LECTURE 3:
INTER-TEMPORAL CHOICE AND THE BUSINESS CYCLE
Reading:
• Sanjay K. Chugh, Modern Macroeconomics, Chapter 3
• William A. Lord (2002), Household Dynamics: Economic Growth and Policy, Chapters 1
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1. INTRODUCTION AND BACKGROUND
INTRODUCTION AND BACKGROUND
In this lecture we will:
• Study inter-temporal choices in the Real Business Cycle model
• Introduce assets and their role in the macroeconomy
• Investigate the relationship between the RBC model and macroeconomic data
• Discuss limitations of the RBC model for understanding assets and asset pricing
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2. BUSINESS CYCLE FACTS AND THE RBC MODEL
INTRODUCTION AND BACKGROUND
• Early business cycle models focused on characterizing fluctuations in the macroeconomy • These early models placed much less attention on asset prices and financial markets
• However, RBC models evolved from an earlier literature on General Equilibrium Theory • General Equilibrium Theory closely studied many important features of finance:
• Inter-temporal choice • Risk
• Insurance
• Asset markets
• Asset prices
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3. THE INTRA-TEMPORAL RBC MODEL
3.A) SIMPLE INTER-TEMPORAL HOUSEHOLDS IN THE RBC MODEL
3.A) INTER-TEMPORAL HOUSEHOLDS IN THE RBC MODEL
Households
1) Choice between Consumption and Saving
• Households must decide how much to consume today, how much to save, and how much
to consume tomorrow
• Because savings earn interest (/returns), the more resources that are saved today, the more resources are available for consumption in the future
• But households are impatient as they discount the value of future consumption more than the value of current consumption
• The optimization problem is to maximize life-time utility subject to an inter-temporal budget constraint
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A TWO-PERIOD MODEL OF CONSUMPTION AND SAVING
• Assumptions:
• Two-periods of life
• Earn (net) real interest rate r on savings S • Future utility is discounted at the rate β
• ExogenousincomeineachperiodY1,Y2
• Households use their savings to smooth consumption across time
• For now we redignore: • Risk
• Inflation
• Different types of assets
• Other financial market participants
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A TWO-PERIOD MODEL OF CONSUMPTION AND SAVING
• A households chooses current consumption C1, future consumption C2, and savings S: max log(C1) + β log(C2)
C1 ,C2 ,S
s.t. C1 + S = Y1 (First period budget constraint)
C2 = Y2 + S(1 + r) (Second period budget constraint) • Combine the within-period budget constraints:
C1+ C2 =Y1+ Y2 1+r 1+r
• This is the inter-temporal budget constraint (or, life-time budget constraint)
Question: Why can we think of (1 + r) as the price of current consumption? Give a mathematical reason and a reason using your intuition.
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HOUSEHOLD CHOICE FOR CONSUMPTION AND SAVING
• The simplified household problem is:
max log(C1) + β log(C2)
C1 ,C2
s.t. C1+ C2 =Y1+ Y2
• The First Order Condition yields: C1
Marginal Utility of Consumption in Period 1
Return on Savings
C2
Marginal Utility of Consumption in Period 2
1+r 1+r
1 = (1+r) × β1 (1)
• This is called the Consumption Euler Equation, which describes efficient inter-temporal consumption choices
• Later, we will see that this equation is fundamental to asset pricing models
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HOUSEHOLD CHOICE FOR CONSUMPTION AND SAVING: SOLUTION
• Rearrange Equation (1) for C2, then substitute into the inter-temporal budget constraint to find C1 and C2:
1 Y2 β(1+r) Y2 C1=1+β Y1+1+r , C2= 1+β Y1+1+r
• To find S substitute either C1 into the first period budget constraint or C2 into the second period budget constraint:
S=βY1− 1 Y2 1 + β (1 + r)(1 + β)
Exercise: Find these solutions on your own!
