \frac{1}{1+i_t}

Can express pricing kernel as

\frac{\beta \lambda_{t+1}}{\lambda_t} = \frac{1}{1+i_t}

If we connect stock-pricing equation with bond-pricing equation

1+r_t = \frac{1+i_t}{1+\pi_{t+1}}

We get fisher’s equation!
- relationship between returns on nominal bonds and real assets

Combining equations, we get the consumption-money optimality condition

\frac{u_2(c_t, M_t/P_t)}{u_1(c_t, M_t/P_t)}=\frac{i_t}{1+i_t}

Money Demand

Suppose $u(c_t, \frac{M_t}{P_t}) = \ln c_t + \ln(\frac{M_t}{P_t})$

With the optimality condition,

\frac{M_t}{P_t} = c_t \cdot (\frac{1+i_t}{i_t})

Real Money Demand depends positively on $c_t$ and negatively on $i_t$
- Since $i_t$ is the opportunity cost of money

Monetary Neutrality

Fed sets $M_t^S$ after consumers make their choices of $c_t$ and $M_t^D$ (and other choices, too)
- If supply $M_t^S$ differs from demanded $M_t^D$ , money shock has occurred

Question: which adjusts ( $P_t$ or $c_t$ ) to ensure consumption-money optimality condition holds?

\frac{M_t}{P_t} = c_t \cdot (\frac{1+i_t}{i_t})

Keynesian view
- $P_t$ cannot adjust because prices are sticky
  - Prices will adjust later, just not in period $t$
- A positive money shock leads to a rise in $c_t$
- Money is not neutral

Classical view
- $P_t$ can adjust because prices are not sticky
- No reason for $c_t$ to adjust (they do reflect optimal choices after all)
- A positive money shock leads to no change in $c_t$
- Money is neutral

Empirical evidence for “how sticky” are prices is very mixed

Money and Inflation in the long run

What determines inflation in the long run?

This relationship basis for the monetarist school of thought
- Milton Friedman’s famous dictum: “Inflation is always and everywhere a monetary phenomenon”
- Basically, a central bank should not worry about/try to control anything other than how quickly the money supply in the economny is growing. Keeping money growth under control will keep inflation under control”

In the long run, monetary policy may have smaller effects on consumption and real GDP

Monetary Policy and Fiscal Policy

Monetary policy and fiscal policy don’t occur in vacuums isolated from each other
- Both occur simultaneously

Conduct of fiscal policy can place restrictions on what monetary policy can do, and vice versa

Policy Coordination (explicit or implicit)
- Fed and Treasury coordination of policy actions amidst crisis
- European coordination on soverign debt crises

Budget constraint effects (balance sheet effects)
- Central bank balance sheet ultimately under control of legislative body
- Macroeconomic analysis has the most to say about this type of fiscal-monetary interaction

Infinite Period Analysis

Two distinct “sides” of the government
- Fiscal authority (Congress/Treasury)
  - Controls government spending $g_t$
  - Collects taxes $T_t$
  - Issues (sells) new bonds (for various financial needs)
  - Receives “profits” from the central bank (because it legally charters the Central Bank)
- Monetary authority (central bank)
  - Controls money supply of economy
  - Turns over any “profits” it earns to fiscal authority

🔥

Note that

B>0

now means the government issues debt (beforehand

B

were net assets ⇒

B<0

meant the government was in debt

Notations

$B_t^T$ - total amount of (one-period) bonds (each with $FV=1$ ) Congress sells in period $t$ , each of which has price $P^b_t$

$B^T_{t-1}$ - total amount of (one-period) bonds (each with $FV=1$ ) that Congress must repay in period $t$

$RCB_t$ - receipts (profits) turned over from the central bank to the fiscal authority in period $t$

$M_t - M_{t-1}$ - the change in the money supply engineered by the central bank during the course of period $t$
- $M_t$ and $M_{t-1}$ individually are stock variables, but $M_t - M_{t-1}$ is a flow variable

Fiscal Authority Budget Constraint in period $t$ :

