Real Business Cycles

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Real Business Cycles

• Motivation
• The Model
• Solving the Model
• Predictions
• Fiscal Policy

2
Readings

• "Understanding Real Business Cycles" by C.
  Plosser, Journal of Economic Perspectives 3, No.
  3 51-77 (Summer 1989)
• Real Business Cycles A New Keynesian
  Perspective by N. Gregory Mankiw, Journal of
  Economic Perspectives, Vol 3, No 3, pp 79-90
  (Summer 1989)
• Williamson, Ch 11.

3
Motivation

• Prior to RBC theory the mainstream idea was that
  aggregate demand caused business cycles (e.g.
  Keynesian IS-LM model)
• Can a dynamic competitive equilibrium (CE) model
  provide an explanation of business cycles?
• In CE without investment, there is no persistence
  (HW assignment 3)
• Investment provides a stronger link of economic
  decisions over time.
• RBC model is the CE model with (i) shocks to
  productivity and (ii) uncertainty and rational
  expectations.

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• Real Business Cycle (RBC) theory originates with
• (i) F. Kydland and E. Prescott
• - 2004 Nobel Winners in Economics
• - Time to Build and Aggregate Fluctuations
• (Econometricia, 1982)
• (ii) J. Long and C. Plosser
• Real Business Cycles (Journal of Political
  Economy, 1983)

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• Productivity for the US economy can be calculated
  using the Cobb-Douglas PF
• y f(K,N) zK0.3N0.7 ? z y/K0.3N0.7
• Historically, the growth rate of productivity (z)
  fluctuates with the business cycle.

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Figure 4.20 The Solow Residual for the United
States
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Figure 5.11 Deviations from Trend in Real GDP
and the Solow Residual
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Table 3.1 The Production Function of the United
States, 19802004
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• RBC model says that business cycles are caused by
  temporary but persistent productivity shocks
• where 0 lt r lt 1 is degree of persistence.
• Productivity shocks are the impulse and
  investment is the propagation mechanism.
• Rational Expectations Households/Firms know all
  variables up to time t and the random process for
  and .

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Numerical Example

• Consider t 20 periods
• There is a one-time shock to et in period 1 where
  e1 10 and et 0 for all other time periods

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• Notice the effect on zt depends on the value of r
  which measures the amount of persistence for the
  shock e.
• r 0 ? purely temporary
• r 0.80 ? temporary but
• persistent

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• r 1 ? permanent

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Timing

• Households Firms are infinitely lived
• In each period t
• (i) Kt is known from last period.
• zt shock is observed.
• (ii) A rational expectation of zt1 is formed.
• (ii) Firms hire labor Ntd and buy capital Kt1.
• (cost of capital rt d)
• (iii) Households supply labor Nts and consume
  ct.
• (wages wt are paid)
• (iv) Markets clear (labor, goods). Firm profits
  paid to households.

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Households

• In each period t households choose ctj,ltj to
• subject to

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• FOC for Utility Maximization

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Firms

• In each period t firms choose Ndt,Kt1 to
• FOC for Profit Maximization

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Market Clearing

• Goods
• Labor

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Social Planner

• Since solution to CE is Pareto Optimal it is
  equivalent to the social planners problem
• subject to

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Productivity Shocks

• Temporary Positive Shock
• Supply ? higher ND and z shifts Ys right.
  Decreases r and shifts NS left ? N ambiguous
  but increase in w.
• Demand ? Higher w increases c ? Yd shifts
  right. No change in future MPK ? no
  (direct) effect on I.
• Overall ? Increase in y and decrease in r
• (C and I increases)

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• There will be persistence Higher I today ?
  Higher future output.
• Future Positive Shock
• Supply ? Current z unchanged ? Ys fixed.
• Demand ? Increases c (from PIH) and
  increase in I ? Yd shifts right.
• Overall ? Increase in y and increase in r

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Functional Forms

• Cobb-Douglas (log) Utility
• Cobb-Douglas Technology
• where 0 lt q lt 1 and 0 lt a lt 1 are the
  elasticities of substitution in utility and
  production functions.

