Clase 2009 Macro II WSU Dr Choi

8/11/2019 Clase 2009 Macro II WSU Dr Choi

1/159

Calibration of a Neoclassical Growth Model

Ultimate Goal of Growth Economics: Understand why poor countries are poor.(still long way to go to understand it)

(But perhaps the most important question of all time?)

Goal here: What do the data suggest on the sources of growth?

Consider a growth model that ts the data.

(Neo-classical) (US, 1960-2005)

"Neoclassical" is just a name of a group of models, like "Marxist" or"Keynesian".

Then think about its implications.

1. Model

Y t = F (inputs ): We want to specify this.

"inputs": labor (human capital), physical capital, ...

t = year (2008, 2009, ...)

Many economists nd the following functional form reasonable. Why? Fit relativelywell to data. (We will see.)

Y t = K t (At Lt )1 : Cobb-Douglas production function

Y t : total output (measured by real GDP)

K t : physical capital stock (typically not in data, sometimes we have in data, but itis constructed from the method that we discuss below)

: a parameter often called "physical capital share" (Why? We will see soon.)

At Lt : "eective units" of labor

Lt : labor input (measured by manhours or #workers)

What determines total output?

K t (stock of physical capital)

Lt (# workers employed)

At (productivity ???. Maybe education, R&D, eciency gains from eliminationof monopoly power)

1


2/159

Sometimes we write Y t = At K t Lt 1 . At in this case is called "total factor produc-tivity (TFP)". These two set-ups are identical. You choose based on what you want

to do with the model. Lt+1 = Lt (1 + gL ): (exogenous) population growth

At+1 = At (1 + gA): (exogenous) productivity growth

I want to emphasize that these two variables grow exogenously . That is, our modeldoes not explain why they grow. So we will not care about how they grow.

But we do care about how K t grows. So we specify it as follows:

K t+1 = I t + (1 )K t

I t : Investment (observed in data)Y t = C t + I t . Out of total output, C t is consumed and I t is invested for futureproductions.

s t = I t =Y t is an investment rate.

: depreciation rate.

So K t+1 = s t Y t + (1 )K t : accumulation of physical capital (a.k.a. law of motion of K t )

2. Calibration

What is , gL , gA and ? What can we learn from this model quantitatively?

We cant just calibrate this model without further assumptions.

First, each variable grows at a constant rate. ("balanced growth path")

This is reasonable. The growth rate of any variable doesnt rise or decline forever.Maybe it does in the short run, but eventually it will be stable.

Second, s t is constant. This is also reasonable from data.

Data: s = 0:20 (average of 1960-2005. For each year t, divide gross domesticinvestment, in NIPA Table 1.1.5, by GDP, in NIPA Table 1.1.)

There is a cross-country variation on s. But the U.S. time-series is stable.

Data: gL = 1:1% per year. (average of 1960-2005, NIPA Table 7.1.)

2


3/159

Data: gY = 3:3% per year. (average of 1960-2005, NIPA Table 7.1. This is thesum of gL and "GDP per capita in chained dollars".)

(1) Calibration of

We investigate LOM of K:

K t+1 = sK t (At Lt )1 + (1 )K t :

So K t+1

K tK t

At Lt= s

K tAt Lt

+ (1 ) K tAt Lt

:

So (gK + ) K tAt Lt

= s K tAt Lt

:

From a gure, K tA t L t is constant.j + +

j ++

j + +

j + +

j + +

j+ ++ K tA t L t

From the production function, Y t , K t and At Lt all grow at the same rate: gY = gK :

So

K t+1 = sY t + (1 )K t :

So (1 + gK )K tY t

= s + (1 )K tY t

:

So gK K tY t

= s K tY t

:

Data: K=Y = 0 :12 (average of 1960-2005, NIPA Table 1.1. This is a ratio of "consumption of xed capital" to Y.)

So0:033

K Y

= 0:20 0:12

3


4/159

Hence, Result: K=Y = 2 :42 Now we measure the value of all U.S. physicalcapital. This was not directly observable But now we measured with a theoretical

model and observations on other variables. Also, Result: = 0:05 Another cool result 5% of physical capital stock depre-ciates per year.

It remains to calibrate and gA .

(2) Calibration of

Y t = K t (At Lt )1

"Interest rate" = marginal product of physical capital

Each unit of K is paid its marginal product. ($ 1 > $(1 + r ))

rt K 1t (At Lt )1

"Phyiscal capital income" = r t K t = K t (At Lt )1 = Y t

Similarly, "labor income" = wt Lt = (1 )Y t

Data: (income share of K ) = 0 :32 (NIPA Table 1.12. This is a fraction of "com-pensation of employees" out of "GDP" minus "propietors income" monus "taxeson production and imports".)

Result: = 0:32

We can also compute the interest rate. Since r t K t = Y t ,

r t = K=Y

= 0:322:42

= 13 :2%

Result: r = 13:2% (Cool!)

(3) Calibration of gA

The most important for our purposes.

Recall: Y t , K t and A t Lt all grow at the same rate.

So gY = gK = gA + gL . (Why? Growth rate of X t is (i) X t+1 =X t 1 or (ii)log(X t+1 ) log(X t ).)

Result: gA = gY gL = 3:3% 1:1% = 2:2%

4


5/159

Calibration completed. Now we discuss quantitative implications.

3. Experiments Assume in 2008, the economy is in a steady state:- Real Per-capita GDP ( Y 2008 =L2008 ) = $45,000 (data)- Population ( L2008 ) = 200 million (data)- Real GDP ( Y 2008 ) = $45,000 x 200 million- Physical Capital Stock ( K 2008 ) = We know K=Y = 2:42. So K 2008 = 2:42Y 2008 =?

(We measured something not observed!)

From 2008 and on,- s = 0:20 from the above calibration. Assume this holds.- So out of Y 2008 ;

(1 s)Y 2008 is consumedsY 2008 is invested. ( I 2008 = sY 2008 :)

- Then

K 2009 = (1 )K 2008 + I 2008 can be obtained. This should be the same as(K=Y )(1 + gY )Y 2008 :

We can do some experiments.

Example: What happens if the investment rate ( s) increases to 0.3 permanently?

Example: What happens if the investment rate is still 0.2, but there is FDI inowso K increased by $100 billion?

4. Growth Decomposition

Now the ultimate question. What are the sources of the U.S. growth?

Y t = K t (At Lt )1

=) Y t =Lt = ( K t =Lt ) (At )1

=) log(Y t =Lt ) = log(K t =Lt ) + (1 )log At=) gY=L = gK=L + (1 )gA

(growth of X = X t+1 =X t 1 log X t+1 log X t )=) gY =L

|{z} (A) Real Per-capita GDP growth= gK=L

| {z } (B ) K/Ls contirubtion+ (1 )gA

| {z } (C ) As contribution5


6/159

We know all these constants, so we are done!

(A) = gY gL = 3:3% 1:1% = 2:2%

(B) = (gK gL ) = (gY gL ) = 0 :32 2:2%(C) = (A)-(B) = 0:68 2:2%

(B)/(A) = 32% (32% of per-capita income growth is due to per-capita K growth orultimately to the savings (or investment).)

(C)/(A) = 68% (68% of it is due to TFP growth In short, we dont know fromthis model!)

Nowadays we dont use this version anymore.

Why? If A rises > Y rises > K rises (since K/Y is constant) > K/L will rise.

That is, K/L may rise just because A rises! This is misleading.

So we dont like K/L. Use K/Y instead. (See Hall and Jones (1999).)

Y t = K t (At Lt )1

=) Y 1t = ( K t =Y t ) (At Lt )1

=) (Y t =Lt )1 = ( K t =Y t ) At 1

=) Y t =Lt = ( K t =Y t ) =(1 )At

=) gY =L

|{z} (A) Real Per-capita GDP growth=

1gK=Y

| {z } (B ) K/Ys contirubtion+ gA

|{z} (C ) As contribution But (B) is 0. So (C)/(A)=100%! (We dont know the fundamental source of growth from thismodel. Need a better model.)

5. Introduction of "People"

The current version assumes that investment rate is simply given.

But it is determined by the "people" (a.k.a. consumers, workers, agents, ...).

It is useful to extend the model so that investment rate is endogenously determined.

"Representative consumer" (innitely lived): The preferences of the entire con-sumers in this economy can be represented by this one, big guy.

6


7/159

u(C t =Lt ): The utility function for each identical consumer.

u(C t =Lt ) = ( C t =Lt )1 =(1 ); > 0. "CRRA" utility function.

(log utility if converges to 0.)

There are Lt consumers. We typically assume that u(C t =Lt )Lt is maximized.

Problem:

maxf C t ;I t g1t =0

1

Xt=0 t (C t =Lt )1

1 Lt

s.t.

K t+1 = I t + (1 )K t ; K 0 given,

Y t = C t + I t ;Y t = K t (At Lt )

1 ;Lt+1 = Lt (1 + gL ); L0 given,At+1 = At (1 + gA); A0 given.

Eliminate Y t and I t :

K t+1 = K t (At Lt )1 C t + (1 )K t :

Now the problem:

maxf K t +1 g1t =0

1

Xt=0 t (K t (At Lt )1 + (1 )K t K t+1 )1

1 1Lt

s.t.

Lt+1 = Lt (1 + gL ); L0 given,At+1 = At (1 + gA); A0 given,

and K 0 given.

7


8/159

Rewriting,

maxK 1 ;K 2 ;:::(K 0 (A0L0)1 + (1 )K 0 K 1)1

1 1L0

+ (K 1 (A1L1)1 + (1 )K 1 K 2)1

1 1L1

+ :::

+ t(K t (At Lt )1 + (1 )K t K t+1 )1

1 1Lt

+ t+1(K t+1 (At+1 Lt+1 )1 + (1 )K t+1 K t+2 )1

1 1Lt+1

+ :::

So the FOC wrt K t+1 is

t C t 1Lt

+ t+1 C t+1 ( K 1

t+1 (At+1 Lt+1 )1 + 1 )

1Lt+1

= 0 :

So

1 = C t+1 =Lt+1

C t =Lt( K 1t+1 (At+1 Lt+1 )

1

| {z } = r t +1 (interest rate bw t and t +1 )

+ 1 )

This equation is often called "Euler eq.". And additional conditions: constraints above.

Lets calibrate.

Recall that before, we made 2 asspts:

(i) Each variable grows at a constant rate.

(ii) s t is constant.

Now we need only (i) because (ii) follows directly from (i) and Euler eq.

To see this,

K 1t+1 (At+1 Lt+1 )1 is const. =) gK = gA + gLY t = K t (At Lt )1 =) gY = gK + (1 )(gA + gL ) =) gY = gLK t+1 = I t + (1 )K t =) gI = gK =) gI = gY =) st = I t =Y t is const.

