Can Taxes on Borrowing Benefit Borrowers and Subsidies Hurt Them?∗ PRELIMINARY AND INCOMPLETE – COMMENTS WELCOME First Version: June 11, 2011 This Version: June 13, 2012

David Rappoport† Yale University

Abstract This paper studies the incidence of taxes and subsidies on debt, when debt is used to purchase long-lived assets. In contrast to the standard incidence result, this paper shows that subsidizing (taxing) debt can hurt (benefit) borrowers. This is explained by the effect of debt policy on both the asset user cost and the interest rate. The subsidy (tax) decreases (increases) the asset user cost for an agent that is financing the asset with debt, and thus it increases (decreases) asset demand. Moreover, by reducing (increasing) the effective interest rate a subsidy (tax) will further increase (decrease) present asset demand. This second effect will push the asset price further up (down) and when the asset price is very sensitive to demand changes borrowers will end up paying more (less) for the asset, relative to the direct subsidy they receive and thus will end worse (better) off. JEL classification: H22, D91, R21, R28.

∗

I specially thank my advisor John Geanakoplos for his advise and constant encouragement. I also benefited from helpful discussions with Drew Fudenberg, Mikhail Golosov, Andrew Metrick, Guillermo Ordoñez and Larry Samuelson and comments from Erica Blom, Don Brown, Juan Eberhard, Rebecca Edwards, Erik Maskin, Noam Tanner, Aleh Tsyvinski and participants at Yale’s Macro Lunch and Theory Workshop, Jerusalem School in Economic Theory and EconCon 2011. A previous version of this paper circulated under the title “Can Borrowers Benefit from Taxes on Borrowing?”. All errors are mine. † Email: [email protected].

1

Contents 1

Introduction

2

An Illustration: The Case of Mortgage Subsidies 7 2.1 Physical Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The Effect of a Mortgage Subsidy . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Understanding the Welfare Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3

Incidence Analysis in a Simple Model of Secured Debt 3.1 The Simple Model . . . . . . . . . . . . . . . . . 3.2 Individuals’ Decision Problem . . . . . . . . . . . 3.3 The Model with Taxes and Subsidies on Debt . . . 3.4 Incidence Analysis . . . . . . . . . . . . . . . . . 3.5 Incidence Analysis with Unsecured Debt Contracts 3.6 Incidence Analysis with Heterogeneous Agents . .

4

5

3

. . . . . .

. . . . . .

. . . . . .

Incidence Analysis in Infinite Horizon 4.1 The Infinite Horizon Economy . . . . . . . . . . . . . . 4.2 Individuals’ Decision Problems . . . . . . . . . . . . . . 4.3 Stationary Equilibrium . . . . . . . . . . . . . . . . . . 4.4 The Infinite Economy with Taxes and Subsidies on Debt 4.5 Incidence Analysis in Infinite Horizon . . . . . . . . . . Conclusion

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

. . . . .

. . . . . .

15 15 17 19 20 24 26

. . . . .

26 26 28 28 29 30 31

References

32

Appendix

34

A Proofs

34

2

1

Introduction

Economic wisdom says that taxes and subsidies introduce distortions and therefore, are undesirable, unless they correct externalities. In contrast, debt has traditionally enjoyed a preferred tax treatment for firms and individuals in many countries. In the case of individuals and mortgage subsidies the argument is that they incentivize homeownership, which has positive externalities as homeowners take better care of their dwellings and are better citizens. But, do debt subsidies aid in the purchase of long-lived assets, like houses? If every prospective home buyer is given more credit, and the supply of houses does not swiftly accommodate the increased demand, wouldn’t all the additional credit go into house prices? And, what if debt subsidies distort an additional decision margin? Can these subsidies make home buyers worse off, and taxes make them better off? Debt subsidies for individuals are economically significant. For instance, U.S. federal programs to subsidize housing are economically significant and they mostly aim at lowering borrowing cost, either directly or indirectly. The federal government directly subsidize housing by letting tax payers deduct interest payments and through explicit guarantees. In 2011 mortgage interest deduction were around $100 billion (Office of Management and Budget, 2012). Indirectly the government subsidizes housing by its implicit guarantee to Government Sponsored Enterprises (GSEs), like Fannie Mae and Freddi Mac. It is estimated that this implicit guarantee translates into interest rate subsidies of around 24 basis points.1 This implicit government guarantee was made more explicit in 2008, when the Treasury Department started providing GSEs with capital to keep them solvent. Between November 2008 and March 2011, government’s net transfers to these institutions totaled $130 billion and it is expected that the government will continue to make net transfers in the next several years (Lucas, 2001). It is, at best, not clear that debt subsidies really help to foster homeownership and that homeownership generates significant externalities (Rosen, 1985; Hilber and Turner, 2010). In fact, looking at the history of tax concessions for housing, debt subsidies seem more of an accident than a carefully designed policy (Bourassa and Grigsby, 2000). Moreover, as Lowenstein (2006) points out the fact that homeownership ratios increased in the second half of the twentieth century in the U.S. is more likely to be explained by the expansion of mortgage credit rather than by the deductions themselves. Nonetheless, as the deductions became widespread at the same time as mortgages, it is widely believed that deductions foster homeownership and the recommendation of presidential advisory panels in the U.S. to eliminate mortgage interest deductions have found no 1

CBO (2001) and see references in Adelino et al (2012).

3

political support.2 In this paper I show that, when debt is used to purchase long-lived assets, debt subsidies and taxes can have exactly the opposite effect that they intended to, with subsidies hurting their recipients and taxes benefiting who pays them. The reason for this is that in this context debt has a dual role both helping to finance the asset and helping to move resources across time. Then, debt policy will affect two “prices”: the asset user cost and the present value of future consumption or the interest rate. Both price movements will affect asset demand in the same direction, opening the scope for asset price movements to undo the intended effect of debt policy, when asset prices are very sensitive to demand changes. To show this result, I proceed in three steps. First, I illustrate the result’s intuition in a stripped-down version of the model, where the economic forces at play are more transparent. Second, I give a formal statement that summarize the main result of the paper, in a simple version of the full model. Finally, I explore the robustness of the result under different modeling assumptions. To facilitate the exposition I describe the stripped-down version of the model in terms of the familiar case of mortgage financed home purchases. The model considers two periods to incorporate credit contracts. Agents have quasilinear preferences, with utility from houses in the first period and linear utility from wealth in the second. Borrowing is limited to only mortgages, which are collateralized by the house. It follows that, the house user cost is determined by its future resale value and the mortgage interest rate; whereas the mortgage interest rate determines the present price of wealth. The model considers the equilibrium in the housing market, where demand arises from the individual demand of atomistic agents and the supply is totally inelastic. The introduction of a mortgage subsidy changes both “prices”, the house user cost and the interest rate. The interest rate decreases, or equivalently the present cost of future wealth rises; whereas, under the assumptions made, the asset user cost increases. So at the end, agents end up paying more for both houses and future consumption and their welfare is reduced. The key to the result is that as the price of future wealth rises agents substitute away from wealth and into houses today, adding to the direct stimulus to house demand from the easing of mortgage conditions. Given that the price is very sensitive to demand movements, the increase in the house price will offset the direct benefit of the subsidy. Next, I show that the general principle described above applies beyond mortgage subsidies and under more general conditions. I consider a a richer model where agents also enjoy consump2

In January 7, 2005, President Bush established a bipartisan panel to advise on options to reform the tax code, that advised to eliminate mortgage interest deduction in favor of 15% home credit. More recently, President Obama asked the President’s Economic Recovery Advisory Board for option to change the tax system, that advised to eliminate mortgage interest deduction in favor of a 12% home credit.

