The Macroeconomic Eﬀects of Mortgage Interest Deduction

(1)

The Macroeconomic Effects of

Mortgage Interest Deduction

Cenkhan Sahin

Faculty of Economics and Business University of Groningen

A thesis submitted for the degree of Msc Finance

(2)

(3)

Acknowledgements

I am grateful for the opportunity provided to me by Jakob de Haan to write my thesis during an internship at De Nederlandsche Bank. I would also like to thank Vincent Sterk who has provided excellent supervision. This research will be continued as a collaborative working paper at De Nederlandsche Bank.

(4)

(5)

3.1.1 Financial Contract . . . 19 3.1.1.1 Households . . . 19 3.1.1.2 Banks . . . 20 3.1.2 Households’ Decisions . . . 21 3.1.3 Banks’ Decisions . . . 22 3.1.4 Firms’ Decisions . . . 23 3.1.5 Exogenous Processes . . . 23 3.1.6 Equilibrium . . . 23 3.2 Calibration . . . 24 3.2.1 Benchmark . . . 24

3.3 Steady State . . . 25

3.3.1 Benchmark . . . 25

3.4 Impulse Response Functions . . . 27

3.4.1 Productivity Shock . . . 27

3.4.2 Housing Preference Shock . . . 27

4 Conclusion 29 Bibliography 31 5 Appendix A 33 5.1 Euler Figures . . . 33

5.1.1 Suppy and Demand of Mortgages . . . 33

5.1.2 Parameter Variation . . . 33

(7)

CONTENTS

6 Appendix B 39

6.1 Parameter Figures . . . 39

6.2 Impulse Responses . . . 45

6.2.1 Income Shock . . . 45

7 Appendix C 53 7.1 Impulse Responses . . . 53

7.1.1 Productivity shock . . . 53

(8)

(9)

List of Figures

1.1 Key figures for mortgages . . . 3

5.1 Demand and supply of mortgages . . . 34

5.2 Demand and supply of mortgages with no default cost . . . 34

5.3 Variation in β . . . 35 5.4 Variation in σ . . . 35 5.5 Variation in µ . . . 36 5.6 Variation in τ . . . 36 6.1 Variation in τ . . . 40 6.2 Variation in β . . . 41 6.3 Variation in η . . . 42 6.4 Variation in σ . . . 43 6.5 Variation in µ . . . 44

6.6 Reponses to an income shock (1) . . . 46

6.7 Reponses to an income shock (2) . . . 47

6.8 Reponses to a housing preference shock (1) . . . 48

6.10 Reponses to a housing dispersion shock (1) . . . 50

7.1 Reponses to an productivity shock (1) . . . 54

(10)

(11)

List of Tables

2.1 Basic Model Calibration . . . 15

2.2 Basic Model Steady State . . . 17

3.1 Extended Model Calibration . . . 25

(12)

(13)

1

Introduction

1.1 Mortgage Interest Deduction

The economic developments during the Great Recession have taught us that hous-ing and mortgage loans are fundamental elements to understand the nature of crises. An important feature that has been left out of the discussion is the tax treatment households enjoy when engaging in mortgage loans. The opportunity for households to deduct mortgage interest from their taxes provides incentives for homeownership since households are easier qualified for mortgages. The role of this mortgage interest deduction in amplifying recessions remains relatively unexplored. Building a general equilibrium model that facilitates an understanding of housing market booms and busts allows us to gauge the importance of mortgage interest deduction and its contribution to mortgage defaults, and ultimately, real activity. For this purpose we construct a dy-namic stochastic general equilibrium (DSGE) model where borrowing and lending by households leads to mortgage defaults in equilibrium. Since our DSGE model features tax-deductable mortgage payments it is appropriate for analysing the macroeconomic effects of mortgage interest deduction.

(14)

interest deduction could lead to a drop in house prices which will endanger households’ ability to repay loans and hurt banks’ balance sheets, and eventually real activity. How-ever, policies that promote homeownership can have downsides where uncreditworthy households are encouraged to engage in mortgage contracts. Opponents of the policy

stress the pro rata increase in governmental budget when house prices increase.1

More-over, the mortgage interest deduction disproportionately benefits taxpayers in the top fifth of the income distribution (Toder, Harris, and Lim 2009). The subsidy rate is larger for individuals in higher marginal tax rate brackets. Since most households who benefit would own homes without the tax treatment, the mortgage interest deduction, accordingly, mostly provides an incentive to live in more expensive homes, and not so much to own instead of rent. Furthermore, it is believed that the tax benefit drives up house prices which makes homeownership less affordable for households with moderate to low incomes. This is particularly the case in condensed countries or in relatively concentrated cities.

To investigate some of these ’claims’ one needs a dynamic general equilibrium set-ting featuring tax-deductible mortgage interest. In such a setset-ting we can examine the impact of mortgage interest deduction on house prices, mortgage default, and real ac-tivity. Our basic model features an endowment economy with two types of agents: households and a government. Households have the opportunity to deduct mortgage interest from their taxes and, simultaneously, act as investors by issuing loans and purchasing mortgage portfolios. The government’s sole aim is to ensure households can deduct mortgage interest from their taxes. In order to model default we introduce idiosyncratic shocks to the value of housing after households have signed a mortgage contract. At maturity, depending on their behaviour, some households will default on their mortagage. We then examine the attributes of mortgage interest deduction. To understand the consequences for the real economy we extend our basic model with competitive firms and financial intermediaries and examine the impact of mortgage de-faults on real activity. The treatment of firms is standard where the decision involves a static maximisation problem. For financial intermediaries, however, we introduce lever-age constraints (or capital requirement constraints) to capture financial frictions. This entails that financial intermediaries cannot convert deposits into loans without

(15)

1.2 Empirical Evidence

ing some conditions. We subject our analysis of dynamic responses to three shocks: income, housing preference, and housing dispersion.

1.2 Empirical Evidence

Figure 1.1 below presents some key series that characterise consumer mortgages in

the US.1 Mortgages, as a percentage of gross domestic product, display steady growth

up to the end of the nineties. There is however enormous growth afterwards. The mortgages as a percentage of residential value display a steady growth after the start of the eigthies. The right panel in figure 1.1 displays the real house price up to the start of the 2008 subprime crisis. It similarly follows a surge after the end of the nineties and witnessed a substantial decline around 2006. This decline has brought the value of the underlying assets for many households and banks dangerously low, resulting in mortgage delinquencies, with banks writing off loans initiated to households that have defaulted. 19700 1980 1990 2000 2007 20 40 60 80 100 120 % Mortgages Mortgages to GDP Mortgages to value 1970 1980 1990 2000 2007 600 700 800 900 1000 1100

Real House Price

Figure 1.1: Key figures for mortgages - Data: Federal Reserve flow of funds.

Although the resulting foreclosures are primarily due to the decline in the value of real estate, it is in the context of macro-prudential (and financial) policy necessary to understand features that could potentially contribute to the building up of financial

1

(16)

imbalances and excessive debt, thereby magnifying the problem. Tax features, such as the mortgage interest deduction, necessarily favour debt over equity and therefore may encourage high leverage.