Now whats the dierence between this version and the previous version?

8


9/159

Now Euler eq. is added. Two more parameters, and , are added.

Nothing else. All the previous calibration results hold. We simply have an additionaleq. w/ and :

We introduced people.

But nothing else.

References

Lucas, Robert E., Jr. (2009), Macroeconomics: A Short Course , unpublished man-uscript, Chapter 5.

9


10/159

Continuous-Time Representation of a Neoclassical GrowthModel

Many economists use continuous-time representations. Once you are accustomed,sometimes you nd it clearer.

Very Informal, sloppy introduction.

1. "dot"

Warning: Not easy to take it at rst.

_x(t) dx(t)=dtx(t)j

j *

j *

j *

j+t

_x(t) is the slope.

So if _x(0) = 2 , then around t = 0, x(t) increases by 2 in 1 period.

Roughly, _x(t) corresponds to xt+1 xt in a discrete-time set-up.

So _x(t)=x(t) is the growth rate.

x(t) d2x(t)=dt2

K t+1 = I t + (1 )K t =) K t+1 K t = I t K t

=) =) _K (t) = I (t) K (t)

For example, _K (2008) = I (2008) K (2008)

: Around 1 July 2008, by how much K increases in one year is the investment in2008 minus a fraction of K:

17


11/159

Lt+1 = Lt (1 + gL ) =) (Lt+1 Lt )=Lt = gL=) =) _L(t)=L(t) = gL

=) A solution for this dierencial equation is L(t) = L(0)egL t .

(Proof: _L(t) = L(0)egL t gL . So...)

Similarly, A t+1 = At (1 + gA) =) =) A(t) = A(0)egA t .

Y (t) = C (t) + I (t) and Y (t) = K (t) (A(t)L(t))1 are the same as before.

Just as A t = A0(1 + gA)t is replaced by A(t) = A(0)egA t ,

t (1 )t is replaced by e t .

max f C t ;I t gP1t=0 t (C t =Lt )11 Lt =) =) maxf C (t );I (t )gR

10 e t (C (t)=L(t)) 11 L(t)dt

2. Solution to the Continuous-Time Version of Optimization

In general,

max u(t)

1

t =0

Z 10 e th( x(t) |{z} state variable; u(t)

|{z} control variable)dt

s.t.x(t) = g(x(t); u(t)); x(0) given.

Set up the Hamiltonian:

H (x(t); u(t); (t)) = h(x(t); u(t)) + (t)

|{z} co-state variableg(x(t); u(t)):

Then FOCs are

(1) H u = 0

(2) H x = (t) _(t)

And of course, dont forget

(3) _x(t) = g(x(t); u(t))

We also have(4) "transversality condition",

which is satised in a lot of economic problems.

18


12/159


13/159

s.t.

_K (t)A(t)L(t) =

K (t)A(t)L(t)

C (t)A(t)L(t)

K (t)A(t)L(t)

= k(t) c(t) k(t)

We knowK (t) = k(t)A(t)L(t)

So

_K (t) = _k(t)A(t)L(t) + k(t) _A(t)L(t) + k(t)A(t) _L(t)

=)_K (t)

A(t)L(t) = _k(t) + k(t)gA + k(t)gL

So the contraint becomes

_k(t) = k(t) c(t) ( + gA + gL )k(t)

Done!

Hamiltonian:

H = [c(t)]1

1 + (t)(k(t) c(t) ( + gA + gL )k(t)):

FOCs are

H c = 0

H k = (t) _(t)

So

c(t) (t) = 0 ;

and (t)( k(t) 1 ( + gA + gL )) = (t) _(t):

So k(t) 1 ( + gA + gL ) = _(t)(t)

= + _c(t)c(t)

20


14/159

Done! Lets recover our original notation:

K (t)A(t)L(t)

1

( + gA + gL ) = ( gA(1 ) gL ) + 0BBB@_C (t)

C (t)

_A(t)A(t)

|{z} gA_L(t)L(t)

|{z}gL 1CCCA K (t)A(t)L(t)

1

( + gA + gL ) = ( gA gL ) + (gC gL )

K (t)A(t)L(t)

1

= + (gC gL )

r = + (gC gL )

All remaining equations are basically the same as before. Use the calibration resultsfor them.

8% = + 2:2%

For example, if = 1 (log utility),

= 8% 2:2% 6%

Well how is this related to our discrete-time version?

1 = C t+1 =Lt+1

C t =Lt (r t+1 + 1 )1 = (1:022) 1:08

If = 1; =

1:0221:08

= 0 :95:

But t was replaced by e t. So e = e 0:06 = 0 :94. So they are the same.

21


15/159

H-School: Human Capital Accumulation

1. A Simple Approach Y (t) = K (t)

|{z} small( A(t)

|{z} What is this?L(t))1

Maybe education? More education > Productivity "

Data: 1 more year in school - > Wage increases by about 10%

"Mincer Regression": log(wage i) = + agei + (years of schooling) i + ...

How can we incorporate this fact?

Here is the simplest possible way. ASSPT: Human capital is accumulated by schooling.

Y (t) = K (t) (A(t) h(t)L(t)

| {z } units of human capital)1

h(t): Human capital per worker

A(t): Everything else (whatever it is...)

Each unit of physical capital ( K (t)) and eective unit of labor ( A(t)h(t)L(t)) receivesthe marginal products.

For example,

1 unit of K after 1 period > Original 1 unit is returned

But units depreciates

Interest r (t) is paid as compensation

r (t) = MPK(t)= K (t) 1(A(t)h(t)L(t))1

Similarly,

1 unit of human capital after 1 period > Original 1 unit remainsSome fraction may depreciate (Forgetting?)

wage rate w(t) is paid as compensation

w(t) = MPH(t)= (1 )K (t) A(t)1 (h(t)L(t))

28


16/159

Worker A Worker B10-year schooling 9-year schooling(S A = 10) (S B = 9 )(Makes 10% more)hA units of human capital hB units of human capitalLabor income: hAw(t) hB w(t)A function f : S i > h ihA = f (S A), S A = 10 hB = f (S B ), S B = 9

Worker A makes 10% more salary:

f (10)f (9)

= 1 :1:

So df (S )=dS = 0 :1f (S ).Why dont we specify as f (S ) = e0:1S ?

QUESTION: Is the improvement in educational attainment important in US eco-nomic growth?

ASSPT: Every worker is identical (with same S ).

According to Barro-Lee data, average schooling years for 25 years old and above:

1960: 8.66 years, 2000: 12.25 years

(Applying 1995-2000 growth rate to 2000-2005, 2005: 12.32.)So h1960 = e0:1 8:66 and h2005 = e0:1 12:25 .

Go back to our growth accounting:

Recall that in the old version: gY=L


1

gK=Y

| {z } (B ) K/Ys contirubtion+

gA

|{z} (C ) As contributionNow Ah is replacing old A. So gA + gh replaces old gA .gY =L


1gK=Y

| {z } (B ) K/Ys contirubtion+ gh

|{z} (C ) hs contribution+ gA

|{z} (D ) As contribution gh = log(h2005 ) log(h 1960 )2005 1960 =

log(e0:1 12 :25 ) log(e0:1 8:66 )2005 1960 =

0:1 12:25 0:1 8:662005 1960 = 0:8%:

29


17/159

So

(A) = 2 :2%

(B) = 0(C ) = 0 :8%

(D) = 1 :4%

CONCLUSION: Education is important! (Contribution is about 1/3)

But still more than one half of growth is unexplained.

REMARK: But is this a correct way to introduce education?

PROBLEMS:

(0) Using cross-sectional result (10% returns to schooling) for time series (assumingeveryone is the same)?

(1) Maybe A " is causing h" (just like A " is causing K/L ")?(2) Other sources of human capital accumulation: On-the-job training? In-hometraining?

(3) Human capital externalities : I become more productive if surrounded by smarterworkers ("static externalities"). I learn faster if surrounded by smarter workers("learning externalities").

Try a more sophisticated model.

2. Two-Sector Model

Contents:

A. Model Description

B. First-Order Conditions

C. Balanced Growth Path

D. Elimination of (t) and (t) E. Recovery of Original Notations

F. Calibration

A. Model Description

30


18/159

In the previous model, the consumer decides fs(t)g (investment rate) (or equiva-lently, f I (t)g) which determines f K (t)g.

Can we try the same for human capital?

Consumer decides fu(t)g (fraction of time devoted to human capital accumulationsuch as schooling, R&D, in-home training, OJT, ...) which determines f h(t)g.

Model:max

f C (t) ;I (t ) ;u (t )g1t =0 Z 10 e t (C (t)=L(t))11 L(t)dt (same)s.t.

Y (t) = C (t) + I (t); (same)

Y (t) = K (t) (A(t) (1 u(t))

| {z } only this fraction of h (t ) is used for Y productionh(t)L(t))1 ;

A(t) = A(0)egA t ; (same)L(t) = L(0)egL t ; (same)_K (t) = I (t) K K (t); (same)

We now add a new constraint:

For H (t) h(t)L(t),

_H (t) = B u(t) |{z} This fraction is for H production H (t) h H (t):

Note: (a) Human capital is the only input. (Physical input, such as school buildings,disregarded We can introduce them of course. Typically 10%.)

(b) CRS. (To easily get the BGP.)

Write this constraint with h(t):

_h(t)L(t) + h(t) _L(t) = Bu(t)h(t)L(t) h h(t)L(t):

So _h(t) + h(t)

_L(t)L(t) = Bu(t)h(t) h h(t):

So _h(t) = Bu(t)h(t) ( h + gL )h(t):

Eliminate Y (t) and I (t),

maxf C (t);u (t )g1t =0 Z 10 e t (C (t)=L(t))11 L(t)dt

31


19/159

s.t.

