**2. The Model**

**2.2 Firm**

**2.2.1 Production of non-housing goods and services**

The budget constraint of renter households is shown below:

###

1

_{c}###

*c*

*, *

_{R t}*r h*

*,*

_{h t}*,*

_{R t}_{}1

###

1

_{yR}###

*w l*

*, , *

_{R t R t}*tr*

*, (17)*

_{R t}where * _{yR}* is the income tax for renter households who receive transfer in a
lump-sum fashion from the government,

*tr*

_{R t}_{,}. Renter households can’t accumulate capital and occupied-housing. We solve the problem by Lagrange multiplier method for maximizing their utility and yield the first-order conditions for consumption (18), renter housing (19), and labor supply (20) where

_{R t}_{,}is the Lagrange multiplier of

**2.2.1 Production of non-housing goods and services **

Producers, each of whom is indexed by *j* , use a Cobb-Douglas

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constant-return-to-scale technology that uses capital and three types of labor as inputs
to produce a non-housing goods, *y**t*

###

*j*for consumption, investment and government expenditure (21). Consumption uses the CES technology (22).

Investment and government uses the CES technology as consumption.

###

1 ,###

,###

,###

^{1}and renter households respectively.

*z denotes the aggregate productivity shock*

*which follows an AR(1) process:*

_{t}###

1 ,log*z** _{t}* 1

*log*

_{z}*z*

*log*

_{z}*z*

_{t}_{}

*(23)*

_{z t}where * _{z}* is the shock parameter of the productivity shock persistence and

_{z t}_{,}is the stochastic shock of productivity. The firm

*j*minimizes the total cost subject to the production function by choosing the quantity of output and input which include three types of labor demand (24)-(26), capital (27) and utilization rate of capital (28):

,

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where * _{t}* is the Lagrange multiplier of the firm which is equal to the marginal cost
as shown in (29):
the elasticity of the utilization rate to the rental rate of capital. The type of the firm is
monopolistically competition; they produce heterogeneous goods and aggregate them
into a homogeneous good using the CES aggregate production function as in
Iacoviello (2005). In order to motivate price stickiness, we assume that firms have
implicit cost when they adjust the nominal price as in Bernanke et al. (1999). Each
firm sells

*y j at the price*

*( )*

_{t}*P j . Aggregate output index and price index are*

*t*

###

given by (30)-(31):

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So, each firm faces the individual demand curve:

###

*t*

###

^{y t}^{,}

which follows an AR(1) process:

###

with the probability*d*. Each firm solves the optimal prices given by:

,

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reset nominal price and the frim index is dropped. Hence, the first-order condition of firms is: period, the aggregate price level can be written as follows:###

^{,}

###

optimizers as a function of the inflation rate:

###

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Combining (36) and (37) yields a forward-looking Phillips curve.

**2.2.2 Capital and housing producers **

To make sure that capital and housing have a single price across agents, we
assume that the accumulations of them are undertaken by perfectly competitive
market such as Bernanke et al. (1999). First, following the setting in Alpanda and
Zubairy (2013), the capital producers purchase the undepreciated part of the installed
capital from patient households at a price *q*_{k t}_{,} , and then, they make new capital
investment goods from the final goods producers at a relative price of 1 to produce the
capital stock. The capital stock will be carried to the next period. The production is

where * _{ik}* is the parameter of the investment adjustment cost of capital.

The installed capital stock that capital producer produce will be sold back to the
patient households at the price *q*_{k t}_{,} . Therefore, the capital producer will maximum
their profit subject to the law of motion of capital:

###

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where the discount rate of patient households is used for discounting future payoff.

For optimization, we can obtain the first-order condition of capital which yields an investment demand:

Similar to capital producer, we also can get the law of motion of housing that is subject to adjustment costs when the investment is made:

###

back to the patient households at the price*q*

_{h t}_{,}. Therefore, the housing producer will maximize their profit subject to the law of motion of housing:

###

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optimization, we can get the first-order condition of housing and yield an investment demand：**2.3 Central bank policy ** **2.3.1 Fiscal policy **

The fiscal authority makes the aggregate transfer to households：

1

where is the level parameter and * _{b}* is the parameter that determines the response
of transfers to government debt. The total tax revenue of the government comes from
the tax of consumption, labor income, rental housing income, capital income, interest

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income and property as following:

###

The accumulation of government debts follows the law of motion as follows:

###

1 1###

^{1}t , t , t ,

where *g is the government expenditure which follows an AR(1) process: *_{t}

###

^{1}

^{,}

log*g** _{t}* 1

*log*

_{g}*g*

*log*

_{g}*g*

_{t}_{}

*(52)*

_{g t}where * _{g}* is the shock parameter of government expenditure persistence,

*g is the*government expenditure level in steady state and

_{g t}_{,}denotes the fiscal policy shock.

