Lecture 1: Introduction - · PDF fileLecture 1: Introduction Fatih Guvenen September 17,...

Lecture 1: Introduction

Fatih Guvenen

September 17, 2015

Fatih Guvenen Lecture 1: Introduction September 17, 2015 1 / 27

Motto for this Class


Four Steps of a Quantitative Project

1 Model specification:I Preferences, technology, demographic structure, equilibrium

concept, frictions, driving forces, etc.

2 Numerical solution:I Programming language, algorithms, accuracy vs speed, etc.

3 Calibration/Estimation:I Simulation-based estimation, global optimization

4 Analyzing the model outputI Policy experiments/counterfactuals, welfare analysis, transitions,

etc.


Model Specification: Partial Equilibrium

V (a,w) = maxc,k 0

⇥u(c, `) + �E(V (a0,w 0)|w)

⇤

c + a

0 = (1 + r)a + w(1 � `)

w

0 ⇠ f (•|w)

Choices, choices:What functional form to choose for u(c, `)?

How about if we also want to model home production? orhousehold preferences?How to specify f (•|w)? There is an entire literature on the choiceof f ().How about if we have other shocks (health, rate-of-return, etc.)?


Model Specification: Preferences

max1X

t=0

�t

2

6664C

1��t

1 � �+ ⇥ (1 � N

t

)1��

1 � �| {z }power separable

3

7775

s.t . C

t

+ K

t+1 � (1 � �)K

t

F

t

(Kt

,Nt

) (⇤t

)

What changes between these four formulations?



max1X

t=0

�t

2

6664(C↵

t

(1 � N

t

)1�↵)1��

1 � �| {z }Cobb-Douglas

3

7775

s.t . C

t

+ K

t+1 � (1 � �)K

t

F

t

(Kt

,Nt

) (⇤t

)




max1X

t=0

�t

2

664C

t

+ (1 � N

t

)1+�

1 + �| {z }GHH ver 1

3

775

1��

s.t . C

t

+ K

t+1 � (1 � �)K

t

F

t

(Kt

,Nt

) (⇤t

)




max1X

t=0

�t

2

664C

t

� N

t

1+�

1 + �| {z }GHH ver 2

3

775

1��

s.t . C

t

+ K

t+1 � (1 � �)K

t

F

t

(Kt

,Nt

) (⇤t

)



Model Specification: Investment

max1X

t=0

�t

U(Ct

, 1 � N

t

)

s.t . C

t

+ (Kt+1 � (1 � �)K

t

) F

t

(Kt

,Nt

) (⇤t

)

What changes between the two formulations?


Model Specification: Investment Adjustment

max1X

t=0

�t

U(Ct

, 1 � N

t

)

s.t . C

t

+ � (Kt+1 � (1 � �)K

t

) F

t

(Kt

,Nt

) (⇤t

)

What changes between the two formulations?


Model Specification: General

max E

(X

t

�t

u (ct

, `t

)

)

s.t . c

t

+ x

zt

+ x

ht

+ x

kt

F (kt

, zt

, st

) (1)z

t

M (nzt

, ht

, xzt

) (2)k

t+1 (1 � �k

) k

t

+ x

kt

(3)h

t+1 (1 � �h

) h

t

+ G (nht

, ht

, xht

) (4)`

t

+ n

ht

+ n

zt

1 (5)h0 and k0 given


Risk Aversion

What is the risk aversion when preferences are of the form:

U(C) =C

1�↵

1 � ↵

Risk aversion is not the curvature of some utility function. It is theanswer to a specific question.

Depending on what question we ask, the risk aversion wemeasure will be different.

Sometimes it will have a simple relationship to the curvature, andsometimes it will not.


What is Risk Aversion?

Start with a static gamble as studied by Pratt(1964, ECMA).Because the problem is static, there is no saving, so Prattassumed the outcome of the gamble would be consumedimmediately:

I bet pays off c + �i

dollars in state i , realized w.p. p

i

.

If the bet is declined, consumption is c minus the risk premium, ⇡.So:

u(c � ⇡) =nX

i=1

p

i

u(c + �i

).



When the risk is small, use the Arrow-Pratt approximation.

