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)

  1. Quantity variables whose natural measurement occurs at a particular moment in time
  1. e.g.
    1. checking account balance
    1. credit card indebtedness
    1. Mortgage loan payoff

Flow variables

  1. Quantity variables whose natural measurement occurs over the course of a given interval of time
  1. e.g.
    1. Income
    1. Consumption
    1. Savings

Fluctuations vs. Trend

fluctuationt,trendx=xtxttrendxttrend=xtxttrend1fluctuation_{t,trend}^x = \frac{x_t - x_t^{trend}}{x_t^{trend}}=\frac{x_t}{x_t^{trend}}-1
1+fluctuationt,trendx=xtxttrend1+ fluctuation_{t,trend}^x=\frac{x_t}{x_t^{trend}} \\

Using trick ln(1+x)x,x1\ln(1+x) \approx x, \forall x \ll 1

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

Present Discounted Value (PDV)

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

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

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

Fisher equation

1+r=1+i1+π1+r = \frac{1+i}{1 + \pi}
  1. ii ⇒ nominal interest rate between periods
  1. rr ⇒ real interest rate between periods
  1. πt\pi_t ⇒ net inflation rate between period t1t-1 and period tt

“Approximate fisher equation”

1+r+π+rπundefinedIn advanced economics, r and π are both generally small=1+ir=iπ1+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=1+i1+πln(1+r)=ln(1+i)ln(1+π)approximatelyriπ1+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

GDP Gross Domestic Product

Three views:

  1. Production
  1. Expenditure
  1. Income

Counting GDP using production

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

Either

  1. Survey very last producer in the production chain
  1. Use value added approach
    1. GDP=VAi=pnqnGDP = \sum VA_i = p_n q_n

Nominal GDP 名义/货币GDP

Nominal Production of good ii:

yi,tnom=pi,t×qi,ty_{i,t}^{nom} = p_{i,t} \times q_{i,t}

Then we calculate nominal GDP by summing over all product categories

NGDPt=iIpi,t×qi,tNGDP_{t} = \sum_{i \in I} p_{i,t} \times q_{i,t}

Growth for Good yy

growthty=ytyt1yt1=ytyt11growth_{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
GDPDeftb=NGDPtRGDPtb=ptpbGDPDef_t^b = \frac{NGDP_t}{RGDP_t^b} = \frac{p_t}{p_b}

Note GDPDefGDPDef captures inflation:

growthtGDPDef=growthtpgrowth_t^{GDPDef}=growth_t^p

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

RGDPtbundefinedReal GDP in year t provided baseline year b=iIpt=b×qt=t=NGDPtGDPDeftb\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 ii:

yi,tb=pi,b×qi,t=yi,ttpi,bpi,ty_{i,t}^b = p_{i,b} \times q_{i,t}=y_{i,t}^t \frac{p_{i,b}}{p_{i,t}}

Growth of RGDP

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

GDPpc(GDP Per Capita) 人均GDP

GDPpct=GDPtPoptGDPpc_t = \frac{GDP_t}{Pop_t}

Counting GDP using expenditure

Y=C+I+G+NXY=C+I+G+NX
🔥
Important Fact: X{C,I,G,NX}dX/dY=0\sum_{X \in \{C, I, G, NX\}} dX/dY = 0

Prince Indices 物价索引

GDP Deflator

GDPDeftb=NGDPtRGDPtb=ptpbGDPDef_t^b = \frac{NGDP_t}{RGDP_t^b} = \frac{p_t}{p_b}

Other Indexes

  1. Consumer Price Index (CPI)
  1. Personal Consumption Expenditure (PEC)

Expenditure

Consumer / Household Theory

🔥
Utility Maximization!