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HOUSEHOLD CHOICE FOR CONSUMPTION AND SAVING: GRAPHICAL ILLUSTRATION
• Rewrite the Euler Equation as: r = C2 − 1 βC1
Y2
• Substitute in the first and second period budget constraints: r = βY1−(1+β)S − 1
• This represents the household’s supply of savings r
Y2 −1 βY1−(1+β)S
S
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HOUSEHOLD CHOICE FOR CONSUMPTION AND SAVING
• When does the household choose to save (i.e. S > 0)? And what makes saving valuable? • Save when income in period 1 (Y1) is larger than income in period 2 (Y2)
• Save more when the interest rate r is high
• Savings transfers resources from periods of high income (when the MU of consumption is low) to
periods of low income (when the MU of consumption is high)
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SAVINGS SUPPLY CURVE
INCREASE IN CURRENT INCOME: Y1 → Y′1
r
Y2 −1 ′Y2 −1 βY1−(1+β)S βY1−(1+β)S
S
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3.B) INTER-TEMPORAL HOUSEHOLDS AND CAPITAL ACCUMULATION IN THE RBC MODEL
3.B) INTER-TEMPORAL HOUSEHOLDS AND CAPITAL ACCUMULATION IN THE RBC MODEL
• Inter-temporal households generate a supply of savings (or a demand for loans!)
• But where do those savings go? What are they used for?
• In the canonical RBC model, households hold physical capital that is then used in production
• “Time to Build and Aggregate Fluctuations” Kydland and Prescott (1982)
• Here, we want to think of capital as a productive asset, but one whose return may fluctuate with the business cycle
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HOUSEHOLD CONSUMPTION CHOICE AND CAPITAL ACCUMULATION
• A household chooses current consumptions C1, C2, and investment in capital I1: max log(C1) + β log(C2)
C1 ,C2 ,I1
s.t. C1 + I1 = Π1 + r1K1 (First period budget constraint)
C2 = Π2 + (1 + r2 − δ)K2 (Second period budget constraint) K2 = I1 + K1(1 − δ) (Capital accumulation equation)
• Households are endowed with capital K1 (cannot be adjusted)
• Households earn (net) real interest r1 , r2 on their capital holdings
• Capital in period 2 is investment in new capital + undepreciated capital from period 1
• In period 2, after production takes place, households consume remaining capital: (1 − δ)K2
• For simplicity, assume households don’t supply labour, but they own firms and receive dividends Π1, Π2
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HOUSEHOLD CONSUMPTION CHOICE AND CAPITAL ACCUMULATION
• Substitute capital accumulation equation into first period budget constraint: C1 +K2 = Π1 +K1(1+r1 −δ)
• Now, substitute the budget constraints into the utility function:
max log(Π1 +K1(1+r1 −δ)−K2)+βlog(Π2 +K2(1+r2 −δ))
K2
• Taking the FOC with respect to K2:
1 =β(1+r2−δ)1 C1 C2
• Which is also a Consumption Euler Equation
• Here, the return on savings/capital holding is (1 + r2 − δ), but otherwise households value
capital in the same way they valued savings in Section 3.a)
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HOUSEHOLD CONSUMPTION CHOICE AND CAPITAL ACCUMULATION
r2
Ksupply
2
K2
Exercise: Solve for the capital supply curve. This is an equation with r2 on the left, which is a function of K2.
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3.C) FIRMS IN THE INTER-TEMPORAL RBC MODEL
3.C) FIRMS IN THE INTER-TEMPORAL RBC MODEL
• Firms produce output using the production technology:
Yt =AtKαt , wheret=1,2
• Where At is technology/productivity; and Kt is the capital inputs of firms Yt
A t K αt
Kt
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FIRM’S CHOICE OF CAPITAL INPUTS
• A competitive firm chooses capital KD2 to maximize profit Π2: Π2= maxA2(KD2)α−r2KD2
KD2
• where r2 is the interest rate or rental rate of capital (taken as given)
• The first order condition yields:
α A 2 ( K D2 ) α − 1 − r 2 = 0
Marginal Product of Marginal Cost of Capital Capital
• Marginal Product of Capital (MPK)= extra output generated by additional capital input • The FOC yields the capital demand curve:
r2 = αA2(KD2 )α−1
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FIRM’S CAPITAL DEMAND CURVE
r2
αA2(KD2 )α−1
K2
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3.D) EQUILIBRIUM IN THE INTER-TEMPORAL RBC MODEL
EQUILIBRIUM DEFINITION
• The Capital Market clearing condition holds:
• The real interest rate r2 ensures that the capital market clears
• Capital supply (by households) is equal to capital demand (by firms) KS2 = KD2
• Aggregate production is determined by technology:
Yt =AtKαt , t=1,2 (2)
• The aggregate resource constraint holds each period:
Y1 = C1 + I1 (3)
Y2 + (1 − δ)K2 = C2 (4) • (Where total resources in period 2 include remaining undepreciated capital)
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EQUILIBRIUM AGGREGATE VARIABLES: GRAPHICAL ILLUSTRATION
r2
r∗2
Y2 Y∗2
KS2
KD2
K2
K∗2
K2
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BUSINESS CYCLE FLUCTUATIONS IN THE RBC MODEL EXAMPLE
Example:
• Consider an increase in productivity in period 1: ↑ A1
• Note that capital in period 1, K1, is fixed
• Neither households or firms can increase the stock capital used in period 1 • What does this mean for capital demand and capital supply in period 2?
