Programming Homework Help

Consider two large open economies, home and foreign (foreign variables that need not

be equal to home variables are denoted by an asterisk). Each economy is inhabited by a

continuum of identical individuals grouped into an aggregate risk sharing household. In each

country there is also a representative nal goods producing rm. International trade occurs

in this nal good. Lifetime utility is given by:

U = Et

X1

t=0

t

lnCt

1 +

N(1+)=

t

and U = Et

X1

t=0

t

lnC

t

1 +

(N

t )(1+)=

,

where: Et is the expectation operator, 2 (0; 1) is the (constant) subjective discount factor,

C and C denote consumption,

> 0 and

> 0 are parameters, > 0 is the Frisch

(marginal value of real wealth held constant) elasticity of labor supply, and N and N

denote labor.

Production in each country is determined by:

Yt = AtZN

t K1

t and Y

t = A

tZ (N

t ) (K

t )1 ,

where: A and A are stochastic technology processes; 2 (0; 1); K and K denote capital;

and Z and Z are scaling parameters. Furthermore:

“

lnAt

lnAt

#

=

“

v

v

# “

lnAt 1

lnA t 1

#

+

“

“t

“t

#

,

where: ; > 0; v; v > 0; Et (“t) = Et (“t

) = 0; and the standard deviations of ” and “

are, respectively, ” and ” . In the preceding, all variables are normalized by the world

population, which consists of a unit mass. Also, the evolution of capital in each country is

given by:

Kt+1 = It + (1 )Kt

and

K

t+1 = I

t + (1 )K

t ,

where I and I denote investment and and are capital depreciation rates. Finally,

changing capital holdings involves a real adjustment cost of the form

2

(Kt+1 Kt)2

for the home country and

2

K

t+1 K

t

2

2

for the foreign country, where and are positive parameters; this adjustment cost means

that the faster adjustments in the capital stock are the more expensive they are. Furthermore,

these costs are symmetric, so that reducing capital is as expensive as expanding it. The way

adjustment costs are written here, replacing depreciated capital does not generate adjustment

costs.

A benevolent world social planner solves the following problem:

max

Ct;C

t ;Nt;N

t ;Kt+1:K

t+1

Et

X1

t=0

t

lnCt

1 +

N(1+)=

t

+(1 )

lnC

t

1 +

(N

t )(1+)=

.

such that:

Ct + It +

2

(Kt+1 Kt)2 + G + NXt Yt,

C

t + I

t +

2

K

t+1 K

t

2

+ G + NX

t Y

t ,

Kt+1 = It + (1 )Kt,

K

t+1 = I

t + (1 )K

t ,

Yt = AtZN

t K1

t ,

Y

t = A

tZ (N

t ) (K

t )1 .

Above, NX is dened as Yt Ct It

2 (Kt+1 Kt)2 G and NX

t is dened as Y

t

C

t I

t

2

K

t+1 K

t

2

G. In addition, is the fraction of the world population that

lives in the home country. So, the benevolent world social planner is weighting the utility of

each country by their relative size in the world economy. Finally, G and G are exogenous

government consumption.