Utility Function

  1. Describes how much “happiness” or “satisfaction” an individual experiences from “consuming”
  1. Marginal Utility uc\frac{\partial u}{\partial c}
    1. extra total utility resulting from consumption of an extra unit of good
  1. Diminishing Marginal Utility
    1. uc>0,2uc2<0\frac{\partial u}{\partial c} > 0, \frac{\partial^2 u}{\partial c^2} < 0

Usual Forms:

  1. Log Utility
    1. u(c1,,cn)=i=1nln(ci)u(c_1, \dots, c_n) = \sum_{i=1}^n \ln(c_i)
  1. More general utility
    1. u(c1,,cn)=i=1nci1σ11σu(c_1, \dots, c_n) = \sum_{i=1}^n \frac{c_i^{1-\sigma}-1}{1-\sigma}
    1. Equals to log utility when σ=1\sigma = 1

Subjective Discount Factor β\beta

Impatience is an potential issue when we think about time

So we develop a simple model of consumer impatience

β(0,1)\beta \in (0,1)
The lower β\beta, the less does individual value future utility

Marginal Rate of Substitution

the quantity of one good(c1c_1) that a consumer can forego for additional units of another good(c2c_2) at the same utility level.
dc1u(c1,c2)c1+dc2u(c1,c2)c2=0MRS=dc1dc2=c2u(c1,c2)c1u(c1,c2)=u2(c1,c2)u1(c1,c2)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:

  1. Labor income
  1. Asset income

Two-period Consumption-savings framework

Notations:

  1. ctc_t ⇒ consumption in period tt
  1. PtP_t ⇒ nominal price of consumption in period tt
  1. YtY_t ⇒ nominal income in period tt
  1. AtA_t ⇒ nominal wealth at the beginning of period t+1t+1 (end of period tt)
  1. ii ⇒ nominal interest rate between periods
  1. rr ⇒ real interest rate between periods
  1. πt\pi_t ⇒ net inflation rate between period t1t-1 and period tt
  1. yty_t ⇒ real income in period tt (=Yt/Pt=Y_t / P_t)

(Lifetime) Budget Constraint

We will assume that A2A_2 is 0 (spend all in period 2 and leave no inheritances)

We will also assume A0=0A_0 = 0 for graphical simplicity

Assume strictly increasing utility with increase in consumption, but diminishing marginal utility in c1,c2,,cnc_1, c_2, \dots, c_n.

Ptct+At=Yt+(1+i)AtP_tc_t + A_t = Y_t + (1+i)A_t

List all of them

P1c1+A1=Y1+(1+i)A0P2c2+A2=Y2+(1+i)A1P_1c_1+A_1 = Y_1 + (1+i)A_0 \\ P_2c_2 + A_2 = Y_2 +(1+i)A_1

LBC:

P1c1+P2c21+iundefinedPDV of all lifetime expenditure=Y1+Y21+i+(1+i)A0undefinedPDV of all lifetime income\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}}
c2=(P1(1+i)P2)c1+(1+iP2)Y1+Y2P2=(1+i1+π2)c1+(1+iP2)Y1+Y2P2\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

c1+c21+r=y1+y21+rc_1 + \frac{c_2}{1+r} = y_1 + \frac{y_2}{1+r}

Optimum

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

u1(c1,c2)u2(c1,c2)undefinedMRS=1+i1+π2undefinedprice ratio=1+r\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’’<0u’>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:

  1. Discount rate
  1. Shape of utility function
  1. Interest rate
  1. 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
  1. Transitory income change dYdY
    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

  1. Preference imply that consumption should be smoothed
  1. 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

s1priv(r,y1,y2)=y1c1(r,y1,y2)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”:

  1. A larger fraction of consumers unable to borrow to pay for their desired early-period consumption ⇒ their consumption falls
    1. Then GDP falls
  1. Consumption in next period actually rises
    1. More saving / less debt taken out by credit-constrained households

Infinite Period Consumption-savings Framework

Notations

  1. ctc_t ⇒ consumption in period tt
  1. PtP_t ⇒ nominal price of consumption in period tt
  1. YtY_t ⇒ nominal income in period tt (assume it falls from the sky)
  1. at1a_{t-1} ⇒ real wealth (stock) holdings at beginning of period tt / end of period t1t-1
  1. πt+1\pi_{t+1} ⇒ net inflation rate between period tt and t+1t+1
  1. yty_t ⇒ real income in period tt