**2.3.2 Monetary policy **

The central bank uses the Taylor rule to target the nominal interest as in Alpanda and Zubairy (2013):

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where is the policy parameter of persistence of Taylor rule, _{} is the policy
parameter of the inflation of Taylor rule, * _{y}* is the policy parameter of the output of
Taylor rule and

_{R t}_{,}denotes the monetary policy shock.

**2.4 Market clearing and equilibrium **

Including labor market, housing market, funds market and commodity market, all of the markets in the model should be cleared. The market clearing condition for non-housing goods market, total non-housing consumption market, total investment and housing market is given by (54)-(57):

*c** _{t}*

*i*

_{t}*g*

**

_{t}*y*

*(54)*

_{t}*c*

**

_{t}*c*

_{P t}_{,}

*c*

_{,}

**

_{I t}*c*

_{,}

*(55)*

_{R t}*i*

**

_{t}*i*

_{k t}_{,}

*i*

_{h t}_{,}(56)

, , ,

*t* *P t* *I t* *R t*

*h* *h* *h* *h* (57)

Using the equations above and those mentioned in the previous sections, we can solve the steady state of this model.

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calibrate the parameters to make the steady-state value of key variables in the model match the data from DGBAS (Directorate General of Budget, Accounting and Statistics) of Taiwan. The specification of parameters is shown in Table 1.It is required that * _{P}*

*to ensure impatient households have greater motivation to borrow. Therefore, the subjective discount factor of patient and impatient households, *

_{I}*and *

_{P}*, are set to 0.9916 and 0.9852, respectively, as in Alpanda and Zubairy (2013). The inverse labor supply elasticity is set to 1, following Alpanda and Zubairy (2013). The loan-to-value ratio in the steady state is set to 0.9 in the baseline case. Furthermore, the disutility of labor, *

_{I}*, is calibrated to 0.6, to generate the labor supply of patient households to be equal to one at the steady state without loss of generality. And the utility level for housing, *

_{l}*, is calibrate to 0.7 in the steady state to generate the total housing value to be around 5.6 times the GDP*

_{h}^{5}.

The parameters of the level and elasticity in the utilization cost, and * _{u}*, are
calibrated to ensure that the utilization is equal to 1 at the steady state. The adjustment
costs of investment,

*and *

_{ik}*, are set to 8 and 30, following Alpanda and Zubairy (2013).*

_{ih} Following the setting of parameters in Iacoviello (2004), the depreciation rate of
housing and capital, * _{h}* and

*, are set to 0.005 and 0.015 respectively, and the capital share in production, , is set to 0.3.*

_{k}Because we do not have data for wages of patient, impatient and renter households separately, we borrow data of the share of buying houses without debts,

5 Based on the data from Directorate General of Budget, Accounting and Statistics, Executive Yuan, R.O.C.-Population and Housing Census in Taiwan and Fujian area (2000).

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with debts and rent houses that Lin and Chen (2005) present to calibrate the wage
share of patient households, impatient households and renter households, * _{P}*,

*and *

_{I}*, to be 0.64, 0.26 and 0.10 respectively. The parameters of transfers to households are calibrated to ensure that the transfers to the three types of households are zero in the steady state. To preserve determinacy of the model while ensuring that government debt doesn’t play a major role in determining the dynamics of the model, the response parameter of transfers to the government bonds,*

_{R}*, is 0.005 as in Alpanda and Zubairy (2013).*

_{b}The tax rates of consumption and interest income, * _{c}* and

*, are specified as 5%, based on the tax’s law of Taiwan. Also, Taiwan has implemented a progressive tax system for individual income taxes, thus we set the labor income tax of three different types of households, *

_{b}*, *

_{yP}*and *

_{yI}*, to be 0.25, 0.25 and 0.05 respectively. The tax rate of capital income is captured using the taxation of capital gains on securities and is set as 15%. Finally, the tax rate of property is set to 1.2% in the baseline case, based on the tax’s law of Taiwan.*

