RBC MODEL Flashcards
In RBC, what are the exogenous shocks?
Exogenous shocks to technology (solow residual)
What are we hoping to discover with RBC analysis?
Whether, given the observed fluctuations in technology, the RBC model can reproduce key business cycle facts for our key variables.
What are these technology shocks?
Hard to interpret -VE shocks. But take broader interpretation e.g. regulation, taxes, weather - anything that affects output for given level of inputs = supply-side shocks.
Household assumptions
one infinitely lived representative household
Max utility
3 choice variables for households
Consumption
Labour supply
savings
Is household problem dynamic / static?
Dynamic - savings today affect consumption tomorrow.
Assumptions about firms
one representative firm selling 1 good
max profits
Assumption about firm’s technology
Production technology with productivity evolving exogenously and stochastically.
How do we make the firm problem static?
assume firms do NOT own capital - they rent it from households.
Assumption about markets
all markets perfectly competitive
Production function
Yt = At F(Kt, ht)
Returns to scale for pf =
CRS
What’s the price of output?
We normalise it to 1.
Firm’s problem
Max profit = AtF(Kt, ht) - wtht - rtKt
2 FOCs for firms
At F’h(Kt, ht) = wt
At F’k(Kt, ht) = rt
Household utility depends on…
U(Ct) + V(lt) - consumption and leisure
Time constraint for households
lt = 1 - ht
Leisure = time spent not working
Normalise time endowment to 1
Dynamic BC for households
Kt+1 - (1-delta)Kt = wtht + rtKt - Ct
Investment = sources of income - consumption
What do household’s maximise? subject to what?
Expected discounted sum of all future instantaneous utilities.
S.t. series of BC - one for each time period.
Rational expectations =
1970s - households predict the future in a rational way. Understand how economy works and use all available info = correct on average.
What assumption does RBC make about household expectations?
Stronger than RE - we assume perfect foresight = we can get rid of the expectations operator.
Lagrangian for households
sum t=0 to infinity B^t [U(Ct) + V(1-ht)] - lamba t [Kt-1 - (1-delta)Kt - wtht - rtKt + Ct]
Why can we not find a solution to household max for most utility forms?
- endogenous variables related in a non-linear way
2. Dynamic system - variables related over time.
2 household optimality conditions
- V’(1-ht) = U’(Ct) wt - optimal labour-leisure
2. U’(Ct) = B(1 + rt+1 - delta) U’(Ct+1) - Euler’s equation
explain the optimal labour-leisure FOC
V'(1-ht) = marginal utility cost of working an extra hour in terms of leisure. U'(Ct)wt = MU gain of working an extra hour in terms of consumption.
What’s the relationship between wt and ht?
Ambiguous - depends on utility form and how calibrate model. SE means higher Wt = higher ht, but IE means higher Wt lower ht.
explain Euler’s equation FOC
Shows optimal consumption-savings decision over time.
U’(Ct) = MU cost of giving up 1 unit of C today
B(1 + rt+1 - delta) U’(Ct+1) = MU gain as can invest in 1 extra unit of K, get return, then consume but discounted by B as in future.
BC of households also gives… IF we assume
The supply of capital
Kt+1 = (1-d)Kt + rtKt + wtht - Ct
assume only way for households to save is to buy capital & rent to firms.
2 Market clearing conditions
htd = ht s Ktd = Kts
Combine our 5 household / firm FOCs to get 3 =
Get rid of prices wt and rt.
- V’(1-ht) = U’(Ct) At Fh(Kt, ht)
- U’(Ct) = B(1 + At+1 Fk(Kt+1, ht+1) - d) U’(Ct+1)
- Kt+1 = (1-d)Kt + AtF(Kt, ht) - Ct
Why can we replace rtKt + wtht with the production fucntion?
Because markets are perfectly competitive = zero profit so all income = paid as remuneration to labour and capital.
How does a computer solve the model? 4 steps.
We want a time series for all endogenous variables in terms of exogenous.