Utility Function

Assume

u(ct,ct+1,)=u(ct)+βu(ct+1)+β2u(ct+2)+u(c_t, c_{t+1}, \dots) = u(c_t) + \beta u(c_{t+1})+\beta^2 u(c_{t+2}) + \cdots

Budget Constraint

ct+at=yt+(1+rt1)at1c_t+a_t=y_t+(1+r_{t-1})a_{t-1}

Consumer Optimum

Construct the lagrangian, we get

L(ct,ct+1,,λt,λt+1,)=u(ct)+βu(ct+1)++λt[yt+(1+rt1)at1ctat]+βλt+1[yt+1+(1+rt)atct+1at+1]+\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

  1. w.r.t ctc_t
    1. u(ct)λt=0u’(c_t)-\lambda_t = 0
  1. w.r.t ata_t
    1. λt+βλt+1(1+rt)=0-\lambda_t + \beta \lambda_{t+1}(1+r_t) = 0
  1. w.r.t. ct+1c_{t+1}
    1. βu(ct+1)βλt+1=0\beta u’(c_{t+1}) - \beta \lambda_{t+1} = 0

Combine them, we get euler’s equation

λt=β(1+rt)λt+1u(ct)=β(1+rt)u(ct+1)\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

🔥
Not in scope for the exam

Notations

  1. ctc_t ⇒ consumption in period tt
  1. PtP_t ⇒ nominal price of consumption in period tt
  1. YtY_t ⇒ nominal income in period tt
  1. yty_t ⇒ real income in period t (=Yt/Pt=Y_t/P_t)
  1. AtA_t ⇒ nominal wealth at the beginning of period t+1t+1 (at the end of period tt)
  1. ata_t ⇒ real wealth at beginning of period t+1t+1 (at the end of period tt)
  1. ii ⇒ nominal interest rate between periods
  1. rtr_t ⇒ real interest rate between periods t1t-1 and tt
  1. πt\pi_t ⇒ net inflation rate between period t1t-1 and period tt
    1. =PtPt11= \frac{P_{t}}{P_{t-1}} -1
  1. StS_t ⇒ nominal price of a unit of stock in period tt
  1. DtD_t ⇒ nominal dividend paid in period tt by each unit of stock held at the start of tt

Budget Constraints

We need infinite budget constraints to describe economic opportunities and possibilities

t,Ptct+Stat=Yt+Stat1+Dtat1\forall t, P_tc_t+S_ta_t=Y_t+S_ta_{t-1}+D_ta_{t-1}

Solving for for lagrangian optimal,

  1. With respect to ctc_t: u(ct)λtPt=0u’(c_t)-\lambda_tP_t = 0
  1. With respect to ata_t: λtSt+βλt+1(St+1+Dt+1)=0-\lambda_tS_t+\beta \lambda_{t+1} (S_{t+1} + D_{t+1}) = 0
  1. With respect to ct+1c_{t+1}: u(ct+1)λt+1Pt+1=0u’(c_{t+1})-\lambda_{t+1}P_{t+1} = 0

Combine

With this we can derive asset pricing

Asset Pricing

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

Why buy an asset?

  1. Pay a dividend in the future
  1. Market value may rise in the future

Microeconomic events affect asset prices

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

St=(βu(ct+1)u(ct))(St+1+Dt+1)(PtPt+1)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+πt+1=Pt+1/Pt1+\pi_{t+1}=P_{t+1}/P_t

St=(βu(ct+1)u(ct))(St+1+Dt+1)(11+πt+1)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

  1. Consumption across time (ctc_t and ct+1c_{t+1}) affects stock prices
  1. Inflation affects stock prices
  1. Any factor (monetary policy, fiscal policy, globalization) that affects inflation and GDP/consumption in principle impacts stock/asset markets

Consumer Optimization

St=(βu(ct+1)u(ct))(St+1+Dt+1)(PtPt+1)S_t =(\frac{\beta u'(c_{t+1})}{u'(c_t)})(S_{t+1}+D_{t+1})(\frac{P_t}{P_{t+1}})
u(ct)βu(ct+1)=(St+1+Dt+1St)(11+πt+1)undefinedAnalogy with Ch 3 & 4, must be (1+rt)\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