4

tion of other goods in the present, have alternative saving opportunities, and have more general preferences. In fact, the only assumption made over agents’ preferences is that asset and wealth are Hicksian substitutes. The main result of the paper is stated in Theorem 2 that formalizes the intuition outlined above. It shows that when the ratio of price elasticities of supply and demand for assets is small relative to the ratio of the elasticity of asset demand with respect to the present price of wealth and the price elasticity of asset demand, then an increase of a borrowing subsidy will hurt agents and an increase in a borrowing tax will benefit them. Finally, I consider the robustness of the result to alternative modeling assumptions. I show that the result extends to the case when a small part of the borrowing is done using unsecured debt. Moreover, I show that the result extends to the case of an infinite period economy, where the asset price change is permanent. In this case agents could benefit from higher asset prices as that will help them borrow more. Nonetheless, agents will still face a higher present cost of wealth and substitute away from it and into asset purchases making the price to rise and offset the debt subsidy they receive, when the asset price is very sensitive to demand changes. Relation to the Literature. As a study of the incidence of taxes and subsidies, this paper draws on insights from both the positive and normative analyses. From a methodological standpoint my paper is related to the tax incidence literature, that highlights the difference between statutory and economic incidence of taxes.3 In this tradition, Harberger (1962) pointed out that equilibrium effects can more than offset the direct effect of taxes.4 The result of my paper is similar, but the economics of both results are quite different. In Harberger’s model it plays a critical role the factor mobility across sectors. In contrast, in my model the critical element is the substitutability between contemporaneous assets and future consumption. My paper is also related to the literature that study the effect of housing subsidies on the user cost of housing.5 In this literature it is commonly assumed that the demand for houses is a decreasing function of the house user cost. This simplification allow these studies to consider the determinants of the user cost in more detail, incorporating maintenance costs, depreciation and a more realistic description of the tax code. I abstract from these consideration and consider instead, the dual role of interest rates on the demand for houses through its effect on both the user cost and the price of future consumption. This allows me to study the equilibrium effects of the substitution 3

See the surveys by Kotlikoff and Summers (1987) and Fullerton and Metcalf (2002). Harberger showed that, corporate taxes can benefit corporations and hurt workers. In a nutshell, if the sector that will have to absorb the capital that will shift away from the corporation sector, housing in Harberger’s model, is more capital intensive, then the wage will fall sharply benefiting the taxed corporations and offsetting the direct effect of the tax. 5 See the survey by Rosen (1985). For a recent study see Poterba and Sinai (2008) and for an earlier contribution see Aaron (1972). 4

5

between assets today and subsequent consumption. The key friction in this paper is that borrowing entails higher interest rates than savings. I follow Geanakoplos (1997) to model collateral in competitive equilibrium and thus my paper relates to the literature that have followed Geanakoplos’ modeling approach (Geanakoplos and Zame, 2010; and Fostel and Geanakoplos, 2011). This literature considers how the use of collateral endogenizes the contracts that are traded in equilibrium and how asset prices are proportional to the credit available to the natural buyers of these assets. The use of debt in general, and mortgages in particular, have received renewed attention after the Great Recession of 2007-2009. Recent papers have made the case for credit policy to limit financial fragility. As Rajan (2010) puts it, “Easy credit has large, positive, immediate, and widely distributed benefits, whereas the costs all lie in the future.” The argument is that by limiting credit in booms will attenuate the subsequent bust in asset prices, which can lead to welfare improvements.6 In contrast, my analysis highlights how easy credit distort the intertemporal allocation of reasources and how this can make credit policy to hurt its intended beneficiaries. Another literature that has followed the Great Recession have analyzed the particular features of the housing sector (Kiyotaki, Michaelides and Nikolov, 2011; Favilukis, Ludvigson and Van Nieuwerburgh, 2010; Stroebel and Floetotto, 2012). In particular, debt subsidies has been challenged in the empirical investigation of Adelino, Schoar and Severino (2012). These authors find, in line with my theoretical analysis, that the increase in house prices from debt subsidies on conforming loans is higher than the subsidies themselves. These studies that measure the welfare losses of debt subsidies, are an important complement of my paper as they provide evidence of the unintended consequences of mortgage subsidies analyzed here. The rest of the paper is organized as follows. In Section 2 I illustrate the main result of the paper in a stripped-down version of the model, that is described in terms of the familiar case of mortgage finances house purchases. In Section 3 I analyze a more general model to study the incidence of taxes and subsidies on borrowing. Section 3.4 states the main result of the paper, that characterizes the conditions under which credit policy hurt its intended beneficiaries. Section 3.5 extends the result to the case where unsecured debt can also be used to finance the purchase of asset and Section 4 extends the result to the case of an infinite horizon economy where the future value of the asset is endogenous. Finally, section 5 provides the final conclusions and discuss the directions for future work. 6

See Bianchi (2010a, 2010b); Bianchi and Mendoza (2011); Geanakoplos (2010); and Jeanne and Korinek (2010).

6

2

An Illustration: The Case of Mortgage Subsidies

In this section I illustrate the main result of the paper in a stylized model that resembles the familiar case of mortgage financed house purchases. The goal of this section is to isolate the economic forces at play when debt subsidies (or taxes) are present. In order to do so, I consider the simplest setup where debt exhibits a dual role, helping both to finance the purchase of houses and to move resources across time. I focus on the equilibrium in the house market when a mortgage subsidy is introduced. In this section I use a stripped-down version of the more complete model of Section 3 to make the intuition of the result as transparent as possible. In subsequent sections I consider richer settings where debt policies generate similar effects, but where the economic forces at play will be characterized by formal theorems and the intuition could be more opaque. The setup of this section will also allow me to contrast the present study with the classical analyses of the incidence of commodity taxation.

2.1

Physical Environment

In order to study debt contracts the model considers two periods: t = 0, 1. Houses are demanded by agents who maximize their preferences given their budget constraint. In period 0 agents enjoy housing, x; while in period 1, agents derive utility from their wealth, w. Agents preferences are represented by u(x) + w (1) The utility from housing, u(x) satisfies standard assumptions.7 A key simplifying assumption made here is that preferences for future wealth are linear. This assumption is relaxed below, but here it will simplify the graphical representation of the welfare effect and facilitate the comparison with the classical studies.8 Agents discount rate for future wealth is set to zero for simplicity. The agent can finance his expenditure with his period 0 income endowment, y and mortgages backed by his housing stock, x. For simplicity in this section I abstract away from considering that the agent may have a future income endowment, which is allowed for in the next section. Each housing unit have an exogenous resale value p¯ in period 1. The mortgage interest rate r M is assumed to be positive. That is, the interest at which agents can borrow is higher than their preferences discount rate, which was normalized to zero. It follows that the agent can borrow up 7

In particular, it is assumed that u is strictly increasing, strictly concave and satisfies Inada conditions. This assumption is made so the compensated demand equals the uncompensated demand; thus, the consumer surplus can be drawn as the area to the left of the uncompensated demand function (cf. Salanie, 2011). On the other hand, this is a key assumption on the classical Marshallian incidence analysis of taxation; thus, will make the comparison more straightforward. 8

7

to

px ¯ 1 + rM

Income and credit may be spent in houses and wealth. Let me denote by p0 the price of a housing unit. For a borrower the mortgage interest rate will determine the present value of period 1 wealth. Then, we can write the budget constraint in period 0 as, p0 x +

1 px ¯ w≤y+ 1 + rM 1 + rM

That is, total expenditure on the left-hand side cannot exceed total income on the right-hand side. Using that preferences are monotonic and rearranging, we get 

1 p¯  w=y x+ p0 − 1 + r 1 + r M M | {z } | {z } user cost ρ0

(2)

present value π0

The first expression in braces, ρ0 , is the user cost per housing unit: the cost of buying a house less its resale value. The second expression in braces, π0 , denotes the present value of future wealth. Equation (2) is the relevant budget constraint for a borrower and can be derived from the sequential budget constraints (in periods 0 and 1) and the borrowing or collateral constraint (see Proposition 1).9 It should be noted that equation (2) does not impose that a home buyer pledges all his housing stock as collateral. As a matter of fact, the assumption that the mortgage interest rate is larger than the preference discount rate, imply that a home buyer wants to finance his house stock with as little debt as possible. This in turn imply that for a home buyer that is considering to increase the size of his house, the relevant trade-off is between a unit of wealth at present price, π0 and an additional unit of house at the user cost, ρ0 , as if he was mortgaging completely the additional house unit. Figure 1 depicts this budget constraint.10 The total demand for houses result from the individual demands of a continuum of mass 1 of identical agents. From above it follows that the individual agent’s problem is simply to maximize utility subject to his budget constraint, equation (2). Thus, we get that the individual demand for housing will be given by, ρ0 u0 (x) = 0 π 9

In this setup the collateral constraint is implied by the non-negativity of future wealth. In this simple setup the house is the only savings device, thus it is not feasible to end up with more wealth than y p¯ . The focus of the analysis is on debt subsidies, which require that debt is in positive demand, so adding another 0 p saving device with an interest rate smaller than r M will not change the analysis. 10

8

Figure 1: Budget Constraint

wealth (w)

y p¯ p0

0

y p0

housing (x)

y ρ0

— budget set, equation (2) in the text.