1.3 Literature

Much of the earlier literature has emphasised the credit market constraints on borrow-ers. Typically the models in the literature incorporate financial frictions with housing as a durable good and provide insight when one side in a contract does not honour debt repayments. Kiyotaki and Moore (1997) provide a model where small shocks to the economy are amplified by credit restrictions and result in large output fluctuations. In this credit cycle model a collateral requirement amplifies business cycle fluctations in a recession due to the fall in the value of capital as collateral thereby affecting borrowing by firms and ultimately hurting the real economy. Iacoviello (2005) assumes that the collateral form of capital is real estate and develops the Kiyotaki and Moore mecha-nism into a dynamic stochastic general equilibrium model and examines fluctuations of real house prices in the United States. Introducing housing as a collateral generates a financial accelarator that seems to give a satisfactory record of VAR evidence. The fi-nancial accelerator has been introduced to the macroeconomics literature by Bernanke, Gertler, and Gilchrist (1996) to address the large fluctuations in aggregate economic activity that seem to arise from seemingly small shocks. The role of financial interme-diaries in these models, however, is surprisingly passive where banks exists merely as a veil. There are models that specifically address financial intermediation, however the shocks triggering dynamic responses usually are exogenous falls in the banks’ net worth thus thereby representing exogenous shocks akin to a fall in capital stock in the real

economy.1 _{Iacoviello (2011) examines the behaviour of households by introducing an}

exogenous repayment shock. The repayment shock remains, however, ad hoc so that a negative repayment shock does not entail that households default on their mortgage since they do not lose their houses. Forlati and Lambertini (2010) develop a DSGE model with housing and risky mortgages and examine the effect of aggregate shocks on the rate of mortgage default. They conclude that economies with lower standard

1 _{Such as Gertler Kiyotaki (2010) and Curdia and Woodford (2009). However these papers remain}

(17)

1.3 Literature

deviation of mortgage risk have a lower steady-state rate of mortgage default. Increases in the riskiness of mortgages hurt borrowers and amplifies the effects of a credit crunch. Our model is relatively different from the models observed in the literature. The model presented in this paper features households that act simultaneously as borrowers and lenders. The decision to engage in a financial contract (e.g. mortgages) hinges on the presence of a fiscal benefit. Our primary aim is to understand the effects of the mortgage interest deduction in the simplest DSGE setting. We do adopt however some similarities with the papers introduced above. Our idiosyncratic shock to the value of housing that borrowers experience at maturity of the contract resembles that of Forlati and Lambertini (2010). However, in their model the standard deviation of mortagage risk follows an exogenous stochastic process. Furthermore, in our extended model we introduce frictions where banks are constrained in issuing liabilities by the amount of capital in their posession, as in Iacoviello (2011). With frictions banks cannot costlessly convert household deposits into firm loans.

We use our model to answer typical questions raised against the mortgage interest

deduction. Do fiscal benefits encourage household investors to issue higher loans?

Does this benefit encourage higher leverage? Does the mortgage interest deduction drive up house prices? Does the elimination of mortgage interest deduction lead to

more mortgage defaults? Since the set-up of our model is rather basic we do not

answer other questions raised in the real estate market. One such observation is that

the subsidy rate is larger for households in higher marginal tax rate brackets.1 _The

latter implies that those who benefit from the deduction would own homes with the tax treatment anyhow and hence provides an incentive to live in more expensive houses thereby rendering its purpose, promoting ownership over renting, in moot. We do not aim to model this. Furthermore, we do not model any mobility aspects of households in relation with the tax benefit thereby not accounting for the regional effects of mortgage interest deduction. Cities with limited housing space tend to experience larger increases in homeownership when the deduction is eliminated compared to cities where the price effect is minimal. Our model does not feature these observations.

1

(18)

(19)

2

Basic Model

2.1 Model Structure

The structure of the model is as follows. There is a representative household that lives forever and consumes housing and non-durables. The model presented here is an endowment economy with perfect insurance among households. Households can si-multaneously issue loans and purchase mortgage portfolios. To model heterogeneity in households in a tractable way, households are divided into borrowers and lenders. Bor-rowers can issue mortgage debt over which they pay interest payments net of mortgage interest deduction. Lenders can purchase mortgage portfolios over which they receive gross returns. Households have incentives to borrow and lend due to fiscal benefits in the form of interest deduction from mortgage payments. In order to understand the role of mortgage interest deduction in a business cycle setting, we introduce idiosyncratic shocks to the value of housing. If the realisation of a shock is below some cut-off value, the loan repayments will exceed the value of the house and therefore, the borrower will default on its mortgage.

(20)

2.1.1 Financial Contract

The financial contract is specified as follows. There is a representative household with a continuum of members indexed by i. At time t, the household members can engage in a one-period mortgage contract with collateral. At time t the household decides on the amount of housing and the interest rate that will be paid on its mortgage in period t + 1. Upon completion of the financial contract, a household member (i.e.

borrower) experiences an idiosyncratic depreciation (appreciation) shock, εi,t+1, to its

housing quality. The idiosyncratic shock εi,t+1 is i.i.d. across household members and

follows a normal distribution with mean one and standard deviation σ. The cumulative

distribution function and the probability density function are denoted by F (εi,t+1) and

f (εi,t+1), respectively. At maturity after one period, borrowers experience idiosyncratic

shocks and choose either to repay the loan or to default. In case of default, the borrower hands over the entire stock of housing to the lender.

Let di,t and ri,t be the amount of mortgage debt and the interest rate on mortgage

debt, respectively. The transfer to the lender at period t + 1 is:

(1 + ri,t)di,t if the borrower repays,

ph,t+1εi,t+1hi,t if the borrower defaults,

where ph,t+1 is the house price at period t + 1 and hi,t is time t durable consumption.

Note that the interest rate is predetermined. The optimal default policy follows from:

(1 + ri,t−1(1 − τ ))di,t−1= ph,tε¯i,thi,t−1, (2.1)

where τ is the fraction of mortgage interest that is tax-deductible and ¯εi,t is a cut-off

value of the idiosyncratic shock for which the borrower is willing to pay the mortgage

debt at the contractual interest rate ri,t−1. Note that the constraint in equation (2.1)

resembles a collateral constraint. If the realisation of εi,t+1 is below the cut-off ¯εt+1,

the borrower defaults. The rate of default is denoted by F (¯εi).

Now we consider the lender’s side of the financial contract, specifically the lender’s participation constraint. Lenders purchase a mortgage portfolio and can fully diversify the idiosyncratic shock and therefore bear only aggregate risk. However, lenders incur

a cost in case the borrower defaults.1 Borrowers, in turn, will reveal their idiosyncratic

1 _{In Bernanke, Gertler, Gilchrist (1999), the cost of default is a monitoring cost for the lender,}

(21)

2.1 Model Structure

shock satisfying equation (2.1). The gross return on a mortgage portfolio for the lender is:

Rm,t+1=

(1 − F (εi,t+1)) (1 + ri,t)mt+ (1 − µ) ph,t+1htGi,t+1

mt

, (2.2)

where mt denotes purchases of the mortgage portfolio and 0 < µ < 1 is a default

cost parameter. The gross return on the mortgage portfolio is equal to the transfer at maturity in case the borrower repays and the housing stock net of default cost in case

the borrower defaults.1 The conditional expectation of the idiosyncratic shock, ε, is

derived in Appendix A and is defined as: Gi,t+1≡

Z εi,t+1

−∞

εdF (ε) .

Gi,t+1represents the expected value of the idiosyncratic shock conditional on the shock

being less than or equal to the cut-off value εi,t+1. The participation constraint of the

lender describes the intertemporal trade-off between current and future consumption. This stochastic Euler equation looks as follows:

uct = Etβuct+1Rb,t+1 , (2.3)

where β is the household discount factor and Rb,t+1= (1 − F (¯ε)) (1 + ri,t) + (1 − µ) ph,t+1hi,t di,t Z ε¯i,t+1 −∞ εdF (ε) .

In the optimum, the utility derived from a marginal unit of current consumption equals the discounted expected value of the utility from the amount of future consumption multiplied by the realised return that the consumer must sacrifice to obtain the extra consumption today. That is, if equation (2.3) holds, the lender is indifferent between consumption today and investment in a mortgage portfolio delivering a risk rated return that is discounted using a stochastic discount factor. The future gross return for the

lender is denoted by Rb,t+1 and is equivalent to Rm,t+1 in equilibrium. A binding

participation constraint ensures the lender’s optimality condition in equilibrium.

problem that justifies the introduction of a monitoring cost. However, the process of taking possession of a mortgaged property as a result of the household’s failure to keep up with payments necessarily entails costs, such as the real estate transfer tax, real estate broker comissions, and yet other remaining costs.

1 _{In the presence of default costs, mortgage contracts may not be optimal contracts. However,}

(22)

Note that the cut-off ¯εi,t enters equation (2.3) exogenously since the default

be-haviour of the borrower is assumed to be known by the lender.1 Note also that the

stochastic discount factor of the lender can implicitly be considered as a stochastic dis-count factor of a financial intermediary that acts as a lender to household borrowers. This will in fact be the case in our extended model.