Atf(kt,nt)A_tf(k_t,n_t)

Takes in production inputs (in real terms)

  1. Fundamental production factors
    1. labor (ntn_t / ll / LL)
    1. capital ktk_t
  1. Intermediate inputs
    1. raw materials
    1. patents
  1. Land

Three Properties:

  1. Atf(kt,nt)i>0\frac{\partial A_t f(k_t, n_t)}{\partial i} > 0 ⇒ the function is always increasing in each factor
  1. 2Atf(kt,nt)i2<0\frac{\partial^2 A_t f(k_t, n_t)}{\partial i^2} < 0 ⇒ the function has decreasing marginal product in the same factor
  1. 2Atf(kt,nt)ij>0\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 AtA_t, changes in AA cause shifts in production function
🔥
Almost always ⇒ we use Cobb-Douglas Production Function Atf(kt,nt)=Atktαnt1αA_t f(k_t, n_t) = A_t k_t^{\alpha} n_t^{1-\alpha}
  1. α(0,1)\alpha \in (0,1) measures capital’s share of output
    1. (1α)(1-\alpha) measures labor’s share of output
    1. US economy: α0.3\alpha \approx 0.3
    1. Chinese economy: α0.15\alpha \approx 0.15

Productivity

3 Concepts

  1. Marginal Product of Labor
    1. mpnt=Atfn(kt,nt)nt=(1α)Atktαntαmp_{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}
  1. Average product of labor
    1. apnt=Atf(kt,nt)nt=Atktαnt1αntap_{n_t} = \frac{A_t f(k_t, n_t)}{n_t} = \frac{A_t k_t^\alpha n_t^{1-\alpha}}{n_t}
  1. Efficiency/total factor of productivity AtA_t

Profits

Ptofit Maximization

maxk,nAtf(nt,kt)Cost(nt,kt)\max_{k,n} A_t f(n_t, k_t) - Cost(n_t, k_t)

Notation:

Labor

Labor Demand

Firm-level demand for labor defined by the relation

wtundefinedwage at time t=(1α)ktαntα=mpnt=(1α)(ktnt)α\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

rt=αktα1nt1α=mpkt=α(ntkt)1αr_t = \alpha k_t^{\alpha - 1}n_t^{1-\alpha} = mpk_t = \alpha(\frac{n_t}{k_t})^{1-\alpha}
📢
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

invt=kt+1ktinv_t = k_{t+1} - k_t ⇒ capital demand and investment demand functions have the same shape

Two-period production schedule

Notation:

So profit maximization with dynamic profit function:

P1f(k1,n1)+P1k1W1n1P1k2+P2f(k2,n2)1+i+P2k21+iW2n21+iP2k31+iP_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,

Solve with first order conditions

With respect to n1n_1:

P1fn(k1,n1)undefinedderivative of f with respect to nP1w1=0fn(k1,n1)w1=0P_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 n2n_2:

P2fn(k2,n2)1+iP2w21+i=0\frac{P_2 f_n(k_2, n_2)}{1+i} - \frac{P_2 w_2}{1+i} = 0

With respect to k2k_2:

P1+P2fk(k2,n2)1+i+P21+i=0P2fk(k2,n2)P1(1+i)+P2P1(1+i)=1(P2P1)(11+i)fk(k2,n2)+(P2P1)(11+i)=1(1+π21+i)fk(k2,n2)+(1+π21+i)=1apply fisher equationfk(k2,n2)1+r+11+r=1fk(k2,n2)=r\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:

Real Interest Rate

Previous lectures:

📢
Now, rr 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:

P1+P2fk(k2,n2)1+i+P21+i=0(FOC on k2)fk(k2,n2)=r-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

Income distribution to LL and KK

(Assume with CRS)

Y=MPL×L+MPK×KY = MPL \times L + MPK \times K

Long Term Economic Growth

Solow Framework

Central Idea:

🔥
Accumulation of kk (capital stock per capita) leads to ever-improving economic standard of living (proxied by yy, GDP per capita)