A(t) = A(0)egA t ;L(t) = L(0)egL t ;_K (t) = K (t) (A(t)(1 u(t))h(t)L(t))1 C (t) K (t);_h(t) = Bu(t)h(t) ( h + gL )h(t):

Dene:

c(t) = C (t)A(t)L(t)

;

k(t) = K (t)A(t)L(t)

:

Then we have a simple form (same as before):

maxf c(t );u (t)g1t =0 Z 10 e t [c(t)A(t)]11 L(t)dt= max

f c(t );u (t )g1t =0 Z 10 e t [c(t)]11 [A(0)]1 | {z } just constantegA (1 )t L(0)

|{z} just constantegL t dt

=) maxf c(t) ;u (t )g1t =0 Z 10 e ( gA (1 ) gL )t [c(t)]11 dt

= maxf c(t );u (t )g1t =0 Z 1

0 et [c(t)]1

1 dt

where gA(1 ) gL ;s.t. the following two constraints:

First constraint:_K (t)

A(t)L(t) =

K (t)A(t)L(t)

((1 u(t))h(t))1 C (t)A(t)L(t)

K (t)A(t)L(t)

= k(t) ((1 u(t))h(t))1 c(t) k(t)

We knowK (t) = k(t)A(t)L(t)

So

_K (t) = _k(t)A(t)L(t) + k(t) _A(t)L(t) + k(t)A(t) _L(t)

=)_K (t)

A(t)L(t) = _k(t) + k(t)gA + k(t)gL

32


20/159

So the constraint becomes

_k(t) = k(t) ((1 u(t))h(t))1 c(t) ( K + gA + gL )k(t):

Second constraint:

_h(t) = Bu(t)h(t) ( h + gL )h(t): (OK)

B. First-Order Conditions

How do we set up Hamiltonian when there are two constraints?

Compare:

max u(t) 1t =0 R

10 e

th(x(t); u(t))dt max u(t),v(t) 1

t =0 R 1

0 eth(x(t); y(t); u(t); v(t))dt

s.t. x(t) = g(x(t); u(t)); x(0) given. s.t. x(t) = g(x(t); y(t); u(t); v(t)); x(0) given.y(t) = f (x(t); y(t); u(t); v(t)); y(0) given.

(x(t): state variable, ( x(t), y(t): state variable,u(t): control variable) u(t), v(t): control variable)Hamiltonian: Hamiltonian:H (x(t); u(t); (t)) = h(x(t); u(t)) H = h+ (t)g(x(t); u(t)) + (t)g + (t)f ( (t): co-state variable) ( (t), (t): co-state variable)(1) H u = 0 (1) H u = 0, H v = 0(2) H x = (t) _(t) (2) H x = (t) _(t), H y = (t) _(t)(3) _x(t) = g(x(t); u(t)) (3) _x(t) = g; _y(t) = f:(4) "transversality condition" (4) "transversality condition"

Set up Hamiltonian:

H = [c(t)]1

1 + (t) k(t) ((1 u(t))h(t))1 c(t) ( K + gA + gL )k(t)

+ (t)[Bu (t)h(t) ( h + gL )h(t)]

FOCs are(1) H c = 0(2) H u = 0

(3) H k = (t) _(t)(4) H h = (t) _(t)

33


21/159

So

(1) c(t) (t) = 0 ;(2) (t)k(t) (1 )(1 u(t)) h(t)1 + (t)Bh (t) = 0 ;(3) (t)[ k(t) 1((1 u(t))h(t))1 ( K + gA + gL )] = (t) _(t);

(4) (t) k(t) (1 u(t))1 (1 )h(t) + (t)[Bu (t) ( h + gL )] = (t) _(t);

(A) _k(t) = k(t) ((1 u(t))h(t))1 c(t) ( K + gA + gL )k(t);

(B) _h(t) = Bu(t)h(t) ( h + gL )h(t):

Before we go further, focus on the BGP.

C. Balanced Growth Path Assume all variables grow at constant rates (or stay at constant levels).

< Step 0: Always start with this > (1) implies

_c(t)c(t)

=_(t)(t)

:

So (1) gc = 1

g :

< Step 1> (B) implies

(B)_h(t)h(t)

gh = Bu(t) ( h + gL )

So u is constant .

< Step 2> (3) implies

(3) k(t) 1((1 u)h(t))1 ( K + gA + gL ) = _(t)(t)

So k(t)=h(t) should be constant, implying that gk = gh .

< Step 3> (A) implies

_k(t)k(t)

|{z} const= ((1 u)

h(t)k(t)

|{z} const (step 2))1

c(t)k(t)

( K + gA + gL ):

So c(t)=k(t) should be constant, implying that gc = gk = gh .

34


22/159

< Step 4> (2) implies

(t)k(t) (1 )(1 u(t)) h(t)1 = (t)Bh (t)

So(2) (t)k(t) (1 )(1 u) h(t)

| {z } const ( k and h grow at same rates)= (t)B

Hence, g = g .

D. Elimination of (t) and (t)

(1), (1):

(1) c(t) = (t); (1) gc = 1

g :

Can be used to eliminate (t).

(2):

(2) (t)k(t) (1 )(1 u) h(t) = (t)B

(2) 1B

k(t)h(t)

(1 )(1 u) = (t)

(t)

(3):

(3) k(t) 1((1 u)h(t))1 ( K + gA + gL ) = _(t)(t)

(3) (1 u)1k(t)h(t)

1

( K + gA + gL ) = g

= + gc

So has been eliminated.

(4):

(4) (t) k(t) (1 u(t))1 (1 )h(t) + (t)[Bu (t) ( h + gL )] = (t) _(t)

So (t)(t)

k(t) (1 u)1 (1 )h(t) + Bu ( h + gL ) = _(t)(t)= + g= + gc

So (t)

(t)k(t)h(t)

(1 u)1 (1 ) + Bu ( h + gL ) = + gc

35


23/159

Using (2),

hk(t)

h(t)

i (1 u)1

(1 )1B hk(t )h(t )i (1 )(1 u) + Bu ( h + gL ) = + gc

So B(1 u) + Bu ( h + gL ) = + gc(4) B ( h + gL ) = + gc

So (2) and (4) are assembled, eliminating = .

(A):

_k(t) = k(t) ((1 u)h(t))1 c(t) ( K + gA + gL )k(t)

So (A) gk _k(t)k(t)

= (1 u) h(t)k(t)

1 c(t)k(t)

( K + gA + gL )

(B):(B) gh = Bu ( h + gL )

Situation: We had 6 eqs. Eliminated and . We now have 4 eqs:

(3) (1 u)1k(t)h(t)

1

( K + gA + gL ) = + gc

(4) B ( h + gL ) = + gc

(A) (1 u)h(t)k(t)

1 c(t)k(t)

( K + gA + gL ) = gk

(B) gh = Bu ( h + gL )

E. Recovery of Original Notations

Recall:

c(t) = C (t)

A(t)L(t);

k(t) = K (t)A(t)L(t)

;

gc = gh ; = gA(1 ) gL

36


24/159

Part 1/4 : Lets start with (3):

(3) (1 u)1 k(t)

h(t)

1

( K + gA + gL ) = + gc

(1 u)1 K (t)

A(t)h(t)L(t)

1

( K + gA + gL ) = gA(1 ) gL + gh

K (t)A(t)(1 u)h(t)L(t)

1

K = + (gA + gh )

Lets look at this eq. carefully.

(LHS ) =

K (t)=Y (t)

| {z } = r (t)

K

We know gc = gk = gh . But gc = gC gA gL since c(t) = C (t)=A(t)L(t). Thisimplies

gC = gA + gh + gL= gK

The production function also implies

gY = gK = gC = gA + gh + gL :

Hence,

(RHS ) = + (gA + gh )= + (gC gL )

Therefore, the eq. nally becomes

r (t) K =

K (t)=Y (t) K

= + (gC gL )

This is Euler equation! We obtained it before. It is obvious that r and K=Y ratioare constant, so

(3) r K = K=Y

K

= + (gC gL )

37


25/159

The rst equality is the denition of r . (Simply, r = MPK.)

The second equality is the Euler eq.

So we did all bunch of computation to reach what we already know.

But we can attempt similar computation for other eqs.

Part 2/4 : Now (4):

(4) B ( h + gL ) = + gc= gA(1 ) gL + (gC gA gL )

So

B h = gA(1 ) + (gC gA gL )= gA + (gC gL )

SoB + gA h = + (gC gL )

Write it with (3):

(3) r K = K=Y

K

= + (gC gL )= B + gA h

Note that B is MP of human capital in human capital production! (Recall (humancapital production) = Bu(t)h(t).)

So it says

(net interest rate) = (MPK in Y production) K = (a function of consumption growth)

= (MPH in H production) + gA h

Interestingly, gA is in MPH only, not in MPK.

We can also manipulate the FOCs to relate (MPH in Y production) to here. I willnot do this here. Maybe at the problem set.

38


26/159

Part 3/4 : Now (A):

(A) (1 u)h(t)k(t)

1 c(t)k(t) ( K + gA + gL ) = gk

A(t)(1 u)h(t)L(t)K (t)

1 C (t)K (t)

( K + gA + gL ) = gC gA gL

Y (t)K (t)

C (t)Y (t)

Y (t)K (t)

K = gC

1 C (t)Y (t)

Y (t)K (t)

K = gC s

K=Y = gC + K

Same as before! This is the law of motion of K.

Part 4/4 : Now (A):

(B) gh = Bu ( h + gL )gY gA + h = Bu

This is about the law of motion of H.

F. Calibration

All equations:s

K=Y = gC + K (same as before)

r K = K=Y

K (same as before)

= + (gC gL ) (same as before)= B + gA h (new)

gY gA + h = Bu (new)

Lets use:

s = 0:20 (same as before)

gL = 0:011 (same as before)

gY = gC = 0 :033 (same as before)

K (K=Y ) = 0 :12 (same as before)

39


27/159

= 0:32 (same as before)

In addition:

h = 0:033

individual depreciation (forgetting) is about 0.013 (Arrazola and de Hevia (2004),...)

2.2% of the labor force retires every year, assuming a workers working lifetime is45 years

(Can we improve?)

u = 0:28

(a) The labor income in educational services and scientic R&D services is 6.7% of GDP

(b) The foregone labor income due to schooling is worth 3.7% of GDP

(c) The foregone labor income due to on-the-job training is worth 5.9% of GDP

(d) The foregone labor income of parents, due to in-home training before formalschooling, is worth 6.9% of GDP

The sum of these four is 23.2% of GDP.

On the other hand, the labor income outside the sectors of educational services andscientic R&D services is 61.1% of GDP.

Therefore, comparing these two values, 23.2% and 61.1%, gives u = 0:28.

So

0:20K=Y

= 0:033 + K with K K=Y = 0 :12

r K = 0:32K=Y

K (same as before)

= + (0:033 0:011) (same as before)= B + gA 0:033 (new)

0:033 gA + 0 :033 = 0:28B (new)

The rst 3 eqs. are exactly the same! We have

K=Y = 2 :42, K = 0:05, r = 13:2%If = 1, then = 0:06

40


28/159

The last 2 eqs. have 2 unknowns, B and gA .

Eliminate gA to have:

0:033 + (B 0:033 0:132 + 0:050) + 0:033 = 0:28B:So B = 0:068

Then,gA = 0:047

What??? This cant be right!

We knowgY gL

| {z } 2.2%

= gA

|{z} 4.7%

+ gh

|{z} ??????

