BUSINESS CYCLES THEORY AND POLICY
MICROFUNDATION
Capital Accumulation: Intro to Macro
(Transformation: units of effective labor input)
Capital growth rate for each unit of effective labor (Solow Model equation):
In Stationary State ALL variables attain constant values, so no growth:
Capital is generated by savings and reduced by depreciation
Higher effective capital stock -> more output in units of effective labor input
In Steady State the effective capital stock is constant, while technological progress and
population increase with constant rates
Population growth higher-> more capital is needed for the ‘new ones’
Savings ratio higher -> each worker can use more capital for production
 Microfoundation needed! Policy changes can affect the endogenously determined
savings rates of optimizing individuals
MEASURING THE BUSINESS CYCLE
“Business cycles are a type of fluctuation found in the aggregate economic activity. A cycle
consists of expansions occurring at about the same time in many economic activities followed
by similarly general recessions, contractions and revivals which merge into the expansion
phase of the next cycle.”
Burns and Mitchell (1946)
 We can decompose a series of observations into trend and cyclical components:
 Extracting a linear time trend from the log of a series removes an exponential growth
trend with constant growth rate from original series:
Small cyclical components
Smooth growth trend
Kaldor stylized facts
Kaldor facts in the Solow Model
FACT 1: Growth of Y/L
FACT 2: Growth of K/L
FACT 3: Constant Interest on Capital
FACT 4: Stability of Y/K
FACT 5: Factor ratios remain approx. constant
 In Summary: Output per worker increases with the rate of technological progress
 Steady state level of output per worker depends
on savings rate, population growth and rate of
technological progress
RBC Stylized Facts
THE REAL BUSINESS CYCLE MODEL
The most standard RBC model is an extension of the Solow Model. There are three
differences however:
 Technological progress is stochastic
 Households choose how much to consume and to save (invest in capital kt)
 Households choose how much labor to supply
Together with factor inputs kt-1 and nt chosen by firms to maximize profits we have
optimizing agents under perfect competition. In period t, all agents know what has
happened up to and including period t. If they agents use this information optimally, they
are forming rational (mathematical) expectations about the future.
The goal is to match salient features of the business cycle using exogenous shifts in total
factor productivity (technology shocks).
There are three traded goods in a perfectly competitive economy:
 Output produced by firms used for consumption and investment
 Labor services supplied by households
 Capital services supplied by households
HOUSEHOLDS
 Households are endowed with an initial capital stock k-1
 Households are endowed with one unit of time. They can use this time to either
work or enjoy leisure
The Optimization Problem:
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##
Intuition of the household optimization:
Utility cost of further reducing
consumption today to finance
additional investment
Optimal capital supply equates
Discounted E(utility) benefit of
the additional consumption in
the future financed by higher
capital income
Intuition of the household consumption smoothing:
PROPAGATION
Intuition of the household labor supply:
^
TECHNOLOGY SCHOCKS AND PRODUCTION (FIRMS)
Stochastic Production
Alternative and Equivalent
 The central question of the model lies on the effect of the productivity shocks on the
remaining variables of the model.
Qualitative: How do productivity shocks affect the remaining variables (Propagation)
Quantitative: How strongly do prod. shocks affect the remaining var (Amplification)
If A=0
MARKET CLEARING AND THE COMPETITIVE EQUILIBRIUM
The market is cleared when:
 The amount of capital services supplied by households has to equal the amount of
capital services demanded by firms.
 The amount of labor services supplied by households has to equal the amount of
labor services demanded by firms.
 The amount of output supplied by firms has to equal the amount of output
demanded by households
An equilibrium is a situation in which:




households maximize,
firms maximize,
markets clear,
households form expectations rationally.
Summary: System of 5 nonlinear difference equations in 5 variables (yt, ct, kt, nt, zt)
I.
Capital market
II.
Labor market
Marginal Rate of Substitution
Marginal Rate of Productivity
III.
Resource constraint (constant returns to scale + perfect competition => zero profits πt = 0)
IV.
Production function
V.
Shock
SOLVING THE REAL BUSINESS CYCLE MODEL
The solution of the RBC Model is a policy function. This function maps the state variables kt1 and zt (pre-determined) into all other variables “resolving” approximately the expectations
in the system above.
Procedure:
1. Find the steady state y, c, n, k, i .
2. Set parameter values (perhaps calibrated, estimated,...).
3. Loglinearize around this steady state to obtain a system of linear difference
equations.
4. Solve the system of linear difference equations with the method of undetermined
coefficients.