Basically, take the first-order Taylor approximation of the LHS, andthe second-order approximation to the RHS (why?) to get:

u(c)� ⇡u

0(c) =nX

i=1

p

i

✓u(c) + �

i

u

0(c) +12�2

i

u

00(c)

◆

= u(c)nX

i=1

p

i

| {z }=1

+ u

0(c)nX

i=1

p

i

�i

| {z }=0

+12

u

00(c)nX

i=1

p

i

�2i

| {z }=var(�

i

)

⇡u

0(c) = �12

u

00(c)⇥ var(�i

) )

⇡ = �u

00(c)

u

0(c)| {z }Absolute risk aversion

⇥ 12

var(�i

)| {z }

.

Amount of risk

(6)



If the gamble is in fixed monetary units, we are talking aboutabsolute risk aversion.

If it is indexed to the average level of the bet, then we are talkingabout relative risk aversion:

u(c(1 � ⇡r )) =nX

i=1

p

i

u(c(1 + �i

)).

The coefficient of relative risk aversion:

RRA(c) = �c

u

00(c)

u

0(c)(7)



In a dynamic model, risk aversion can be as simple as what wehave seen so far or it can be as complex as you can imagine.

Why? Because in a dynamic context it does not usually makesense to assume that you have to consume the outcome of thebet immediately.

For example, a worker who loses his job will usually have theoption to borrow to smooth consumption.

Or somebody who has a windfall gain from an inheritance, doesnot have to spend all of it in the current period. And so on.

So, in general, risk aversion will depend on the market structureand the type of gamble that is offered, so it can mean differentthings.


What is Risk Aversion?In a dynamic model, individuals can “typically” use financial markets tosmooth consumption relative to income, so we should think aboutwealth/income bets:

V (!(1 � ⇡r )) =nX

i=1

p

i

V (!(1 + �i

)).

⇡r = �!V

00(!)

V

0(!)| {z }Absolute risk aversion

⇥ 12

var(�i

)| {z }

.

Amount of risk

(8)

Result: If (i) preferences are separable over time, and (ii) themarket structure is such that (i.e., markets are complete) theenvelope condition is V

0(!) = u

0(c) @c

@! , then:

�!V

00(!)

V

0(!)= �c

u

00(c)

u

0(c),

where we used Euler’s theorem that @c

@!! = c.Fatih Guvenen Lecture 1: Introduction September 17, 2015 17 / 27

Risk Aversion

This explanation also makes it clear that this result is more specialand limited than it looks.

Because we know that in many models the marginal utility ofconsumption is not equated across dates and states, most notablywhen markets are incomplete—which is most of the models thiscourse covers!

In such cases, immediately consuming the outcome of the betcannot be any greater than finding the state with the highestmarginal utility and consuming in that state.

So wealth will have (weakly) higher marginal utility than currentconsumption yielding an inequality:

�w

V

00(w)

V

0(w)� �c

u

00(c)

u

0(c)= �. (9)


Non-Separable Utiliy

A second case of interest is when preferences aretime-non-separable, e.g., Epstein-Zin preferences or habitformation.

In this case, even if markets are complete, risk aversion may differ(sometimes substantially) from the curvature of the utility function.

With incomplete markets it is not clear what w should be. Wealthgambles are not too meaningful if most of your cash on handcomes from labor income.

If it is literally financial wealth, risk aversion may be zero ornegative as measured by (9), since w could be zero or negative.

If we think that it should include wealth labor income, so it iscash-on-hand, then how do we discount future earnings? Ingeneral, the formula above is not very useful in incompletemarkets models as a measure because of these difficulties.


Dynamic Setting: Risk Aversion with LaborSupply

Consider a one-shot wealth gamble of size A

t

:

a

t+1 = (1 + r

t

)at

+ w

t

`t

� c

t

+ A

t

�"t+1.

Pay risk premium A

t

µ to avoid gamble.Because the fee is proportional to the size of the gamble, theproper measure—relative risk aversion—is defined as

lim�!0

2µ(�)�2 = �

A

t

Et

V11(a⇤t+1; ✓t+1)

Et

V1(a⇤t+1; ✓t+1)

where ✓ is the exogenous state of the individual’s problem and a

⇤

is the optimal choice of assets tomorrow as a function of today’sstate variables (a

t

, ✓t

).Typically, we define A

t

to be a fraction of household’s wealth attime t .Question: what is the proper definition of wealth in a dynamicmodel with labor supply.