The model is closed by making an additional key simplifying assumption, that is, that houses are supplied inelastically with a total supply of houses of S x . This assumption will be relaxed below, but here it greatly simplifies the graphical representation of the welfare effect. Then, equilibrium between supply and demand for houses requires that u0 (S x ) =

ρ0 π0

Note that once we solve for the relative price ρ0 /π0 we can solve the house price, using the definition of the user cost, ρ0 and the present value, π0 ρ0 = (1 + r M )p0 − p¯ π0

2.2

The Effect of a Mortgage Subsidy

Now I consider the case where a proportional subsidy τ on mortgage borrowing is introduced. The subsidy increases the available funds by τ for a given mortgage repayment in period 1, therefore an agent will be able to borrow up to (1 + τ) px ¯ 1 + rM since the mortgage determines the present value of wealth for a borrower the subsidy will change 1+τ the present value of wealth to 1+r . Let me use a superscript τ to denote the prices with a subsidy M 9

wealth (w)

Figure 2: Budget Constraint with Subsidy y p¯ p0 y p¯ pτ

y pτ

y p0

0 housing (x) budget set with a subsidy τ

y y ρτ ρ0

budget set with no subsidy.

τ, then we can write the budget constraint in period 0 as, (1 + τ) p¯  1+τ pτ − x+ w=y 1 + r 1 + r M M | {z } | {z }



user cost ρτ

(3)

present value πτ

This budget set is depicted in Figure 2. Note that the subsidy changes both the house user cost and the present value of wealth. The present value increases by the same proportion as the subsidy. On the other hand, the user cost is affected both by the change in available credit and by the change in house prices. For a given house price, the mortgage subsidy reduces the house user cost, as it increases available credit per house unit, thus it will increase the demand for houses and increase the equilibrium house price, so pτ > p0 . The increase in house prices make agents who don’t borrow worse off, as they will end up with less houses in period 0 and thus less wealth in period 1. In fact, consider the case of an agent that was consuming y/p0 house and y p/p ¯ 0 wealth before the introduction of the subsidy. After the introduction of the subsidy the maximum amount of wealth that can be afforded without borrowing is y p/p ¯ τ and that requires to buy y/pτ houses in period 0. As shown in Figure 2 the previous bundle consist of less house units and less wealth units, so we conclude that agents who don’t borrow too much will end up worse off by the introduction of the debt subsidy. In general, the user cost, ρτ could increase or decrease upon the introduction of the debt subsidy. Nonetheless, in this case given that the supply of houses is totally inelastic, the user cost will

10

increase.11 Thus, the maximum house stock that can be purchased drops, i.e. y/ρτ < y/ρ0 and we have that the budget set with a subsidy lies to the left of the original budget set, as depicted in Figure 2. The solid green line represents the budget set with a subsidy τ; whereas the dashed gray line represents the budget set with no subsidy. This lines are parallel as the relative “prices” of house and wealth remain the same in equilibrium when the house supply is totally inelastic. In fact, in the presence of the subsidy the agent will maximize his utility subject to the budget constraint (3), from where we see that his demand for housing will be given by, u0 (x) =

ρτ πτ

And market equilibrium requires that ρτ ρ0 0 = u (S ) = x πτ π0

(4)

That is, that the ratio of house user cost and wealth present value remains constant. It follows that ρτ = (1 + τ)ρ0

⇔

pτ = (1 + τ)p0

Figure 2 presents graphically the main result of the paper, when the house price is very sensitive to demand changes the demand expansion induced by the debt subsidy will cause a large price effect, that will have a negative effect that is larger than the benefit of the subsidy, in terms of agents’ expenditure, and agents will end up worse off upon the introduction of a debt subsidy.

2.3

Understanding the Welfare Effect

To shed further light into the intuition for why the subsidy makes home buyers worse off, lets consider the change in consumer surplus and its decomposition into the contribution of housing and wealth expenditure. In this case where more than one price change with the policy intervention, we need to consider the Hicksian consumer surplus or the equivalent variation (Slesnick, 1998).12 The Hicksian surplus is the analogue of the consumer surplus but using the compensated or Hicksian demand functions. The equivalent variation is the difference between the expenditure needed to achieve the after-subsidy utility level and the expenditure needed to achieve the original utility 11

Theorem 2 below shows that it is sufficient that the ratio of the price elasticities of supply and demand for housing is lower than a specific bound, characterized there. 12 As it is seen below, in this case the consumer surplus is a line integral and is not single valued when the uncompensated demands are integrated. In contrast the compensated price effects are symmetric, which guarantee that the line integral is path independent.

11

level, at the original prices, and corresponds to the Hicksian surplus using the after-subsidy utility level. To formally define the equivalent variation, lets introduce some notation for the indirect utility and expenditure functions. Let v(ρ, π, y) be the indirect utility function that gives the maximum utility that can be achieved by an agent with income y, who faces a house user cost ρ and present value of wealth, π. And let e(ρ, π, v) be the expenditure function that gives the minimum expenditure necessary to obtain utility level v when house user cost and present value of wealth are ρ and π, respectively. Then the equivalent variation, EV from the introduction of the subsidy is given by

EV(τ) = e ρ0 , π0 , vτ − e ρ0 , π0 , v0
where vτ = v(ρτ , πτ , y and v0 = v(ρ0 , π0 , y). The equivalent variation can be visualized in Figure 2, that depicted the budget sets with and without the subsidy. In this case of a totally inelastic house supply the equilibrium demands will be on a vertical line, so we can calculate the equivalent variation as the vertical distance between the two budget sets times the original price of wealth. From the budget constraint we have that demand for future wealth is the difference between income and housing “user-expenditure” divided by the present price. Thus, given that the ratio between the house user cost and the present value of wealth remains constant, we get wτ − w0 =

τy y − ρτ S x y − ρ0 S x − =− τ τ 0 π π π

So,

τy 1+τ An alternative way to compute the equivalent variation is using the Hicksian surplus that equals the area to the left of the compensated demand functions. This calculation will allow us to decompose the welfare effect into the contribution of housing and wealth expenditure. To see the equivalence between the Hicksian surplus and the equivalent variation, first note that income equals expenditure before and after the introduction of the subsidy, that is y = e(ρ0 , π0 , v0 ) = e(ρτ , πτ , vτ ). Then, using the Envelope Theorem and the Fundamental Theorem of Calculus for line integrals, we get Z h i 0 0 τ τ τ τ EV(τ) = e(ρ , π , v ) − e(ρ , π , v ) = x(ρ, π, vτ )dρ + w(ρ, π, vτ )dπ EV(τ) = (wτ − w0 ) π0 = −

Γ

τ

τ

where Γ is a path from (ρ , π ) to (ρ , π ). The Theorem allows us to consider any path from the
original to the after-subsidy “prices”.13 Lets consider the following path (1 + t)ρ0 , (1 + t)π0 where 13

0

0

The important assumption is that the compensated demand functions are the gradient of the expenditure function,

12

the ratio between the house user cost and the present value of wealth remains constant, then we get EV(τ) = −ρ