5. Observe the effect of a single productivity shock—impulse response analysis
1) Steady State: all variables are constant
Equilibrium of the quarterly RBC – Model
Steady State equations
2) Setting parameter values
Productivity shocks zt are the cyclical
components of the log Solow residuals
log TPF estimated as an AR(1) process
zt= pzt-1 + ut
Factor share of capital
3) Log linearizing around the steady state
Linear system of equations in log deviations
I)
III)
Capital Market
Resource Constraint
II)
IV)
Labor Market
Production Function
4) Solving the linear system
-
To solve the syste we need policy funtions: control variables as function of state variables
The capital we chose today becomes a state variable next period
-> endogenous state variable
We don’t seek to explain our technology schock in the model but rather it´s repercusions
-> exogenous state variable
5) Analyzing the model
-
Consumption has a different shape: It´s maximum deviation is
when capital dev. and output dev. cross each other
We get rich suddenly, capital is constantly built and we
Invest some of our output. The built capital stock is going to provide us
with consumption even though technology shock dissipates
The New Keynesian Model
The Household Optimization Problem
S
Aggregate Price Dynamics:
Aggregate price level is given by:
Log-Linearization around a zero-inflation steady state (Π = 0, π = 1)
Inflation results because reoptimizing firms choose a price
which differs from the average price in the previous period
Optimal price setting
v
Log-Linearization and market clearing
New Keynesian Phillip’s and IS Curves
Real Marginal Costs
If firm a lowered the prices in t, demand will increase and
therefore the output will be larger than average output in the
economy. The marginal cost will increase.
If firm was able to adjust prices and increased them; demand
decreases, WPN increases, MC decreases.
Same logic for equation (11)
*NKPC
Plug this in the equation derived from (28) FOC:
Side Note: Under the assumption of
constant returns to scale (α = 0)
marginal cost is independent of the
level of production and, hence, it is
common across firms
Weighted average of current conditions
(mc and inflations) with the price
differential we would expect tomorrow.
We want to choose our optimal price change today as our monopolistic price setting would tell us to set it
today and we are going to weight that with the optimal price we would expect to choose tomorrow if we
were able to reset prices. The decision of tomorrow is embedded in today´s.
*IS Curve
Recall log lin Euler:
In the RBC model: a tech shock led us to wish more capital accumulation.
Now New Keynes: r leaves the household indifferent.
If agents´ impatience change, that changes the natural rate of interest that must exist, so agents (on average)
have no incentive to start accumulating assets/capital but rather stay indifferent and keep purchasing bonds.
_______________________________________________________________________________________
We later show, how the NK-Output-Gap is related to the liquidity preference and the money supply in the
economy. So far, we saw how the NKPC represents the supply side of the economy between and the pricing
decision of firms.
The IS curve represents the demand side of the model, hence the investment-saving decisions of agents.
Equilibrium under an interest rate rule
We want to analyze the conditions needed for an equilibrium. This is, what is necessary so
forward-looking variables like inflation and output oscillate back to their natural level after a
monetary policy shock.
Combining the non-policy block with the Taylor-type interest rate rule delivers
Policy seeks to contra rest the inflationary psychology in the economy i.e the self-fulfilling
effect of inflation.
If firms and consumers expect higher future inflation, then it can become a selffulfilling prophecy. If workers expect future inflation, they are more likely to bargain for
higher wages to compensate for the increased cost of living. If workers can successfully
bargain for higher wages, this will contribute towards inflation. Also, most consumers will
spend their money on a product immediately if they think its price is going to increase
shortly.
To pop the expected inflation bubble (decrease pressure on possible self-fulfilling inflation)
the key mechanism by which the models work depends on the coefficient on expected
inflation in the Taylor rule is to be greater than one. Thus, a rise in expected inflation leads
to a raise in the real interest rate. This demotivates spending, leading to a decrease in output
that eventually gives way to a decrease in inflation.
Suppose a contractionary monetary policy, so a positive monetary policy shock where
nominal i increases.
What happens with the real interest rate after the shock?
How does the money demand/supply react?
We use this to determine the change
in the money supply required to
bring about the desired change in
the interest rate.
The sign of the change in money
supply can be ambiguous.
In our model inflation expectations
(fisher effect) do not outweigh the
liquidity effect
Liquidity effect: increase in the money supply can reduce the real interest rate,
more money, i.e. more liquidity, tends to lower the price of money which is
equivalent to lowering the interest rate
= 0.6 – 0.15 = 0.45
Tech shock under Taylor Rule: The Central Bank tries to stimulate output
to close gap by lowering i. Here we don’t have this reaction. Employment
decreases in both examples (household enjoys leisure thanks to tech
shock) but here we have the added effect of a lower output level, hence
less labor needed.
Suboptimality:
 Optimal monetary policy
Implications of optimal monetary policy
 Simple policy rules (Welfare loss)
 Simple policy rules
Evaluation