What might be wrong?

(A) Physical input in H production? Not enough.

(B) B is also increasing. (Technology for H production is improving?)

(C) Human capital externalities?

We will stop here.

We dont know how to clearly distinguish gh from gA .

See Rangazas (2005, ...) for a recent treatment.

3. Human Capital Externalities

We gain from each other. But how much?

Static Externalities (Lucas (1988)): Goods production is now

Y (t) = K (t) (A(t)(1 u(t))h(t)L(t))1 h(t)

|{z} staticexternalities;

where

h(t): his/her own human capital stock,

h(t): average human capital stock in this economy, taken as given by individualconsumer

41


29/159

How large is ? No consensus.

Micro study: Mincer regression with additional explanatory variable (average edu-cational attainment of the state, etc.).- log (wage) i = + 1(age)i + 2(education )i + 3(education of co-workers )i + :::

- Acemoglu and Angrist (2000), Ciccone and Peri (2006): is small, perhaps negli-gible

- Moretti (2004), Liu (2008): is large!

Learning Externalities (Tamura (1991)): Human capital accumulation is now

_h(t) = B [u(t)h(t)]1 h(t) ( h + gL )h(t):

How large is ? Still no consensus.

But this may be the key to understand the puzzle above.

Micro study: Borjas (1992, 1995): Large!

Why the govt subsidizes schooling?

(1) Externalities

(2) Imperfect Capital Market

42


30/159

A-School: R&D and Technological Advances

1. H School vs. A School

Y (t) = K (t) (A(t)h(t)L(t))1

H School A SchoolTo learn knowledge, everyone Once technology is found,

spends time/resources. it is (almost) free.Schooling, OJT, ... are emphasized. R&D, patent system, ... are emphasized.

They are not necessarily mutually exclusive. Often dicult to distinguish betweenthe two. They just emphasize dierent things. (Perhaps they are the same.)

So what is the view of this A school?

2. Price Decline in Capital Goods

Better computers, better telecommunication and transportation, robotization of assembly lines, ...

Machine becomes cheaper!

Lets look at price decline of capital goods compared to consumption goods.

Data: NIPA Table 1.X

1960 2000 2005General Price Index 21.044 100.00 113.039Price index for Investment Good ( s = 20%) 29.619 100.00 111.381

21.044= 29.619*0.2 +X*0.8Price index for Consumption Good (80%) 18.900 100.00 113.454

Growth of Price index for Investment Good: log(111 :381) log(29 :619)45 = 2.9%

Growth of Price index for Consumption Good: log(113 :454) log(18 :900)

45 = 4.0% According to data, investment goods become cheaper compared to consumptiongoods!

Model: Consumption goods and investment (or physical capital) goods are separate.

44


31/159

Model:max

f C (t );I (t )g1t =0

Z 1

0e t

(C (t)=L(t))1

1 L(t)dt (same)

s.t.

Y (t) = C (t) + I (t) (same)_K (t) = q (t)I (t) K (t) (dierent!)

For example,

Y (t)100 apples

= C (t)80 apples

+ I (t)20 apples

(same)

_K (t) = q (t)

|{z} =1 =2

I (t)

| {z } 20 applies transformed to 10 computersK (t) (dierent!)

So consumption goods and capital goods are dierent. I (t) units of consumptiongoods are transformed to q (t)I (t) units of capital goods.

=) q (t) is the "productivity" of capital goods production1=q (t) is the "price" of capital good (machine) compared to consumption good(apple)

Y (t) = K (t) (A(t)L(t))1 (same)A(t) = A(0)egA t (same)L(t) = L(0)egL t (same)q (t) = q (0)egq t (new!)

Eliminating I (t) and Y (t) :

maxf C (t)g1t =0 Z 10 e t (C (t)=L(t))11 L(t)dt

s.t.

_K (t) = q (t)[K (t) (A(t)L(t))1 C (t)

| {z } ]

= I (t)

K (t)

A(t) = A(0)egA t

L(t) = L(0)egL t

q (t) = q (0)egq t

45


32/159

How to eliminate A(t), L(t) and q (t)?

Dene:

c(t) = C (t)

q (t) =(1 )A(t)L(t);

k(t) = K (t)

q (t)1=(1 )A(t)L(t):

Then we have a simple form:

maxf c(t)g1t =0 Z 10 e t [c(t)q (t) =(1 )A(t)]11 L(t)dt= maxf c(t )g1t =0 Z

1

0e t

[c(t)]1

1 [q (0)](1 ) =(1 ) | {z } just constantegq [(1 ) =(1 )]t [A(0)]1

| {z } just constantegA (1 )t L(0)

|{z} just constantegL t dt

=) maxf c(t )g1t =0 Z 10 e ( gq [(1 ) =(1 )] gA (1 ) gL )t [c(t)]11 dt

= maxf c(t )g1t =0 Z 10 e t [c(t)]11 dt

where gq[(1 ) =(1 )] gA(1 ) gL

s.t._K (t)

q (t)1=(1 )A(t)L(t) =

q (t)[K (t) (A(t)L(t))1 C (t)]q (t)1=(1 )A(t)L(t)

K (t)

q (t)1=(1 )A(t)L(t)

= K (t) (A(t)L(t))1

q (t) =(1 )A(t)L(t)

C (t)q (t) =(1 )A(t)L(t)

K (t)

q (t)1=(1 )A(t)L(t)

= K (t)

q (t) =(1 )[A(t)L(t)]

C (t)q (t) =(1 )A(t)L(t)

K (t)

q (t)1=(1 )A(t)L(t)

= K (t)

q (t)1=(1 )A(t)L(t) C (t)

q (t) =(1 )A(t)L(t)

K (t)q (t)1=(1 )A(t)L(t)

= k(t) c(t) k(t)

How to deal with LHS? Since K (t) = k(t)q (t)1=(1 )A(t)L(t),

_K (t) = _k(t)q (t)1=(1 )A(t)L(t) + k(t) 1

1q (t)1=(1 ) 1 _q (t)A(t)L(t)

+ k(t)q (t)1=(1 ) _A(t)L(t) + k(t)q (t)1=(1 )A(t) _L(t)

46


33/159

To make the LHS,

_K (t)q (t)1=(1 )A(t)L(t) = _k(t) +

11 k(t)

_q (t)q (t)

|{z} = gq+ k(t)

_A(t)A(t)

|{z} = gA+ k(t)

_L(t)L(t)

|{z} = gL So the constraint is

_k(t) + gq1

k(t) + gAk(t) + gL k(t) = k(t) c(t) k(t):

So _k(t) = k(t) c(t) + gq1

+ gA + gL k(t)

Hamiltonian:

H = [c(t)]1

1 + (t) k(t) c(t) +

gq1

+ gA + gL k(t) :

FOCs are

H c = 0

H k = (t) _(t)

So

c(t) (t) = 0 ;

and (t) k(t) 1 + gq1

+ gA + gL = (t) _(t):

So k(t) 1 + gq1

+ gA + gL = _(t)(t)

= + _c(t)c(t)

47


34/159

Done! Lets recover our original notation:

k(t)1

+ gq1 + gA + gL = +

_c(t)c(t)

K (t)q (t)1=(1 )A(t)L(t)

1

+ gq1

+ gA + gL = gq[(1 ) =(1 )] gA(1

+ 0BBB@_C (t)

C (t)

1

_q (t)q (t)

|{z} gq_A(t)

A(t)

|{zgA_L(L(

|g K (t)q (t)1=(1 )A(t)L(t)

1

+ gq1

+ gA + gL = (1 )

1 gq gA(1 ) gL

+ _C (t)

C (t)

1

gq gA gL

K (t)q (t)1=(1 )A(t)L(t)

1

+ gq1

= 1

gq + _C (t)

C (t) gL

K (t)q (t)1=(1 )A(t)L(t)

1

= + 1 21

gq + _C (t)C (t)

gL! Note: If q (t) = 1 (and hence gq = 0), then this becomes exactly the same as theone-sector model we considered previously.

Note: The rst term of the LHS is, again, the interest rate!

Rewriting this FOC and all constaints:

(1) K (t)

q (t)1=(1 )A(t)L(t)

1

= + 1 21

gq + _C (t)C (t)

gL!(2) Y (t) = K (t) (A(t)L(t))1

(3) _K (t) = q (t)I (t) K (t)(4) Y (t) = C (t) + I (t)

Consider the BGP in which all varaibels grow at constant rates.

In (1), K (t )q(t )1=(1 ) A(t )L(t) is constant, implying that

(A) gK = 11

gq + gA + gL

48


35/159

In (2),gY = gK + (1 )gA + (1 )gL

So these two results imply

(B) gY = gK + (1 )gA + (1 )gL

= 1

1gq + gA + gL + (1 )gA + (1 )gL

= 1

gq + gA + gL

So gY is dierent from gK ! That is,

gY =

1

gq + gA + gL

= gq + 11

gq + gA + gL

| {z } = gK= gK gq In (3),

_K (t)K (t)

= q (t) I (t)Y (t)

Y (t)K (t)

= s(t)q (t)Y (t)K (t)

Notice that q (t) Y (t )K (t) is constant because we know gY = gK gq: So s(t) is constant!

Calibration: Use the following data

s = 0:20 (same as before)

gL = 0:011 (same as before)

gY = gC = 0 :033 (same as before)

K (K=Y ) = 0 :12 (same as before)

= 0:32 (same as before)

q (t) is the units of capital goods that can be bought by one unit of consumptiongood.

= (consumption good price) / (capital good price )

49


36/159

So gq = ( growth rate of consumption good price ) (growth rate of capital goodprice)

= 4:0% 2:9% = 1:1% Write the above equations on BGP:

(B):

gY = 1

gq + gA + gL

0:033 = 0:321 0:32

0:011 + gA + 0 :011

So gA = 0:017

(A):

gK = gY + gq= 0 :033 + 0:011 = 0:044

We can go further as usual to calibrate , r , etc.

But we will not do that. For growth accounting, this is enough:

Method 1:

Y t = K t (At Lt )1

=) Y t =Lt = ( K t =Lt ) (At )1

=) gY=L

| {z} (A) Real Per-capita GDP growth= 2 :2%= gK=L

| {z } (B) K/Ls contirubtion= (gK gL ) = 1 :1%+ (1 )gA

| {z (C ) As contribution= remaining 1:1% Method 2:

Y t = K t (At Lt )1

=) Y 1t = ( K t =Y t ) (At Lt )1

=) (Y t =Lt )1 = ( K t =Y t ) At 1

=) Y t =Lt = ( K t =Y t )=(1 )

At=) gY=L

| {z} (A) Real Per-capita GDP growth= 2 :2%=

1

gK=Y

| {z } (B) K/Ys contirubtion= 1 (gK gY ) = 0 :5%(contribution from cheaper

capital goods)

+ gA

|{z(C ) As contribu= remaining 150


37/159

Recall that when we measured h(t) by h = e0:1 S , we obtained gh = 0:8%:

So

gY=L

| {z} (A) Real Per-capita GDP growth=

1gK=Y

| {z } (B ) K/Ys contirubtion+ gh

|{z} (C ) hs contribution+ gA

|{z} (D ) As contribution (B): 0.5% (0.5/2.2=22.7%) contribution from cheaper capital goods (cheapercomputers, machines, ...)