τ

Z

τ

x(ρ(t), π(t), v ) dt − π

0

Z

0

0

τ

w(ρ(t), π(t), vτ ) dt

0

The first term is the contribution of housing expenditure to the equivalent variation, whereas the second term represents the contribution of wealth expenditure, that will be denoted respectively with EV x (τ) and EV w (τ). The two simplifying assumption will prove useful here to simplify the calculations. On the one hand, linear utility from wealth imply that the uncompensated and compensated demands for housing are the same, i.e., x(ρ(t), π(t), vτ ) = x(ρ(t), π(t), y) = S x On the other hand, a totally inelastic supply of housing allowed us to consider an integration path were relative “prices” remain constant and thus the compensated demand for wealth will remain constant. In fact it equals the uncompensated demand for wealth that is consistent with the given indirect utility level, that is w(ρ(t), π(t), vτ ) = vτ − u(x(ρ(t), π(t), vτ )) = vτ − u(S x ) = wτ and from above we had that

y ρ0 S x y − ρτ S x = τ− 0 wτ = πτ π π Now it is straightforward to compute the contributions of housing and wealth expenditure to the equivalent variation. The contribution of housing expenditure equals EV x (τ) = −τρ0 S x This is represented graphically in Figure 3, that plots housing demand as a function of the user cost. Since the subsidy increases the house user cost the contribution of housing expenditure to the equivalent variation from the debt subsidy is negative.14 Given that the ratio of the user cost to the present value of wealth remains constant along the integration path and so does the (compensated) demand for houses; therefore, the contribution of housing expenditure equals minus the area of a rectangle with width equal to housing demand, and height equal to the change in house user cost. This rectangle corresponds to the gray area in Figure 3. which follows from the Envelope Theorem. 14 This is the opposite as the standard case of a subsidy on a single commodity, where the price paid by the subsidized

13

house user cost

Figure 3: House Expenditure Contribution to the Equivalent Variation from the Debt Subsidy

S ρτ

−EV x

0

D(τ )

ρ

D

0

Sx

housing units house demand with subsidy τ house demand with no subsidy contribution of housing expenditure to the equivalent variation.

house supply

On the other hand, the contribution of wealth expenditure equals EV w (τ) = −

τy + τρ0 S x 1+τ

Figure 4: Supply and Demand for Future Wealth: Effect of the Debt Subsidy

wealth price

πτ π0

0

−EV w

y−ρ τ Sx πτ

wealth units

uncompensated wealth demand with subsidy τ compensated wealth demand with subsidy τ uncompensated wealth demand with no subsidy compensated wealth demand with no subsidy contribution of wealth expenditure to the equivalent variation.

consumers is non-increasing.

14

The second term shows that the increase in housing expenditure is matched one-for-one with a reduction in wealth expenditure. Moreover, the first term shows that the increase in the present value of future wealth brought about by the subsidy reduces the future value of income and thus the consumer spend less in future wealth. Figure 4 displays the contribution of wealth expenditure as the area to the left of the compensated demand for wealth, along the integration path. As argued above the compensated demand for wealth along this path remains constant. Thus, the area that corresponds to the equivalent variation, corresponds to a rectangle with height equal to the change in the present value of wealth and width equal to wealth demand wτ . Again, as the present price of wealth rises the contribution to the equivalent variation is the negative of this area. In this section I illustrated the mechanism behind the main result of the paper. Debt policy changes two “prices” for borrowers who use debt to finance the purchase of long-lived assets: the asset user cost and the present value of future consumption or the interest rate. This exercise considered several simplifying assumptions that we relax in what follows.

3

Incidence Analysis in a Simple Model of Secured Debt

In this section I consider a more general model of secured debt to state the main result of the paper. The model incorporates consumption of other goods in the first period and an additional saving technology. The model also considers that the agent may receive income in the second period. In addition, the only assumption that will be made on agents’ preferences will be that future wealth and asset services are substitutes. As in the previous section, the model captures a dual role for secured debt, helping both to finance the purchase of houses and to move resources across time. Subsection 3.5 considers the case of unsecured debt and Subsection 3.6 the case of agent heterogeneity.

3.1

The Simple Model

There are two periods in order to consider debt contracts. Time periods are indexed by t = 0, 1. There is as continuum of mass 1 of identical agents. In period 0 agents derive utility from an asset, x and a numeraire consumption good, c. In period 1 agents derive utility from wealth, w. Agents’ preferences are represented by u(c, x, w) (5) with u being strictly increasing, strictly concave and satisfying Inada conditions in each argument. 15

Agents have an income endowment y0 and y1 , received in period 0 and 1, respectively. In addition an agent can use secured debt to finance his period 0 purchases. It is assumed that financial frictions makes debt contract not enforceable so borrowers will pledge their asset as collateral. Lenders, on the other hand, can seize as much of the collateral as will make them whole, but not more and debt is nonrecourse. For tractability, I restrict attention to secured debt contracts and abstract away from other forms of borrowing.15 A secured debt contract specifies a promised payoff and a collateral securing that promise. The asset is the only eligible collateral and it has an exogenous value in period 1 equal to p. ¯ The model allows for the possibility that agents promise more (or less) than the value of the underlying collateral, but in this simple environment, where there is no uncertainty, we can restrict attention to the case where agents only borrow on contracts that promise to pay exactly the future value of the collateral.16 Therefore, the borrowing decision can be reduced to the choice of the fraction of the asset holding pledged as collateral denoted by f . Then the collateral constraint is simply: f ≤1

(6)

Thus a secured debt contract is a promise to repay p¯ f x in period 1. It is assumed that debt contracts p¯ f x , when he pledges a fraction have a yield r M > 0. Therefore, a borrower will be able to borrow 1+r M f of an asset holding x. An agent can spend his income and credit on consumption, assets and savings. Savings corresponds to a storage technology and the utilization of the this technology is denoted by b . Each unit stored in this technology yields a unit of wealth in period 1. In other words, the return of the storage technology is zero. The key assumption made here is that the return on the alternative saving technology is strictly lower than the return of a secured debt contract.17 This assumption captures the fact that normally deposit rates are lower than borrowing rates. Moreover, it will prevent the case where agents borrow on their asset to invest in the storage technology and it will generate an incentive for asset buyers to borrow as little as possible. Thus, the budget constraint in period 0 is given by c + b + p0 x ≤ y0 +

p¯ f x 1 + rM

(7)

where p0 denotes the price of the asset. 15

Subsection 3.5 considers the possibility of unsecured debt contracts. Formally, for any equilibrium, there is another equilibrium where agents only trade contracts that promise exactly the value of the collateral and with the same allocations of goods. 17 The positive interest rate differential makes borrowing against the risk-free asset suboptimal. 16

16

By choosing asset holdings x, secured borrowing on a fraction f and storage b, an agent in period 1 will have wealth equal to his income, plus the proceeds of the storage technology, plus the value of assets, minus the repayment of debt contracts. That is, w ≤ y1 + b + px ¯ − p¯ f x

(8)

It is assumed that neither consumption, nor the storage technology utilization, nor asset holdings can be negative, as to rule out other forms of borrowing. In addition, agents cannot end up in debt, so their remaining wealth cannot be negative. Finally, agents cannot save in secured debt contracts, so f is bounded below by zero. The model is closed by assuming that there is an exogenous supply of houses, X s (p) that only depends on the asset price and not on the credit conditions. The price elasticity of supply is given S by ζ x,p . In this environment a competitive equilibrium can be defined as follows. Definition 1 (Competitive Equilibrium) A competitive equilibrium consist of an asset price {p0 }  and allocations x, c, w, f such that:  1. x, c, w, f is the solution to the agent’s problem taking prices as given 2. the asset market clears

3.2

Individuals’ Decision Problem

With the maintained assumptions we can express the problem of an agent as, max u(c, x, w)

c,x, f,b,w

s.t.

(9)

equations (6), (7) and (8) c, x, b, w, f ≥ 0

The solution to the agent problem can be characterized by the following proposition: Proposition 1 (Asset Financing Pecking Order) Given that the yield on secured debt is higher than the return on savings, i.e. r M > 0, agents will finance their asset purchases according to the following pecking-order: internal funds are used first, and only when they are depleted secured debt will be used. It follows that we can describe the budget set for borrowers and savers as:

17

Figure 5: Budget Constraint in the Simple Model y˜0 + y1

f =0

wealth (w)

y1 + y˜0 p¯ p0

f =1 y1

y˜0 p0

0

housing (x)

y˜0 ρ0

— budget set for a fixed c ≤ y0 , y˜ 0 = y0 − c, see Proposition 1.

