Real Business Cycle Theory

Question 1

RBC/Walrasian theory

Model assumptions (5)

Answer 1

Competitive markets
Flexible prices
No externalities
No asymmetric info
No missing markets (imperfections/frictions)

Question 2

RBC: So what drives these shocks (2)

B) What doesn’t influence the shocks

Answer 2

Tech shocks mainly
Gov spending shocks (less tho)

B) money! Since REAL, no money, and even if there was, since flexible prices mean still no real changes. (A criticism of this approach)

Question 3

Baseline RBC model setup:

Closed economy with price-taking firms and price-taking households (since competitive markets assumption)

Households live forever (forward looking)

Question 4

CD function for RBC firms (simple)

Answer 4

Yt = Kt to the α (AtLt) to the 1-α

(Firms use K,L and A, tech is labour-augmenting)

Important later as tech increases productivity, increasing wage and thus labour hours!

Question 5

Capital accumulation/resource constraint (Kt+1)

(Hint: capital in next period = …)

B) how to simplify using output equation

Answer 5

Kt+1 = Kt + It - δKt
K next period = initial K + investment - depreciation

B) output equation Y=C+I+G (rearrange to I)
= Kt + [Yt - Ct - Gt] - δKt

Question 6

Given competitive market assumption, labour and capital are paid their marginal products.

Real wage (w) expression

Return to capital i.e real interest rate (r) expression

Answer 6

Differentiate the CD function with respect to labour to get MPL (real wage) which is:

wt = (1-a)K to the a (AtLt) to the -a At

Find MPK, then finally subtract depreciation for return to capital!

rt = a(AtLt/Kt) to the 1-a - δ

Question 7

So that was firms.

Now households: what do they aim to do

Answer 7

Maximise LIFETIME utility

Question 8

Household utility expression pg 5

Answer 8

U = Σ e to the -pt u(ct, 1-Lt) Nt/H

-p: discount rate (higher p=more impatient)

1 - Lt is leisure (so Lt is labour hours)
N is population
H is no. of households

Question 9

What does population (Nt) grow at?

B) expression (lnNt = …)

C) how do we find N

Answer 9

Population (Nt) grows at rate n

B)
lnNt = Nbar + nt

Nbar is starting value of population

Find level of N by logs Nt = e to the Nbar + nt

Question 10

Technology grows at rate…

B) expression (lnAt = …)

C) random shock expression

Answer 10

Tech grows at rate g every period…

B)
lnAt = Abar + gt + A~t
Abar is starting value of tech
A~t: random shocks.

C)
A~t = pa A~t-₁ + ε
ε is white noise (shocks are uncorrelated)\
p: persistence of shock is (higher=more persistent)

Question 11

Log form of the household utility function

B) so what do they get utility from (2)

Answer 11

ut = ln Ct + b ln(1-Lt)

Where b is a leisure preference parameter

B)
So get utilty from consumption and leisure

Question 12

Government spending expression: what does it grow at

B) expression (lnGt = ….)

C) shock to gov spending expression

Answer 12

Grows at rate n+g (since if tech growth, gov may spend more, and population growth they need to spend more too)

B)
ln 𝐺𝑡 = 𝐺- + (𝑛 + 𝑔)𝑡 + 𝐺~𝑡
G~t: shock

C)
G~t = pg G~t-₁ + ε

ε: white noise i.e shocks uncorrelated
p: persistence of shock

Question 13

1-person household lives for 1 period, no initial wealth.

What is the utility function and budget constraint (i.e how much they can consume)

Answer 13

u = ln C + b ln(1-L)

same as usual (don’t need to add T since one period)

BC:
c = wL
(Consumption = wage x labour hours)
(No + R (non-labour income as assume no initial wealth)

Question 14

Use Lagrange to solve

Step 1: set up Lagrange expression ℒ

Answer 14

It’s just utility and budget constraint added together (BC rearranged to wL - c) x λ

ℒ = ln 𝑐 + 𝑏 ln (1 − ℓ) + 𝜆(𝑤ℓ − 𝑐)

Question 15

2. Do FOC of Lagrange for:

A)how much they want to consume

B) how much to work (the intensive margin)

Do expressions for both

Answer 15

i) for consumption, FOC with respect to C to get…

1/c - 𝜆 = 0

ii) for intensive margin, FOC with respect to L
-b/1-L + 𝜆w = 0

(Its easy just looks messy!)

Question 16

Then eliminate λ by…. (Working on pg7)

RMB final answer

Answer 16

To get

-b/1-L + 1/L = 0

Question 17

So what does this expression represent
-b/1-L + 1/L = 0

B) why doesn’t it include real wage w

c) when does this result from b not hold

Answer 17

We get household labour supply

b)
as 1-person household with no initial wealth, income and substitution effect cancel each other out following a change in real wage

(e.g if W increases, sub effect tells households to work more (sub leisure for work). income effect tells households to work less since earn the same amount for less hours. So overall remains constant)

C) substituion and income effect cancelling each other out, making labour supply not influenced by real wage: no longer holds if we extend this model to more than one time period

Question 18

So 1 period 1 person with no initial wealth, labour supply is not influenced by real wage (sub/inc effect offset each other)

This does not hold in a 2 period model; we’ll see.

