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:

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.

E [∑t=0

∞

β t U (c t , lt )]=E ¿

Each person must satisfy a time constraint: 1=lt+Nt

- Planner’s problem is then much like the version of the Ramsey model with labour supply added (see earlier notes).

(McCandless solves the model in this form)

- 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.

(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 models follows from our earlier models

Households: intertemporal optimization

- Euler equation will determine slope of consumption path;

- Wealth and financial resources determine level of consumption along the path (assume consumption is a normal good).

- Consumption determines savings which finance investment in capital.

- 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 stories (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

Capital accumulation: - reflects consumption and savings decisions of households and

investment / capital demand of businesses.

Response to shocks in these models:

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“In general, we can think of the effects of shocks as working through wealth and intertemporal substitution effects” Romer p.

214

- 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.

- McCandless pp. 120-124 (technology shock with persistence in two different models)

- Three of these shocks are considered in the notes:

- Temporary technology shock

- Permanent technology shock

- Temporary but persistent technology shock.

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(1) Temporary (one period) positive productivity shock:

- Heijdra p. 525-527, Fig. 15.7 (handout)

- 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.

- Handout diagram:

- all variables are deviations from the original steady state (s.s.)

- time since the shock is on the horizontal axis.

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Period 0: Initial response to the Temporary Shock

- Capital is fixed in period 0, productivity is higher: economy is wealthier (ability to produce output is higher).

- Labour is more productive so competitive wages will be higher too.

- intertemporal substitution suggests that labour supply rises.

(this 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.

- Households are slightly wealthier due to the shock: consumption rises(effects is small since the shock is only one period)

- Saving and investing shifts some of the extra output to the future by creating more K (i.e. smoothing). Extra labour supply contributes to this.

- Price variables:

- Wage is higher in period 0 (labour is more productive which generates more labour demand)

- Interest rate (r) and rental price of K are higher.

(K is more productive due to shock and the extra labour supply so FK is higher, boosting K demand recall that FK-

r )

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Transition to the steady state after a temporary shock (period 1 and after):

- Shock disappears in period 1; K stock has risen vs. period 0 (via extra saving and investment in period 0).

- Labour productivity has fallen back toward its original value (not quite as low since K is higher in period 1); wages fall toward pre-shock

level.

- Employment? Wages are slightly above old s.s. level (work more!) but wealth effect says work less.

- In Heijdra’s diagram, the wealth effect dominates the substitution effect leaving labour supply & employment lower than in the initial steady state.

- How will consumption (C) and capital (K) evolve?

- Technically: the economy is on the old, pre-shock saddlepath in period 1.

i.e. higher K and C than the steady state, then gradually move downthe saddlepath to the steady state (↓C, ↓K).

- 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. The Euler equation suggests

that the time path of C is downward sloping.

- Savings and investment is below s.s. level due to low return on K.

- K falls 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) rise back toward their s.s. levels.

- 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: this will change the model’s steady state.

How? Expect C to be higher and 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 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 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)

(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. McCandless p. 111-112

- 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=ZKN1- and you can measure Y, K , N and then you can infer the value of the productivity

term (Z) from data on a specific economy (as in Assignment 3).

( lnZ = lnY – lnK – (1-) lnN )

Then can find the stochastic process that best fits the resulting pattern of ‘Z’’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.

- Prescott (1986) interpretation? fluctuations of the kind seen in actual data are predicted by a competitive, Walrasian 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.

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Heijdra Table 15.5 p. 533: S.D. – std. deviation, ‘corr’ – correlation.

US Actual RBC I RBC IIS.D. corr(X,Y) S.D. corr(X,Y) S.D. corr(X,Y)

Y 1.76 1.0 1.35 1.0 1.76 1.0C 1.29 .85 0.42 .89 0.51 .87I 8.60 .92 4.24 .99 5.71 .99K 0.63 .04 0.36 .06 0.47 .05N 1.66 .76 0.70 .98 1.35 .98Y/N 1.18 .42 0.68 .98 0.50 .87

- He has two versions of the RBC model. Discuss RBC I here.

- RBC I is much like the model in Romer and the model in the impulse response exercises.

- RBC I successes:

- output variability lower but not too different from actual.

- I is more volatile and C is less volatile than Y.

- correlations between Y and C, I and K are close.

- RBC I problems:

- employment variability: - actual data Y and N similar volatility- RBC I: much less variation in L

- productivity puzzle: high correlations between Y/N and output in RBC models (implies pro-cyclical wages). The variables are

only more weakly related in the actual data.

- propagation problem: tendency for output responses to mimic shock.

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- Results can indicate successes and weaknesses of the approach.

- suggest areas where model must be adapted to better approximate reality.

(RBC II is an attempt to do this, i.e. to address employment volatility)

- Romer on evaluation pp. 219-220): is it a reasonable exercise?

- models are very simplified so expect them to miss.

- isn’t the model constructed to fit the data? e.g. choice of parameter values.

- is matching moments really a good test?

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- Some reactions by proponents of RBCs to the results:

Heijdra: “Given the extremely simple structure … of the model the match between actual and model-generated

moments is quite impressive”

Prescott (1986) “Theory Ahead of Business Cycle Measurement”Federal Reserve Bank of Minneapolis Quarterly Review.

“The match between theory and observation is excellent but far from perfect”

“We ask whether these artificial economies display fluctuations with statistical properties similar to those which

the American economy has displayed... They do.”