• if an agent uses the savings technology, i.e. b > 0 then he will not borrow, i.e. f = 0 and his demands will lie on the budget line given by
c + p0 − p¯ x + w = y0 + y1 • if an agent borrows, i.e. f > 0 then he will not use the savings technology, i.e. b = 0 and his demands will lie on the budget line given by p¯  1 1 c+ p − x+ w = y0 + y1 1 + rM 1 + rM 1 + rM 

0

The proof is relegated to the appendix. The argument uses the first order conditions to establish that a saver, who uses the savings technology, will choose not to borrow; and conversely a borrower will choose not to use the savings technology. This in turn imply that an agent will first tap his internal funds and only when funds have been depleted he will use secured debt. Thus, a borrower on the margin will trade-off an additional unit of the asset for other goods, as if he was financing this last unit using as much debt as possible. This budget set is depicted in Figure 5. As savers get a lower rate of return, than the cost of funds for borrowers, the budget set for a given consumption choice exhibits a kink at the point where asset holdings are maximal without borrowing. Figure 5 also shows another important feature of secured debt, namely that the possibility of using secured debt is not enough to borrow against future non-pledgeable income. Thus, agents

18

borrowing using secured debt will never end up with less wealth than their future income endowment. In other words, they will not be able to be net seller of future wealth and in that sense they could be consider “savers”.18 From Proposition 1 it follows that the present value of wealth is 1 for a saver and 1+r1 M for a borrower. Lets denote π0 this present price, then the user cost of the asset will be given by p0 − π0 p¯ that will be denoted by ρ0 . Using this notation we can rewrite the date 0 budget constraint as: c + ρ0 x + π0 w ≤ y0 + π0 y1 ≡ y0

(10)

where y0 corresponds to the present value of the income endowment at price π0 . Then, the individual decision problem reduces to maximize utility subject to the budget constraint (10). Since the objective is strictly concave and the budget set is convex we can express the agents’ (Walrasian or uncompensated) demand functions as a function of the “prices” (ρ0 , π0 ) c(ρ0 , π0 , y0 ) ,

x(ρ0 , π0 , y0 )

and

w(ρ0 , π0 , y0 )

(11)

The upshot of equation (11) is that the demand for the asset will depend on both its user cost, ρ0 (the cost of buying an asset less its discounted resale value) and the present price of wealth π0 . Both of these “prices” will be different for savers and borrowers as we saw above.19 On the demand side of the asset market, we have a continuum of identical agents, therefore, we can express the aggregate demand for houses as, X(p0 , y0 ) = x(ρ0 , π0 , y0 ) where x(ρ0 , π0 , y0 ) represent the asset demand of the representative agent. Therefore, an equilibrium corresponds to a price, p0 such that supply and demand for the asset are in equilibrium X s (p0 ) = X(p0 , y0 )

3.3

The Model with Taxes and Subsidies on Debt

In this section I introduce a proportional tax or subsidy on secured debt. For that, let me define τ ∈ ’ as a proportional subsidy, with negative values representing a proportional tax. For simplicity I will refer to τ as the subsidy with the understanding that negatives values are allowed and will 18 19

In this paper savers are agents who utilize the savings technology and h i borrowers are agents who use debt contracts. Note that when x = yp˜00 there is a range of values for π0 ∈ 1+r1 M , 1 consistent with that asset demand.

19

turn it into a tax. The subsidy increases available funds for a given debt repayment by τ. Thus, the period 0 budget constraint, now will be given by, c + b + pτ x ≤ y0 +

(1 + τ) p¯ f x 1 + rM

(12)

where pτ is used to denote the asset price in the presence of a subsidy τ. It follows from Proposition 1 that the subsidy will have no direct effect on savers and thus for them the present value of wealth will remain at 1. On the other hand, for borrowers the subsidy will change the present value of 1+τ . Let me denote by πτ the present value of wealth in the presence of the subsidy and wealth to 1+r M ρτ = pτ − πτ p¯ the asset user cost in the presence of the subsidy. Thus, as before Proposition 1 imply that we can describe the budget set by c + ρτ x + πτ w ≤ y0 + πτ y1 ≡ yτ As before, we can express the individual demands as functions of the “prices” (ρτ , πτ ). Then c(ρτ , πτ , yτ ) ,

x(ρτ , πτ , yτ )

and

w(ρτ , πτ , yτ )

A subsidy will weakly increase the demand for the asset. If the representative agent is saving then the subsidy will have no effect, but if the representative agent is borrowing the subsidy will stimulate the demand for assets and thus weakly increase the asset price, i.e. pτ ≥ p0 if τ > 0. This will weakly reduce the consumption possibilities of agents who are not borrowing too much. To understand the welfare effect in the case when agents are borrowing an arbitrary amount we need to characterize the equilibrium asset price effect and how it affect affect individual welfare, what we turn to next.

3.4

Incidence Analysis

In this section I analyze the welfare effect of the subsidy on secured debt. I focus on the local effect of changing the borrowing subsidy, equal to the derivative of the indirect utility function τ τ τ with respect to the subsidy, dV(ρdτ,π ,y ) . Note that this welfare measure is the infinitesimal analogue

20

of the equivalent variation considered in section 2.20 By the Envelope Theorem we have that,21  d pτ dV(ρτ , πτ , yτ ) p¯ f  = −λ − x dτ dτ 1 + rM

(13)

where λ > 0 is the Lagrange multiplier of the period 0 budget constraint (equation 12) and equals the marginal value of period 0 income. The term in parenthesis corresponds to the change in the average asset user cost for an agent that borrows against a fraction f of his asset holdings x. From equation (13) we can see that for an agent who don’t borrow a subsidy (or a tax) will have no welfare effect as we expected. In fact, this agent is not borrowing, i.e. f = 0 and the welfare effect will depend on the price change upon the change in the subsidy. But the original asset price will remain to be the equilibrium price after the the subsidy has changed. Indeed, at that price the supply will remain unchanged and at the same time the demand for the asset will remain unchanged as the agent will face a present value of wealth of 1 and the same user cost as before. The welfare effect for borrowers is more interesting. In this case the agent will gain from the subsidy itself, but he will loss from the increase in the asset price. The magnitude of the asset price increase will depend on the asset price elasticity of supply and the asset price elasticity of demand with respect to both the asset user cost and the present value of wealth. To calculate the asset price change upon a change in the debt subsidy we totally differentiate with respect to the subsidy τ the asset market equilibrium condition dX S (pτ ) d pτ dX(pτ , yτ ) dx(ρτ , πτ , yτ ) = = dp dτ dτ dτ

(14)

The total effect of the subsidy on the individual asset demand will be given by: dx(ρτ , πτ , yτ ) ∂x(ρτ , πτ , yτ ) dρτ ∂x(ρτ , πτ , yτ ) dπτ ∂x(ρτ , πτ , yτ ) dyτ dπτ = + + dτ ∂ρ dτ ∂π dτ ∂y dπ dτ τ

From the expression for the present value of the income endowment, we have that dy = y1 . In dπ 1+τ addition, as we saw above for a borrower the present value of wealth, πτ equals 1+rM , so dπτ 1 = = π0 dτ 1 + rM

and

dρτ d pτ = − π0 p¯ dτ dτ

The changes in the (uncompensated) demand for assets due to the change in “prices” (ρ, π) can be τ

τ

τ

Recall that decreasing the borrowing subsidy is equivalent to increasing a tax on borrowing, so − dV(ρdτ,π ,y ) measures the local welfare effect of changing the tax on borrowing. 21 Note that we are invoking the Envelope Theorem on the original problem (9). 20

21

expressed in terms of the compensated price elasticities and the income elasticity of demand by the Slutzky equation. For that let, ζ x,ρ and ζ x,π be the compensated price elasticities of asset demand with respect to the user cost, ρ and the present price, π, respectively. Let also ζ x,y be the wealth elasticity of asset demand. Moreover, I make the following assumption: Assumption 1 (Substitutability between wealth and assets) It is assumed that wealth and assets are substitute, that is the derivative of the compensated asset demand with respect to the present price of wealth is non negative. Then the Slutzky equations are xζ x,ρ x2 ζ x,y ∂x(ρτ , πτ , y) =− τ − ∂ρ ρ y

and

∂x(ρτ , πτ , y) xζ x,π wxζ x,y = τ − ∂π π y

Note that we have used Assumption 1 when writing down the last equation, as it assumes that the derivative of the compensated asset demand with respect to the present price, π is positive. Then, we have that the total effect of the subsidy on asset demand can be express by:  xζ  xζ x,ρ  d pτ d pτ  xζ x,y dx(ρτ , πτ , y) x,π 0 0 =− τ − pπ ¯ + pπ x + ¯ fx− dτ ρ dτ 1+τ dτ y