The loan to value ratio is an important indicator of the riskiness of a mortgage loan. The higher the ratio, the higher the risk that the value of the collateral in case of foreclosure will be insufficient to cover the remaining principal of the loan. The loan to value ratio, ψ, is defined as:

ψi,t+1≡

di,t

ph,t+1hi,t

. (2.4)

It is worth repeating the borrower’s decision variables in the contract. The borrower signs a mortgage contract with the lender and decides on the choice of durable and non-durable consumption, the mortgage debt, the mortgage interest rate, and the cut-off value of the idiosyncratic shock such that the lender’s participation constraint (and the household budget constraint) are satisfied. By symmetry, all borrowers make the same choices in equilibrium. Note that it is assumed the household will engage a mortgage loan and not finance the durable good with their own funds. De facto all households are ex ante credit worhty. However, in equilibrium, a fraction of the households will default due to the idiosyncratic shock on the housing value.

(23)

2.1 Model Structure

2.1.2 Households

A representative household derives utility in the following way:

u (ct, ht) = ln ct+ ηzh,tln ht,

where ctand htdenote time t consumption of housing and durables, respectively, zh,tis a

time t housing preference shock, and η is a housing preference parameter.1 Households

have an incentive to borrow and lend from each other on account of the mortgage interest deduction. This fiscal benefit will induce households to sign loan contracts. The budget constraint of the representative household is:

ct+ ph,t(ht− ht−1) + Rd,tdt−1+ mt+ Tt= yt+ Rm,tmt−1+ dt, (2.5)

where ytdenotes wage income, Ttis a lump-sum tax, and Rd,tis the gross interest rate

paid on mortgage debt in period t for debt issued at time t − 1. Rd,t+1 is denoted by:

Rd,t+1=

(1 − F (εt+1)) (1 + rt(1 − τ ))dt+ ph,t+1htGt+1

dt

. (2.6)

In equation (2.6) gross interest payments on mortgage debt equal the transfer at period t + 1 to the lender net of tax-deductible mortgage interest and the value of collateral.

Households choose sequences of non-durable consumption {ct}, durable

consump-tion {ht}, mortgage debt {dt}, mortgage portfolio {mt}, mortgage interest rate {rt},

and the cut-off value of the idiosyncratic shock {εt} to solve:

max E0

∞

X

t=0

βt{ln ct+ ηzh,tln ht}

subject to the budget constraint (2.5), the participation constraint (2.3), and the gross returns on the mortgage portfolio and debt, equations (2.2) and (2.6), respectively. The first order condition for the amount of housing is:

ph,t ct = ηzh,t ht + βEt ph,t+1 ct+1 (1 − Gt+1) + βEt λ2,t ct ct+1 Υt+1 , (2.7) where Υt+1 ≡ (1 − µ) ph,t+1 dt Gt+1+ νt (1 + rt(1 − τ ))dt h2_tph,t+1 f (εt+1), νt ≡ (1 + rt) − (1 − µ) (1 + rt(1 − τ )),

(24)

and λ2,t is the lagrange multiplier on the participation constraint. The right hand

side of equation (2.7) represents the shadow value of housing and consists of three terms. The first is the direct utility gain from consuming a marginal unit housing. The second term is the utility derived from the discounted value of the house in period t + 1. The final term captures the fact that the borrower can more easily satisfy the lender’s participation constraint. In equation (2.3) (and implicitly in equation (2.1)) a household with higher durable consumption is relatively less likely to default, and less likely to incur a default cost. At the optimum, the shadow value of housing must be

equal to the utlity derived from ph,t marginal units of non-durables.

The Euler equations for mortgage debt and mortgage portfolio purchases are given by, respectively: 1 ct = βEt (1 − F (εt+1)) (1 + rt(1 − τ )) ct+1 + βEt λ2,t ct ct+1 ht dt Υt+1 (2.8) 1 ct = βEt Rm,t+1 ct+1 (2.9) The Euler condition for mortgage debt needs some explanation. In equation (2.8) an extra unit of consumption today by the borrower must equal the right hand side which consists of two terms. The first term is the repayment at maturity adjusted for a default probability and mortgage interest deduction. The second term captures the additional burden of satisfying the lender’s participation constraint.

The first order condition for the interest rate on mortgage debt is:

(1 − F (εt+1)) (1 − τ ))dt ct+1 = λ2,t ct ct+1 h (1 − F (εt+1)) − υt+1 i , (2.10) where υt+1 ≡ (1 − τ )νtψt+1f (¯εt+1).

Finally, the first order condition for the cut-off value of the idiosyncratic shock is given by:

ph,tεtht−1 = (1 + rt−1(1 − τ ))dt−1,

(25)

2.1 Model Structure

2.1.3 Government

The role of the government in this economy is to ensure that mortgage interest is deductible from taxes. The government budget constraint is:

(1 − F (εt+1)) rtτ dt= Tt. (2.11)

2.1.4 Exogenous Processes

The three shocks of interest: income, housing preference, and housing dispersion shocks evolve according to the following laws of motion:

ln yt= ρyln yt−1+ εy,t, εy,t ∼ N (0, σy), (2.12)

ln zh,t = ρhln ht−1+ εh,t, εh,t ∼ N (0, σh), (2.13)

ln zσ,t = ρσln zσ,t−1+ εσ,t, εσ,t ∼ N (0, σσ). (2.14)

The housing preference shock is in essence a consumption shock for housing representing a shift in demand for housing. The housing dispersion shock, given a certain cutoff level, will lead to defaults. The standard deviation of the idiosyncratic shock is denoted by σ. An increase in the standard deviation of the idiosyncratic shock will disperse the distribution of the underlying asset. Due to a given cutoff level, an increase in σ will

lead to more defaults. zσ,t enters the model via the relation σt= σ ln zσt.

2.1.5 Equilibrium

Aggregate housing is fixed and normalised to one:

ht= 1. (2.15)

Equilibrium in the mortgage market requires:

dt= mt. (2.16)

The aggregate resource constraint is:

yt= ct− µph,t+1htGt+1.

A competitive equilibrium are laws of motion for ct, dt, mt, Rd,t, Rm,t, rt, Tt, ph,t, ht,

εt, yt, F (ε), λ2,t, ψt satisfying the total system of equations (2.1)-(2.16) and the cdf

(26)

2.2 Calibration

2.2.1 Benchmark

By calibration the parameters of the model are chosen such that the model replicates outcomes for which sufficiently robust information is available. Below the calibration is presented for a quarterly model. The parameter values are presented in Table 2.1.

The household discount factor is set equal to 0.99, which implies a steady state annual real interest rate of 4 percent. The default cost here is calibrated to 10 percent of the housing value. We motivate this value by accounting for three occurrences in housing markets. To begin, during a foreclosure residents sell their houses typically below the median price of houses adjacent to the foreclosed property. According to Cagan (2006) the median foreclosure price in 2006 was 12 percent lower in comparison

to houses in the same area that were previously not foreclosed.1 A second matter is the

real estate transfer tax, which a buyer incurs ipso facto on the privelege of transferring real property. The magnitude of this tax differs internationally but also nationally. The realty tax transfer in the U.S., for example, ranges from as low as 0.01 percent of the total value of the transfer in some states to 4 percent in others (Federation of Tax

Administrators, 2006).2 _{A further source of the default cost is the process of reselling,}

where resellers, who in case of foreclosures buy and resell houses below market value, and add to the loss of the initial seller.

The weight of housing in the utility, η, measures the stock of housing over annual output. We set it equal to 0.05 in order to achieve a suitable steady state target. The mortgage interest deduction, τ , is calibrated to 40 percent. We motivate this value by setting the subsidy equal to a typical marginal tax rate of 40 percent. The standard deviation of the idiosyncratic shock is calibrated to approximately 0.07, which delivers a suitable steady state target for the default rate of 0.92 percent per annum. The steady state target of Forlati and Lambertini (2010) is 2.3 percent per annum. We aim to achieve a lower default rate as we do not intend to replicate default rates during the sub-prime crisis in 2008. The persistence of the exogenous process parameters are set equal to 0.9.

1 _{Note also that the monitoring cost in Bernanke, Gertler, Gilchrist (1999) is calibrated to 0.12.}

2

In the Netherlands the real estate transfer tax was 6 percent up to June 2011. The Dutch

(27)

2.3 Steady State

Table 2.1: Basic Model Calibration

Description Parameter Value

Discount factor β 0.99

Housing preference η 0.05

Standard deviation idiosyncratic shock σ 0.072

Default cost µ 0.1

Tax deductible mortgage interest τ 0.4

Exogenous process parameter income ρy 0.9

Exogenous process parameter housing ρh 0.9

Exogenous process parameter default ρσ 0.9

2.2.2 Low Mortgage Interest Deduction

In order to see the effects of a change in the mortgage interest deduction in the steady state, we vary τ . Specifically, we set it equal to τ = 0.2 and τ = 0.1 . Our main interest is to understand at what level of deduction the rate of default is sufficiently close to or equal to zero. The remaining parameter values in the benchmark calibration do not change.