Assumptions

  1. CRS(Constant Return to Scale)
  1. Exogenous(外源的) TFP(Total Factor Productivity) Growth
  1. Exogenous Savings Rule
  1. Closed Economy ⇒ Investment = Saving
  1. no “consumption-savings optimality condition”

Question

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

Notation

Aggregate Production Function

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

Yt=Ktα(XtNt)undefinedEfficiency units of labor1αY_t = K_t^{\alpha} \cdot {\underbrace{(X_t \cdot N_t)}_{\mathclap{\text{Efficiency units of labor}}}}^{1-\alpha}

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

If we define At=Xt1αA_t = X_t^{1-\alpha} (total factor productivity), by rearranging equation, we get

Yt=AtKtαNt1αY_t = A_t \cdot K_t^{\alpha} \cdot N_t^{1-\alpha}

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

YtXtNt=(KtXtNt)α(XtNtXtNt)1αyt=ktα\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

Kt+1=Kt+ItDepreciationt=Kt+ItδKtK_{t+1} = K_t + I_t - Depreciation_t = K_t + I_t - \delta K_t

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

In per-capita terms:

Kt+1Nt+1Nt+1Nt=ItKtKtNt+(1δ)KtNtKt+1Nt+1(1+gt+1N)=ItKtKtNt+(1δ)KtNtkt+1kt(1+gt+1N)=ItKt+(1δ)gt+1kln(1+[ItKtδ])gt+1NItKtδgt+1N\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(0,1)s \in (0,1) of output in every period
total savings of economy=syt=sktα\text{total savings of economy} = s \cdot y_t = s \cdot k_t^\alpha
⚠️
Assumption 2: Fraction δ\delta of physical kk depreciates every period
  1. Physical wear out of equipments
  1. Chips and wires frying

So general case:

total break-even investment=(grX+grN+δ)kt\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)
Yt=Ct+ItYtCt=ItsYt=Stnet=It=Kt+1(1δ)KtY_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 Kt+1K_{t+1} in normalized form

Kt+1XtNt=sktα+(1δ)kt=Kt+1Xt+1Nt+1Xt+1Nt+1XtNt=kt+1(1+grX)(1+grN)\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 ktk_t transitions over time)

kt+1=sktα(1+grX)(1+grN)+(1δ)kt(1+grX)(1+grN)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=kt+1=kk = k_{t+1} = k^*:

k=s(k)α(1+grX)(1+grN)+(1δ)k(1+grX)(1+grN)=[s(1+grX)(1+grN)(1δ)]α1\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=kα=[s(1+grX)(1+grN)(1δ)]α1αy^* = k^{* \alpha} = [\frac{s}{(1+gr_X)(1+gr_N)-(1-\delta)}]^{\frac{\alpha}{1-\alpha}}
ln(y)=ln(s)α1αln((1+grX)(1+grN)(1δ))α1αln(s)α1αln(grX+grN+δ)α1α\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}
🔥
Population growth lowers GDPpc, and the savings rate increases it.

Transitional Dynamics

How does economy converge to steady state?

Transitional Dynamic Equilibrium

If kt<kk_t < k^*, does economy converge towards kk^*?

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

kt+1=sktα(1+grX)(1+grN)+(1δ)kt(1+grX)(1+grN)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

max{ct,kt+1}t=0t=0βtu(ct)s.t.t=0βtλt[ktα(ct+kt+1(1δ)kt)]=0\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

Golden Rule

🧙🏽‍♂️
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+δ(s)y(k(s))=y(k(s))δk(s)y(k(s)) = c+i = c+sy(k(s)) \\ y(k(s)) = c+\delta(s) \\ y(k(s)) = y(k(s)) - \delta k(s)
cs=y(k(s))kk(s)sδk(s)s=(y(k(s))kδ)k(s)s=(fkδ)k(s)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”)

c(smax)s=(fk(k(smax))δ)k(smax)s=0\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

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

Simplify (with long-run assumptions):

1β=1+r\frac{1}{\beta} = 1+r

Now a second interpretation of rr

Which came first, β\beta or rr?