(15)

where we used that a borrower will not use the saving technology (Proposition 1) so debt repayment, p¯ f x equal period 1 income plus the resale value of the asset minus wealth. The first term in equation (15) represents the change in demand from a change in the asset user cost. The second term represents the increase in asset demand caused by an increase in the present price of wealth, where the sign of the effect is a consequence of Assumption 1 that asset and wealth are substitutes. Finally, the third term represent the total income effect, which depends on the direct effect of the subsidy and the indirect effect brought about by the increase in the asset price. This two income effects has opposite signs. The direct effect of the subsidy increases income by as much as the agent was originally borrowing and will have a positive effect on asset demand. In contrast, the indirect effect is negative as asset expenditure rises by the increase in the asset price. Substituting back into in (14) we get the magnitude of the price increase d pτ = dτ

p p¯ ζ ρ(1+r M ) x,ρ

+

S + ζ x,p

p p¯ f xp ζ + (1+r ζ x,y 1+τ x,π M )y p px ζ + y ζ x,y ρ x,ρ

This expression for the price effect can be expressed in terms of the price elasticity of asset demand using the following result. Theorem 1 (Price Elasticity of Asset Demand) The compensated price elasticity of asset demand 22

equals the compensated asset demand elasticity with respect to the user cost times the maximum leverage allowed for asset purchases, i.e. the ratio of asset price to asset user cost. That is, ζ x,p =

p ζ x,ρ ρ

The proof of the theorem is relegated to the appendix, but its intuition is straightforward. Lets consider the case of a mortgage financed house purchase to illustrate. For a home buyer a percentage increase in home prices will be associated with a bigger increase in the house user cost, with a constant of proportionality equal to the maximum leverage that can be used in the house purchase. For instance, suppose a house sells for 100 dollars and its user cost is 20 dollars. The leverage on   the house investment is 5, equal to the ratio of the house value to the downpayment 100 . From 20 Proposition 1 we know that for any borrowers the relevant marginal price is the user cost. Now suppose the house price increases 2%, to 102 dollars, with the amount that can be borrowed against the house remaining constant. Then the user cost will increase to 22 dollars, or 10%, which is the relevant price for the home buyer. Thus, ignoring the income effects, the home buyer will reduce his demand for houses by a percent equal to 10% times the elasticity of housing demand to the user cost. This percentage reduction needs to be equal also to 2% times the price elasticity of housing demand, from where the result follows. Using Theorem 1 we can rewrite the previous expression for the effect of a subsidy on the asset price, using the price elasticity of asset demand. That is: d pτ = dτ

p¯ ζ 1+r M x,p

+

p ζ 1+τ x,π

S +ζ ζ x,p x,p +

+

p¯ f xp ζ (1+r M )y x,y px ζ y x,y

(16)

Now we are ready to state the main result of the paper. Theorem 2 If the representative agent is a borrower, Assumption 1 holds and S ζ x,p

ζ x,p

<

1 − f (1 + r M ) p ζ x,π + f (1 + τ) p¯ f ζ x,p

(17)

Then an increase in taxes on borrowing will benefit borrowers and an increase in subsidies on borrowing will hurt them. The proof is relegated to the appendix. The key is that the bound on the ratio of the price elasticity of supply and demand imply that the average user cost is increasing in the subsidy, thus agents will end up paying a higher user cost for the asset upon an increase in the subsidy and therefore end up worse off. On the other hand, an increase in taxes on borrowing is equivalent to a 23

reduction in subsidies on borrowing, thus this policy will reduce the average user cost and benefit agent who purchase houses. Recall that Theorem 1 imply that the price elasticity of asset demand is amplified by the use of leverage, thus the use of leverage on the demand side reduces the ratio of the price elasticities of supply and demand making it more likely that debt policy will have the opposite affect as in the standard case. The two terms in the upper bound for the ratio of supply and demand price elasticities (condition 17) represent two economic forces behind the result. The first term, 1−f f represents the non-linear effect of debt subsidies. Individual decisions on the margin are fully distorted but the average subsidy will differ from the marginal one for agents that are not borrowing on their total asset holdings, i.e. f < 1. This effect in itself opens the possibility that subsidies will hurt borrowers for small enough ratios of the price elasticities of supply and demand for the asset. The second term represents the effect of changing the present price of wealth or the interest rate. This second effect depends on the ratio of the elasticities of asset demand with respect to the present price of wealth and the price of the asset. By Assumption 1 the asset and future wealth are substitutes, therefore by increasing the present price of wealth the agent will substitute away from wealth and into more assets, stimulating asset demand. This demand was already stimulated by the increase in credit available to asset buyers, so by further stimulating it the subsidy will cause a significant asset price increase leaving borrowers spending more on the asset relative to the subsidy they receive and therefore leaving them worse off. It is important to note the role of Assumption 1. For example, if assets today and wealth tomorrow are complements then the second term of condition (17) will be negative, placing all the burden in the nonlinear effect of debt policy.22 Next I consider the robustness of the result to the absence of unsecured debt and the presence of heterogeneous agents.

3.5

Incidence Analysis with Unsecured Debt Contracts

One of the key assumptions made in the previous analysis was that there was no unsecured debt available to borrow against future income endowment. The analysis can be extended to incorporate this and in this subsection I briefly discuss the implications of the use of unsecured debt for the previous result. With unsecured debt, an agent will be able to sell part of his future income endowment, thus an increase in the present price of wealth, which he sells on net, will benefit him. 22

Complementarity should not be confused with the fact that a higher level of assets, ceteris paribus, will provide a higher level of wealth tomorrow. In fact, the latter is the case here as asset’s resale value directly contribute to wealth. Instead, complementarity is used in the Hicksian sense that the enjoyment of wealth is enhanced by the level of assets thorugh complementarity in preferences.

24

Nonetheless, as this price change stimulates the current demand for the asset it is still possible that he end up worse off when the asset price is very sensitive to demand changes. The model can easily incorporate unsecured debt, assuming that this debts contracts are enforceable and considering that they entail an interest rate that is strictly higher than the interest rate on secured debt.23 Let rU denote the interest rate on unsecured debt contracts, then it is assumed that rU > r M . Let also d be the promised payoff on unsecured debt contracts. Then, the period 0 and 1 budget constraints, in the presence of a subsidy τ can be written as c + b + pτ x ≤ y0 +

(1 + τ) p¯ f x (1 + τ)d + 1 + rM 1 + rU

w ≤ y1 + b + px ¯ − p¯ f x − d We can generalize Proposition 1 to this environment. In this case agents will only use unsecured debt after exhausting their self financing and secured debt borrowing. If the representative agent was not using unsecured debt the effect of a debt subsidy is the same as before. Alternatively, if the representative agent is using unsecured debt and this debt is subsidized by the same proportion τ then we can compute the effect on the asset price by an infinitesimal change in the subsidy in the same way as we did before. Moreover, we can state the following result which extends Theorem 2 to this case. Theorem 3 If the representative agent is borrowering using unsecured debt contracts, Assumption 1 holds and S  pτ x ζ ζ x,p ¯ d  px d −1 x,π (18) < − + ζ x,p (1 + τ) ζ x,p 1 + rU 1 + r M 1 + rU Then an increase in taxes on borrowing will benefit borrowers and an increase in subsidies on borrowing will hurt them. Note that in this case there is no nonlinear effect brought about by the debt subsidy. In fact, in this case every agent is borrowing more than the present resale value of the asset. Moreover, in order for the upper bound provided in Theorem 3 to be positive it must be that the substitution between wealth and assets is strong enough, or that unsecured borrowing is small relative to the value of the asset. Formally, the ratio of the present value of unsecured debt to the total asset cost needs to be smaller than the ratio of the elasticities of asset demand with respect to the present 23

A positive interest rate spread between unsecured and secured debt in consistent with the model as it should reflect the expected cost of enforcing unsecured contracts or, alternatively, the expected lower recovery rates of unsecured credit.