(C ): 0.8% (0.8/2.2=36.4%) contribution from more schooling

(D): 2.2%-0.5%-0.8%=0.9% (0.9/2.2=40.9%) Still missing. (???)

So what causes capital goods to become cheaper? Maybe R&D? Greenwood, Hercowitz and Krusell (AER 1997) separate K into structure (build-ings) and equipment (computers, machines, ...). Structure price didnt decline.Equipment price declined a lot.

Eaton and Kortum (EER 2001) measure gains from importing (cheaper) capitalgoods.

51


38/159

International Knowledge Diusion

Main Paper: Lucas (2009, AEJ Macro)

Data

Sachs and Warners (1995) openness criterion

1. Eective protection rates less than 40%

2. Quotas on less than 40% of imports

3. No currency controls or black markets in currency

4. No export marketing boards

5. No socialistic govt

Good: Criterion for openness policy

cf. ((import)+(export))/GDP

Bad: Arbitrary criterion

0-1.

Open economies converge!

1. (Mechanical) Model of Catch-Up Growth A Leading Economy (Economy A):

Output per worker: yA(t) = hA(t)

|{z} Human Capital Stock per worker:

_hA(t) = hA(t). (So _hA (t )hA (t ) = :)

A Catch-up Economy (Economy B, with lower hB (t))

Output per worker: yB (t) = hB (t):

Human capital stock per worker: _hB (t) = [hB (t)]

1

[hA(t)](There is a knowledge ow from Economy A.)

Growth rate of B: _yB (t)yB (t) = _h B (t )h B (t ) = hhA (t )hB (t)i= hyA (t)yB (t )i:

Higher if hA (t )hB (t ) is higher

Higher if is higher. If = 0, then (Bs growth rate) = (As growth rate).

52


39/159

is the growth rate of per-capita income for Economy A. = 2% is reasonable.

Use A=UK or US. Then pick up any (open) economy B. Solve for in _yB (t )yB (t ) =

hyA (t )yB (t )i Try various values for . = 0:67 ts well with the data. Look at the data.

Problem: Some "open" economies (Thailand, Malaysia, Indonesia, Sri Lanka) aredown there.

2. Dual Economy Model

Motivation: (i) Agricultural employment shares decline as per-capita GDP grows.(ii) Some "open" economies (Thailand, Malaysia, Indonesia, Sri Lanka) are downthere.Employment Share of Agriculture in 1960:Hong Kong: 8% Indonesia: 75%South Korea: 66% Thailand: 84%

There are two sectors, "city" and "farm".

Labor is allocated to "city" (fraction 1 x) and "farm" (fraction x).

City production: yB;city = (1 xB )hB

Farm production: yB;farm = AB (hB )

|{z} spillover( xB

|{z} raw labor) (Land )1

| {z } =1 (const):

= psi: "Spillover" from "city" to "farm"The source of farms productivity growth

0 < < 1 : land is xed

Problem: Maximize yB = yB;city + yB;farm .

maxxB (1 xB )hB + AB (hB ) (xB ) :FOC:

hB = AB (hB ) (xB ) 1:

So xB =(hB )1

AB !1=( 1)53


40/159

= 0:6 (share of labor)

Then we can connect xB to yB . That is, since

(hB )1 = AB (xB ) 1

So hB = AB (xB ) 11=(1 ) ;

we can have an equation relating xB to yB .

= 0:75 (with A = (some unknown number)) ts the data.

Look at the data.

Original Model Dual Economy Model

_hB (t) = [hB (t)]1

[hA(t)] _hB (t) = 24 1 x(hB (t))

| {z } fraction of workers in city 35

[hB (t)]1

[hA(t)]

= 1 hB (t)1

A B

1=( 1)

[hB (t)]1 [hA(t)]

= zeta

= 0:6, = 2%; = 0:67, = 0:75 and = 1 (new!)

How will we apply this model?

Given: yB (1960) and xB (1960) for some initial year (such as 1960), we can solve for

hB (1960) and AB in the following equation system:

yB

|{z} data= (1 xB

|{z} data)hB + AB (hB )

0:75

z}|{ ( xB |{z} data)

0:6

z}|{ xB

|{z} data=

(hB )1

AB !1=( 1) Obtain hA(1960) with this method, too.

Then the growth of hB

is obtained:

_hB (t)hB (t)

=

|{z} 0:02241

h1BAB!1=( 1)35 | {z } known

1

z}|{ 2664hA(t)hB (t)

| {z } known3775

0:67

z}|{

54


41/159

This will give hB (1961): Then xB (1961) and yB (1961).

Then we can have a prediction on f hB (t)g and f xB (t)g and f yB (t)g forever:

Look at the data.

Remaining problems include "Why openness matters?"

55


42/159

EconS 502, S. Choi, Spring 2009, WSU

FIGURE: Income Doubling Times

(Lucas (2009), Figure 7.)


FIGURE: Income and Growth Rates, 112 Economics (1960-2000)



43/159


FIGURE: 25 European Economics (1960-2000)



FIGURE: 16 Asian Economics (1960-2000)



44/159


FIGURE: Model vs. Data



FIGURE: Agricultural Employment Shares: XS (1980)



45/159


FIGURE: Agricultural Employment Shares: TS (1980)






46/159





FIGURE: Hong Kong: Predictions vs. Observations



47/159


FIGURE: South Korea: Predictions vs. Observations



FIGURE: Indonesia: Predictions vs. Observations



48/159


FIGURE: Thailand: Predictions vs. Observations



49/159

Fundamental Equation of Asset Pricing

Derivation of Fundamental Eq. is in a separate note.

0. Review

Period 0 > Period t

Period 1 > Period t+1

Fundamental Eq.:

P it

|{z} Price of Asset i at Period t

= E t [m t+1 X it+1

|{z} Stochastic Payo of Asset i at Period t+1

] .....(1)

wherem t+1

u0(C t+1 )u0(C t )

= u0(Y t+1 )

u0(Y t ) (in equil).

For example, if u(C ) = C 1 =(1 ), > 0, 6= 1 (CRRA) and u(C ) = log C when ! 1; then

u0(C ) = C :

So

m t+1 = C t+1C t

= Y t+1Y t

:

We may also write

1 = E t 2664m t+1

X it+1P it

|{z} Gross Return3775

= E t 24m t+1 (1 + rit+1

|{z} Stochastic Net Return on Asswet i between t and t +1

)35 .....(2)1. Risk-Free Rate

Data: There is no risk-free asset in reality. The U.S. govt security is perhaps close.

r f t+1 (risk-free rate between t and t + 1 ) is, on average, 1%.

But again, (i) ination risk, (ii) default risk.

58


50/159

Calibration. Consider a risk-free asset that pays 1 unit at t + 1 . From (1),

P f t = E t [m t+1 ] = E t "

Y t+1Y t #

We know Y t +1Y t is about 1.02 on average. (Per-capita GDP growth is 2% on average.)

But the point of asset pricing is that the future is uncertain. We introduce someuncertainty.

Assumption: = 0 :95, = 2.

Data: Mehra and Prescott (1985)

(average of Y t+1 =Y t 1)= 0 :018. (standard deviation of Y t+1 =Y t 1)= 0 :036 (rst-order serial correlation of Y t+1 =Y t 1)= (Y t+1 =Y t 1; Y t+2 =Y t+1 1) =

0:14

There are two "states".

Y t+1Y t

1 = 0:018 + 0:036 = 0:054 (state 1: "good")0:018 0:036 = 0:018 (state 2: "bad")

Assume a "Markov chain." ( t +1 s state depends on ts state only and nothing else.)

Transition matrix = P = prob 1! 1 prob 1! 2prob 2! 1 prob 2! 2We make it symmetric.

P = prob1! 1 prob 1! 2prob 2! 1 prob 2! 2 = p 1 p1 p p =

0.43 0.570.57 0.43

(You should be able to recover serial correlation= 0:14.)

So

P f t = E t " Y t+1Y t #If state 1 at t =) P f 1

|{z} Price when today is "good"= (0:43)(1:054) + (0:57)(0:982) = 0 :929248

If state 2 at t =) P f 2 = (0:57)(1:054) + (0:43)(0:982) = 0 :911048

59


51/159

Result:

If this year ( t) is "good" ( t is state 1), the risk-free rate between this year ( t)and the next ( t + 1 ) is:

1 + r f 1 = 1P f 1

= 1

0:929248 = 1:076139

If this year is "bad" ( t is state 2):

1 + r f 2 = 1P f 2

= 1

0:911048 = 1:097637

= 1 = 1 = 2 = 2 = 0:95 = 0:99 = 0:95 = 0:99

r f 1 6.5% 2.2% 7.6% 3.3%r r 2 7.6% 3.2% 9.8% 5.3%

average 7.1% 2.7% 8.7% 4.3%

Risk-free Rate Puzzle: Risk-free rate is ridiculously high! Data: 1%.

2. Stock Returns

Endowment economy.

The representative consumer "owns" this economy.

The economy pays f Y t g as dividends.

What is the price of this economy ( W : wealth)?

(That is, what is the price of all companies?)

What is the "return"?

Data: rW t+1 (stock return between t and t + 1 ) is, on average, 8-9%.

Problem: Not all companies are in the stock market.

Set-Up:

+-+-

t t+1

At the beginning of t: Dividends paid

At the end of t: Assets traded

60


52/159


53/159

So

P 1P 2 = ( I A)

1A 11

if (I A) is invertible.

