RBC Model Simulation Results: Impulse Responses to Shocks - How do RBC model economies behave in response to shocks? - Good examples of this type of exercise: Heijdra Foundations of Modern Macroeconomics Ch. 15 Section 15.5 Romer Advanced Macroeconomics Ch. 5, Section 5.6 McCandless The ABCs of RBCs Ch. 6, Section 6.5 - The results in these sources are derived using the same type of method as Uhlig: start with a version of the Ramsey model, derive first-order conditions and the steady-state, create a log-linearized version of the model, find the law of motion for the model and then generate the impulse responses. - But the models are slightly more complicated versions of the Uhlig’s model. - What do the more typical first generation RBC models add? - Labour - Government 1
Transcript
RBC Model Simulation Results: Impulse Responses to Shocks
- How do RBC model economies behave in response to shocks?
- Good examples of this type of exercise:
Heijdra Foundations of Modern Macroeconomics Ch. 15 Section 15.5Romer Advanced Macroeconomics Ch. 5, Section 5.6 McCandless The ABCs of RBCs Ch. 6, Section 6.5
- The results in these sources are derived using the same type of method as Uhlig:
start with a version of the Ramsey model, derive first-order conditions and the steady-state, create a log-linearized version of the model, find the
law of motion for the model and then generate the impulse responses.
- But the models are slightly more complicated versions of the Uhlig’s model.
- What do the more typical first generation RBC models add?
- Labour - Government
- Notation:
Y = output (GDP); Z = productivity; K = capital;
N = employment; w = wages; r = interest rate
I = investment; C = consumption; l = leisure
FK = marginal product of capital = depreciation rate
FN = marginal product of labour
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(1) Labour and the Labour-Leisure Choice:
Production:Yt= F(Kt,Nt) = Zt K t
a N t1−a K= capital, N=labour,
Z= productivity parameter
Utility: is often additively separable in consumption (c) and leisure (l). e.g. with log-utility:
E [∑t=0
∞
β t U (c t , lt )]=E ¿
Each person must satisfy a time constraint: 1=lt+Nt
- One-decision maker problem is then much like the version of the Ramsey model with labour supply added (see Overhead Set 8)
- Decentralized version: Labour demand and labour supply conditions linking behavior to wage rates (w).
Labour demand: value of marginal product equals the wage
- So ptFN = wt with FN≡∂F/∂N and typically output price=p=1.
- Wage changes: driven by changes in productivity (Fn).
Labour supply: via the household’s intertemporal optimization problem.
- See MaCurdy (1981) on dynamic labour supply and intertemporal substitution (Overhead Set 3)
(Heijdra and Romer both set the model up with competitive markets)
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(2) Government:
- both Heijdra and Romer add government spending and taxation in a limited way.
- Government spending of Gt in period t, financed by lump-sum taxes of Tt (so the government budget has: Gt=Tt in each
period).
(lump-sum taxes? No disincentive effects)
- Government spending is assumed to be subject to shocks (just like the productivity shocks in Uhlig’s model)
- follow autoregressive form, i.e. show some persistence.
- Government spending does nothing of value in those models: ‘G’ does not build capital (so its not infrastructure), it
doesn’t provide any goods or services to the household nor does it pay any transfers to the household.
(odd view of government? odd for a general equilibrium model!)
- So a rise in ‘G’ raises ‘T’ and leaves households poorer.
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- Behavior in these simulation models follows from our earlier models
Households: intertemporal optimization
- Euler eqn. determines the consumption path slope between periods
- Wealth and financial resources determine level of consumption along the path (assume consumption is a normal good).
- Consumption determines saving which finances capital investment.
- Labour supply: marginal condition governing division of time between labour and leisure in each period.
- leisure is a normal good: more leisure, less labour supply if more wealthy (i.e. if more lifetime resources)
- intertemporal substitution (see MaCurdy; Lucas and Rapping): work more in periods when the wage is high.
Firms: - produce and supply output.
- demand capital and labour to produce output.e.g. higher productivity boosts input demand; higher input prices tend to decrease input demand.
- link between productivity of capital (FK≡∂F/∂K) and the interest rate (r), i.e. FK-=r (r is the return on savings and investment)
Capital accumulation: - reflects consumption and savings decisions of households and
investment / capital demand of businesses.