2 period RBC model: what is the notable difference?

Answer 18

Now include real interest rate r to account for multiple periods (Intertemporal - households can borrow and save)

Question 19

Budget constraint for the 2 period model

pg 8

Answer 19

𝑐₁ + c2/1+𝑟 = 𝑤₁ℓ₁ + w2l2/1+𝑟

LHS Lifetime consumption = RHS Lifetime income
(No debt outstanding or leftover saving after death!)

Question 20

Step 1: Set up lagrangian problem (ℒ ) pg 8

Hint: there’s an e in it

Answer 20

ℒ = ln 𝑐₁ + 𝑏 ln₁(1−ℓ₁) + 𝑒 to the −𝜌 [ln𝑐₂ + 𝑏ln(1 − ℓ₂] +
𝜆[𝑤₁ℓ₁ + 1/1+r 𝑤₂ℓ₂− 𝑐₁−1/1+r 𝑐₂]

Tip: lagrange is always the BC rearranged

We add discount value (-p) since we discount future as we are impatient. Since although face the same real interest rate, we have our different preferences e.g I am v impatient.

Question 21

Recall in 1 period model we choose consumption and how much to work (intensive margin)

They do same, but now only focus on labour supply, this time present (L1) and future labour supply (L2)

So what expressions for ℓ₁, ℓ₂ ? pg9 (simple)

Answer 21

L1: -b/1-L1 + λw1 = 0

L2: -e to the -p x b /1-L2 + 1/1+r λw2 = 0

(Simple, but just RMB its negative signs for both at the beginning (idk why))

Question 22

Then eliminate λ to get labour supply expression

Final equation:

Answer 22

(FIND WORKING LATER)

1-ℓ₁/1-ℓ₂ = 1/𝑒 to the −𝜌 (1 + 𝑟) x 𝑤₂/𝑤₁

Question 23

Intuition of this labour supply expression
1-ℓ₁/1-ℓ₂ = 1/𝑒 to the −𝜌 (1 + 𝑟) x 𝑤₂/𝑤₁

Answer 23

𝑤₂/𝑤₁ is relative wage ;

E.g if w2 higher, work more in period 2! Instead increase leisure in period 1, but reduce in period 2!

Question 24

Intertemporal elasticity of substitution

B) what value is it usually

Answer 24

how responsive HH will be to changing their labour allocation following a change in wages

it is usually 1

Question 25

real interest rate is also in the expression. influence of a change in real interest rate e.g a rise in the rate today

Answer 25

increases attractiveness of working today, since can save it to earn more interest to fund consumption in period 2

Question 26

Now introduce uncertainty into model; what is meant by this? B) We can use Euler equation for introducing this. Suppose household reduces current consumption by (Δc) and uses this to increase consumption in period t+1. If behaving optimally, what should a change like this do to expected utility?

Answer 26

We dont know future r and w. B) Expected utility shoiuld be unchanged. (since reducing now to use later? Ignoreeee C) original utility expression; see the sum function so there’s 2 terms (first one for MU₁ loss, 2nd one for MU₂ gain) 0 = −[𝑒 to the −𝜌𝑡 𝑁𝑡/𝐻 1/𝑐𝑡] ∆𝑐 + 𝐸𝑡[𝑒 to the −𝜌(𝑡+1) 𝑁𝑡+1/𝐻 𝑒−𝑛 (1 + 𝑟𝑡+₁)1/𝑐𝑡+₁] ∆𝑐 First part is MU today, 2nd part is expected MU. First part is negative as shows giving away consumption now reduces MU today. 2nd part is positive shows positive expected MU as a result

Question 27

Rearranging this we can get main Euler equation (just rmb this main one) pg 12 B) key: so what does our marginal utility depend on? (2)

Answer 27

1/Ct = e to the -p Et[1/ct+1 (1+rt+₁) 1/Ct is MU today 1/Ct+1 is MU future B) So the way households decide consumption over time depends on real interest rate and impatience (p)

Question 28

So that was consumption overtime (reducing consumption Δc now for greater consumption in future. We can also capture relationship between consumption and labour supply: Consider a household increases labour supply by a small amount Δℓ in order for consumption spending. If behaving optimally, what should this change do to expected utility B) expression

Answer 28

Leave expected utility unchanged. 0 = -[e to the -pt Nt/H b/1-ℓ𝑡] Δℓ + [e to the -pt Nt/H 1/Ct]WtΔℓ KEY IMPORTANT: First term is marginal cost i.e working another hour (hence why - at start) Second term - marginal benefit from consumption! Focus on the highlighted blue bit only since discount factor e-pt Nt doesnt matter since loooking at consumption vs labour supply all in time t. (Previous page looked at consumption overtime)

Question 29

We can simplify this expression since only period t as mentioned. Pg12

Answer 29

Ct/ 1 -lt = Wt/b Euler equation of Labour and consumption relationship (intratemporal) Euler equation of Consumption overtime 1/Ct = e to the -p Et[(1/Ct+1) (1+rt+1)] (Intertemporal) So we get 2 key equations

Question 30

So that was model set up. Assume G=0 (no government) and δ=1 (depreciation rate is 100%, unrealistic but needed to solve this model) And 2 state variables, capital stock and tech. 2 possible results (both very unrealistic!)