Labor Market

Income YY

Ct+AtAt1=iAt1+YtC_t + A_t - A_{t-1} = iA_{t-1} + Y_t

Two perspectives on income types:

How important is financial wealth?

Income Flows

Measuring financial wealth

Ct+At+1=At+iAt+YC_t + A_{t+1} = A_t + iA_t + Y
t=YoBYoDct(1+r)tYoB=a0aYoD(1+r)YoDYoB+t=YoBYoDyt(1+r)tYoB\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}}

Measuring “human” wealth (from labor income)

Apply our present value(PV) calculation by just summing up discounted salaries over lifecyle
PVBirth(All Y)=age=0Age at deathYage(1+π1+i)age=age=0AaDyage(1+r)age=y1(11r)AaD+1111+r=y1+r(11+r)AaDry1+rr\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

Those choices are part of the household’s labor supply

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

Mincer model of education and wages

Wt=ptEStW_t = p_t^{E} \cdot S_t

ln(Wt)=ln(pE)+ln(St)=ln(pE)+ln(St)=ln(pE)+sln(1+a)+ln(S0)ln(pE)+as+ln(S0)\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+nn=TsT = s+n \equiv n = T-s

If we assume no discount

maxs[(Ts)W(s)+s0]\max_s[(T-s) \cdot W(s) + s \cdot 0]

FOC reveals

W(s)+(Ts)W(s)=0W(s)=(Ts)W(s)-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 Wt(s)=ptESt=ptES0(1+a)sW_t(s) = p_t^E \cdot S_t = p_t^E \cdot S_0(1+a)^s

ptES0(1+a)s=(Ts)S0(1+a)sln(1+a)1/ln(1+a)=Ts1/ln(1+a)1/aTs=1/ap_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)

📢
Consumer’s decision problem: maximize utility subject to budget constraint - bring together both cost side and benefit side
maxu(s,l)s.t.Pc+Wl=W\max u(s,l) \\ \text{s.t.} \\ Pc+Wl = W

At the optimal choice

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

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

Price changes

Government

Government Spending

Household income goes into savings, taxes, consumption

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

Therefore

(TG)=(IS)+(EM)(T-G) = (I-S)+(E-M)

Bonds

Government Bonds

Gov. Budget Constraint

Gt+Bt=T+it1Bt1+Bt1G_t + B_t = T+ i_{t-1}B_{t-1} + B_{t-1}

📢
Useful way to look at “debt burden” in data: B/Y-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

Dynamic Model of the Gov.

Notation

Period-t government budget constraint

gt+bt=(1+r)bt1+ttgt+btbt1undefinedsavings during period t=tt+rbt1undefinedasset income during period tg_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):

g1+g21+r=t1+t21+r+(1+r)b0g_1 + \frac{g_2}{1+r} = t_1 + \frac{t_2}{1+r} + (1+r)b_0

Consumer LBC with tax introduced

c1+c21+r=y1t1+y2t21+r+(1+r)a0c_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

c1+c21+r=y1g1+y2g21+r+(1+r)(a0+b0)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,

s1nat=s1priv+s1gov=y1t1c1+t1g1=y1c1g1s_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.

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)

Taxation

🔥
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!
The Laffer curve
T=tW(1l(t))Tt=W(1l(t))+tW(1l(t))tT = 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}

Monetary Policy

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

Roles of money

🔥
It’s hard to conceptually model “money”

Bonds

Key relationship between price of a bond and nominal interest rate

Ptb=FVt+11+itP_t^b = \frac{FV_{t+1}}{1+i_t}

Inverse relationship between PbP^b and ii:

it=1Ptb1i_t = \frac{1}{P_t^b} - 1

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

Types of money

Unconventional monetary policy

Infinite Period Monetary Policy

MIU(Money-in-the-utility) function

DEMAND holdings of nominal money

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

In consumer optimization it’s MDM_D everywhere, where MSM^S is determined by the central bank.