Response to shocks in these models:“In general, we can think of the effects of shocks as working through
wealth and intertemporal substitution effects” Romer p. 214
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- What shocks are typically examined?
- Technology shocks the most common in the literature.- “technology shock” or “productivity shock”: same thing – a shift in
the production function.
- Also have government spending shocks in the Romer and Heijdra.
- Shocks could be permanent or temporary. If temporary they may also differ in their persistence.
- Specific discussions of what is happening in response to a shock:
- Romer pp. 211-215 a good discussion of what is happening (technology shock with persistence; also has a government spending
shock).
- Heijdra pp. 523-531 looks at three types of technology shocks (temporary; permanent and a “realistic” shock i.e. persistent with parameters
chosen via Solow-growth accounting exercise. Also a good discussion of the intuition behind the observed patterns.
- Three of these shocks are considered in the notes:
- Useful for illustrating the workings of these models.
- Economy starts in the steady state with Euler equation and the dynamic resource constraint satisfied.
- A 1% (.01) rise in the technology parameter occurs in period 0; it then falls back to its initial value in period 1 and stays there
(no persistence: unlike Uhlig’s example)
- No long-run effect: the economy will eventually return to its original steady state.
- Diagrams next page (from Heijdra Ch.15):
- all variables are deviations from the original steady state (s.s.)
- time since the shock is on the horizontal axis.
(these diagrams are like those in Overhead Set 8 showing how y, c, i and k evolve after a shock except that here variables are measured as deviations from their initial
steady state values – so their values are 0 when at the initial steady state values)
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Period 0: Initial response to the Temporary Shock
- Capital is fixed in period 0 (so in the capital panel the deviation from its s.s. value is 0 in period 0), productivity is higher (deviation is .01 in
period 0): economy is wealthier (it’s ability to produce output is
higher).
- Labour is more productive so competitive wages will be higher too (real wage is .005 in period 0 in the real wage panel)
- intertemporal substitution suggests that labour supply rises due to the high wage (employment deviation is .015 in period 0).
(substitution overpowers a small offsetting wealth effect that boosts demand for leisure)
- Higher productivity, more labour supply and fixed K mean actual output rises in period 0 (.02 higher in period 0 than pre-shock)
- Households are slightly wealthier due to the shock: consumption rises (to .0014). This effect is small since the shock lasts only one period: most of the rise in output is smoothed.
- Saving and investing rises to just under .01 in period 0. This shifts some of the extra output to the future by creating more K (i.e. this is how smoothing is done). Extra labour supply contributes to this.
- Price variables:
- Wage is higher by .005 in period 0 (labour is more productive which generates more labour demand)
- Return to investment i.e. interest rate (r) is higher in period 0 (.05)(K is more productive due to shock and the extra labour
supply)
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- it will fall sharply next period: this affects C now via Euler.
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Period 1 and later: Transition to the steady state after a temporary shock
- Shock disappears in period 1 (Productivity deviation back to 0); K stock has risen vs. period 0 (via extra saving and investment in period 0).
- Labour productivity falls back toward its original value (not all the way since K is higher in period 1 than pre-shock); this means wages fall toward their pre-shock level.
- Employment (N)? Wages are slightly above old s.s. level (work more!) but wealth effect says work less.
- In the diagram, the wealth effect dominates the substitution effect leaving labour supply & employment slightly lower than in the initial steady state.
- How will consumption (C) and capital (K) evolve?
- Technically: the economy is on it’s old, pre-shock saddlepath in period 1. i.e. the productivity shock has disappeared so the steady state is same
as pre-shock but K and C are higher than in the steady state. The economy will gradually move down the saddlepath to the s.s.(↓C, ↓K).
(like pts. in quadrant II of the earlier Ramsey phase diagram of Ramsey notes)
- Consumption response over time reflects interest rates (via Euler equation)- K is higher in period 1 than pre-shock and so FK (and the interest
rate) is lower than its s.s. level since higher K means lower FK. The Euler equation suggests that the time path of C is
downward sloping.
- Period 1 savings & investment is below s.s. level due to low return on K.
- K starts to fall back towards its s.s. value, as K falls FK and the interest rate rise back toward their s.s. levels.
- The time path of C flattens out as FK (r) rises back toward its s.s. level.
- Wages are gradually falling back toward their steady state level due to the effect of falling K on labour productivity.