Answer 30

1. Constant savings rate. Sub/income effects offset each other (i.e sub effect- e.g improvement in tech,capital is more productive so invest more in it (thus they increase saving) Income - capital is more productive so need less of it to produce the same amount (reduce saving) 2. ℓ (labour hours) constant E.g improvement in A (tech) – raise current wages (since more productive), so increases labour supply but this increases savings, expected future interest rates fall (to trigger economic activity) and this lowers labour supply since less incentive to work.

Question 31

Problems with this baseline model (4)

Answer 31

Result 1: believing constant savings rate implies C&I are equally volatile. Result 2: it believes labour hours constant: but IRL labour hours are pro-cyclical (rise with GDP) Results are dependent on what value of pa we pick. (Very key criticism; lacks own internal propogation mechanism) Real wage is strongly pro-cyclial in this model (it =MPL!) but data suggests weak or acyclial correlation.

Question 32

Consider effect of capital and labour hours on a tech shock (pg 14 diagram) Assume pa=0.95 (very persistent, 1 is permanent)

Answer 32

Capital stock: increases above it’s long run equilbirum because tech shock makes capital is more productive, so firms invest more in it. Labour hours - jumps upward rapidly, more productive following tech and so they are willing to work more. However it declines and then goes below the long-run value, then will return to normal.

Question 33

Effect on consumption and output too? B) which one has smaller effects, and why C) how can we also see investmetn

Answer 33

Consumption: increases as since rmb labour hours jumped rapidly. Output: Jumps up very high too (almost 0.9) supports the idea of tech shocks driving most the changes. B) consumption doesn’t rise as high as output as they consider leisure and also saving - highlights consumption smoothing C) We can also see investment - gap between consumption and output. Shows volatile

Question 34

Effect on real wage and real interest rate

Answer 34

Wages: initial jump (since more productive following tech shock, hence why also labour hours jumps rapidly) then slow fall; since slow return back to 0 cannot be a main driver behind the fall in labour hours! (Supports our argument of saying marginal utility depends on real interest rate (and persistence) Real interest rate: small jump then falls back quickly and dips below long run. SINCE QUICKLY, MEANS IT contributes to fall in labour hours, Since if interest rates falls, labour supply falls since incentive to work falls since earn less on savings.

Question 35

Key: So what causes the fall in labour hours?

Answer 35

Not wages since they jump and then fall back slowly. So it is the fall in the interest rate which falls back and below quick! (Fall in interest rate reduces incentive to increase labour hours now!)

Question 36

So for both shocks, what are 2 effects occuring (same thing as income and sub effects) B) which tends to be stronger

Answer 36

Wealth effect (if tech improves prod, increase wages today, but now can work less for the same amount so reduce labour hours (INCOME EFFECT) Intertemporal-substitution effect (if tech improves prod and wages, work more today while high wages!) (SUB EFFECT) B) intertemporal-sub effect tends to be stronger, i.e people increase their hours following a positive shock.

Question 37

Criticisms of RBC models (4)

Answer 37

No formal way to calibrate (estimate parameters) During great depression - people weren’t CHOOSING to reduce labour hours, they were laid off! RBC doesn’t explain this. RBC empirical performance in the labour market is wrong. Labour input and productivity and far too closely correlated Money not included - Studies show money does actually have real effects

Question 38

Seminar questions Given Y=K to the a AL to the 1-a Show MPK positive but DMR

Answer 38

Show positive by MPK>0 MPK = aK to the a-1 AL to the 1-a >0 Show DMR by SOC<0 MPK’ = a(a-1)K to the a-2 AL to the 1-a <0

Question 39

Show MPL positive but DMR (note, this one not as simple - hint chain rule stuff)

Answer 39

Show positive as MPL>0 MPL = (1-a)K to the a AL to the -a At > 0 Show DMR as MPL’<0 MPL’ = -a(1-a)K to the a AL to the -a-1 At²

Question 40

IES formula

Answer 40

IES = d(relative wage)/ dw2/w1 X (w2/w1)/(1-L1)/(1-L2) Which is just d(1-L1/1-L2)/ dw2/w1

Question 41

But what is d(1-L1/1-L2)/ dw2/w1 Recall in 2 period model relative labour supply 1-L1/1-L2 = 1/e to the -p (1+r) X w2/w1 So when we differentiate with respect to W2/W1 (d(1-L1/1-L2)/ dw2/w1) what is the answer (simple)

Answer 41

d(1-L1/1-L2)/ dw2/w1 = 1/e to the -p (1+r) Simple! w2/w1 disappears!

Question 42

So prove IES =1 starting with the relative labour supply expression

Question 43

Now prove IES=1 starting with relative labour supply expression (working pg9) B) what does it mean to have IES=1

Answer 43

IES=1, a 1% increase in relative wage (means W2 relatively higher) so they increase their leisure by 1% too!