2.3 Steady State

2.3.1 Benchmark

The results are presented in Table 2.2. In the benchmark scenario the annual default rate is close to 1 percent, that is 0.92 percent. The loan-to-value ratio is a little over 79 percent. This somewhat exceeds the average U.S. loan-to-value ratio between 1973 and 2008, which is close to 76 percent. The annual mortgage interest rate paid by borrowers is 4.16 percent. The annual gross return for a lender with a debt-portfolio is 4.025 percent whereas the gross return for a mortgage portfolio is 4.0404 percent.

(28)

show a steady decline. The interest rate on mortgages declines as well from an initial annual rate of 4.16 percent to 4.04 percent with the mortgage interest deduction of 10 percent. The house price shows a decline of 25.2 percent annually when the mortgage interest deduction is lowered from 40 percent to 10 percent. This suggests that in the presence of the mortgage interest deduction house prices are higher than in a situation this fiscal benefit would not be present. Moreover, lowering the benefit will cause a significant drop in houseprices. What is interesting however is the consumer default rate which declines as the amount of mortgage interest deduction households can use is lowered. By lowering the mortgage interest deduction to a fourth of its previous value, the annual rate of default becomes only 0.12 percent. With lower tax treatment, the loan-to-value ratio declines to 74.57 percent. The riskiness of mortgages declines since the risk, that the value of the collateral in case of foreclosure will be insufficient to cover the remaining principal of the loan, declines.

These results suggests that in the absence of the mortgage interest deduction fewer households will default on their contract. This, in fact, does hold since the probability of default is very close to zero if the fiscal treatment is completely abolished. More-over once the mortgage interest deduction is eliminated, the rate on mortgages paid by households will exactly equal the gross returns on debt- and mortgage portfolios. Obviously, this makes sense since in equations (2.2) and (2.6) the elimination of τ leads to the equality of both equations.

2.3.3 Euler Figures and Parameter Variation

The steady state can be analysed using the Euler equations of the borrower and lender. Moreover figures 5.3 to 5.6 in Appendix A present variations in the parameters for the Euler equations. These figures facilitate our further understanding of the demand and supply equations. Figures 6.2 to 6.5 in Appendix B present the steady state as a function of the parameters.

2.4 Impulse Response Functions

(29)

2.4 Impulse Response Functions

Table 2.2: Basic Model Steady State

Description Variable Benchmark τ = 0.2 τ = 0.1

Consumption c 0.9986 0.9997 0.9999

Debt-Mortgage portfolio d, m 5.6996 4.4861 4.0201

Gross return debt Rd 1.0062 1.0081 1.0091

Gross return mortgage Rm 1.0101 1.0101 1.0101

Mortgage interest rate r 0.0104 0.0102 0.0101

Tax T 0.0236 0.0091 0.0041

House price ph 7.2092 5.8720 5.3913

Housing h 1 1 1

Cut-off value shock ε¯ 2.2216 2.1608 2.1223

Income y 1 1 1

Default probability F (¯ε) 0.0023 0.0007 0.0003

Lagrange multiplier λ2 31.2150 36.5152 37.4227

Loan to value ratio ψ 0.7906 0.7640 0.7457

Notes: Displayed values are quarterly results.

and simulate the effects of a one percent change in the standard deviation of income, housing preferences, and housing dispersion.

2.4.1 Income Shock

Benchmark. The responses to an income shock are presented in figures 6.6 to 6.7

in Appendix B. A sudden one-percent increase in wage income results in a rise of

non-durable consumption, debt and mortgage demand. House prices increase, the

loan-to-value ratio decreases and there is a decrease in mortgage defaults.

2.4.2 Housing Preference Shock

Benchmark. Shocks that originate in housing markets can affect the real economy.

To illustrate, responses to a positive housing preference shock, generating a sudden

increase of one percent of zh, are presented in figures 6.8 to 6.8 in Appendix B. The

model predicts an increase in household non-durable consumption, debt and mortgage

demand. Moreover, following a positive shock house prices rise as well. The rise

(30)

household mortgages. This can be seen via equation (2.4) and the associated increase in leverage displayed in figure 6.9.

2.4.3 Housing Dispersion Shock

Benchmark. The responses to a housing dispersion shock are presented in figures

(31)

3

Extended Model

3.1 Extended Model with Banking and Production

In this chapter the basic model in Chapter 2 is extended by two additional agents representing a banking sector and a production sector. The set-up of the financial contract changes slightly to account for the presence of banks in the economy. In the model presented below banks perform the role of lending to households in terms of mortgages.

3.1.1 Financial Contract

3.1.1.1 Households

Households can sign a mortgage contract with the bank with full collateral. After the

contract is signed households experience an idiosyncratic shock εi,t+1 to their housing

value. The gross interest rate paid on mortgage debt in period t for debt issued at period t − 1 is:

Rm,i,t=

(1 − F (εt)) (1 + rm,i,t−1(1 − τ ))mi,t−1+ ph,thi,t−1Gi,t+1

mi,t−1

. (3.1)

The left hand side of (3.1) equals the transfer at period t to the bank net of tax-deductible mortgage interest and the value of collateral. Households in turn place

(32)

3.1.1.2 Banks

The gross return on a mortgage portfolio for the bank is denoted by:

Rb,t=

(1 − F (εt)) (1 + rm,i,t−1)mb,t−1+ (1 − µ)ph,thi,t−1Gi,t+1

mb,t−1

, (3.2)

where mb,t denotes bank purchases of mortgage portfolio. The gross return on the

mortgage portfolio in (3.2) is equal to the transfer at maturity in case households repay and the housing stock net of default cost in case households default.

Banks engage in financial contracts with households if their participation constraint is satisfied: 1 − χλb,tcb,t= Etβb cb,t cb,t+1 Rb,t+1 , (3.3)

where βb is the bank discount factor, cb denotes bank consumption, χ is a borrowing

constraint parameter, and λb,t is the lagrange multiplier on the bank borrowing

con-straint. The borrowing constraint remaines ad hoc for the moment but will be explained in the bank’s optimisation problem below.

Note that as in the basic model the cut-off value of the idiosyncratic shock, ¯ε, enters

the participation constraint exogenously, and is therefore only a household decision variable outside this equation. That is:

(33)

3.1 Extended Model with Banking and Production

3.1.2 Households’ Decisions

There is a representative household with a continuum of members of mass unity. The household has iso-elastic utility:

u (ct, ht) =    c1−ρ_t −1 1−ρ + ηzh,th1−ρt −1 1−ρ − zw,tn1+θt 1+θ , if ρ 6= 1, ln ct+ ηzh,tln ht− zw,tn 1+θ t 1+θ if ρ = 1.