_{yR}## ‧

**Parameter ** **Description ** **Value **

*P* Discount factor-Patient household 0.9916

*I* Discount factor-Impatient household 0.9852

*d* Probability of remain pricing 0.75

Inverse labor supply elasticity 1

Utilization cost elasticity 5

*l* Level for labor in utility 0.6

*h* Level for housing in utility 0.7

*y* Elasticity of substitution between different goods 5

Capital share in production 0.3

*P* Labor share in production-Patient household 0.64

*I* Labor share in production-Impatient household 0.26

*R* Labor share in production-Renter household 0.10

*h* Housing depreciation rate 0.005

*k* Capital depreciation rate 0.015

*u* Utilization cost level 0.025

*ih* Housing investment adj. cost 30

*ik* Capital investment adj. cost 8

*P* Transfer share-Patient household 0.002

*I* Transfer share-Impatient household 0.002

*R* Transfer share-Renter household 0.002

*b* Response of transfer to gov. bonds 0.005

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*c* Consumption tax rates 0.05

*b* Interest income tax rate 0.05

*k* Capital income tax rate 0.15

*yP* Labor income tax rate-Patient household 0.25

*yI* Labor income tax rate-Impatient household 0.25

*yR* Labor income tax rate-Renter household 0.05

Shock parameter-persistence

Persistence of Taylor rule 0.7239

*z* Persistence of productivity shock 0.5

*g* Persistence of government expenditure shock 0.9021

Persistence of property tax shock 0.9999

Persistence of LTV shock 0.9999

*P* Persistence of markup shock 0.6802

*h* Persistence of housing preference shock 0.8795
Taylor rule parameters

Parameter for inflation 5

*y* Parameter for output 0

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We first consider that the central bank conducts contractionary monetary policy in the attempt to reduce housing prices. Figure 3 outlines the impulse response functions of key macroeconomic variables under a 1% increase in the interest rate. First, impatient households reduce their borrowing by about 1% when they face higher interest rate and lower their demand for housing and consumption. The decline in the demand for housing leads to a fall in housing price, and leads to a fall in home equity of borrowers.

Second, faced with lower discounted value of future returns due to higher interest
rate, patient households reduce their consumption and the investment purchase of
house and capital. This will lead to a fall in the capital price. Although the fall of
housing prices may make patient households increase their own-occupied housing, the
overall decline in demand leads to a fall in production and wages and leads to a
reduction in the aggregate output, *y** _{t}*, and inflation,

*. In particular, the inflation rate falls about 4%. Therefore, rental households reduce their consumption and demand for rental housing increase cause the decline in interest rate of housing.*

_{t}Finally, the decline in debts is stronger than the decline in the overall output; thus, the loan-to GDP ratio falls about 0.35% and the decline in the demand for housing leads to a fall in housing price about 0.014%.

**4.2 Fiscal Policy **

In this section, we investigate how exogenous and near-permanent changes in raising the property tax rate would affect housing price, household’s debt and other

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macroeconomic variables. We will evaluate which policy is more effective to reduce the price of housing and lowers the loan-to GDP ratio. We let the property tax rate tax rate. We assume that the property tax rate is raised from 1.2% to 1.56% annually, a 0.36 percentage point increase annually. The housing demand falls because the cost of holding house is increased, and the lower housing demand leads to a fall in the nominal house prices.

For impatient households, the increased cost of holding housing leads to a decline in their demand for housing. The fall in nominal housing price and lowered housing demand means lower home equity of borrowers, and thus, impatient households reduce their borrowing and consumption.

For patient households, they increase their consumption and reduce the capital
purchase, so the price of capital falls.^{6} Because of the decline in saving, the real
interest rate rises upon impact. The decline in saving leads to the decline in the supply
of credit and tightens the borrowers’ constraint.

For renter households, the demand for rental housing increases because of the decrease in the interest rate of housing. Therefore, renter households increase their consumption because they have lower cost of rental housing.

Overall, the increase in the property tax rate results in lower aggregate output and debt. However, the decline in debt is stronger than the decline in overall output, and

6 Cause non-residential investment declines along with residential investment in the short-run.

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thus, the loan-to GDP ratio falls about 2%.