Use = 0:95; = 2: Then,

P 1P 2 =

14:268014:1436

But P t = P W t =Y t : So P W 1(Y )P W 2(Y ) =

14:2680Y 14:1436Y

Now think about returns.1 + r st+1 =

Y t+1 + P W t+1P W t

(1) Good in t, Good in t + 1 :

1 + r st+1 = Y t+1 + 14:2680Y t+1

14:2680Y t

= 1 + 14:2680

14:2680Y t+1Y t

= 1 + 14:2680

14:2680 1:054= 1 :1279

(2) Good in t, Bad in t + 1 :

1 + r st+1 = Y t+1 + 14:1436Y t+1

14:2680Y t

= 1 + 14:1436

14:2680Y t+1Y t

= 1 + 14:1436

14:2680 0:982

= 1 :0423

62


54/159

(3) Bad in t, Good in t + 1 :

1 + rs

t+1 = Y t+1 + 14:2680Y t+1

14:1436Y t

= 1 + 14:2680

14:1436Y t+1Y t

= 1 + 14:2680

14:1436 1:054

= 1 :1378

(4) Bad in t, Bad in t + 1 :

1 + r st+1 = Y t+1 + 14:1436Y t+1

14:1436Y t

= 1 + 14:143614:1436

Y t+1Y t

= 1 + 14:1436

14:1436 0:982

= 1 :0514

Average of all these four is 9.0%. (Roughly match with the data.)

But risk-free rate was predicted to be at 8.7%. The prediction on the "premium" istoo low.

Equity Premium Puzzle : The premium is only 0.3%. In reality, it is 7-8%. Two puzzles are probably related.

3. Comovement of Consumptions and Payos

(1):

P it = E t [m t+1 X it+1 ]

= E t [m t+1 ]E t [X it+1 ] + covt [m t+1 ; X it+1 ]

But (2):

1 = E t [m t+1 (1 + r f t+1 )]

= (1 + r f t+1 )E t [m t+1 ]

(Note: r f t+1 is the return between t and t + 1 . But we know it at t. So it comes outof E t .)

63


55/159

SoE t [m t+1 ] =

11 + r f t+1

The above eq. becomes

P it = E t [X it+1 ]1 + r f t+1

+ covt [m t+1 ; X it+1 ]

"Decomposition" of asset price:

First Term: Expected value of future payment, discounted by risk-free rate

(Expected value " =) Price ")

Second Term:

covt [m t+1 ; X it+1 ] = covt u0(C t+1 )u0(C t )

; X it+1

= covt u0(Y t+1 )u0(Y t )

; X it+1

= covt " Y t+1Y t ; X it+1#So this cov " =) Price "

Economic interpretation: If Y t +1Y t and X it+1 tend to "co-vary",

(If this asset pays more when the economy is good)

covt hY t +1Y t ; X it+1i is be highcovt Y t +1Y t ; X

it+1 is low

covt [m t+1 ; X it+1 ] is low(We dont need more when the economy is already good.This asset is less attractive.This is a cheap asset!)

Vice versa.

"Good" assets pay more in a recession. We want to hold them for insurance.So they are more expensive.

64


56/159

Another way to see the same implication (this time with r ):

1 = E t [m t+1 (1 + r it+1 )]= E t [m t+1 ]E t [1 + r it+1 ] + covt [m t+1 ; 1 + r

it+1 ]

= E t [1 + r it+1 ]

1 + r f t+1+ covt [m t+1 ; 1 + r it+1 ]

SoE t [1 + r it+1 ] = 1 + r

f t+1 (1 + r

f t+1 )covt [m t+1 ; 1 + r

it+1 ]

So

E t [r it+1 ] = rf t+1 (1 + r

f t+1 )covt [m t+1 ; 1 + r

it+1 ]

| {z } " risk adjustment"= r f t+1 (1 + r f t+1 )covt "Y t+1Y t ; 1 + r it+1# Economic Interpretation: If Y t +1Y t and r

it+1 tend to "co-vary"

(If this asset has higher return when the economy is good)

covt hY t +1Y t ; 1 + r it+1i is highcovt

Y t +1Y t ; 1 + r

it+1 is low

E t [r it+1 ] is high

(Expected return should be high!)

"I dont want to hold this asset. It pays more when theeconomy is already good. I will hold it only if you pay more on average."

65


57/159


U.S. Rate of Return on 3-Month T-Bill (1952Q1-2003Q2)

(Real, 206 Quarterly Obs)

-30.0%

-20.0%

-10.0%

0.0%

10.0%

20.0%

30.0%

1 9 5 2 Q 1

1 9 5 5 Q 1

1 9 5 8 Q 1

1 9 6 1 Q 1

1 9 6 4 Q 1

1 9 6 7 Q 1

1 9 7 0 Q 1

1 9 7 3 Q 1

1 9 7 6 Q 1

1 9 7 9 Q 1

1 9 8 2 Q 1

1 9 8 5 Q 1

1 9 8 8 Q 1

1 9 9 1 Q 1

1 9 9 4 Q 1

1 9 9 7 Q 1

2 0 0 0 Q 1

2 0 0 3 Q 1

Return on 30-DayTreasury Bill

Average (0.3%/Q)


U.S. Rate of Returns on Stocks (1952Q1-2003Q2)(Real, 206 Quarterly Obs)

-30.0%

-20.0%

-10.0%

0.0%

10.0%

20.0%

30.0%

1 9 5 2 Q 1

1 9 5 5 Q 1

1 9 5 8 Q 1

1 9 6 1 Q 1

1 9 6 4 Q 1

1 9 6 7 Q 1

1 9 7 0 Q 1

1 9 7 3 Q 1

1 9 7 6 Q 1

1 9 7 9 Q 1

1 9 8 2 Q 1

1 9 8 5 Q 1

1 9 8 8 Q 1

1 9 9 1 Q 1

1 9 9 4 Q 1

1 9 9 7 Q 1

2 0 0 0 Q 1

2 0 0 3 Q 1

Return on 30-DayTreasury BillAverage (0.3%/Q)

Return on Stocks(VW)Average (2.1%/Q)


58/159


59/159


HOW TO UNDERSTAND THE MOVEMENT OF r

1. Federal Funds Rate

2. Interest Rates on the U.S. Government Securities

3. Interest Rates on Corporate Bonds

4. Stock Returns


FEDERAL FUNDS RATE

- Banks maintain certain levels of reserves, as reserves with the Fed or as

cash.

- Federal funds rate : Interest rate at which banks (and other institutions)

lend and borrow for reserves with the Fed, usually overnight.

-

The Fed has a target federal funds rate.- This target is determined in the meeting. But Taylor rule predicts (?) the

target as iF rF whererF 2% 0.5 2% (: inflation rate)

(Other versions have unemployment rate or GDP growth rate.)


60/159


Figure: Taylor Rule: Actual and Predicted

(Data: Economic Report of the President, 2008.)






4. Stock Returns


61/159


INTEREST RATES ON THE U.S. GOVERNMENT SECURITIES

1. Low growth rate is expected / Risks are reassessed

- 1/ 1 r EY/Y - we see r .- Intuition: More private securities expected to default

Demand for less risky assets Interest rate on govt

securities



2. People become more risk-averse (?)

- Intuition: Higher demand for less risky assets r - Utility function: uC C / 1

Relative risk aversion: (Example: If 0, risk-neutral.)- 1/ 1 r EY/Y - If , e.g., from 0 to 1,

0: RHS = 1: RHS = (0.5/1.054+0.5/0.982) = 0.984

- So r ?????


62/159


MORE ON CRRA UTILITY FUNCTION

- uC C / 1 . What is ?- You have

m units of consumption goods. Consider a lottery,

x, with

Ex 0 and varx : You receive an additional fraction x of m units. You lose if x is negative.

- Your expected utility: E u m 1 x - Suppose for some , we have um 1 E u m 1 x.

= risk premium

m 1 = certainty equivalence



- uC C / 1 .- What is in the context of um 1 E u m 1 x?- LHS = u m m um m um u m

= um m m m um m m ( is relatively small)

- RHS = E u m mx E um mx um u m = um mumEx u m E x = um 0 m

- So roughly, /2. is proportional to certainty equivalence!


63/159



- Good! If , then more risk averse.- But that is not all about . It has another meaning.- Elasticity of Substitution

Utility: uC,C Elasticity of Substitution:

% / % /

% / % / =

If high: As 1 becomes more expensive ( p/p ), the consumer willmore sensitively react to replace 1 by 2 ( C/C )!



- Elasticity of Intertemporal Substitution Utility: uC uC Elasticity of Intertemporal Substitution:

% / %

% / % / =

If high: As interest rate rises ( 1 r ), the consumer will moresensitively react to save more ( C/C )!

For uC C / 1 ,

EIS =


64/159



- To conclude,

If , then more risk averse. ( is a measure of certainty equi.) If , then less sensitively react to a rise in interest rate. ( 1/ is an

EIS.)

- Risk aversion and intertemporal substitution are separate concepts.

- But our CRRA utility links them!

- No problem in growth because there was no uncertainty anyway.

- But now the uncertainty is our focus!

- Epstein and Zin (1991): Utility function that separates them.




- Intuition: Higher demand for less risky assets r - Utility function: uC C / 1

Relative risk aversion: Elasticity of intertemporal substitution: 1/

- If , consumers become more risk averse. r consumers need higher interest rate to save. r

- The latter may dominate. So it is possible that r .- What we need is the former only.


65/159



3. Govt runs larger fiscal deficits

- Higher supply of the U.S. govt securities Must be more

attractive so that they can be sold r 4. The Fed decreases Federal Funds Rates.

- Depends on how you view the monetary policy. Perhaps no

real effects?

- If real effects: More funds available, so demand . r .


Figure: INTEREST RATES ON THE U.S. GOVERNMENT SECURITIES

Note: Right figure is measured as deviation from 1990-2007 average.

(Source: IMF WEO, Oct 2008, p. 22)


66/159






4. Stock Returns


INTEREST RATES ON CORPORATE BONDS


- More private securities expected to default lower demand

for corporate bonds r Corporate Spread (=Risk Premium = Difference between

r and promised

r ) 2. People become more risk-averse (?)

- Lower demand for risky assets r CorporateSpread


67/159



3. Govt runs larger fiscal deficits

-

Higher supply of the U.S. govt securities r

U.S.

govt securities more attractive (if everything else is the same)

Perhaps lower demand for corporate bonds r - It also depends on whether this believed to stimulate the

economy. If so, maybe r (an offsetting effect).



4. The Fed decreases Federal Funds Rates.


real effects?

- If real effects: More funds available, so demand . So

r .- If this is believed to stimulate the economy,


68/159


Figure: SOVEREIGN AND CORPORATE BOND SPREADS

Note: Mostly against London Interbank offered rate.







4. Stock Returns


69/159


STOCK RETURNS

1. Growth rate is indeed low

- Y AK H - r AK H - If productivity A is indeed low, then r . (That is, dividend

. So stock return .)


- Lower demand for stocks Stock prices

r D P /P is lowered.


