LECTURE 2:
EQUILIBRIUM (REAL) BUSINESS CYCLES
Reading:
• Sanjay K. Chugh, Modern Macroeconomics, Chapter 2
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1. INTRODUCTION AND BACKGROUND
INTRODUCTION AND BACKGROUND
In this lecture we will be studying:
• A brief history of attempts to understand business cycles • The Real Business Cycle (RBC) model
• A simple static/intra-temporal business cycle model
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INTRODUCTION AND BACKGROUND
• The business cycle theory of the early 20th century aimed to quantitatively analyse economic fluctuations using mathematical and statistical approaches to economics
• This research agenda was led by Ragnar Frisch and Jan Tinbergen, the first winners of the Nobel Prize in Economics
• This work on business cycles begun before John Maynard Keynes became one of the most well-known names in macroeconomics
time
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INTRODUCTION AND BACKGROUND
• More recently, the general equilibrium Real Business Cycle (RBC) theory began as an attempt to quantitatively explain macroeconomic fluctuations via stochastic (i.e. random) shifts in aggregate production technology
• This followed the tradition of Classical and Neo-Classical Economics
• Economic agents behave as if they make choices subject to budget constraints
• Macroeconomic outcomes are determined by market clearing
• The “classical dichotomy”: nominal variables do not affect real variables
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INTRODUCTION AND BACKGROUND
• The RBC model structure follow from Optimal Growth Theory (e.g. the Solow-Swan model)
• RBC models incorporate Neo-Classical growth with stochastic shifts or shocks as the
driving force behind cyclical macroeconomic fluctuations
• The RBC research agenda uses the stochastic growth model to try to explain fluctuations
that can be quantitatively assessed
• Another aim of RBC economists was to build small laboratories in which government policies could be tested
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INTRODUCTION AND BACKGROUND
• Modern macroeconomic models used at central banks are rooted in the RBC framework • Federal Reserve Board (“FRB/US”); Norges Bank (“NEMO”); Swedish Riksbank (“RAMSES II”);
Reserve Bank of Canada (“TOTEM”); Reserve Bank of Australia (“MSM”); Reserve Bank of New
Zealand (“KITT” and “NZSIM”)
• RBC models have evolved into Dynamic Stochastic General Equilibrium (DSGE) models
• E.g. “Impulse Response Functions” for a monetary policy shock (Smets and Wouters, 2007):
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INTRODUCTION AND BACKGROUND
• Inspired by Robert Lucas (1977), Kydland and Prescott (1982) aimed to study growth and fluctuations in a single model framework asking the following question:
• “Can business cycle fluctuations occur as a natural consequence of the competitive economy where agents make optimal inter-temporal resource allocation decisions in response to stochastic shifts in technology and preferences?”
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INTRODUCTION AND BACKGROUND
• “Can business cycle fluctuations occur as a natural consequence of the competitive economy where agents make optimal inter-temporal resource allocation decisions in response to stochastic shifts in technology and preferences?”
• If answer is No (as most economists at the time believed): ⇒ Market co-ordination failure
⇒ Large welfare losses from market outcomes
⇒ Role for active macroeconomic stabilization policy (e.g. Keynesian stimulus)
• “In business cycles we don’t have market clearing … and we do have involuntary unemployment … we are losing output … it’s not a moving equilibrium” (James Tobin)
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INTRODUCTION AND BACKGROUND
• “Can business cycle fluctuations occur as a natural consequence of the competitive economy where agents make optimal inter-temporal resource allocation decisions in response to stochastic shifts in technology and preferences?”
• If answer is Yes (as RBC economists believed): ⇒ Business cycles are “efficient”
⇒ Negligible welfare costs from market outcomes
⇒ Active stabilization policies can be disruptive/destabilizing
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INTRODUCTION AND BACKGROUND
• Why Real business cycles?
• ‘Real’ as opposed to ‘nominal’ or monetary forces
• Why are Real business cycles efficient?
• No economic frictions to distort optimal decisions
• Modern DSGE models incorporate many nominal/monetary features: • Price rigidity, nominal shocks, monetary and fiscal policies
• Modern DSGE models incorporate many economic rigidities/frictions: • Imperfect competition, search frictions
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2. BUSINESS CYCLE FACTS AND THE RBC MODEL
2.A) BUSINESS CYCLE FACTS: ‘STYLIZED FACTS’
• Stylized facts: simplified (statistical) descriptions of the macroeconomy • Of what use are these stylized facts? (Why not more ‘complex’ facts?)