The household utility changes only with respect to labour supply compared with the basic model in the previous chapter. The household budget constraint is as follows:

ct+ ph,t(ht− ht−1) + st+ Rm,tmt−1+ Tt= wtnt+ Rs,t−1st−1+ mt (3.4)

where stdenotes period t household bank deposits, Rm,t denotes the period t gross rate

on mortgage debt, wt is the wage rate, nt is the labour supply, Rs,t−1 is the period

t − 1 return on household deposits from period t − 1 and mt denotes the issuance of

new mortgage debt.1

We assume that the household utility presented in Chapter 2 changes only with

respect to a labour supply decision and we therefore set ρ = 1. A representative

household chooses sequences of non-durable consumption {ct}, durable consumption

{ht}, labour supply {nt}, mortgage debt {mt}, mortgage interest rate {rm,t}, and the

cut-off value of the idiosyncratic shock {εt} to solve:

max E0 ∞ X t=0 βt ( ln ct+ ηzh,tln ht− n1+θ_t 1 + θ )

such that its budget constraint (3.4) and the bank participation constraint (3.3) are satisfied. The first order condition for the amount of housing is, similarly:

ph,t ct = ηzh,t ht + βEt ph,t+1 ct+1 (1 − Gt+1) + βbEt λh,t cb,t cb,t+1 Υt+1 , (3.5)

where λh is the lagrange multiplier associated with the participation constraint (3.3)

and as earlier: Υt+1 ≡ (1 − µ) ph,t+1 mt Gt+1+ νt (1 + rm,t(1 − τ ))mt h2 tph,t+1 f (εt+1), νt ≡ (1 + rm,t) − (1 − µ) (1 + rm,t(1 − τ )). 1

Note that the notation has somewhat changed compared with the basic model. Whereas previously

mortgage debt was denoted by dt, the notation has changed to mt. We also denote the mortgage interest

(34)

The first order condition for labour supply is standard:

nθ_t = wt

ct

. (3.6)

The Euler equations for mortgage debt and mortgage portfolio purchases are given by, respectively: 1 ct = βEt (1 − F (εt+1)) (1 + rm,t(1 − τ )) ct+1 + βbEt λh,t cb,t cb,t+1 ht mt Υt+1 (3.7) 1 ct = βEt Rs,t ct+1 . (3.8)

Finally the first order condition for the interest rate on mortgage debt is:

0 = βEt (1 − F (εt+1)) (1 − τ ) mt ct+1 − β_bEt λh,tcb,t[(1 − F (εt+1)) − υt+1] cb,t+1 . (3.9) 3.1.3 Banks’ Decisions

The budget constraint of the banks is as follows:

cb,t+ Rs,t−1sb,t−1+ mb,t+ kb,t+1= sb,t+ Rb,tmb,t−1+ rk,tkb,t+ (1 − δ)kb,t (3.10)

where cb,t denotes bank consumption, kb,t+1 is capital investment in firms, rk the

re-turns on capital investment, and δ is the depreciation rate of capital. Banks typically face frictions in maturity transformation, which can be captured by a bank capital constraint. As in Iacoviello (2011), banks are constrained in issuing liabilities by the

amount of capital in their posession.1 The bank capital constraint is introduced as a

standard borrowing constraint:

Rs,tsb,t≤ χ(mb,t+ kb,t+1), (3.11)

where χ denotes a bank leverage constraint. For every unit of liability, the bank has to hold (1 − χ) units assets. The left hand side in (3.11) denotes bank liabilities and the right hand side denotes the bank’s base of collateral assets.

A representative bank chooses sequences of consumption {cb,t}, deposit purchases

{sb,t}, mortgage purchases {mb,t}, and capital investments {kb,t+1} to solve the

prob-lem: max u (cb,t) = ∞ X t=0 β_bt{ln c_b,t}

1 _{This constraint can be motivated in terms of an external regulatory requirement by an authority}

(35)

3.1 Extended Model with Banking and Production

subject to the budget constraint (3.10) and the borrowing constraint (3.11). The re-spective first order conditions are:

1 cb,t = βbEt Rs,t cb,t+1 + λb,tRs,t, (3.12) 1 cb,t = βbEt Rb,t+1 cb,t+1 + χλb,t, (3.13) 1 cb,t = βbEt [rk,t+1+ (1 − δ)] cb,t+1 + χλb,t. (3.14) 3.1.4 Firms’ Decisions

There is a continuum of firms of mass unity. Each firm produces output using a Cobb-Douglas production technology:

yi,t = γk_i,tα1nα_i,t2 (3.15)

A representative firm chooses sequences of labour {ni,t} and capital investment {ki,t}

to solve the problem:

max π (ni,t, ki,t) = yi,t− wi,tni,t− (1 + rk,i,t)ki,t

where rk,i,t denotes the return on capital.

It follows that firms choose capital and labour to satisfy, respectively:

rk,t = α1 yi,t ki,t , (3.16) wt = α2 yi,t ni,t . (3.17) 3.1.5 Exogenous Processes

The housing preference, and housing dispersion shocks evolve according to (2.13) and (2.14), respectively. The productivity shock evolves according to:

ln zγ,t = ργln zγ,t−1+ εγ,t, εγ,t∼ N (0, σγ). (3.18)

3.1.6 Equilibrium

Equilibrium in the mortgage market requires:

(36)

Equilibrium in the deposit market requires:

si= sb

Equilibrium in the capital market requires:

ki= kb

A competitive equilibrium are laws of motion for c, h, n, m, s, ph, Rm, Rb, Rs, rm, cb,

y, k, rk, w, T , λh, λb, F (¯ε), and ψ satisfying the total system of equations consisting

of (2.4), (2.11), (2.15), and (3.1)-(3.18).

3.2 Calibration

3.2.1 Benchmark

The benchmark calibration of the model is presented in Table 3.1. The household discount factor is set equal to 0.99, whereas the discount factor of banks is set to 0.98. This implies that banks are more impatient. The weight of housing in the utility remains unchanged at 0.05. The persistence of the exogenous process parameters are set equal to 0.9. The inverse Frish elasticity of labour supply is set to 1. The rate of depreciation of capital, δ, is set to 0.01, implying an annual depreciation rate of 4 percent. The capital share in production is set to 33 percent. These values are typically observed in the litarature. The bank leverage constraint parameter, χ, is set to 0.9.

In order to obtain a a suitable steady state target for the default rate, we calibrate the following parameters as follows. The default cost here is calibrated to 5 percent of the housing value. The mortgage interest deduction is calibrated to 48 percent. This is somewhat higher than our value in Chapter 2 where τ was set to 40 percent. The standard deviation of the idiosyncratic shock is calibrated to 0.041.

(37)

3.3 Steady State

Table 3.1: Extended Model Calibration

Description Parameter Value

Discount factor households β 0.99

Discount factor banks βb 0.98

Housing preference η 0.05

Standard deviation idiosyncratic shock σ 0.041

Default cost µ 0.05

Mortgage interest deduction from taxes τ 0.48

Exogenous process parameter productivity ργ 0.9

Exogenous process parameter housing ρh 0.9

Exogenous process parameter default ρσ 0.9

Inverse Frish elasticity of labour supply θ 1

Capital depreciation rate δ 0.01

Capital share in production α1 0.33

Labour share in production α2 1 − α1

Technology parameter γ 1

Bank leverage constraint χ 0.9

3.3 Steady State

3.3.1 Benchmark

The results are presented in table 3.2. The annual mortgage interest rate paid by households is 4.56 percent. The annual default rate for the benchmark calibration in Table 3.2 is 1.18 percent. The leverage ratio is in addition close to the bank leverage constraint parameter χ.

(38)

previous model, in the presence of the mortgage interest deduction house prices are higher than in a situation this fiscal benefit would not be present. Lowering the fiscal benefit will therefore cause a huge drop in houseprices. The probability of default declines as the amount of mortgage interest deduction households can use is lowered. By lowering the mortgage interest deduction to 10 percent, the annual rate of default becomes very close zero. With lower tax treatment, the leverage ratio declines to 77.36 percent.

Our results do not qualitatively differ from the basic model in chapter 2. In the absence of the mortgage interest deduction fewer households default on their contract and house prices are lower.

Table 3.2: Extended Model Steady State

Variable Benchmark τ = 0.2 τ = 0.1 c 2.8226 2.8309 2.8336 h 1 1 1 n 0.9170 0.9143 0.9134 m 19.6499 13.3359 10.9605 s 66.5838 60.8145 58.6502 ph 22.2834 15.6159 14.1683 Rm 1.0059 1.0090 1.0101 Rb 1.0112 1.0112 1.0112 Rs 1.0101 1.0101 1.0101 rm 0.0114 0.0112 0.0112 cb 0.1662 0.1518 0.1464 y 3.5425 3.5322 3.5287 k 55.0794 54.9183 54.8646 rk 0.0212 0.0212 0.0212 w 2.5883 2.5883 2.5883 T 0.1075 0.0300 0.0123 λh 3.67761 3.81164 3.51671 λb 0.06016 0.06586 0.06829 F (¯ε) 0.00294 0.000371 0.0000 ψ 0.8818 0.8540 0.7736

(39)

3.4 Impulse Response Functions

The impulse response functions are presented next. To illustrate the magnitude and persistence of shocks in this economy, we assign values to the parameters of the econ-omy and simulate the effects of a one percent change in the standard deviation of productivity, housing preferences, and housing dispersion.