25

price of wealth and the asset price. That is ζ x,π (1+τ)d 1+rU

(1+τ)d

ζ x,ρ ζ x,p > τ = τ p x ρ x

or

ζ x,π 1+r > τU ζ x,p p x

Theorem 3 shows that the use of unsecured debt makes the conditions, under which the effect of debt policy is the opposite as in the standard case, more demanding. But it is still possible that borrowers will get hurt by debt subsidies if the asset price is very sensitive to demand changes and unsecured debt is small relative to total asset cost.

3.6

Incidence Analysis with Heterogeneous Agents

In this subsection I consider the case of heterogeneous agents, who differ on their income endowment, so some agents are savers, some are borrowers and some are constrained. [COMPLETE]

4

Incidence Analysis in Infinite Horizon

In this section I consider an economy with an infinite number of periods. In this environment the house price in the future is endowment and it is shown that the main result extend to this case.

4.1

The Infinite Horizon Economy

Time is infinite and is denoted by t = 0, 1, . . . There are two types of agents: borrowers and lenders, with a mass of 1 of identical agents of each type. Lenders are assumed to live forever and discount the future at rate 1+r1 M . Their preferences are linear in consumption and have “deep pockets”, so they will be willing to lend any amount at interest rate r M . I use superscript L to denote lender’s variables. Their endowment in period t is ytL , assumed to be large. In each time period lenders consume ctL and they can buy secured debt contracts mtL at price q0M,t , that pay 1 unit of consumption in the next period. Thus, lenders problem is given by

26

max ctL ,mtL

s.t.

ctL

X t≥0

ctL (1 + r M )t

(19)

L + q0M,t mtL ≤ ytL + mt−1

ctL ≥ 0 Borrowers live for two periods and superscript t denotes the variables of the generation born in period t. Borrowers are born in period t ≥ 0 are endowed with ytt units of consumption, that they can consume or spend in houses. When old borrowers like to have wealth denoted by wtt+1 that they consume. As in the simple model individual borrowers can pledge their house as collateral to borrow, using mortgages. In this case the problem of borrowers of generation t is given by max u(ctt , xtt , wtt+1 )

ctt ,xtt , ftt ,wtt+1

s.t.

(20)

ctt + p0t xtt ≤ ytt + q0M,t p0t+1 ftt xtt ftt ≤ 1 wtt+1 ≤ p0t+1 xtt − p0t+1 ftt xtt ctt , xtt , ftt , wtt+1 ≥ 0

where p0t+1 ftt xtt is the amount promised to be repaid in the next period in consumption units. I abstract away from the possibility to save on an alternative device that carries a lower interest rate than the borrowing rate, r M . As long as this agent borrows there is no loss of generality in making this assumption as follows from Proposition 1. In period 0, there are house suppliers who supply houses according to the aggregate supply S function, S x (p00 ) with a price elasticity of supply given by ζ x,p Suppliers consume all the proceeds from house sales. In this environment a competitive equilibrium can be defined as follows.  Definition 2 (OLG Equilibrium) A competitive equilibrium consist of prices p0t , q0M,t t≥0 and   allocations for lenders ctL , mtt t≥0 and borrowers xtt , ctt , wtt+1 , ftt t≥0 such that:  1. ctL , mtt t≥0 is the solution to the lender’s problem taking prices as given  2. xtt , ctt , wtt+1 , ftt is the solution to the borrower’s problem of generation t taking prices as given 3. the house market clears 27

4. the mortgage market clears 5. the consumption market clears

4.2

Individuals’ Decision Problems

Since lenders are assumed to have deep pockets, from the lenders optimality condition it follows that the price of mortgages in equilibrium equals their discount factor. Then, q0M,t =

1 1 + rM

And lenders demand for consumption will be given by ctL

=

ytL

+

L mt−1

mtL − 1 + rM

As in the simple model, I collapse the sequential budget constraint of each generation into a single budget constraint at the time they are young. Solving for pt+1 ftt xtt we obtain: ctt

p0t+1  t 1 xt + wtt+1 ≤ ytt + − 1 + r 1 + r M M | {z } | {z } 

p0t

user cost ρ0t

present price π0t

where ρ0t is the asset user cost in period t and π0t the (present) price at time t of a unit of wealth in period t + 1. Then, we can express the demand of borrowers as ctt (ρ0t , π0t ; ytt ),

4.3

xtt (ρ0t , π0t ; ytt )

and

wtt+1 (ρ0t , π0t ; ytt )

Stationary Equilibrium

It is assumed that each generation of borrowers has the same income endowment, y and I consider the stationary equilibrium, where the price of houses relative to consumption is constant and equal to p0 . Thus, we have that π0 =

1 1 + rM

and

 ρ0 = 1 −

1  0 p = (1 − π0 )p0 1 + rM

In this stationary equilibrium every generation t ≥ 0 will have demands for consumption, housing and wealth that depend on the house user cost, the present price of wealth and income endowment. 28

That is c(ρ0 , π0 , y),

x(ρ0 , π0 , y)

and

w(ρ0 , π0 , y)

The house suppliers, on the other hand, will supply S x (p0 ) and the market clearing conditions can be expressed as c(ρ0 , π0 , y) + w(ρ0 , π0 , y) + cL = c(ρ0 , π0 , y) + p0 S x (p0 ) + cL = y + yL x(ρ0 , π0 , y) = S x (p0 )

4.4

The Infinite Economy with Taxes and Subsidies on Debt

As in the static case I abstract away from the funding of subsidies and the use of tax proceeds. Thus, a subsidy τ will affect only the budget constraint of agents when young, which will now be ctt + pτt xtt ≤ ytt +

(1 + τ)pτt+1 ftt xtt 1 + rM

Once more, we can rewrite the budget constraint for each generation when young as ctt + ρτt xtt + πτt wtt+1 ≤ ytt with ρτt

=

pτt

(1 + τ)pτt+1 − 1 + rM

and

πτt =

1+τ 1 + rM

Then, we can express the demand of borrowers as ctt (ρτt , πτt , ytt ),

xtt (ρτt , πτt , ytt )

and

wtt+1 (ρτt , πτt , ytt )

In the case of a stationary equilibrium these reduce to c(ρτ , πτ , y), with

1+τ π = 1 + rM τ

x(ρτ , πτ , y)

and

and

w(ρτ , πτ , y)


1+τ  τ ρ = 1− p = 1 − πτ pτ 1 + rM 

τ

29

4.5

Incidence Analysis in Infinite Horizon

I proceed as before to sign the welfare effect of a subsidy (or tax) on mortgages. The focus is on the local effect of changing the borrowing subsidy, that corresponds to the derivative of the indirect utility function with respect to the subsidy. Using the Envelope Theorem on borrowers’ problem (20) we have  d pτ  d pτ τ
 dV(ρτ , πτ , y) τ
τ dπ τ = −λ 1 − fπ − p f x − λπ − 1− f dτ dτ dτ dτ 
d pτ dπτ  = −λ 1 − πτ − pτ f x dτ dτ

(21)

where we use that the ratio of marginal values of wealth between young and old equals the present price of wealth.24 In order to compute the the magnitude of the house price change we fully differentiate the house market clearing condition τ dx(ρτ , πτ , y) 0 τ dp = S x (p ) dτ dτ

(22)

with the total effect of an infinitesimal subsidy on individual asset demand given by: dx(ρτ , πτ , y) ∂x(ρτ , πτ , y) dρτ ∂x(ρτ , πτ , y) dπτ = + dτ ∂ρ dτ ∂π dτ From the expression for the present value and the asset user cost in the stationary equilibrium we have that
d pτ 1 dρτ dπτ τ dπτ = and =− p + 1 − πτ dτ 1 + rM dτ dτ dτ As before It is assumed that Assumption 1 holds so we can use the Slutzky equations, Theorem 1 and the budget constraint at old age, to obtain an expression for the derivative of house demand with respect to the subsidy in terms of price derivatives and price and income demand elasticities. Substituting this expression into the left hand side of equation (22) we get an expression the subsidy effect on the house price d pτ = dτ

f pτ x ζ x,y y (1−πτ ) ζ x,p p

ζ x,p + 1 S ζ pτ x,p

+

+ +

1 ζ πτ x,π (1−πτ )x ζ x,y y

1 1 + rM

(23)

That is, if λ is the marginal value of wealth when young, or the Lagrange multiplier of the flow budget constraint when young, and λ0 is the marginal value of wealth when old, or the Lagrange multiplier of the flow budget constraint when old, then λ0 = πλ. 24

30

Now we are ready to state the following result. Theorem 4 (Infinite Horizon) If the representative agent is a borrower, assets and wealth are substitutes and S ζ x,p (1 − f )(r M − τ) (r M − τ) ζ x,π < + (24) ζ x,p f (1 + r M ) (1 + τ) f ζ x,p Then, an increase in subsidies (taxes) on borrowing will hurt (benefit) borrowers. The proof is relegated to the Appendix.