Risk Aversion with Labor Supply, Cont’d

One definition that makes sense is the (properly) discounted valueof future resources, either based on future consumption alone:

A

t

⌘ (1 + r

t

)�1Et

1X

⌧=t

m

t ,⌧c

⇤⌧

where m

t ,⌧ ⌘ �⌧�t

u1(c⇤⌧ , `

⇤⌧ )/u1(c

⇤t

, `⇤t

) is individual’s stochasticdiscount factor; or including future values of leisure time:

A

t

⌘ (1 + r

t

)�1Et

1X

⌧=t

m

t ,⌧�c

⇤⌧ + (`� `⇤⌧ )

�.


4 Cases:1 Power separable specification:

u(ct

, `t

) =c

1��t

1 � �� ⌘

`1+�

1 + �,

with �, ⌘,� > 0. We have

RRA

c =�

1 + ��

w`c

⇡ 11� + 1

�

,

if we assume c ⇡ w`.For example, if � = 2 and � = 1, RRA

c is �/3.12 Cobb-Douglas specification:

u(ct

, `t

) =(c1��

t

(1 � `t

)�)1��

1 � �,

with � 2 (0, 1), we have

RRA

cl = �,

since consumption and leisure act as a single compositecommodity subject to the same risk aversion.

1RRA

cl is not well defined since ` can be scaled up and down to generate anynumber.


4 Cases

3. King-Plosser-Rebelo (KPR) preferences:

u(ct

, `t

) =c

1��t

(1 � `t

)�(1��)

1 � �,

with �,� > 0, and �(1 � �) < � for concavity. Here we have

RRA

cl = � + �(� � 1).

Note that RRA

cl > � as long as � > 1. Thus, leisure margin doesnot always reduce risk aversion, it can also increase as in thisexample.

4. GHH preferences: Risk aversion is � since labor supply onlyvaries with wages and cannot be used to insure any fluctuations inconsumption.


Aversion to Higher Order RiskWhy did the Arrow-Pratt derivation not consider higher orderterms?Usual answer: first moment > second moment > third moment > ..Is this true?Taylor’s Theorem: If f

n exists for all n in an open interval I

containing a then for each positive integer and for each x in I,

f (x) = f (a) + f

0(a)(x � a) +f

00(a)

2(x � a)2 +

f

000(a)

2(x � a)3 + ...

+f

(n)

n!(x � a)n + R

n

(x)

where

R

n

(x) =f

(n+1)(c)

(n + 1)!(x � a)n+1 for some c2[a,x] .

Key question is what happens to f

(n+1)(c) as n grows. For apolynomial f with n 2 Z+ it goes to zero. But for a function with anegative exponent, it will get larger if c < 1.


Aversion to Higher Order Risk

U(c(1 � ⇡)) = E⇣

U(c(1 + �)⌘

Fourth order Taylor Approximation to the RHS: U(c)� U

0(c)c⇡ =

= E✓

U(c) + U

0(c)c� +12

U

00(c)c2�2 +16

U

000(c)c3�3 +1

24U

0000(c)c4�4◆.

Second term on the RHS is zero when E(�) = 0, and rearrangingyields:

⇡ = �12

u

00(c)c

u

0(c)⇥ m2 �

16

u

000(c)c2

u

0(c)⇥ m3 +

124

u

0000(c)c3

u

0(c)⇥ m4,

(10)

where m

n

denotes the n

th central moment of �.Fatih Guvenen Lecture 1: Introduction September 17, 2015 25 / 27

Higher Order Risk Aversion

To convert these into statistics that are more familiar, writem2 = �2

� , m3 = s� ⇥ �3� , where s� is the skewness coefficient, and

m4 = k� ⇥ �4� , where k� is kurtosis.

With this notation, and assuming a CRRA utility function withcurvature ✓, we get:

⇡⇤ =✓

2⇥ �2

� �(✓ + 1)✓

6⇥ s� ⇥ �3

� +(✓ + 2)(✓ + 1)✓

24⇥ k� ⇥ �4

�

which can also be written as:

⇡⇤ =✓

2⇥ �2

� ⇥1 +

13(✓ + 1)

✓�s� ⇥ �� +

14(✓ + 2)k� ⇥ �2

�

◆�.


Risk Premium: Skewness and Kurtosis

Let � be a static gamble. And ⇡ is the risk premium to avoid it:

U(c ⇥ (1 � ⇡)) = EhU(c ⇥ (1 + �))

i.

Risk Premium (⇡)

Gamble: �A �B

Mean 0.0 0.0Standard Deviation 0.10 0.10Skewness 0.0 –2.0Excess Kurtosis 0.0 27.0

Premium 4.88% 22.15%


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