- Guess and verify method to find policy functions:Ct = C(Kt, At) ht = h(Kt, At)
- Sub into K accumulation to get time series for Kt
- Sub Kt into policy functions = time series for Ct and ht
- Then find series for Yt, wt, rt
2 advantages of quantitative analysis of RBC
- can estimate magnitudes as well as directions of effect = useful for policy
- calculate effects that are ambiguous e.g. wt and ht
calibration =
assigning values to parameters so that the model is consistent with some empirical facts.
Ideal calibration targets are…
- Informative on parameters
2. Not about model’s prediction of interest
4 common practices for calibration
- direct empirical measures
- LR features
- calibrate some variables to prediction of interest, then evaluate others
- Try some values & explore robustness
Functional form for production fucntion
F(kt, ht) = Kt^alpha ht^1-alpha
Function form for utility function
U(Ct) + V(1-ht) = log(Ct) + theta log(1-ht)
6 parameters of our model
alpha, theta, delta, B, [At]t=0 to infinity
Theta in the utility function measures
relative taste for leisure
2 parameters we use direct measures for and what are they?
delta - estimate efficiency loss of K = 0.1 /year
alpha = 1/3 = capital share of income
What 2 parameters do we use LR features to calibrate?
B = 1 / 1 + r bar where r bar = 0.016 / quarter is average return to K. theta = h bar = 0.2 - proportion of day spent working.
How do we calibrate At?
Postulate a stochastic process
LogAt = plogAt-1 + €t estimate from time series reg. Or At from data on solow residual.
2 ways we can evaluate RBC model performance
- feed time series for At from the data into the model & compare time series of our variables to data
- Simulate model using stochastic process for At & compare
Output model vs data
Good fit for volatility and timing, but not surprising given we’re using At from data and output is a direct function of At.
Consumption model vs data
Direction of changes mostly good
But model underpredicts volatility
Investment model vs data
Volatility well predicted
But model moves too early
Labour input model vs data
Bad fit in terms of timing, and underpredicts volatility.
So in terms of volatility how well does RBC do?
Good for high volatility of It
Underpredicts Ct and ht = not enough amplification of shock.
So in terms of persistence how well does RBC do?
weaker autocorrelation in model = not enough propagation of shock
So in terms of co-movement how well does RBC do?
Correctly predicts directions for all but rt
But overstates magnitude of co-movement
Mechanisms through which a positive shock t At affects our key variables.
For given factor supplies, MPK and MPL rise –> rt and wt rise. Higher income for household = Ct rises, but some saved = Kt+1 rises too. Shock amplified by higher wt and rt causing higher ht (SE>IE). Shock propagates over time as higher Kt+1 –> higher MPLt+1 etc.
3 criticisms of RBC
- unclear what tech shocks rly are
- Failure to explain labour market volatility and in model no invol U which is mainly why we care about BC
- extreme policy implications - implies no policy needed.
Why is no stabilisation policy needed according to RBC?
Fluctuations are efficient responses by households and firms. Agents choose to work less in a recession - no invol U. Perfect comp, perfect info so market = pareto efficient by first welfare theorem.
What is the Classical Dichotomy?
real and nominal sides of economy completely separate. We describe equilibrium all in real terms - w and r are really w/p and r/p with p=1 so they’re in terms of output.
Money =
an asset accepted in payment for goods and services.
Money demand according to QTM.
Mtd V = Pt Yt
V = velocity of money = how many times a unit is used for a transaction each quarter. exogenous and constant.
PtYt = nominal GDP.
Money supply =
Set exogenously by CB: Mts = Mt bar
Equilibrium QTM
Mt bar V = Pt Yt
LHS all exogenous, Yt determined by RBS = can find Pt endogenously.
What does QTM imply?
Exogenous increase in Ms –> 1-1 increase in price level.
gtM = Pi t + gty
Money is neutral - change in MS = no effect on GDP, only causes inflation.
Does money neutrality hold empirically?
Yes 1-1 relation in LR.
But our model is SR, and it does NOT hold in SR - money has real effects.
How do we determine the SR effect of money policy?
Vector Auto-regression (VAR) models
Zt = c + A1Zt-1 + … + €t
Zt = vector of real GDP, inflation, indicator of MS.
