Exercise $(100$ points$)$

Consider a two$-$period small open economy with one representative household who owns L units of

labour in each period and derives utility from aggregate consumption in each period. The household$$s

utility function is

U$($C$1,$C$2)=$ log C$1+$ log C$2$

where Ct is aggregate consumption and $>0$ is the discount factor. Labour is supplied to the market

by the household, who derives no utility from it so labour supply is fixed at Lfor both periods. The

household also owns the representative firms in each period and gets profits from them. The budget

constraints are

P$1$C$1+$ B $$ w$1$L$1+$ $1$

P$2$C$2$ w$2$L$2+$ $2+(1+$ i$)$B

where Pt is the price level, wt is the wage, i is the nominal interest rate and Lt is actual labour

employed. B is nominal bonds holdings by the household. It may be a negative number $($i$.$e$.$ the

household takes debt$).$ t is nominal profits. These are all expressed in home currency.

Production in this economy is handled by representative firms $1$ and $2.$ Firm t produces in period t

using only labour with the technology

Yt $=$ L$3/4$

t

where Lt is the amount of labour used in production and Yt is the total amount produced. Firm tis

perfectly competitive and chooses Lt and Yt so as to maximise profits, given by t $=$ PtYt$$ wtLt$.$

The consumption good in each period is traded in the international markets with its international

price being P$$

t $.$ The economy also has access to the international market for bonds. The international

nominal interest rate is i$.$ These are all expressed in a common foreign currency.

In period t$,$ one unit of foreign currency is worth Et units of home currency. Take Et as exogenously

determined.

Assume the nominal wage is downwardly rigid in period $1$: w$1$ w$1.$ Take w$1$ as exogenously

determined.

Consider labour as nontradable, so that under full employment:

L$1=100,$ L$2=100$

Because of wage rigidity, there may be unemployment in period $1.$ In that case, w$1=$ w$1$ and L$1$