In money market equilibrium,

MtPt=MtDPt=MtSPt\frac{M_t}{P_t} = \frac{M_t^D}{P_t}=\frac{M_t^S}{P_t}

Notation

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

Ptct+PtbBt+Mt+Stat=Yt+Mt1+Bt1+Stat1+Dtat1P_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}

Lagrangian function:

L=t=tβttu(ct,Mt/Pt)+t=tβttλt[Yt+(St+Dt)at1+Mt1+Bt1PtctStatMtPtbBt]\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 ct,at,Bt,c_t, a_t, B_t, \dots

Stock-pricing equation (from equation 2)

StundefinedPeriod-t stock price=(βλt+1λt)undefinedpricing kernel(St+1+Dt+1)undefinedFuture return\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}}

Bond-pricing equation (from equation 3)

Ptb=βλt+1λtP_t^b = \frac{\beta \lambda_{t+1}}{\lambda_t}
📌
Price of short-term bond is the pricing kernel

Note:

Ptb=11+itP_t^b = \frac{1}{1+i_t}

Can express pricing kernel as

βλt+1λt=11+it\frac{\beta \lambda_{t+1}}{\lambda_t} = \frac{1}{1+i_t}

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

1+rt=1+it1+πt+11+r_t = \frac{1+i_t}{1+\pi_{t+1}}

Combining equations, we get the consumption-money optimality condition

u2(ct,Mt/Pt)u1(ct,Mt/Pt)=it1+it\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(ct,MtPt)=lnct+ln(MtPt)u(c_t, \frac{M_t}{P_t}) = \ln c_t + \ln(\frac{M_t}{P_t})

With the optimality condition,

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

Monetary Neutrality

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

Money and Inflation in the long run

What determines inflation in the long run?

Monetary Policy and Fiscal Policy

Infinite Period Analysis

🔥
Note that B>0B>0 now means the government issues debt (beforehand BB were net assets ⇒ B<0B<0 meant the government was in debt

Notations

Fiscal Authority Budget Constraint in period tt:

Ptgt+Bt1Tundefinedtotal outlays=Tt+PtbBtT+RCBtundefinedtotal inflows\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 tt:

PtbBtM+RCBtundefinedTotal outlays=Bt1M+MtMt1undefinedTotal inflows\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

PtbBtM+PtMBSMBSt+PtFANNIEBtFANNIE+RCBt=Bt1M+MBSt1+Bt1FANNIE+MtMt1P_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)

PtbBtM+RCBt=Bt1M+MtMt1P_t^bB_t^M + RCB_t = B_{t-1}^M+M_t-M_{t-1}

(2)

Ptgt+Bt1T=Tt+PtbBtT+RCBtP_tg_t+B_{t-1}^T=T_t+P_t^bB_t^T+RCB_t

Ptgt+Bt1TBt1M=Tt+Ptb(BtTBtM)+MtMt1P_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}

Therefore, consolidated flow government budget constraint (GBC)

Ptgt+Bt1=Tt+PtbBt+MtMt1P_tg_t + B_{t-1} = T_t + P_t^bB_t+M_t-M_{t-1}

Active / Passive Policy

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

Lifetime consolidated GBC

First we have Period-t consolidated GBC

Ptgt+Bt1=Tt+PtbBt+MtMt1P_tg_t + B_{t-1} = T_t + P_t^b B_t + M_t - M_{t-1}

divide by PtP_t to put into real terms

gt+Bt1Pt=TtPt+PtbBtPt+MtMt1Ptundefinedsrtg_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}

Define bt=Bt/Pt,tt=Tt/Ptb_t = B_t / P_t, t_t = T_t / P_t,

We have

Bt1Pt=srtundefinedRevenue generated by monetary authority actions+(ttgt+Ptbbt)undefinedrevenue generated by fiscal authority actions\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}}
bt11+πt=srt+(ttgt+Ptbbt)\frac{b_{t-1}}{1+\pi_t} = sr_{t} + (t_{t} - g_t + P_t^bb_t)

Combine all periods (use fisher’s equation)

Bt1Pt=s=0tt+sgt+sx=1s(1+rt+x)+s=0srt+sx=1s(1+rt+x)\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

Inflationary Finance (FTI / FTPL)

Central Bank Independence