**4.3 Macroprudential policy: lowering the LTV ratio on debt **

The control on the LTV ratio has been considered as one of the primary macroprudential policies. In this section, we investigate how exogenous and near-permanent changes in lowering the LTV ratio would affect housing price, household’s debt and other macroeconomic variables. We let regulatory LTV follows AR(1) process as:

###

1###

1 ,*t* *t* *t*

_{} (59)

Figure 5 plots the impulse response functions under a negative shock to the LTV ratio. We lower the LTV ratio from 0.9 to 0.7, a 20% decrease. In this case, the direct effect of lowering the LTV ratio is on the impatient households’ (borrowers) borrowing constraint. Due to the lower amount of their borrowing, impatient household reduces their consumption and demand for housing. The decline in the housing demand leads to the decline in housing price and overall housing investment.

For renter households, the lowering LTV ratio leads to a fall in the interest rate of housing, thus they increase their demand for rental housing. Given the substitutability between housing and consumption, renters decrease their consumption about 3%.

For patient households, lowering the LTV ratio also decreases the inflation rate,
which prompts the central bank to reduce the interest rate, *R** _{t}*. The lowered interest
rate will make patient households increase their consumption and lower their saving.

But the decline of renter households’ consumption is stronger in the beginning, which leads to the initial decline in the aggregate consumption. The decline in housing price

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incentivizes patient households to increase their holding of housing.

Lowering the LTV ratio results in lower debts. The decline in debt is stronger than the increase in the overall output and thus the loan-to GDP ratio falls about 5%.

**4.4 The effects of three policies **

Figure 6 shows the effects of the three policies on the output, households’ debt, households’ debt-to-GDP ratio and housing prices for a 5 year horizon. In the aggregate output, tightened monetary policy leads to a fall in the aggregate output. But raising the property tax and lowering the LTV ratio lead to a rise in the aggregate output.

As for the households’ debt, tightened monetary policy causes a 1% fall in households’ debt, smaller than its decline under rest of the two policies. Furthermore, tightening the LTV ratio has greater impact on households’ debts than raising the property tax rate due to the fact that it is a targeted policy instrument for the housing sector.

As for the households’ loan-to-GDP ratio, tightened monetary policy shock leads to a stronger decline in debts than in the overall output. Therefore, the loan-to-GDP ratio falls about 0.35%. Other two policies also result in the decline in the loan-to-GDP ratio. Finally, regarding the housing price, increasing the property tax rate has the most significant effect among all three. Note that, although tightening the LTV ratio increases housing price by only 0.003%, it still has a negative effect on housing price after 2 periods and it has a much more persistent effect than tightening monetary policy in suppressing housing price.

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Figure 3 Impulse responses to a 1% innovation in the Taylor rule.

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Figure 4 Impulse responses to a 0.36 % persistent increase in the property tax rate.

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Figure 5 Impulse responses to a 20 % persistent decline in the regulatory LTV on loans.

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Note: contractionary monetary policy (red line), fiscal policy (blue line) and macroprudential policy (black line).

Figure 6 Comparing the effects of the three policies on household’s loan-to-GDP ratio.

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**5. Conclusion **

In this paper, we examine the effects of tightening monetary policy, raising the property tax and lowering the LTV ratio on housing price and loan-to-GDP ratio with a dynamic stochastic general equilibrium (DSGE) model. In this study, we follow Alpanda and Zubairy (2013) to set up housing and capital producer, three types of infinitely-lived households (patient households, impatient households and renters), a goods firm and a government sector. Different from the model of Alpanda and Zubairy (2013), we modify the tax system to let this model fit Taiwan’s situation, and include the nominal rigidity in price and monetary policy in the model.

We calibrate the parameters to let the steady-state values of this model to match the long-term situation in Taiwan. We find out that, the monetary policy has smaller effect in suppressing the price of housing and lowering the loan-to-GDP ratio.

Furthermore, tightening the LTV ratio has greater impact on households’ loan-to-GDP ratio and price of housing than increasing the property tax rate.

We conclude this paper by pointing out some related issues for future research.

First, we can consider different debts including long-term mortgages and flow household debts. Also, we can consider an alternative central bank’s monetary policy which responds to the change in price of housing. Furthermore, we can consider more fiscal policies that can affect housing price in their effects on stabilizing housing prices.

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