\underbrace{P_tg_t+B_{t-1}^T}_{\text{total outlays}} = \underbrace{T_t + P_t^bB_t^T + RCB_t}_{\text{total inflows}}

Monetary authority budget constraint in period $t$ :

\underbrace{P_t^b B_t^M + RCB_t}_{\text{Total outlays}} = \underbrace{B_{t-1}^M + M_t - M_{t-1}}_{\text{Total inflows}}

Monetary Authority

Until late 2007, around all of Fed’s assets were Treasury securities

During 2007-2009, fell to $\approx 25\%$ of Fed’s assets
- Since then has grown to $\approx 50\%$

Government Sponsored Enterprises(GSE)
- e.g. Freddie Mac and Fannie Mae

Mortgage Backed Securities (MBS)
- Fed purchasing “bad debt” backed by mortgages(按揭/房屋抵押贷款) from financial firms

P_t^bB_t^M+P_t^{MBS}MBS_t+P_t^{FANNIE}B_t^{FANNIE}+RCB_t = B_{t-1}^M+MBS_{t-1}+B_{t-1}^{FANNIE}+M_t-M_{t-1}

Consolidated Government Budget

If we view two sides of the government as one consolidated entity

(1)

P_t^bB_t^M + RCB_t = B_{t-1}^M+M_t-M_{t-1}

(2)

P_tg_t+B_{t-1}^T=T_t+P_t^bB_t^T+RCB_t

⇒

P_t g_t + B_{t-1}^T - B_{t-1}^M = T_t + P_t^b(B_t^T-B_t^M)+M_t-M_{t-1}

$B_T$ is the total quantity of fiscally-issued bonds
- $B^M$ is purcahsed by the central bank on the open market

$B^T - B^M$ is total quantity of fiscally-issued bonds held by the private sector
- Define this as $B = B^T - B^M$

Therefore, consolidated flow government budget constraint (GBC)

P_tg_t + B_{t-1} = T_t + P_t^bB_t+M_t-M_{t-1}

highlights the short-run limits that fiscal policy places on monetary policy and vice-versa

Active / Passive Policy

A policy authority is active if every instrument at its disposal can be completely freely chosen, without any concern for the consolidated government budget constraint

A policy authority is passive if not every instrument at its disposal can be completely freely chosen, without any concern for the consolidated government budget constraint
- Passive authority must engage in policy in such a way as to make sure the consolidated government budget balances

At the beginning of period $t$ , $B_{t-1}, M_{t-1}$ are both fixed.
- Then fiscal authority sets $g_t, B_t, T_t$
- Monetary authority sets $M_t$
- How will consistency between them be guaranteed?
  - Active Fiscal Policy
    - Suppose fiscal authority sets all of its policy instruments (all three of them) with no concern for the consolidated flow GBC
    - Monetary authority must react by setting $M_t$ to ensure the consolidated GBC holds
  - Active Monetary Policy
    - Suppose monetary authority sets all of its policy instruments ( $M_t$ ) with no concern for the consolidated flow GBC
    - Fiscal authority must react by setting at least one of $(g_t, T_t, B_t)$ to ensure the consolidated GBC holds
- Game-theory undertones
- Policy pressure is implicit and through market forces

🔥

Core issue: There are limits or restrictions that each policy-setting authority places on the actions of the others

Analysis:
- the period-t choices of one policy authority restrict the choices of other policy authority in period $t$

In reality:
- Not only period $t$ , but also $t+1, t+2, t+3, \dots$

Emphasizes that the limits may not be realized immediately, but can occur later (in the economy’s / government’s lifetime)

Requires analyzing the PDV (present discounted value) version of the consolidated GBC

Lifetime consolidated GBC

First we have Period-t consolidated GBC

P_tg_t + B_{t-1} = T_t + P_t^b B_t + M_t - M_{t-1}

divide by $P_t$ to put into real terms

g_t+\frac{B_{t-1}}{P_t} = \frac{T_t}{P_t}+\frac{P_t^b B_t}{P_t}+\underbrace{\frac{M_t - M_{t-1}}{P_t}}_{sr_t}

$sr_t$ - seignorage revenue (铸币税收入) - real quantity of resources the government raises for itself through the act of money creation