STOCK RETURNS


- Lower demand for risky assets Stock prices

r D P /P is lowered4. Govt runs larger fiscal deficits

- Higher supply of the U.S. govt securities r U.S.govt securities more attractive (if everything else is the same)

Perhaps lower demand for stocks Stock prices

- It also depends on whether this is believed to stimulate the

economy. If so, maybe Stock prices (an offsetting effect).


70/159


STOCK RETURNS

5. The Fed decreases Federal Funds Rates.


real effects?

- If real effects: More funds available, so demand . So Stock

price .

- If this is believed to stimulate the economy,


Figure: STOCK MARKET INDEX



71/159


72/159

So t is given in macroeconomy. Each asset is dierent becasue it is dierent.

2. "CAPM" as a Special Case of (2)

Assume = 1 (log utility).

Previously: We analyzed a one-period asset. A "wealth" portfolio: Pay P W t . ReceiveY t+1 + P W t+1 .

Now: Analyze a wealth portfolio as a multi-period asset. (Dierent ways, sameresult.) Pay P W t . Receive Y t+1 , Y t+2 , ...

maxa E t " 1

X =0

log(C t+ )#s.t.C t = Y t aP W t ;

C t+1 = Y t+1 + aY t+1 = (1 + a)Y t+1 ;C t+2 = (1 + a)Y t+2 ;:::

Unconstrained problem:

maxa

log(Y t aP W t ) + E t " 1

X =1 log((1 + a)Y t+ )#

FOC:

P W t1

Y t aP W t+ E t " 1X =1 1(1 + a)Y t+ Y t+ # = 0:

In equil, a = 0. SoP W tY t

+1

X =1

= 0 :

So

P W t = Y t1

X =1 = Y t 1 using P1 =0 = 1 =(1 ) for 0 < < 1.

83


73/159

So

1 + rW t+1 =

Y t+1 + P W t+1P W t =

Y t+1 + Y t+1 1Y t 1 =

1 Y t+1Y t :

So E t [1 + r W t+1 ] = E t1

Y t+1Y t

But then,

m t+1 = u0(Y t+1 )

u0(Y t ) =

Y t+1 =Y t

= 1

1 + r W t+1

Conclusion: By assuming log utility, the stochastic discount factor, mt+1 , can bereplaced by a function of r W t+1 ! So for any asset i,

(2) 1 = E t [m t+1 (1 + r it+1 )] = E t1 + r it+11 + r W t+1

(We dont need to worry about consumption growth or endowment growth.)

This representation is called CAPM (capital asset pricing model).

CAPM is often written as

E t [1 + r it+1 ] = it +

it

|{z} CAPM beta

E t [1 + r W t+1 ]

3. Arrow-Debreu Market

Today is time 0.

At time 0, consumers trade "claims" to future consumption goods for all possiblefuture histories.

q 0t (s t ): price at 0, in units of time-0 consumption goods, of a security that pays 1

unit of time- t consumption good if history st

(s0; s1;:::) occurs and 0 otherwise. Example: st = 1 ("good") > s t+1 = 1 ("good") (w/prob 43%)

s t+1 = 2 ("bad") (w/prob 57%)

s t = 2 ("bad") > s t+1 = 1 ("good") (w/prob 57%)

s t+1 = 2 ("bad") (w/prob 43%)

84


74/159

Example:

Suppose s0 = 1 (todays state is #1).

q 00(s0 = 1

| {z } known) = 1 (The price of todays consumption good is, of course, 1 unit of

todays consumption good.)

q 01(s0 = 1

| {z } known; s1 = 1

| {z } possible future scenario): Price of a security paying 1 at time 1 if s1 = 0

and 0 otherwise

q 01(s0 = 1

| {z } known

; s1 = 1; s2 = 0

| {z } possible future scenario

): ...

The problem of consumer j is to maximize

E 0" 1Xt=0 t u(c jt )#; 0 < < 1;or,

maxf cjt (s t )g

1

Xt=0 Xs t t u(c jt (s t )) prob(s t js0) BC: You sell all your claims to future contingent endowments. Then you buy allfuture contingent consumptions.

1

Xt=0 Xs t q 0t (s t )c jt (s t ) 1

Xt=0 Xs t q 0t (s t ) y jt (s t ) | {z } time- t endowment contingent on st | {z } value today of all j s future contingent endowment Lagrangian:

1

Xt=0 Xs t t u(c jt (s t )) prob(s t js0) + j " 1

Xt=0 Xs t q 0t (s t )y jt (s t )1

Xt=0 Xs t q 0t (s t )c jt (s t )# FOC wrt c jt (s t ) :

t u0(c jt (s t )) prob(s t js0) = j q 0t (s t )

for all i, t and s t .

85


75/159

So

q 0t (st ) = t u0(c jt (s t )) prob(s t js0)= j

1 = u0(c j0(s0))= j

So

q 0t (st ) = t

u0(c jt (s t ))u0(c j0(s0))

prob(s t js0)

Looks familiar? Related to the Fundamental eq.:

(1) P 0 = E 0 u0(C 1)u0(C 0)

X 1

In a competitive equilibrium of this Arrow-Debreu economy,

(i) Each consumer i solves the problem.

(ii) The solution f c jt (s t )g is feasible, i.e.,

X j c jt (s t ) X j y jt (s t ), for all t and s t : In this set-up, any asset can be broken into a set of Arrow-Debreu securities.

Example: The price of a risk-free security that pays 1 in period 1:

Xs1 q 01(s0; s1) Example: The price of a security that pays d(s t ) when period- t state is s t is

1

Xt=0 Xs t q 0t (s t )d(s t )

86


76/159

Term Structure of Interest Rates

Consider risk-free assets.

There is no ination (as usual).

time 0 time 1 time 2

++++

"We are here

$1 - - - - - - - - - - - > $1 + r (1)0

$1=(1 + r(1)0 ) - - - - - > $1

$1=(1 + r (2)0 ) - - - - - - - - - - - - - - - - - - - > $1

$1=(1 + r (1)1 ) - - - - - > $1

(not observable)

Notation: r (i)t : risk-free rate from t to t + i

(Our previous notation, r f 1 , is the same as r(1)0 here.)

Problem : How are r (1)0 , r(2)0 and r

(1)1 related?

1. Theory

(1):

P 0 = E 0[m1X 1]; where m1 = u0(Y 1)u0(Y 0)

:

r (1)0 satises

(A) 1

1 + r (1)0= E 0[m1] = E 0

u0(Y 1)u0(Y 0)

:

What about r (2)0 ? Write the Fundamental Eq. with a two-period asset:

maxa

u(Y 0 aeP 0) + E 0hu(Y 1) + 2u(Y 2 + aeX 2) +

3u(Y 3) + :::iWe have

eP 0 = E 0[em2 eX 2]; where em2 = 2 u0(Y 2)u0(Y 0)

:

91


77/159

r (2)0 satises

(B) 1

1 + r(2)0

= E 0[

em2] = E 0 2

u0(Y 2)

u0

(Y 0) Or, if we stick to a one-period version of Fundamental Eq.,

11 + r (2)0

= E 0[m1 (?)]

After one period, this two-period asset becomes exactly the same as a one-periodasset:

(C) 1

1 + r(2)0

= E 0

"m1

1

1 + r(1)1

# = E 0

" u0(Y 1)

u0

(Y 0)

1

1 + r(1)1

#= E 0 u0(Y 1)u0(Y 0) E 0" 11 + r (1)1 #+ cov0" u0(Y 1)u0(Y 0) ; 11 + r (1)1 #=

11 + r (1)0

E 0" 11 + r (1)1 # | {z } pure expectation

+ cov0" u0(Y 1)u0(Y 0) ; 11 + r (1)1 # | {z } risk adjustment

If we understand cov0 u0 (Y 1 )

u 0 (Y 0 ) ; 1

1+ r (1)1, then we have a prediction on r (1)1 given ob-

served r (1)0 and r (2)0 .

2. Application: Business Cycle and Term Structure

Calibrate (A) and (B). Leave (C).

= 0 :95, = 2:

There are two "states".

Y t+1Y t 1 = 0:018 + 0:036 = 0:054 (state 1: "good")0:018 0:036 = 0:018 (state 2: "bad")

More sophisticated environment: Drop an assumption of Markov transition.

At the beginning of time 0, we have "some" information that predicts future GDPgrowth:

92


78/159

Period 1 Period 2Case 1 "good": 75%, "bad": 25% "good": 75%, "bad": 25%Case 2 "good": 25%, "bad": 75% "good": 25%, "bad": 75%Case 3 "good": 75%, "bad": 25% "good": 25%, "bad": 75%Case 4 "good": 25%, "bad": 75% "good": 75%, "bad": 25%

... (More)

Problem : How are r (1)0 and r(2)0 related?

Case 1:

(A) 1

1 + r (1)0= E 0[m1] = E 0

u0(Y 1)u0(Y 0)

= E 0"Y 1Y 0 #= 0 :95 0:75(1:054)

2+ 0 :25(0:982)

2

= 0 :888

So r (1)0 = 12 :7%

And

(B) 1

1 + r (2)0= E 0[em2] = E 0

2 u0(Y 2)u0(Y 0)

= 2E 0"Y 2Y 1 Y 1Y 0 #(Assume 1s state and 2s state are indendent)

= 0 :952

[0:75 0:75 (1:054 1:054)2

+ 0 :25 0:25 (0:982 0:982) 2

+ 2 0:25 0:25 (0:982 0:982) 2]= 0 :788

So (1 + r (2)0 )1=2 1 = 12:7%

Summary:Annual risk-free rate Annual risk-free rate

on one-year security on two-year security(r (1)0 ) ((1 + r

(2)0 )1=2 1)

Case 1 (good/good) 12.7% 12.7% FlatCase 2 (bad/bad) 5.0% 5.0% FlatCase 3 (good/bad) 12.7% 8.7% Downward SlopingCase 4 (bad/good) 5.0% 8.7% Upward Sloping

... (More)

93


79/159

Draw the yield curve.

Fact: (1) The "yield curve" is typically upward sloping. (?)

(2) Downward-sloping "yield curve" tends to predict a recession.

94


80/159


TERM STRUCTURE OF INTEREST RATES

http://stockcharts.com/charts/YieldCurve.html


TERM STRUCTURE OF INTEREST RATES

(Source: The Economist , Jan 5th 2006.)