3.4.1 Productivity Shock

The impulse responses to a negative productivity shock are presented in figures 7.1 to 7.3 of Appendix C. A sudden one-percent decline in productivity results in a decline of output, non-durable consumption, and labour supply. On impact house prices drop, the leverage ratio rises and there is an increase in mortgage defaults.

Shocks that originate in housing markets can be captured with a housing preference shock. Dynamic responses to a negative housing preference shock, generating a sudden

decline of one percent of zh, are presented in figures 7.4 to 7.6 in Appendix C. The

model predicts a decrease in household consumption and mortgage demand. Output and labour supply rise. Following a negative shock house prices decline. The fall in house prices reduces the house value and consequently there is default on household mortgages.

(40)

(41)

4

Conclusion

In this paper we have developed a DSGE model in which we analyse the impact of mortgage interest rate deductions on macro-economic dynamics. Our basic model shows a decline of 25.2 percent in house prices annually if the mortgage interest deduction is lowered from a benchmark rate of 40 percent to 10 percent. This suggests that in the presence of the mortgage interest deduction house prices are higher than in a situation this fiscal benefit would not be present. Moreover, lowering the benefit causes a significant drop in houseprices. In addition, with a lower tax treatment, the riskiness of mortgages declines. This is because the risk that the value of the collateral in case of foreclosure will be insufficient to cover the remaining principal of the loan falls. The results from our extended model are qualitatively the same.

(42)

(43)

Bibliography

[1] Bernanke, Ben, Mark Gertler, and Simon Gilchrist (1996). ”The financial acceler-ator and flight to quality”. Review of Economics and Statistics, 78, 1-15.

[2] Bernanke, Ben, Mark Gertler, and Simon Gilchrist (1999). ”The nancial accelerator in a quantitative business cycle framework”. In Handbook of Macroeconomics, ed. J. B. Taylor and M. Woodford, vol. 1 of Handbook of Macroeconomics (Elsevier) chapter 21, 1341 1393.

[3] Cagan, C. 2006. ”A Ripple, Not a Tidal Wave: Foreclosure Preva- lence and Fore-closure Discount”. Study by First American Real Estate Solutions (November), 117.

[4] Curdia, Vasco, and Michael Woodford (2009). ”Conventional and Unconventional Monetary Policy”. Working paper, FRBNY and Columbia.

[5] Gertler, Mark, and Nobuhiro Kiyotaki (2011). ”Financial Intermediation and Credit Policy in Business Cycle Analysis”. Working paper, NYU and Princeton. [6] Forlati, Chiara and Luisa Lambertini (2010). ”Risky Mortgages in a DSGE Model”.

Center for Fiscal Policy, EPFL, Chair of International Finance (CFI) Working Paper No. 2010-02.

[7] Iacoviello, Matteo (2005). ”House Prices, Borrowing Constraints and Monetary Policy in the Business Cycle”. American Economic Review, 95(3), 739-764. [8] Iacoviello, Matteo (2011). ”Financial Business Cycles”. Working Paper.

(44)

of Imputed Rental Income”. American Economic Review: Papers and Proceedings, 98(2): 8489.

(45)

5

Appendix A

5.1 Euler Figures

5.1.1 Suppy and Demand of Mortgages

The Euler conditions in equations (2.8) and (2.9) reflect the demand and supply of mortgages by households. In the steady state these are, respectively:

1 β = (1 − F (¯εi))(1 + r(1 − τ )) + (1 − F (¯εi))(1 − τ )Υ(ψ) (1 − F (¯ε)) − υ(ψ) (5.1) 1 β = (1 − F (¯ε)) (1 + r) + (1 − µ) G ψ (5.2)

where ψ denotes steady state leverage and r the steady state interest rate for mortgages. Note that Υ and υ are both functions of ψ. Equation 5.1 can therefore be rewritten as:

1 = β    (1 − F (¯εi))(1 + r(1 − τ )) + (1 − F (¯εi))(1 − τ ) h_(1−µ)G ψ + ψν(1 + r(1 − τ ))f (¯ε) i (1 − F (¯ε)) − (1 − τ )νψf (¯ε)   

Equations (5.1) and (5.2) can then be solved to determine the equlibrium leverage and interest rate. Figure 5.1 presents the demand and supply of mortgage loans. It is important to note that in the demand equation the borrower takes into account the associated risk premium. With an elimination of the default cost the figure changes to 5.2 where the calibration in table 2.1 is used except for the fiscal benefit τ .

5.1.2 Parameter Variation

(46)

0.4 0.5 0.6 0.7 0.8 0.9 1 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 Interest Rate Leverage Euler Equation Borrower

Euler Equation Lender

Figure 5.1: Demand and supply of mortgages - Euler Equations (5.1) and (5.2).

0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.0105 0.011 0.0115 0.012 0.0125 0.013 Interest Rate Leverage Euler Equation Borrower Euler Equation Lender

Figure 5.2: Demand and supply of mortgages with no default cost - Euler

(47)

5.1 Euler Figures 0.4 0.5 0.6 0.7 0.8 0.9 ï0.03 ï0.02 ï0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Varying Beta Interest Rate Leverage Borrower beta=0.97 Borrower beta=0.98 Borrower beta=0.99 Lender beta=0.97 Lender beta=0.98 Lender beta=0.99

Figure 5.3: Variation in β - Equations (5.1) and (5.2) as a function of β.

0.4 0.5 0.6 0.7 0.8 0.9 1 ï0.04 ï0.03 ï0.02 ï0.01 0 0.01 0.02 0.03 0.04 0.05

Varying Euler Equations for Sigma

Interest Rate Leverage Borrower sigma=0.04 Borrower sigma=0.08 Borrower sigma=0.12 Lender sigma=0.04 Lender sigma=0.08 Lender sigma=0.12

(48)

0.4 0.5 0.6 0.7 0.8 0.9 1 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026

0.028 Varying Euler Equations for Mu

Interest Rate Leverage Borrower mu=0.01 Borrower mu=0.015 Borrower mu=0.02 Borrower mu=0.025 Borrower mu=0.03 Lender mu=0

Figure 5.5: Variation in µ - Equations (5.1) and (5.2) as a function of µ.

0.4 0.5 0.6 0.7 0.8 0.9 1 ï0.04 ï0.03 ï0.02 ï0.01 0 0.01 0.02

Varying Euler Equations for tau

Interest Rate Leverage Borrower tau=0 Borrower tau=0.2 Borrower tau=.4 Lender tau=0 Lender tau=.2 Lender tau=.4

(49)

5.2 Supplementary Derivation

The definition of the truncated normal distribution is: f (x; µ, σ, a, b) = 1 σφ x−µ σ Φ b−µ σ − Φ a−µ_σ ,

where the standard normal pdf and cdf are denoted by φ (·) and Φ (·), respectively. Thus, if x follows a truncated normal, then the expectation of x is given by:

E (x) = Z b a xf (x; µ, σ, a, b) dx, = Z b a x 1 σφ x−µ σ Φb−µ_σ − Φ a−µ_σ dx, = 1 σ Φ b−µ σ − Φ a−µ_σ Z b a xφ x − µ σ dx.

We also know from the properties of the truncated normal distribution that:

E (x) = µ + φ a−µ_σ − φb−µ σ Φ b−µ σ − Φ a−µ_σ σ.

Combining the above two expressions gives:

Z b a xφ x − µ σ dx = σ Φ b − µ σ − Φ a − µ σ µ + σ2 φ a − µ σ − φ b − µ σ , = σ Φ b − µ σ − Φ a − µ σ µ + σ 2 √ 2π e−12 (a−µ)2 σ2 − e− 1 2 (b−µ)2 σ2 . We are interested in:

Z b a x1 σφ x − µ σ dx = Φ b − µ σ − Φ a − µ σ µ+√σ 2π e−12 (a−µ)2 σ2 − e− 1 2 (b−µ)2 σ2 .

Letting a go to minus infinity gives:1

Z b −∞ x1 σφ x − µ σ dx = Φ b − µ σ µ − √σ 2πe −1 2 (b−µ)2 σ2 .