• Important for understanding simple empirical properties of aggregate fluctuations
• Provide a metric/measure against which an economic model can be evaluated
• Use facts to help choose model parameters via “calibration”: ensures that model variables
match the means, variances, and auto-correlations of variables in the data
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2.A) BUSINESS CYCLE FACTS: ‘STYLIZED FACTS’
Key stylized facts for industrialized economies
• Fluctuations in output of similar magnitude to fluctations in hours worked and employment • Consumption of non-durables+services strongly pro-cyclical, fluctuates less than output
• Investment strongly pro-cyclical, fluctuates more than output (∼ 3 times more volatile)
• Productivity tends to be pro-cyclical
• Government expenditure is largely uncorrelated with output • Real interest rates and real wages tend to be acyclical
• Asset prices are strongly pro-cyclical
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2.A) BUSINESS CYCLE FACTS: ‘STYLIZED FACTS’
How do we document ‘empirical facts’?
• Detrend the (log) time series with secular or growth components
• Compute summary statistics from detrended data (i.e. cyclical components)
Summary Statistics
1) For individual variables/time series, compute: • Mean
• Standard deviation (i.e. volatility)
• Autocorrelation (i.e. persistence)
2) For pairs of variables (e.g. consumption and GDP)
• Relative standard deviation: S.D.(xt)/S.D.(yt)
• Cross correlation (co-variance) at various leads/lags:
• Corr(xt+j,yt) for j positive (lead) or negative (lag)
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2.A) BUSINESS CYCLE FACTS: ‘STYLIZED FACTS’
Source: King and Rebelo (1999)
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2.B) AN INTRODUCTION TO RBC MODELS
Key RBC model features:
• Optimizing, rational agents, with rational expectations
• Intra-temporal and Inter-temporal substitution is the key mechanism
• These model features are based on microeconomic theory (i.e. ‘micro-founded’)
• Competitive factor and product markets, and market clearing (i.e. equilibrium) • Business cycles arise due to responses to random shocks
• An economic agent’s response to these shocks is rational
• Main theoretical model/mechanisms for generating growth cycles
Question: Given rational responses to shocks, is there any role for government stabilization policies?
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3. SIMPLE INTRA-TEMPORAL RBC MODEL
3.A) INTRA-TEMPORAL HOUSEHOLDS IN THE RBC MODEL
3.A) INTRA-TEMPORAL HOUSEHOLDS IN THE RBC MODEL
Households
1) Consumer Choice between Work and Leisure
• Households must decide how much to work (in order to earn income) and how much
leisure to enjoy.
• The more a household works, the more income they have to spend, but the less leisure time they can enjoy (there are only so many hours in a day!)
• Leisure is a normal good
• The static optimization problem is to maximize utility subject to a static budget constraint and a time endowment constraint
Question: If leisure is a normal good, what does this imply about its relationship with income?
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HOUSEHOLD CHOICE BETWEEN WORK AND LEISURE
• A household’s problem is to choose consumption (C) and leisure (L): max U(C) + V(L)
C,L
subject to C = w · Ns + Π (budget constraint)
L + Ns = 1 (time endowment)
• Where:
• Ns is hours worked, or the amount of labour supplied by the household
• L+NS =1referstothetotaltimeavailableinaday
• Π are the dividends paid out by the firms owned by households (more on this later!)
• U′(C) > 0,U′′(C) < 0 implies diminishing marginal utility of consumption
• V′(N) > 0,V′′(N) < 0 implies diminishing marginal utility of leisure. Thus, households do not
spend all of their time on leisure!