Prescott (2004) views the models as a success:

“conditional on labour supply elasticity close to 3, TFP shocks are the major contributor to fluctuations in the

period 1954-1981 in the US” p. 383 (TFP – productivity)

3? 1% rise in wages raises labour supply by 3% (people must be very responsive to wages)

- he also notes that this finding seems to be robust to a number of extensions.

“We learned that business cycle fluctuations are the optimal response to real shocks. The cost of a bad shock

cannot be avoided and policies that attempt to do so will be counterproductive ...” Prescott p. 389

- So supporters think the exercise says that Walrasian, competitive models with shocks can give results like actual cycles and that

these results have big implications for policy.

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Basic RBC Model: Conclusions

- Romer pp. 226-229.

- RBC models: extend Ramsey model to allow for fluctuations

- Microfoundations: - some logical appeal– link to microeconomics, less ad hoc. - free from the Lucas critique

- Can generate cycles in a competitive general equilibrium framework. - implications: do you really need approaches that treat recessions as

coordination or market failures?

- Welfare theorems: suggest that the outcomes in such models are efficient. Does this mean stabilization policy is undesirable?

Some Problems with basic RBC models:

- They do not match some key features of observed cycles.

- see section on moments e.g. employment too stable.

- Technology shocks are a key source of fluctuations in these models.

- shocks of about 1% per quarter could give reasonable sized cycles.

- examples of actual shocks of this type: hard to pinpoint.

- Are positive shocks advances in technological or organizational knowledge?

- What are negative shocks? forgetting technology? Maybe organizational, legal or regulatory changes

- Inability to identify such shocks raises questions regarding their existence.

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- Microeconomic foundations: do basic RBC use the right ones?

- evidence that many markets depart from Walrasian competitive models without imperfections (labour markets, credit markets)

- labour literature: small intertemporal elasticity of labour supply.

- this is a main source of employment variation in RBC models.(see Prescott quote: elasticity of 3.0)

- limited ability to explain such variation with realistic elasticities.

- Alternative microfoundations?

- different models of how people interact economically.

- drop the representative agent assumption: many different people.

- imperfect competition models of price and wage setting

- search and matching models of labour markets.

- models with information imperfections in financial markets.

- In RBC models dynamics of output follow shock dynamics quite closely.

- can get realistic cycles only if assume shocks that look like the cycles.

- Empirical evidence that demand shocks (including nominal or monetary shocks) are associated with business cycles.

- not consistent with basic RBC models.

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Extensions:

- There have been many extensions to the basic RBC model. - Motivated by shortcomings of the original models. - Computational issues: place limits on what extensions are workable.

- Indivisible labour - Workers either have 0 hours or a some standard number of weekly hours.

- by making jobs have only standard hours get more employment adjustment.

- this is done in the RBC II model in the table from Heijdra (see p.18)

- Assumptions aimed at giving more complex dynamics.- Add adjustment costs for capital or adjustment costs for employment. - Preferences with habit formation.

- Additional sources of shocks:

- Government as a source of shocks:

- Government spending (G) shocks.

- Taxes (output tax is like a negative productivity shock: reduces share of Y going to households and businesses)

- Regulatory shocks: like productivity shocks – changes in rules that make business easier or harder.

- Romer: adding G shocks can reduce the correlation of Y/L and L. Heijdra p. 538 notes this too (G shocks dampen supply

shifts) – less wage response to productivity shocks.

- Financial systems with confidence or risk shock (inspired by 2008 financial crisis)

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- Money (with money growth shocks)

- Large literature suggests that monetary shocks have real effects.

- Need to introduce money into the basic model with some microfoundations story justifying its presence.

- Changes in money supply can then be an additional source of shocks.

- Rigid wages and/or prices: a more radical change – no longer Walrasian.

- typically assumes price setting e.g. monopolistic competition.

- some kind of rigidity in price and/or wage setting assumed.

- this is at the root of New Keynesian DSGE models.

- important if monetary shocks are going to have real effects.

- Some versions are working with a quite different labour market model.

- use a matching model of a labour market in place of the competitive labour market.

- Heterogenous households in place of identical households.

e.g. credit constrained households and households who can borrow and lend subject to an intertemporal budget constraint.

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Macroeconomics in 2018:

- Most research in macroeconomic theory adopts a DSGE model approach.

- Ph.D. programs tend to teach RBC and its DSGE extensions.

- Forecasting, day-to-day discussions of the macroeconomy, macro policy:

- older style macroeconomics is still influential.

- sometimes a mix of the two approaches is used, e.g. central banks often have old-style forecasting models and DSGE forecasting

models.

- Heated disagreements over right way to proceed.

Some supporters of the DSGE approach:

Christiano, Eichenbaum and Trabandt (2018) “On DSGE Models”http://faculty.wcas.northwestern.edu/~lchrist/research/JEP_2017/DSGE_final.pdf

R. Reis (2017) “Is Something Really Wrong with Macroeconomics?”https://personal.lse.ac.uk/reisr/papers/17-wrong.pdf

Prominent Critics:

J. Stiglitz (2018) “Where Modern Macroeconomics went Wrong” https://www.ineteconomics.org/research/research-papers/where-modern-

macroeconomics-went-wrong

P. Romer (2016) “The Trouble with Macroeconomics”https://paulromer.net/wp-content/uploads/2016/09/WP-Trouble.pdf

A more balanced view from another big name macroeconomist:

O. Blanchard (2016) ‘Do DSGE Models have a Future?” https://piie.com/system/files/documents/pb16-11.pdf

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