81/159


82/159

(where: q j : Period-0 price of Asset # j , Period-0 price of an Arrow-Debreu securitypaying 1 at state j at period 1 and 0 otherwise, equivalent to q 01(s = j ) in our

previous notation.) Consumer 2 solves a symmetric problem:

maxc21 ;c

22

13

log(x21) + 23

log(x22)

s.t.

q 1c21 + q 2c22 q 1e21 + q 2e22

= q 2

Market clearing:

c11 + c21 = e

11 + e

21 = 1;

c12 + c22 = e

12 + e

22 = 1:

Solution 1 :

Lagrangian for Consumer 1:

L1 = 23

log(c11) + 13

log(c12) + 1 q 1 q 1c11 q 2c12

The FOCs for Agent 1 are

23

1c11

= 1q 1;

13

1c12

= 1q 2;

q 1c11 + q 2c12 = q 1: (constraint itself)

Eliminating 1,

2c12c11 =

q 1q 2 :

The constraint becomes

q 2c12 = (1 c11)q 1:

So c121 c11

= q 1q 2

:

97


83/159

Finally

2c12c11 =

c121 c11 :

So 2(1 c11) = c11:

So c11 = 23

:

(Whats wrong if c12 = 0? Then corner solution. That is, log(0) in the utility.)

The FOCs for Agent 2 will similarly give c22 = 23 :

From market clearing, c12 = c21 = 13 .

Finally, the contraint gives q

1q2 = 1 : Solution 1 (Tricky Version):

For Agent 1:

If State #1 occurs, good! If State #2 occurs, bad.

It is obvious that Agent 1 sells Asset #1 and buys Asset #2.

The only problem is how many units of Asset #1 Agent 1 will sell (and howmany units of Asset #2 Agent 1 will buy)

Agent 1s problem:max

23

log(1 D) + 13

log

(D: Agent 1s delivery to Agent 2, if State #1 occurs at period 1.

: Agent 2s delivery to Agent 1, if State 2 occurs at period 1.)

Subject to the budget constraint:

q 1D

|{z} Agent 1s sales

= q 2

|{z} Agent 1s purchases

:

This is a symmetric problem, so WE KNOW q 1 = q 2.

Eventually the problem is

max 23

log(1 D) + 13

logD

98


84/159

FOC:23

11 D

+ 13

1D

= 0 :

So

2D = 1 D:

So D = 13

:

2. Default and Punishment

New Set-up: You dont have to provide a "full" delivery.

Example: You sold Now state #1 occured. You deliver1 unit of Asset #1 1 unit of consumption good

(You are obliged to to (in a complete market)

deliver 1 unit of consumption But you may default now!

good if state #1 occurs.) ! 2/3 units of consumption goodDefault on 1/3 unit. But receive

punishment.

Problem 2 : Agent 1s problem: Maximize

23264

log(1 D 111

| {z } = c11) ( ' 11 D 111

| {z } amount defaulted)+375

+ 13264

log( 12 K 22

| {z } = c12)375

' i j (varphi ): Agent is sales of Asset # j(i.e., if ' 11 = 13 , then Agent 1 sold

13 units of Asset #1. So he promised to

deliver 13 units of consumption goods if state 1 occurs and 0 unit if state 2occurs.)

Di

sj : Agent is delivery of consumption goods, at state s, regarding Asset # j(i.e., if D111 = 19 , then Agent 1 delivered only 19 units of consumption goods

after state 1 has occured, regarding Asset #1. This means he defauted on 29units.)

Notice a plus sign in ( ' 11 D 111

| {z } amount defaulted)+ . If ' 11 D 111 is negative, this agent is

paying even more than he promised. This will not happen!

99


85/159

: Default penalty per unit of defaulted consumption goods (e.g., legal pun-ishment)

i j : Agent is purchases of Asset # j(i.e., if 12 = 23 , then Agent 1 bought

23 units of Asset #2. If Asset #2 doesnt

default at all, he is delivered 23 units of consumption goods if state 2 occurs.)

K sj : Asset # j s delivery rate at state s(i.e., if K sj = 0 :5, then only 50% of promised 23 units are delivered.)

Subject to the budget constraint:

q 1 ' 11

|{z} Agent 1s sales

= q 2 12

|{z} Agent 1s purchases

:

This is a symmetric problem, so WE KNOW q 1 = q 2.

Eventually the unconstrained problem is

maxD; '

23

log(1 D) (' D)+ + 1324log(' K |{z} not the control of Agent 1

)35where I omitted all superscripts and subscripts since there are no confusions.

Dicult because "+"!

So consumer chooses ' at period 0 and then D at period 1.

STEP 1: Lets say ' is determined, and see how Agent 1 chooses D .

If ' = 0, then obviously D = 0.

If 0 < ' < 1, then D is between 0 and ' . (The agent doesnt gain at all by deliveringmore than what is promised!)

jjj ' 1+######### D

j + # =) (' D)+

j # +

j # ******* +

100


86/159

j # * +' #* +

j + =) log(1 D)

Solutions are one of the following three:

(i) D = 0

(ii) D is a solution to maxD log(1 D) (' D) and this D is between 0 and ':

(iii) D = ':

Lets discuss these solutions in detail.

Consider: maxD> 0 log(1 D) (' D): (without "+".)

The rst derivative is 11 D + :

(a) If 11 D + = 0 for some D < 0, then it is obvious that D = 0 : (Show by gure.)

Solution: D = 1 1

, If D = 1 1 < 0, then it is obvious that D = 0 :

, If < 1, then D = 0 .

(b) If 11 D + = 0 for some 0 < D < ' , then it is obvious that D = 1 1 : (Show

by gure.)

Solution: D = 1 1

, If 0 < 1 1 < ' , then it is obvious that D = 1 1 :

, If 1 < < 11 ' , then D = 1 1 :

(c) If 11 D + = 0 for some D > ' , then it is obvious that D = ' . (Show bygure.)

, If ' < 1 1 , then it is obvious that D = ':

, If 11 ' < , then D = ': (Full delivery! No Default!)

STEP 2: What is ' ?

(a) If < 1, then D = 0 . By symmetricity, K = 0. Nobody delivers anything.Nobody purchases any assets. ' = 0:

101


87/159

(b) For and ' satisfying 1 < < 11 ' , we have D = 1 1 : The problem is

max'23 log(1 1 +

1) (' 1 +

1) +

13 [log(' K )] :

FOC is23

+ 13

1'

= 0:

So 2 ' = 1:

So ' = 12

:

But this is meaningful if it satises 1 < < 11 ' . That is,

< 1

1 12 :

So < 12 1

2:

So < 22 1

:

So 2 1 < 2:

So < 32

:

Summary: If 1 < < 32 , then D = 1 1 and ' = 12 :

(c) For and ' satisfying 11 ' < , we have D = ': The problem is

max'

23

[log(1 ' )] + 13

[log(' K )] :

FOC is23

11 '

+ 13

1'

= 0:

So ' = 13

:

Same as the perfect market! But this is meaningful if it satises 11 ' < . That is,

11 13

< :

So > 32

:

102


88/159

Conclusion:

Punishment D ' K

> 32 High 13 13 1 Same as Perfect Market!1 < < 32 Medium 1

1 12

1 11

2= 2( 1)

< 1 Low 0 0 0 No Market

103


89/159


PICTURE: Hyperinflation in Germany, 1923-24

A woman in Germany feeds her tiled stove with moneThe money is worth less than firewood. (Source: Wikipedia


90/159


PICTURE: Hyperinflation in Hungary, 1945-46

Inflation rate: 1.3 10 16% per month. (Prices double every In 1946, 410 29 pengos became 1 forint by currency re

(Source: Wikipedia.com)


91/159


PICTURE: Hyperinflation in Yugoslavia, 1989-94

500 Billion Yugoslavia Dinar Banknote, 1993. The largest nominaofficially printed in Yugolavia.

(Source: National Bank of Serbia ( www.nbs.yu)


92/159


PICTURE: Hyperinflation in Zimbabwe, 2007-

A dinner at a hotel restaurant costs more than 1 billion Zimbabwe dto US$52.


93/159


94/159


FIGURE: Money Growth Rate and Inflation Rate

(110 Countries, 30-Year Average of 1960-90)

(Source: Lucas (1995), Figure 1. Taken from McCandless and We

MV=PY (QUANTITY THEORY OF MONEY)


95/159


FIGURE: Money Growth Rate and Inflation Rate

(U.S. Time Series, Smoothed)

(Source: Benati (2005), Chart 4(d).)


96/159

Cash-In-Advance Model

Growth Long-run Trend of outputAsset Pricing "Uncertainty" added

Monetary policy "Money" added

"Cash-in-Advance Model": Lucas and Stokey (1987, Econometrica)

A simple way to introduce money People need cash to buy goods

1. Set-Up

Endowment economy, Representative consumer

A. Endowment and Money Supply

Y t : (stochastic, exogenous) endowment

No storage =) In equilibrium, C t = Y t

M t : (stochastic, exogenous) money supply

=) ! t : (stochastic, exogenous) gross money growth rate, satisfying M t = ! t M t 1(injection of new money: (! t 1)M t 1. New money is lump-sum transferred

to the consumer.) How to describe an evolution of stochastic, exogenous variables:

s t = ( Y t ; ! t ): Macroeconomic "Shock". Assume s t is rst-order Markov.

F ( s0

|{z} next period: s

|{z} this period) is conditional distribution.

B. Asset Market (Bond Market)

QBt : $-price of one-period nominal bond

++time t time t + 1

Pay $ QBt Receive $1

N t : # of nominal bond held by the Representative consumer between t and t + 1

In equilibrium, Net demand=0 =) N t = 0

114


97/159


98/159

Constraint 2 - Cash-In-Advance constraint : Need cash to consume (even thoughit is for your endowment)!

P t C t M t ;for all t.

(articial constraint Doesnt explain why we need money.)

In equilibrium , the market clears:

N t = 0Y t 1 = C t 1

=) M t 1

|{z} money demand= M t 1

|{z} money supply3. Bellman Equation

Further assumption: We look for an equilibrium where

P t = M t p(s t ): Goods price will be proportional to MQBt = q b(s t ) : Bond price will not depend on M

(Reasonable!)

Constraint 1 - Budget constraint :

M t + QBt N t M t 1 + N t 1 + P t 1(Y t 1 C t 1) + ( ! t 1)M t 1:

So M t + q b(s t )N t M t 1 + N t 1 + M t 1 p(s t 1)(Y t 1 C t 1) + ( ! t 1)M t 1:

Normalize by M t .

So M tM t

+ q b(s t ) N tM t

M t 1M t

M t 1M t 1

+ N t 1M t 1

+ p(s t 1)(Y t 1 C t 1) + ( ! t 1) :

Dene m t = M tM t and n t = N tM t

:

m t + q b(s t )n t 1! t

[m t 1 + nt 1 + p(s t 1)(Y t 1 C t 1) + ( ! t 1)]

| {z

Date post:	02-Jun-2018
Category:	Documents
Upload:	aqpperu
View:	220 times
Download:	0 times

Clase 2009 Macro II WSU Dr Choi

Documents