1 Model
The model presented in this final project is an open economy real business cycle model along the lines of Schmitt-Grohé and Uribe (2003). The model is the real business cycle version of the framework described in the NK small open economy lecture notes. In this model households can trade with the rest of the world, both goods and financial assets. The small-open-economy assumption implies that domestic households take the international interest rate as given. Domestic and foreign households consume the same good. The international financial market is complete: domestic households can trade a complete set of Arrow-Debreu securities with the rest of the world.
1.1 Households
Consider an economy with infinitely many identical households. The representative household maxi- mizes discounted lifetime expected utility as follows:
where ct and ht are the household consumption and labor supply at time t , respectively. 0 < β < 1 is a discount factor. The disutility of labor supply is controlled by a scale parameter κL > 0 and the inverse of the Frisch elasticity φ > 0. When maximizing lifetime utility Eq. (1), the household faces the following budget constraint in every period t :
ct + it +Et (st+1bt+1) = r kkt−1 + bt + wt ht − Tt +Γt , (2)
k = (1 − )k + 1 − κI it − 1 2 i , (3)
t δ t −1
2 it −1
1
where bt denotes a random variable indicating the number of assets purchased in period t to be deliev- ered in each state of period t + 1. The variable st,t+1 is the period-t price of an asset that pays one unit of good in a particular state of period t + 1, divided by the probability of occurrence of that state given information available in period t . Notice that the utility function is non-separable between consumption and labor. Γt represents dividend income, or (real) profit received from the ownership of firms. r k is a gross return of capital. Tt represents a lump-tax (or transfers from the government). wt is the real wage rate. [1 − κI ( it − 1)2] indicates the investment adjustment costs. δ is a depreciation rate.
2 it −1
1. Please first write out the Lagrangian function faced by households. Suppose λt and qt
are lagrangian multipliers associated with constraints (2) and (3), respectively.
2. Please also derive the FOCs with respect to consumption, bonds, and labor from your Lagrangian problem.
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