Households’problem

For households of vintage v, their life-time utility function at time t is assumed to have the following form

U_tv = 1 X i=0 i _{log c}v t+i + log 1 h v t+i where cv

t and hvt denote individual consumption and labour supply. and are

respectively the discount factor and the weight controlling the relative impor- tance between consumption and leisure.

The budget constraint facing the individual is

v 1t+1+ v 2t+1=r1t v1t+r2t v 2t+l v t c v t

for all t where lv

t denotes generation v’s labour or human income during period

t. It is given by the product of labour supplyhv

t and real wage (nominal wagegt

over CP I,pt).

lv_t = gt

pt

hv_t

v

t denotes generationv’s net holding of a particular asset at the end of period

t 1. We label the home equity asset1and the foreign equity asset2. So v

1tand v

2t are respectively their holding of home and foreign assets. r1t and r2t denote

the two assets’ gross rate of return which we will de…ne later. So the budget constraint states that households can save by investing in the two assets.

It is useful to de…ne the gross wealth as the sum of all holdings across asset

w_tv = v_1t+ v_2t

so above constraint can also be written as

wv_t+1=r2twtv+ v

1trxt+lvt c v t

where rxt=r1t r2t is the excess return of asset 1 over asset2.

The household’s problem is to choose optimalcv_t,lv_t, vts to maximize their life-

time utility,Uv

t;subject to all intertemporal budget constraints. Their behaviour

can thus be described by the following …rst-order conditions

t = (cvt) 1 (cv_t) 1 = Et h r1t+1 cvt+1 1i (cv_t) 1 = Et h r2t+1 cvt+1 1i hv_t = 1 pt gt cv_t

where t is the Lagrangian multiplier associated with time-t budget constraint.

Foreign households maximize the utility function of the same form. However, their budget constraints read

st 1vt+1+ v 2t+1 =st(r1t 1vt +r2t v 2t) +l v t c v t

where st denotes the real exchange rate at timet. That is the price of the home

consumption basket in terms of the foreign consumption basket. It appears in the constraint because we adopt the following convention. Apart from the asset- related variables (including foreign asset holding tv and return r2t) which are

denoted in terms of the home country consumption basket, all the other variables are in terms of the local consumption basket. The related …rst-order conditions are obtained similar to those of home country.

For the reasons we elaborated in the last chapter, even though households’ optimization is undertaken at the individual level, we care about the model’s per capita outcome (in other words, aggregate behaviour). By the demographic assumptions we have made at the beginning of this section, apart from the Euler

equations, all the other equations above are linear relations which means they can be easily aggregated into the related per capita version by our aforementioned method of population-weighted averaging. For example, per capita labour income is just the product of real wage and per capita labour supply

lt=

gt

pt

ht

Per capita budget constraints for the two countries are

(1 +n) ( 1t+1+ 2t+1) =r1t 1t+r2t 2t+lt ct

st(1 +n) 1t+1+ 2t+1 =st(r1t 1t+r2t 2t) +lt ct

where (1 +n) emerges because we assume new generations are born with no assets so t

t= 0 following Weil(1989). This is also familiar from Section3:3:2of

Chapter 3.

Aggregating the Euler equations is a little bit more di¢ cult. Section3:3:5of Chapter 3 described how we deal with them in more detail. To put it simply, because to solve the model we still use the standard method of log-linearization around the (per capita) steady states, we do not need the model in exact form if we can obtain their linearized form. So for the Euler equations, rather than to obtain the exact (per capita) form and then linearize them, we linearize them …rst and then aggregate them. The resulting linearized equations are omitted here and can be found in Section 4:B of the Appendix to this chapter.

To end this subsection, let us de…ne asset returnsr1t and r2t

r1tz1t=dt+ (1 +n)z1t+1

r2tz2t=

d_t st

+ (1 +n)z2t+1

where z1t and z2t denote the prices of asset 1 and 2 (in terms of the home

consumption basket) at the end of period t 1. dt is the dividend paid by the

home …rm whiled is that paid by the foreign …rm. stconverts the latter into the

home consumption basket. Note the asset returns are de…ned at the per capita aggregate level. ( In fact all the aforementioned variables are per capita variables as is indicated by the absence of the superscript v:)