Printing money is a source of income for the government
- Unimportant in the US and other developed countries
- Can be important i developing countries (poorly-developed tax collection systems and corruption)

Define $b_t = B_t / P_t, t_t = T_t / P_t$ ,

We have

\frac{B_{t-1}}{P_t} = \underbrace{sr_t}_{\text{Revenue generated by monetary authority actions}} + \underbrace{(t_t - g_t + P_t^b b_t)}_{\text{revenue generated by fiscal authority actions}}

$\frac{B_{t-1}}{P_t}$ - REAL value of government debt that must be repaid at the start of period t

$sr_{t}$ - Revenue generated by monetary authority actions

$(t_t - g_t + P_t^b b_t)$ - Revenue generated by fiscal authority actions

\frac{b_{t-1}}{1+\pi_t} = sr_{t} + (t_{t} - g_t + P_t^bb_t)

Combine all periods (use fisher’s equation)

\frac{B_{t-1}}{P_t} = \sum_{s=0}^\infin \frac{t_{t+s} - g_{t+s}}{\prod_{x=1}^s (1+r_{t+x})} + \sum_{s=0}^\infin \frac{sr_{t+s}}{\prod_{x=1}^s (1+r_{t+x})}

Ricardian vs. Non-ricardian Policy

A ricardian fiscal policy is in place if the fiscal authority sets its planned sequence of tax and spending policy to ensure that present-value consolidated GBC is balanced.

A non-Ricardian fiscal policy is in place if the fiscal authority sets its planned sequence of tax and spending policy without regard for whether present-value consolidated GBC is balanced.

Inflationary Finance (FTI / FTPL)

FTI
- effects of inflationary finance felt as a long and sustained (not necessarily very sharp) rise in inlfation

FTPL
- effects of inflationary finance felt as a short-lived but very sharp rise in inflation
- Many historical episodes in developing countries of FTPL
- Little empirical evidence for developed countries

Central Bank Independence

A mechanism to limit the reliance of government on seigniorage

Private sector will not expect the government to “print money” to balance budgets
- Less inflation

Generic Terms

Stocks vs. Flows

Fluctuations vs. Trend

Present Discounted Value (PDV)

Fisher equation

Fisher Effect

Capital and Investment

GDP Gross Domestic Product

Counting GDP using production

Nominal GDP 名义/货币GDP

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Real GDP(实际GDP)

GDP Deflator(平均物价指数)

GDPpc(GDP Per Capita) 人均GDP

Counting GDP using expenditure

Prince Indices 物价索引

GDP Deflator

Other Indexes

Expenditure

Consumer / Household Theory

Utility Function

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Marginal Rate of Substitution

Income Types

Two-period Consumption-savings framework

(Lifetime) Budget Constraint

Optimum

Consumption Smoothing

Interest Rate Changes

Credit Crunch

Infinite Period Consumption-savings Framework

Notations

Utility Function

Budget Constraint

Consumer Optimum

Infinite Period Consumption-savings Framework with Asset Pricing

Notations

Budget Constraints

Asset Pricing

Consumer Optimization

Production

Modeling Production

Production Function

Productivity

Profits

Labor

Labor Demand

Capital Demand

Two-period production schedule

Real Interest Rate

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Long Term Economic Growth

Solow Framework

Assumptions

Question

Notation

Aggregate Production Function

Savings Supply

Transitional Dynamics

Transitional Dynamic Equilibrium

Neoclassical Framework

Transitional Dynamics

Long-run Theory of Macro

Labor Market

Income @import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')YYY﻿

Income Flows

Measuring financial wealth

Measuring “human” wealth (from labor income)

Labor Supply

Mincer model of education and wages

Tradeoff between Labor and Leisure (Consumption-Leisure Framework)

How do micro-level consumption/leisure choices change as real wage changes

Price changes

Government

Government Spending

Bonds

Government Bonds

Gov. Budget Constraint

Ricardian Equivalence

Dynamic Model of the Gov.

Growth for Good $y$

Subjective Discount Factor $\beta$

Income distribution to $L$ and $K$

Income $Y$