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BUSINESS CYCLE FLUCTUATIONS IN THE RBC MODEL: EXAMPLE: ↑ A1
r2
KS2
r∗2 KS′
r∗∗ 2 KD 22
Y2 Y∗∗
Y∗2
K2
2
K∗ K∗∗ K2 22
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4. MAPPING THE RBC MODEL TO MACROECONOMIC DATA
4. MAPPING THE RBC MODEL TO MACROECONOMIC DATA
• Total Factor Productivity, At, is the primary driver of fluctuations in the RBC model • But how do we measure TFP?
• Consider the aggregate production function, with typical estimate of α = 0.3:
• To estimate At, rearrange:
Y = A K0.3N0.7 tttt
At= Yt K0.3 N0.7
• And measure At using data on Yt, Kt, Nt
tt
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MEASURING TFP SHOCKS
Source: Otto (1999)
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MEASURING TFP SHOCKS
Source: King and Rebelo (1999)
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CALIBRATING THE RBC MODEL:
HOW WELL DOES IT DO AGAINST THE DATA?
Source: King and Rebelo (1999)
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CALIBRATING THE RBC MODEL:
HOW WELL DOES IT DO AGAINST THE DATA?
• Y, C, I, N: not volatile enough relative to the data
• Y, C, I, N are far too correlated with each other (corr≈1) relative to the data • Why?
• There is only one shock in the simple RBC model
• So macro variables in the model cannot easily move independent of each other
• One reason: not enough “internal propagation” in the model (see Cogley and Nason, 1995)
• Instead, all persistence and auto-correlation in the model driven by persistence and
auto-correlation in At
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RBC MODEL IMPLICATIONS
• Business cycles are due to “real” shocks (e.g. TFP or technology shocks)
• Productivity, real wages, employment, consumption, and investment are all pro-cyclical • Markets are always in equilibrium.
• Prices and wages always adjust (flexibly) to ensure this equilibrium is efficient
• No involuntary unemployment in the model (why?)
• Money neutrality holds: changes in money supply do not affect real variables
• Government stabilization policies tend to be counter-productive (why?)
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5. LIMITATIONS OF RBC MODELS
5.A) LIMITATIONS OF THE RBC MODEL FOR UNDERSTANDING THE MACROECONOMY
• How do we measure TFP shocks? Solow Residuals?
• Do we really have frequent regressions in technological progress that cause recessions?
• What is the role of fiscal and monetary policy in the evolution of the macroeconomy?
• Most macroeconomists now convinced that money neutrality only holds in the long run
• Real wages are not pro-cyclical in the data. What does this imply?
• “Real Wages and the Business Cycle”, Abraham and Haltiwanger (JEL, 1995)
• “Short-Run Equilibrium Dynamics of Unemployment, Vacancies, and Real Wages”, Pissarides,
(AER, 1985)
• To answer these questions, will typically need a DSGE model that incorporates price
stickiness, wage stickiness, and policy shocks
Question: How might we characterize a pandemic or a lockdown as a “technological shock”?
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5.B) LIMITATIONS OF THE RBC MODEL FOR UNDERSTANDING ASSET MARKETS
• Most models in the RBC literature are solved using linear approximations to the model
• These linear approximations study deviations of the model from a well-defined steady state of the model economy
• But linear approximation means agents solve their problems under certainty equivalence: • Certainty equivalence ⇐⇒ agents behave as if there is no risk!
• But risk is one of the primary reasons for holding financial assets:
• We often want to insure against risks by holding financial assets that pay out if certain
undesirable states of the world eventuate (e.g. unemployment, fire, theft, death)
• In equilibrium, agents often want to share or smooth risks e.g. you payout when I am doing
poorly, and I payout when you are doing poorly
More to say about this later!
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