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• A CE is ct,Nt,Kt1 solving

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Special Case d 1

• 100 depreciation ? It Kt1
• Guess Nt N constant
• Method of Undetermined Coefficient plug into
  equilibrium conditions and verify guess by
  solving for N, f1, and f2.

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• Result

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Predictions of Special Case

• Persistent Cycles in GDP
• Volatility of C and I
• MODEL DATA
• C,I procyclical C,I procyclical
• Var (ct) lt Var (yt) Var (ct) lt Var (yt)
• Var (It) lt Var (yt) Var (It) gt Var (yt) (X)
• Labor Market
• Average Productivity (yt/N) is procyclical,
• Real Wages wt zFN(Kt,N) procyclical

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• Real interest rate r is countercyclical.
• Problem No fluctuations in N!
• Can be resolved by d lt 1
• Substitution effect gt income effect
• Higher MPK magnifies productivity shock.
• No analytical solution. Need to use numerical
  methods.

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Form of Solution

• In each period t, the models state variables
  are
• Solution for each period t are functions of the
  models state variables given K0 , , and et

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Effect of One Time Productivity Shock
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Steps to Solving RBC Model

• (1) Solve for solutions of c(K,z), n(K,z),
  k(K,z)
• (2) Calibrate Parameters d 0.25, a 0.3, b
  0.99 (4 annual real interest rate), persistence
  r 0.8, ect.
• (3) Simulate Model to Generate Artificial Data
• (4) Compare Artificial Economy with Real Economy.

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Figure 10.3 Small shocks and large cycles
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GDP
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Consumption
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Investment
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Employment
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(No Transcript)
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Compare with BC Facts

• Explains persistent fluctuations in Y, C, I.
• C and I are procyclical, C is less volatile than
  Y, I more volatile than Y.
• N is procyclical but model still understates
  volatility.
• Labor productivity (Y/N) is procyclical (too
  much)
• Price Level is countercyclical (?)
• Correlation between N and productivity (and w) is
  close to one (too large).

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Figure 11.3 Average Labor Productivity with
Total Factor Productivity Shocks
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Table 11.1 Data Versus Predictions of the Real
Business Cycle Model with Productivity Shocks
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Hours-Wage Correlation
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Shortcomings

• Still not enough volatility in N. Need higher
  intertemporal substitution effect relative to
  income effect.
• N and w correlation too large.
• Money is neutral.

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Adding Government Spending Shocks to RBC Model

• Firms
• Households
• and
• Government BC

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• Market-Clearing
• Labor Nd Ns
• Goods yt Ct It Gt

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Temporary DG

• Supply Side Effects
• Increase in G ? Increase in T
• Small decrease in PDV of lifetime income
• Small shift of NS and Ys right
• Demand Side Effects
• Higher G ? shifts Yd right by DG/(1-MPC).
• Higher T ? small negative income effect
  (consumption smoothing) ? Yd left by
  MPCDT/(1-MPC).
• Since DT DG, Shift Yd DG
• Overall Shift Yd gt Shift Ys ? Increase Y and
  r ? lower C and I

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• Evidence
• (1) Procyclical G
• (2) Wartime government spending and Interest
  Rates
• (3) G and I

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The Growth Rate of U.S. Real Gross Domestic
Product since 1870
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Figure 4.5 Gross and net investment, 19292002
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Government Expenditures Investment
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RBC Model w/ Government Spending Shocks
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Hours-Wage Correlation
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Hours-Wage Correlation
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• Suggested Reading
• G. Hansen and R. Wright
• V. Li, Can Market-Clearing Models Explain U.S.
  Labor Market Fluctuations? Economic Review,
  Federal Reserve Bank of St. Louis (July 1999).

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RBC Debate Plosser vs Mankiw

• Interpretation of Productivity Shocks (Solow
  Residuals)
• Labor Hoarding
• Aggregate Demand Shocks affect
  Productivity
• Labor Supply Elasticity of Substitution
• Optimality of Business Cycles
• - Stabilization policy
• Cyclical behavior of prices and neutrality of
  money Shocks
• Are prices pro or counter-cyclical?
• Phillips Curve trade-off
• Internal vs External Consistency