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- Summary of Temporary Shock Transitional effects:
Period 0 Period 1 and Later
Z 1% higher Back to initial value in period 1 and later
K Fixed in period 0 K rises in period 1, then extra K depreciates
C Rises (wealth effect) Gradual decline to s.s. value
I Rises (C smoothing) Below s.s. in period 1 (low FK), return to s.s.
Y Rises (shock, ↑N) Returns to close to s.s. level in period 1 (slightly higher K offset by lower N)
N Intertemporal subst. Falls below s.s. level (slightly) in period 1 –overpowers wealth effect if wealth effect dominates. Returns to s.s.
slowly.
w Rises (productivity Falls once shock reverses but higher than shock) pre-shock as long as K above steady state value.
r Rises due to shock Shock over, r is lower than before since K is above s.s. level. r rises gradually
as K shrinks.
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(2) Permanent Positive Productivity shock: Heijdra p. 528-530
- Productivity rises permanently by .01: this will change the model’s steady state.
How? C will be higher and employment (N) lower due to wealth effects.
Expect K to be higher given higher productivity (via FK=+ the s.s. condition from Euler)
Y higher (more K, higher productivity overpower lower N).
Higher w (via more K and higher productivity)
Higher r (FK) due to higher productivity during the transition but back to level consistent with FK=+ in the s.s. (via more K)
More I in s.s. to support the new higher s.s. level of K.
(similar to the Ramsey model phase diagram with a productivity shock)
- Impact (Period 0) and transition to steady state?
- Generally: wealth effects will be larger than with the temporary shock (encouraging more C and less N);
intertemporal substitution effects weaker than the temporary shock (since productivity rise and resulting wage rises are not limited to period 0).
- Impact: productivity higher, K same, labour productivity and w higher.
- Simulation: N rises slightly (substitution>wealth effect).
- Output and r are rising (N and productivity higher).
- C rises immediately: wealth effect overpowers substitution effect - note: less need to save in order to smooth than with a temporary shock since it is a permanent rise in productivity.
- I: rises too (higher return; some of extra output went to C, some to I)
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- Transition to the new steady state:
- Move to the new higher C, higher K steady state.
- Euler equation: FK and r are above old s.s. level (level where C is constant over time)
- so C is rising over time, fuelled by extra saving, I and so growing K.
- as K rises, FK and r fall so C rise in C path becomes flatter as it approaches new s.s. level.
- N falls eventually below old s.s. level (via wealth effects): slow to take effect (must build up K-wealth to allow this)
- Initially w and r rose due to the productivity shock. - w continues to rise as extra K boosts labour
productivity and wealth effects limit N supply.
- r falls due to diminishing returns as K grows to new higher s.s.
- I rises and then declines: reflects higher productivity initially then diminishing returns as K grows.
- K rises incrementally due to extra I in response to higher productivity but plateaus due to diminishing returns.
(idea that the K rise is workers buying more leisure due to greater wealth)
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Persistent technology shock:
- This is like the shock in Uhlig’s simple model.
- Romer pp. 211-215 has a good discussion of the response. McCandless does a simulation of a case like this. Results differ slightly due to how the
productivity term enters the production function (bigger effect in McCandless than Romer). - Economy starts in a steady state.
- 1% rise in the technology parameter above its steady state value.
i.e. a temporary rise in t where the technology parameter evolves according to:
Zt=ψ Z t−1+ t so there is persistence.
with 0<<1 the shock eventually dies out.
- The following diagram shows the response of a typical RBC model to such a shock (lambda is the productivity shock – Z, L is labour):
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- the diagram is derived from the model in McCandless’ Ch. 6 (Hansen’s model: see also McCandless’ Figure 8.1 in next notes)
(Romer’s Figure 5.2-5.4 show responses to similar model to a similar shock but with a slightly different production function)
- horizontal axis: time period (shock occurs in period 0), parameters of the model are chosen consistent with time periods being quarters.
- vertical axis: deviation of the indicated variable from its steady state value.
- labels? L=N is labour, lambda=Z is productivity.
- Blue line (lambda) is the shock: .01 in period 0 then gradually dying out.