1 _{From the properties of the normal distributions we know that if x follows a normal distribution}

with mean µ and standard deviation σ, then its pdf is given by: 1

σφ x−µ

(50)

(51)

6

Appendix B

6.1 Parameter Figures

(52)

0.32 0.34 0.36 0.38 0.4 0.9986 0.9988 0.999 0.9992 0.9994 c _tau 0.32 0.34 0.36 0.38 0.4 5 5.2 5.4 5.6 5.8 d _tau 0.32 0.34 0.36 0.38 0.4 5 5.2 5.4 5.6 5.8 m tau 0.32 0.34 0.36 0.38 0.4 1 1.005 1.01 1.015 1.02 Rd tau 0.32 0.34 0.36 0.38 0.4 1 1.005 1.01 1.015 1.02 Rm tau 0.32 0.34 0.36 0.38 0.4 0.0102 0.0103 0.0103 0.0104 r _tau 0.32 0.34 0.36 0.38 0.4 0.016 0.018 0.02 0.022 0.024 T _tau 0.32 0.34 0.36 0.38 0.4 6.4 6.6 6.8 7 7.2 7.4 ph tau 0.32 0.34 0.36 0.38 0.4 − 0.5 0 0.5 1 1.5 2 h _tau 0.32 0.34 0.36 0.38 0.4 2.19 2.195 2.2 2.205 2.21 2.215 2.22

eps bar tau

(53)

6.1 Parameter Figures 0.988 0.99 0.992 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.999 0.999 c beta 0.988 0.99 0.992 4 4.5 5 5.5 6 d beta 0.988 0.99 0.992 4 4.5 5 5.5 6 m beta 0.988 0.99 0.992 1.006 1.0065 1.007 1.0075 1.008 1.0085 Rd _beta 0.988 0.99 0.992 1.01 1.0105 1.011 1.0115 1.012 1.0125 Rm _beta 0.988 0.99 0.992 0.01 0.0105 0.011 0.0115 0.012 0.0125 0.013 r beta 0.988 0.99 0.992 0.02 0.021 0.022 0.023 0.024 T beta 0.988 0.99 0.992 5.5 6 6.5 7 7.5 ph _beta 0.988 0.99 0.992 − 0.5 0 0.5 1 1.5 2 h beta 0.988 0.99 0.992 2.205 2.21 2.215 2.22 2.225 2.23 eps b ar beta 0.988 0.99 0.992 0 0.5 1 1.5 2 y beta 0.988 0.99 0.992 0 0.5 1 1.5 2 zh beta 0.988 0.99 0.992 1.8 2 2.2 2.4 2.6 2.8 3 x 10 − 3 def beta 0.988 0.99 0.992 15 20 25 30 35 lambda2 beta 0.988 0.99 0.992 0.784 0.786 0.788 0.79 0.792 0.794 0.796 leverage beta

(54)

(55)

6.1 Parameter Figures 0.02 0.04 0.06 0.08 0.9986 0.9988 0.999 0.9992 0.9994 0.9996 0.9998 c sigma 0.02 0.04 0.06 0.08 5 5.5 6 6.5 7 d sigma 0.02 0.04 0.06 0.08 5 5.5 6 6.5 7 m sigma 0.02 0.04 0.06 0.08 1 1.005 1.01 1.015 1.02 Rd sigma 0.02 0.04 0.06 0.08 1 1.005 1.01 1.015 1.02 Rm _sigma 0.02 0.04 0.06 0.08 0.0101 0.0102 0.0102 0.0103 0.0103 0.0104 0.0104 r sigma 0.02 0.04 0.06 0.08 0.018 0.02 0.022 0.024 0.026 0.028 0.03 T sigma 0.02 0.04 0.06 0.08 6.8 7 7.2 7.4 7.6 7.8 ph sigma 0.02 0.04 0.06 0.08 − 0.5 0 0.5 1 1.5 2 h sigma 0.02 0.04 0.06 0.08 2.2 2.25 2.3 2.35 2.4 2.45 2.5

eps bar sigma

(56)

(57)

6.2 Impulse Responses

6.2.1 Income Shock

The responses to an income shock are presented in figures 6.6 to 6.7.

Dynamic responses to a positive housing preference shock are presented in figures 6.8 to 6.8.

(58)

5 10 15 20 25 30 35 40 0 0.005 0.01 0.015 c 5 10 15 20 25 30 35 40 0 2 4 6 8 x 10 − 3 d 5 10 15 20 25 30 35 40 0 2 4 6 8 x 10 − 3 m 5 10 15 20 25 30 35 40 − 10 − 5 0 5 x 10 − 4 R_d 5 10 15 20 25 30 35 40 − 15 − 10 − 5 0 5 x 10 − 4 R_m 5 10 15 20 25 30 35 40 − 0.2 − 0.15 − 0.1 − 0.05 0 r 5 10 15 20 25 30 35 40 − 0.1 − 0.05 0 0.05 0.1 T 5 10 15 20 25 30 35 40 0 0.005 0.01 p_h 5 10 15 20 25 30 35 40 − 8 − 6 − 4 − 2 0 x 10 − 3 eps_bar

(59)

6.2 Impulse Responses 5 10 15 20 25 30 35 40 0 0.002 0.004 0.006 0.008 0.01 0.012 y 5 10 15 20 25 30 35 40 − 0.4 − 0.3 − 0.2 − 0.1 0 def 5 10 15 20 25 30 35 40 − 0.014 − 0.012 − 0.01 − 0.008 − 0.006 − 0.004 − 0.002 0 lambda2 5 10 15 20 25 30 35 40 − 0.01 − 0.008 − 0.006 − 0.004 − 0.002 0 leverage

(60)

5 10 15 20 25 30 35 40 − 2 0 2 4 x 10 − 6 c 5 10 15 20 25 30 35 40 0 2 4 6 8 x 10 − 4 d 5 10 15 20 25 30 35 40 0 2 4 6 8 x 10 − 4 m 5 10 15 20 25 30 35 40 − 4 − 2 0 2 x 10 − 6 R_d 5 10 15 20 25 30 35 40 − 4 − 2 0 2 x 10 − 6 R_m 5 10 15 20 25 30 35 40 − 4 − 2 0 2 x 10 − 4 r 5 10 15 20 25 30 35 40 0 2 4 6 8 x 10 − 4 T 5 10 15 20 25 30 35 40 0 0.5 1 x 10 − 3 p_h 5 10 15 20 25 30 35 40 − 10 − 5 0 5 x 10 − 4 eps_bar

(61)

6.2 Impulse Responses 5 10 15 20 25 30 35 40 0 0.002 0.004 0.006 0.008 0.01 0.012 z_h 5 10 15 20 25 30 35 40 − 0.04 − 0.03 − 0.02 − 0.01 0 0.01 def 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 x 10 − 3 lambda2 5 10 15 20 25 30 35 40 − 10 − 8 − 6 − 4 − 2 0 2 x 10 − 4 leverage

(62)

(63)

6.2 Impulse Responses 5 10 15 20 25 30 35 40 0 0.05 0.1 0.15 0.2 def 5 10 15 20 25 30 35 40 − 0.015 − 0.01 − 0.005 0 lambda2 5 10 15 20 25 30 35 40 − 3 − 2 − 1 0 1 x 10 − 3 leverage 5 10 15 20 25 30 35 40 0 2 4 6 8 x 10 − 4 sigma 5 10 15 20 25 30 35 40 0 0.005 0.01 0.015 z_sigma

(64)

(65)

7

Appendix C

7.1 Impulse Responses

7.1.1 Productivity shock

The impulse responses to an productivity shock are presented in figures 7.1 to 7.3.

Dynamic responses to a negative housing preference shock are presented in figures 7.4 to 7.6.

