The representative household

There is a representative household in the small open economy. The representative household maximizes expected lifetime utility, which is de…ned over consumption, Ct, and labor hours,

4In contrast, McCallum (2006, 2007) does not have capital accumulation in his model.

5However, the qualitative results of this paper will not change if the alternative speci…cation is adopted.

The results are available upon request.

6While incomplete exchange rate pass-through can have important implications for welfare comparisons of monetary policy regimes (e.g. Devereux and Engel, 2003; Corsetti and Pesenti, 2005), we will leave the case of incomplete pass through for future research.

L_t. Period utility function is speci…ed as separable in consumption and labor hours:

E₀

t=1X

t=0

tU (C_t; L_t) ; (1)

U (C_t; L_t) = C_t¹ 1 1

L¹⁺_t

1 + ; (2)

where Et is the expectations operator conditional on time t information; 2 (0; 1) is the subjective discount factor, 0 is the inverse of Frisch labor supply elasticity, and > 0 is a preference parameter.

The representative household owns the capital stock, Kt, in the small open economy. The capital stock evolves according to the law of motion:

K_t+1+1 2

fK^t+1 K_tg²

K_t = (1 ) K_t+ I_t; (3)

where It is the gross investment. ¹₂ ^fKt+1_K^Ktg2

t with 0 is a capital adjustment cost.

2 (0; 1) is the depreciation rate of the capital.

In addition to choosing consumption, labor hours, capital stock and investment, the representative household also holds a risk-free domestically traded domestic currency de-nominated bond, At+1 and a risk-free internationally traded foreign currency denominated bond, Bt+1. The budget constraint for the representative household is:

A_t+1+ e_tB_t+1+ P_tC_t+ P_tI_t= A_tR_{t 1}+ e_tB_tR^f_{t 1}+ R^k_tK_t+ W_tL_t+ D_t; (4)

where et is the nominal exchange rate, expressed as the number of unit of domestic currency required to purchase one unit of foreign currency. Pt is a price index for the domestic

…nal goods, to be de…ned formally below. Rt is the nominal interest rate on domestically traded bond. R^f_t is the nominal interest rate on the internationally traded foreign currency

denominated bond. R_t^k is the nominal rental rate of capital. Wt is the nominal wage rate.

D_t is the dividend from owning domestic …rms.

Following Kollmann (2002), we assume that the interest rate at which the domestic representative household can borrow or lend foreign currency fund, R^f_t, is subjected to a

"spread" from the foreign nominal interest rate, R_t. The "spread" is assumed to be a decreasing function of the net foreign asset position of the domestic economy:

R_t^f = R_t B_t+1

P_t^xQ^x_t; (5)

where P_t^xQ^x_t is the nominal value of the small open economy’s export in foreign currency term, to be de…ned formally below. The parameter > 0 captures the extent of …nancial integration. A lower value of corresponds to a higher degree of integration with the inter-national …nancial markets. The spread term also plays the role of "closing" the small open economy model, to ensure that the small open economy model has a stationary equilibrium (Schmitt-Grohé and Uribe, 2003). R_t is assumed to follow an exogenous process in this model.

The …rst order conditions for the representative household’s maximization problems are:

L = W_t Equation (6) equates the marginal disutility and marginal bene…t of labor hours.

Equa-tion (7) is the domestic bond’s Euler equaEqua-tion. EquaEqua-tion (8) is the Euler equaEqua-tion for internationally traded bond. Following Kollmann (2002), a term, '_t, is exogenously imposed on the Euler equation for internationally traded bond, so that up to a log-linear approxi-mation, equations (7) and (8) imply, Et e^_t+1 = ^R_t R^_t^f '^_t, where e_t e_t=e_{t 1} and a hat on a variable denote log deviation of that variable from its steady state. The term '_t, can be interpreted as an uncovered interest parity (UIP) shock, which is designed to capture deviations from the UIP condition. Equation (9) is the capital Euler equation.

2.2 Firms

There is a continuum of monopolistically competitive domestic intermediate goods …rms, indexed by i 2 [0; 1]. The production function for an intermediate goods …rm i is:

Y_i;t = ^#1 h

where Yi;t is the output of …rm i, Ki;t and Li;t denote the capital stock and labor hours used by …rm i, respectively. t is an exogenous economy-wide technology process. tKi;t Li;t1

is the domestic value added in the production. Q^m_i;t is the amount of imported goods used as input by …rm i. 2 (0; 1) is a parameter that determines the share of domestic value added in the production. # > 0 is the elasticity of substitution between domestic value added and imported goods in the production. The parameter 2 (0; 1) determines the share of rental income in domestic value added.

Firm i chooses Ki;t, Li;t and Q^m_i;t by solving a cost minimization problem:

min R^k_tK_i;t+ W_tL_i;t+ P_t^mQ^m_i;t; (11)

where P_t^m is the price of the imported goods in domestic currency term. The …rst order

where M Ct is the Lagrange multiplier associated with the constraint (12), which can also be interpreted as the nominal marginal cost.⁷

Following the literature, we assume that the intermediate goods are aggregated into composite …nal domestic goods, Yt, via the Dixit-Stiglitz aggregator:

Y_t =

where v is the elasticity of substitution between di¤erent varieties of intermediate goods.

Cost minimization leads to the following demand function for Yi;t:

Y_i;t = P_i;t

7Given the structure of the model, nominal marginal cost will be equalized across …rms, so there is no subscript i on M Ct.

The composite …nal domestic goods is demanded as consumption and investment goods in the domestic market as well as exported:

Y_t= C_t+ I_t+ Q^x_t; (19)

where Q^x_t is the export demand. Following McCallum (2006, 2007), McCallum and Nelson (1999) and Kollmann (2002), we assume that Q^x_t depends on the ratio of export price in foreign currency term, P_t^x, relative to the foreign price level, P_t:

Q^x_t = P_t^x

P_t ; (20)

where > 0 is the elasticity of substitution for export; > 0 is a scaling factor. P_t is an exogenous process in this model. Following McCallum (2006, 2007) and McCallum and Nelson (1999), we assume that …rms cannot price discriminate across markets, so that the export price in foreign currency is simply the …nal domestic goods price divided by the nominal exchange rate, P_t^x = P_t=e_t. Similarly, following McCallum (2006, 2007) and McCallum and Nelson (1999), the price for imported goods in domestic currency term is simply the product of foreign price level and nominal exchange rate, P_t^m = e_tP_t.

We assume that price adjustment for the intermediate goods …rms is staggered à la Calvo (1983). Each period, each intermediate goods …rm i faces a random probability of (1 ), 2 [0; 1], of resetting its price, P^i;t. If Pi;t is not reset, it is updated by the steady state in‡ation rate, , according to the rule Pi;t = P_{i;t 1}. Let ~P_i;t denote the new price that is reset in period t. After resetting the price at period t, there is probability that the price has not been reset at period t+ , and hence Pi;t+ = P~_t. Using the demand function, (17),

and the price updating rule, the optimization problem for …rm i in the domestic market is:

t+ is a discount factor for evaluating pro…t streams, T C( ) is the total cost as a function of the demand. The …rst order condition for the optimization problem is:

P~t=

Market clearing for the labor, capital and imported goods markets requires the supplies, Lt, K_t and Q^m_t , equal the sum of demand from all domestic intermediate goods …rms:

L_t=

Since the domestic bonds, At, is assumed to be traded only domestically, its net supply is zero in equilibrium:

A_t = 0: (25)

De…ning Y_t^A R1

0 Y_i;tdi, equation (17) can be aggregated across …rms as:

Y_t^A= u_tY_t; (26)