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- Romer p.214 describes what drives the response:“A positive technology shock implies that the economy will be
more productive for a while. This increase in productivity means that households’ lifetime wealth is greater, which acts to increase their consumption and reduce their labor supply. But there are also two reasons for them to shift labor supply from the future to the present and to save more. First, the productivity increases will dissipate over time, so that this is an especially appealing time to produce. Second, the capital stock is low relative to technology, so the marginal product of capital is especially high.”
- Immediate response:- Rise in productivity raises r immediately (r=FK-).
- K is more productive, payoff to saving and investment is higher:so more K will be created (can think of some of this as C
smoothing, i.e. the shock and additional wealth it creates fades as time passes – so create extra K now to transfer some of this
gain to the future).
- L=N will also rise immediately (labour is also more productive, wages will be higher – intertemporal substitution likely
dominates wealth effects); use extra L to build K.
- Y is higher (more N and higher productivity)
- C is higher as well (why? higher Y, higher lifetime wealth); limited: FK is very high too (as high as it will be)
- r is higher: productivity shock increases K productivity (no wage in McCandless picture but expect it to rise too since labour
is more productive due to the shock, see Romer Fig. 5.4).
- Later periods:- the shock eventually dies out so move back toward the initial s.s.
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- Y: high initially, declines as the shock fades away and L=N declines due to weaker intertemporal substitution effect (decline slowed initially by rising K). Moves eventually toward s.s.
- L: labour supply and employment; rises immediately then declines.- pattern reflects wealth and intertemporal substitution effects.
- productivity and wages are temporarily higher after shock.- work more while shock is still strong.- partly just wage vs. value of time; partly because extra
wages can be saved at high r. - long-run? shock increases lifetime wealth – lower labour
supply in the long-run.
- K, C and r are quite intertwined. - K rising initially (smoothing the shock; also high initial r). - as shock fades, rising K sets off diminishing returns, r
falls (eventually below s.s. level) and K declines as well.
- C response? - lifetime wealth higher due to shock so tendency to raise C
now.- also want to save to transfer some of shock-induced wealth to
future (create new K): smoothing.- r is high after shock encouraging a rise in saving.- Over time Euler equation relates r to C path:
- r high but falling early in transition, C rising.- r low but rising later in period, C falling.
- Euler equation gives the gradual response in C and K to the expected pattern in r (r high now but will return to steady state value)
- Model gives a boom with a positive persistent productivity shock.
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Evaluation and Moments: How Successful are RBC Models?
- Romer pp. 218-220 is good on this. Heijdra p. 531-533.
- How do you judge the validity and usefulness of RBC models?
- The method suggests making comparisons between simulated data based on the models and de-trended (filtered) real world data.
- The basic RBC model assumes that technological shocks are the source of business cycles.
- To compare performance of model to actual, real-world economies you need to impose the pattern of real world technological process on the model.
- How? based on “growth accounting” and measurement of the Solow
residual
i.e. if production is Cobb-Douglas Y=AKN1- and you can measure Y, K , N and then you can infer the value of the productivity
term (A) from data on a specific economy (as in Assignment 3).
( lnA = lnY – lnK – (1-) lnN )
Then can find the stochastic process that best fits the resulting pattern of ‘A’’s
e.g. in Romer AR parameter () .95 and std. deviation of quarterly is 1.1 percent)
- You can then input this into the RBC model and generate time paths of all variables.
- The results (variances and correlations between endogenous variables) can then be compared to data on the same variables from actual,
real world data.
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- Results of these comparisons (see notes on specific sources).
- Romer, Table 5.4:US data RBC model
Y: Output (std. deviation) 1.92 1.30
Ratio of the std. deviation of the variable to the std. deviation of outputConsumption 0.45 0.31Employment 0.96 0.49Investment 2.78 3.15
Correlation of N andY/N -0.14 0.93
- Output fluctuations:-somewhat smaller in the RBC model than actual data.
- order of magnitude is however similar (see also Heijdra next page)
- Prescott (1986) interpretation? fluctuations of the kind seen in actual data are predicted by a competitive, GE model.
- Consumption volatility relative to output volatility is similar in actual data and RBC models.
- in both actual and RBC model consumption is considerably less volatile than output (smoothing!)
- Investment is much more volatile than consumption or output in both the actual data and the RBC model.
- Employment in the RBC model is far less volatile than actual employment.
- Labour Productivity (Y/N): uncorrelated in US data but positively correlated in the model.