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OPTIMAL CHOICE BETWEEN WORK AND LEISURE
• For tractability, let’s simplify functional forms:
U(C) = log(C), V(L) = b log(L)
• Now substitute the time endowment and the budget constraint into the utility function(s):
max log(wNs +Π)+blog(1−Ns) NS
• Taking First Order Conditions yields: w1−b1=0
Marginal Cost of Labour Supplied
C
1 − Ns
Marginal Benefit of Labour Supplied
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THE LABOUR SUPPLY CURVE
• We can then write the Labour Supply curve as:
w=b
C
1 − Ns
w
Marginal Benefit In Consumption Units
Marginal Cost In Consumption Units
bCs 1−N
Ns
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3.B) FIRMS IN THE SIMPLE RBC MODEL
3.B) FIRMS IN THE SIMPLE RBC MODEL
• Firms produce output using the production technology:
Y = A · F(Nd)
• Where A is the exogenous level of technology; and Nd is labour inputs demanded by firms • Production function has diminishing marginal product: F′ (N) > 0, F′′ (N) < 0
• We often simplify to the Cobb-Douglas production function: Y = A(Nd)1−α
Y
A(Nd )1−α
Nd
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FIRM’S CHOICE OF LABOUR INPUTS
• A competitive firm chooses labour Nd to maximize profit Π (returned to households) Π = max A(Nd)1−α − wNd
Nd
• where w is the wage or cost of hiring labour (and is taken as given)
• The first order condition yields:
(1 − α)A(Nd)−α − w
= 0
• Marginal Product of Labour (MPN) = extra output generated by one additional labour input
• The FOC yields the labour demand curve:
w = (1 − α)A(Nd)−α
Marginal Product of Labour
Marginal Cost of Labour
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FIRM’S LABOUR DEMAND CURVE
w
(1 − α)A(Nd)−α
Nd
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FIRM’S LABOUR DEMAND CURVE
AN INCREASE IN PRODUCTIVITY: A → A′
w
(1 − α)A′(Nd)−α (1 − α)A(Nd)−α
Nd
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3.C) EQUILIBRIUM IN THE SIMPLE RBC MODEL
3.C) EQUILIBRIUM IN THE RBC MODEL
• What is meant by Equilibrium?
• An equilibrium is a set of prices and allocations in the economy such that no agents are willing
to change their behaviour
• These equilibrium conditions are usually summarized by a set of market clearing conditions:
• Given wages:
• Households do not want to supply any more or any less labour to the market
• Firms do not want to hire any more or any less labour from the market
• Given prices:
• Households do not want to consume more or less goods from the market
• Firms do not want to provide any more or less goods to the market
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EQUILIBRIUM DEFINITION
• The Labour Market clearing condition holds:
• Labour supply is equal to labour demand Ns = Nd:
b C =w=(1−α)AN−α (1) 1−N
• Aggregate production is determined by technology:
Y = AN1−α (2)
• The aggregate resource constraint holds:
Y=C (3) 24
EQUILIBRIUM AGGREGATE VARIABLES: Y, C, N, w
• First, substitute Equation (3) into (2), and substitute this into Equation (1): AN1−α −α
b1−N =(1−α)AN • Second, rearrange and solve for N:
N = (1 − α)
b + (1 − α)
(4) w = (1−α)A b+(1−α) (5)
• Finally, use (4) and (2) and (3) to solve for Y and C:
(1−α) 1−α
Y=C=A b+(1−α) (6)
Exercise: Find these solutions on your own!
• Third, substitute into either the labour supply or demand curve to find w: (1−α) −α
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EQUILIBRIUM AGGREGATE VARIABLES: GRAPHICAL ILLUSTRATION
w
NS
w∗
Y Y∗
ND
N
N∗ N
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BUSINESS CYCLE FLUCTUATIONS IN THE RBC MODEL
• Macroeconomic fluctuations in early RBC models were driven entirely by changes in aggregate productivity
• In our simple RBC model, changes in productivity A can drive fluctuations in each of the aggregate variables: C, Y, w, N
• Note: In our simplified model, N is unaffected by A. It turns out this is a special case that arises due to the assumption of log-utility over consumption. This is not a general result!
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EQUILIBRIUM AGGREGATE VARIABLES: AN INCREASE IN PRODUCTIVITY: A → A′
w NS′ NS
w∗∗ w∗
Y Y∗∗ Y∗
ND′ ND
N
N∗ N
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EQUILIBRIUM AGGREGATE VARIABLES: AN INCREASE IN PRODUCTIVITY: A → A′
• An increase in productivity causes an increase in output
• The increase in output is associated with an increase in the demand for labour
• In equilibrium, the increase in output is associated with an increase in consumption
• The increase in consumption is associated with an increase in the labor supply curve • Wealthier households require a higher wage to induce the same amount of labor supply
• An increase in labor demand and a decrease in labor supply cause an increase in wages • However, the effect on labour/employment is ambiguous
The change in labour depends on the relative strength of two forces
(1) The effect of rising productivity of labour demand
(2) The effect of rising productivity on household wealth and thus willingness to supply labour
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4. LIMITATIONS OF THE SIMPLE (INTRA-TEMPORAL) RBC MODEL
4. LIMITATIONS OF THE SIMPLE (INTRA-TEMPORAL) RBC MODEL
• In our simple RBC model, households and firms only make static or intra-temporal decisions
• But these households do not care about the future! • So the model lacks:
• inter-temporal decisions
• Household savings
• Productive capital
• Financial assets (or asset prices!)
• A relationship between the past and the future
• A serious characterization of aggregate dynamics
Coming Up Next: The inter-temporal RBC model
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