(66)

10 20 30 40 50 60 70 80 ï 4 ï 3 ï 2 ï 1 0 x 10 ï 3 c 10 20 30 40 50 60 70 80 ï 6 ï 4 ï 2 0 2 x 10 ï 3 n 10 20 30 40 50 60 70 80 ï 4 ï 2 0 2 4 6 8 x 10 ï 3 m 10 20 30 40 50 60 70 80 ï 5 ï 4 ï 3 ï 2 ï 1 0 1 x 10 ï 3 s 10 20 30 40 50 60 70 80 ï 4 ï 3 ï 2 ï 1 0 x 10 ï 3 p_h 10 20 30 40 50 60 70 80 ï 3 ï 2 ï 1 0 1 2 x 10 ï 4 R_m 10 20 30 40 50 60 70 80 ï 3 ï 2 ï 1 0 1 2 x 10 ï 4 R_b 10 20 30 40 50 60 70 80 ï 3 ï 2 ï 1 0 1 2 x 10 ï 4 R_s 10 20 30 40 50 60 70 80 ï 0.03 ï 0.02 ï 0.01 0 0.01 r_m

(67)

7.1 Impulse Responses 10 20 30 40 50 60 70 80 ï 5 ï 4 ï 3 ï 2 ï 1 0 1 x 10 ï 3 c_b 10 20 30 40 50 60 70 80 ï 0.015 ï 0.01 ï 0.005 0 y 10 20 30 40 50 60 70 80 ï 5 ï 4 ï 3 ï 2 ï 1 0 x 10 ï 3 k 10 20 30 40 50 60 70 80 ï 15 ï 10 ï 5 0 5 x 10 ï 3 r_k 10 20 30 40 50 60 70 80 ï 0.01 ï 0.008 ï 0.006 ï 0.004 ï 0.002 0 w 10 20 30 40 50 60 70 80 ï 0.03 ï 0.02 ï 0.01 0 0.01 T 10 20 30 40 50 60 70 80 0 0.01 0.02 0.03 lambda_h 10 20 30 40 50 60 70 80 ï 1.5 ï 1 ï 0.5 0 0.5 1 x 10 ï 3 lambda_b 10 20 30 40 50 60 70 80 0 1 2 3 4 x 10 ï 8 def

(68)

10 20 30 40 50 60 70 80 0 2 4 6 8 x 10 ï 3 leverage 10 20 30 40 50 60 70 80 ï 0.012 ï 0.01 ï 0.008 ï 0.006 ï 0.004 ï 0.002 0 gamma 10 20 30 40 50 60 70 80 ï 0.012 ï 0.01 ï 0.008 ï 0.006 ï 0.004 ï 0.002 0 z_gamma

(69)

7.1 Impulse Responses 10 20 30 40 50 60 70 80 ï 1 ï 0.5 0 0.5 1 x 10 ï 7 c 10 20 30 40 50 60 70 80 ï 5 0 5 10 x 10 ï 8 n 10 20 30 40 50 60 70 80 ï 1 ï 0.8 ï 0.6 ï 0.4 ï 0.2 0 x 10 ï 5 m 10 20 30 40 50 60 70 80 ï 2 ï 1.5 ï 1 ï 0.5 0 x 10 ï 6 s 10 20 30 40 50 60 70 80 ï 1 ï 0.8 ï 0.6 ï 0.4 ï 0.2 0 x 10 ï 3 p_h 10 20 30 40 50 60 70 80 ï 1 ï 0.5 0 0.5 1 x 10 ï 9 R_m 10 20 30 40 50 60 70 80 ï 1.5 ï 1 ï 0.5 0 0.5 1 x 10 ï 9 R_b 10 20 30 40 50 60 70 80 ï 1 0 1 2 3 x 10 ï 8 R_s 10 20 30 40 50 60 70 80 ï 1 ï 0.5 0 0.5 1 x 10 ï 7 r_m

(70)

10 20 30 40 50 60 70 80 ï 20 ï 15 ï 10 ï 5 0 5 x 10 ï 7 c_b 10 20 30 40 50 60 70 80 ï 4 ï 2 0 2 4 6 x 10 ï 8 y 10 20 30 40 50 60 70 80 ï 4 ï 2 0 2 4 x 10 ï 8 k 10 20 30 40 50 60 70 80 ï 1 ï 0.5 0 0.5 1 x 10 ï 7 r_k 10 20 30 40 50 60 70 80 ï 4 ï 2 0 2 4 x 10 ï 8 w 10 20 30 40 50 60 70 80 ï 1 ï 0.8 ï 0.6 ï 0.4 ï 0.2 0 x 10 ï 5 T 10 20 30 40 50 60 70 80 ï 4 ï 3 ï 2 ï 1 0 x 10 ï 5 lambda_h 10 20 30 40 50 60 70 80 ï 15 ï 10 ï 5 0 5 x 10 ï 7 lambda_b 10 20 30 40 50 60 70 80 0 2 4 x 10 ï 9 def

(71)

7.1 Impulse Responses 10 20 30 40 50 60 70 80 ï 2 0 2 4 6 8 10 x 10 ï 4 leverage 10 20 30 40 50 60 70 80 ï 0.012 ï 0.01 ï 0.008 ï 0.006 ï 0.004 ï 0.002 0 z_h

(72)

10 20 30 40 50 60 70 80 ï 3 ï 2 ï 1 0 1 2 3 x 10 ï 7 c 10 20 30 40 50 60 70 80 ï 2 ï 1 0 1 2 3 x 10 ï 7 n 10 20 30 40 50 60 70 80 ï 4 ï 3 ï 2 ï 1 0 x 10 ï 5 m 10 20 30 40 50 60 70 80 ï 6 ï 4 ï 2 0 x 10 ï 6 s 10 20 30 40 50 60 70 80 0 0.5 1 1.5 2 2.5 x 10 ï 7 p_h 10 20 30 40 50 60 70 80 ï 4 ï 2 0 2 4 x 10 ï 9 R_m 10 20 30 40 50 60 70 80 ï 4 ï 2 0 2 4 x 10 ï 9 R_b 10 20 30 40 50 60 70 80 ï 2 0 2 4 6 8 x 10 ï 8 R_s 10 20 30 40 50 60 70 80 ï 4 ï 2 0 2 4 x 10 ï 7 r_m

(73)

7.1 Impulse Responses 10 20 30 40 50 60 70 80 ï 6 ï 4 ï 2 0 2 x 10 ï 6 c_b 10 20 30 40 50 60 70 80 ï 1.5 ï 1 ï 0.5 0 0.5 1 1.5 x 10 ï 7 y 10 20 30 40 50 60 70 80 ï 1.5 ï 1 ï 0.5 0 0.5 1 1.5 x 10 ï 7 k 10 20 30 40 50 60 70 80 ï 2 ï 1 0 1 2 x 10 ï 7 r_k 10 20 30 40 50 60 70 80 ï 1 ï 0.5 0 0.5 1 x 10 ï 7 w 10 20 30 40 50 60 70 80 ï 4 ï 3 ï 2 ï 1 0 x 10 ï 5 T 10 20 30 40 50 60 70 80 ï 1.2 ï 1 ï 0.8 ï 0.6 ï 0.4 ï 0.2 0 x 10 ï 4 lambda_h 10 20 30 40 50 60 70 80 ï 4 ï 3 ï 2 ï 1 0 1 x 10 ï 6 lambda_b 10 20 30 40 50 60 70 80 ï 5 0 5 10 15 x 10 ï 9 def

(74)

10 20 30 40 50 60 70 80 ï 4 ï 3 ï 2 ï 1 0 x 10 ï 5 leverage 10 20 30 40 50 60 70 80 0 2 4 6 x 10 ï 4 sigma 10 20 30 40 50 60 70 80 0 0.002 0.004 0.006 0.008 0.01 0.012 z_sigma

(75)

8

Non-Technical Summary

(76)

I herewith declare that I have produced this paper without the prohibited assistance of third parties and without making use of aids other than those specified; notions taken over directly or indirectly from other sources have been identified as such.

The thesis work was conducted from April 2011 to July 2011 under the supervision of Vincent Sterk at De Nederlandsche Bank and Frank Smets at the University of Groningen.

Cenkhan Sahin,

The Macroeconomic Eﬀects of Mortgage Interest Deduction

The Macroeconomic Effects of

Mortgage Interest Deduction

Cenkhan Sahin

Acknowledgements

Contents

List of Figures

List of Tables

1

Introduction

1.1

Mortgage Interest Deduction

1.2

Empirical Evidence

1.3

Literature

2

Basic Model

2.1

Model Structure

2.2

Calibration

2.3

Steady State

2.4

Impulse Response Functions

3

Extended Model

3.1

Extended Model with Banking and Production

3.2

Calibration

3.3

Steady State

3.4

Impulse Response Functions

4

Conclusion

Bibliography

5

Appendix A

5.1

Euler Figures

5.2

Supplementary Derivation

6

Appendix B

6.1

Parameter Figures

6.2

Impulse Responses

7

Appendix C

7.1

Impulse Responses

8

Non-Technical Summary