2 The Model

2.1 Household Preferences and Endowments

Taiwan is a highly open economy, widely involving in international trades in goods and capital markets. Until the end of 2011, the export is 70% of the GDP and the import is 65% of the GDP, rising from 60% and 55% of export and import shares respectively in 2001. The …nancial market structure is well structured. In the end of 2011, 78% of the external funds are from the …nan-cial intermediary, and 22% is from the direct …nance including the stock and bond markets. Thus, a small-open-economy DSGE model with an established

…nancial sector can well characterize Taiwan’s economy.

Households own capital for rental and supply labor to both these export and home-goods …rms. Capital for rental to the …rms depreciates at the rate . When households accumulate or decumulate capital beyond the steady state level, they pay adjustment costs. The following law of motion is speci…ed for capital, with adjustment costs given by AC_t, and is the adjustment cost parameter.

We assume that investment goods are both domestically produced and im-ported from abroad, and that the price Pⁱ is the relevant price for these goods.

The investment variable is a CES aggregate of these two investment goods:

I_t=

The parameters _iand (1 _i) are the relative shares of foreign and domestic goods in the overall investment index, while iis the price elasticity of demand for each investment component. The variable K is the steady state level of the capital stock for domestic goods producing …rms.

The demand for each investment component is a function of their relative price:

The index P_t^f is the price of imported goods, in domestic currency, while P_t^x is the price of domestic goods-producing forms (which can be exported, or used for domestic consumption and domestic investment). The overall price index for investment goods is given by the following equation:

P_tⁱ= (1 _i) (P_t^x)^{1 1}+ _i P_t^{f 1 i}

1 1 i

(6) The household consumption at time t, C_t; is a CES bundle of both domestic consumption goods, C_t^d and imported consumption goods, C_t^f:

Ct=

The demand for each component of consumption is a function of the overall consumption index and the price of the respective component relative to the general price level, P :

C_t^d = (1 ₁) P^x

The parameters ₁and (1 ₁) are the relative shares of foreign and domestic goods in the overall consumption index, while 1is the price elasticity of demand for each consumption component.

Domestically-produced goods are composed of both non-traded services C_t^h and home-produced traded goods C_t^x (some of which are consumed domesti-cally). The following CES aggregator is used for domestically-produced con-sumption goods:

5

C_t^d= (1 ₂)¹² C_t^h The relative demands for the home non-traded goods and the export goods are given by the following equations:

C_t^h = (1 ₂) P_t^h

where the parameters ₂ and (1 ₂) are the shares of the export and non-traded goods in domestic production of consumption goods, and ₂is the price elasticity of demand.

The domestically-produced price index is given by the following CES aggre-gator:

P_t^d=h

(1 ₂) P_t^{h 1 2}+ ₂(P_t^x)^{1 2}i₁¹₂

(13) In the same manner, the overall price index, of course, is a CES function of the price of foreign and domestic consumption goods:

Pt= (1 ₁) P_t^{d 1 1}+ ₁ P_t^f

1 1 1

1 1

(14) In addition to buying consumption goods, households put deposits Mt in the bank and receive dividends from the export and non-traded or home-goods producing …rms. Total dividends is given by _t, with _t = ^x_t + ^h_t: The household pays taxes on labor income W_tL_tand on consumption _cC_t: The following equation gives the household budget constraint (P_t^f is the price of imported goods):

W_tL_t+ (1 + R^m_{t 1})M_{t 1}+ _t+ R^k_tK_t (15)

= P_tC_t(1 + _c) + M_t+ W_tL_t+ P_tⁱI_t+ P_tⁱ I_t K ² 2K_t^x

!

We assume that government spending G is bundled with consumption for utility in CES aggregator. We do this to indicate that there is a reason for government spending to take place, that such spending creates externalities for consumption, in the form of services which enhance household marginal utility (such as law enforcement and communication services):

Cet= _CC_t ^{+ (1 _C)G_{t 1}^{

1

{ (16)

6

However, household utility does not simply come from the current consump-tion bundle. Rather, habit persistence applies to this consumpconsump-tion index when it enters the speci…c utility function, so that the relevant consumption index is de‡ated by the Habit Stock, H_t. The Habit stock is a function of the lagged average consumption bundle, raised to the power %, the habit persistence para-meter:

Ht= eC

%

t 1 (17)

Overall utility is a positive function of the consumption bundle, the habit stock and a negative function of labor:

U ( eCt=Ht+; Lt) = Z_t^c

Ce_t=H_t ¹

1 ^L

L^1+$_t

1 + $ (18)

The parameter is the relative risk aversion coe¢ cient, while is the disu-tility of labor, and $ is the Frisch labor supply elasticity. The variable Z_t^C is a shock to the utility of consumption and evolves according to the following process:

ln(Z_t^c) = _cln(Z) + ^c_t (19)

c

t N (0; ²_c ) (20)

The household chooses the paths of consumption, labor, deposits, investment and capital, to maximize the present value of its utility function subject to the budget constraint and the law of motion for capital. Thus, the objective function of the household is given by the following expression:

M ax

fC^t;Lt;Mt;It;KtgE_t X1

=0

U ( eC_t+=H_t+; L_t+) (21) where the parameter represents the constant, exogenous discount factor.

This optimization is subject to the two constraints:

1.

The variable R^k_t is the rental rate for capital to the goods-producing …rms , R^m_t is the return on deposits held at banks, while Wtis the nominal wage rate.

The household optimization is represented by the intertemporal Lagrangean:

7

(24)

Note that there are two Lagrange multipliers, one, t+, is the marginal utility of income, while Qt+i, known as Tobin’s Q, is the shadow price of capital.

Optimizing the Bellman equation with respect to the decision variables Ct; Lt; Mt; I; Kt yields the following set of First-Order Conditions for the

The …rst equation, Eq. (25), simply tells us that the marginal utility of wealth is equal to the marginal utility of consumption divided by the price level. The second equation, Eq. (26), states that the marginal disutility of labor is equal to the after tax marginal utility of consumption provided by the after-tax wage. The third equation is the Keynes-Ramsey rule for optimal saving: the marginal utility of wealth today should be equal to the discounted marginal utility tomorrow, multiplied by the gross rate of return on saving (in the form of deposits).

The equation for Tobin’s Q tells us that the value of capital today is the discounted marginal utility of capital tomorrow, multiplied by the return to capital, in addition to the reduced value of adjustment costs in the future (due to the higher level of capital) and the discounted value of capital tomorrow, net of depreciation.

Finally, the investment equation tells us that investment will be equal to the steady state investment, K, when ^Qt

t = P_tⁱ: Any increase in Tobin’s Qt, relative to the marginal utility of income and the price of investment goods, will trigger increases in investment.

8