5

Conclusion

In this paper I showed that debt policy can have the opposite effect as it intended to, with borrowing subsidies hurting borrowers and taxes benefiting them. Theorem 2 formalizes this result, giving precise sufficient conditions for it. Crucially, that contemporaneous assets and future consumption are Hicksian substitutes. Moreover, it was required that the ratio of price elasticities of supply and demand for assets is small relative to the ratio of the elasticity of asset demand with respect to the present price of wealth and the price elasticity of asset demand. The result was shown to be robust to allowing for the use of unsecured debt, provided unsecured borrowing is small relative to the total value of asset expenses. In addition, the result was shown to be robust to the case where asset price changes are permanent. This was studied in an infinite horizon economy populated by overlapping generations of borrowers. Directions for future work include considering the case of financial assets, that in contrast to housing do not provide a direct service to the owner. From the analysis presented here it is straightforward to consider that case, as the main mechanisms did not rely on the fact that agents derived utility from asset services, rather it required that debt policy have an effect on both the interest rate and the asset user cost. Future work also considers including uncertainty into the analysis. Finally, this analysis calls for a quantitative evaluation of the welfare losses associated to different debt subsidies programs. These evaluations are left for future research.

31

References [1] The mortgage interest deduction and its impact on homeownership decisions, SERC Discussion Papers 0055, Spatial Economics Research Centre, LSE, Sep 2010. [2] Aaron, H.J., Shelter and subsidies: who benefits from federal housing policies?, Studies in social economics, Brookings Institution, 1972. [3] Adelino, Manuel, Antoinette Schoar and Felipe Severino, Credit Supply and House Prices: Evidence from Mortgage Market Segmentation, NBER Working Paper 17832, http://www. nber.org/papers/w17832, 2012. [4] Bianchi, J., Credit externalities: Macroeconomic effects and policy implications, American Economic Review 100 (2010a), no. 2, 398–402. [5]

, Overborrowing and systemic externalities in the business cycle, American Economic Review (2010b), forthcomming.

[6] Bianchi, J. and E. Mendoza, Overborrowing, financial crises and ’macroprudential’ policy, IMF Working Paper WP/11/24, 2011. [7] Bourassa, Steven C. and William G. Grigsby, Income tax concessions for owner-occupied housing, Housing Policy Debate 11 (2000), no. 3, 521–546. [8] Favilukis, Jack, Sydney C. Ludvigson and Stijn Van Nieuwerburgh, The Macroeconomic Effects of Housing Wealth, Housing Finance, and Limited Risk-Sharing in General Equilibrium, SSRN eLibrary (2010). [9] Fostel, A. and J. Geanakoplos, Endogenous Leverage: VaR and Beyond, Cowles Foundation Discussion Paper No. 1800, 2011a. [10] Fullerton, D. and G.E. Metcalf, Tax incidence, Handbook of Public Economics (A. J. Auerbach and M. Feldstein, eds.), Handbook of Public Economics, vol. 4, Elsevier, 2002, pp. 1787–1872. [11] Geanakoplos, J., Promises Promises, Santa Fe Institute Studies in the Sciences of Complexity-proceedings volume, vol. 27, Addison-Wesley Publishing Co, 1997, pp. 285– 320. [12]

, Solving the present crisis and managing the leverage cycle1, FRBNY Economic Policy Review 16 (2010), no. 1.

[13] Geanakoplos, J. and W.R. Zame, Collateral equilibrium: Some general theorems and specific examples, mimeo, 2010. [14] Harberger, A., The incidence of the corporation tax, Journal of Political Economy 70 (1962), 215–40. 32

[15] Jeanne, O. and A. Korinek, Excessive volatility in capital flows: A pigouvian taxation approach, American Economic Review 100 (2010), no. 2, 403–07. [16] Kiyotaki, Nobuhiro, Alexander Michaelides and Kalin Nikolov, Winners and Losers in Housing Markets, SSRN eLibrary (2011). [17] Kotlikoff, L.J. and L.H. Summers, Tax incidence, Handbook of Public Economics (A. J. Auerbach and M. Feldstein, eds.), Handbook of Public Economics, vol. 2, Elsevier, 1987, pp. 1043–1092. [18] Lowenstein, Roger, Who needs the mortgage-interest deduction?, New York Times, March 5th . [19] Deborah Lucas, The budgetary cost of fannie mae and freddie mac and options for the future federal role in the secondary mortgage market, Tech. report, Testimony before the Committee on the Budget U.S. House of Representatives, June 2001. [20] Office of Management and Budget, Analytical perspectives, Tech. report, Office of Management and Budget, 2012. [21] Congressional Budget Office, Interest rate differentials between jumbo and conforming mortgages, 1995-2000, Tech. report, Congressional Budget Office, May 2001. [22] Poterba, James and Todd Sinai, Tax expenditures for owner-occupied housing: Deductions for property taxes and mortgage interest and the exclusion of imputed rental income, The American Economic Review 98 (2008), no. 2, pp. 84–89. [23] Rajan, R., Fault lines: How hidden fractures still threaten the world economy, Princeton University Press, 2010. [24] Harvey S. Rosen, Housing subsidies: Effects on housing decisions, efficiency, and equity, Handbook of Public Economics, vol. 1, 1985, pp. 375 – 420. [25] Salanié, Bernard, The economics of taxation, 2nd ed., The MIT Press, 2011. [26] Slesnick, Daniel T., Empirical approaches to the measurement of welfare, Journal of Economic Literature 36 (1998), no. 4, 2108–2165. [27] Stroebel, Johannes and Max Floetotto, Government intervention in the housing market: Who wins, who loses?, Department of Economics, Stanford University, 2012.

33

Appendix A

Proofs

Proof of Proposition 1: We can write the agents problem as max u(c, x, w)

c,x, f,b,w

f ≤1

(µ) p ¯ f x (λ0 ) c + b + p0 x ≤ y0 + 1 + rM w ≤ y1 + b + px ¯ − p¯ f x (λ1 ) s.t.

c, x, b, w, f ≥ 0

(η• )

By the Inada condition assumed on each variable of the utility function we have that c, x, w > 0 and we can dispense of those constraints. It follows that the FOC are (c)

0 = uc − λ0

(b)

0 = −λ0 + λ1 + ηb 
p¯ f  0 0 = u x − λ0 p − + λ1 p¯ − p¯ f 1 + rM px ¯ 0 = λ0 − λ1 px ¯ + ηf − µ 1 + rM 0 = uw − λ1

(x) (f) (w) First, note that b>0

⇔

ηb = 0

⇔

uw = uc

η f = µ + uc

r M px ¯ >0 1 + rM

⇒

f =0

Thus, if b > 0 then ( f ) imply that

Using that f = 0 and solving for b from the budget constraint in period 0 and 1, we get
c + p0 − p¯ x + w = y0 + y1 On the other hand, if f > 0 then η f = 0 and from ( f ) we have uc >

uc µ = uw + ≥ uw 1 + rM px ¯

⇒

b=0

Using that b = 0 and solving for p¯ f x from the budget constraint in period 0 and 1, we get  p¯  1 1 c + p0 − x+ w = y0 + y1 1 + rM 1 + rM 1 + rM

34

This last expression imply that agents only use secured debt contracts when their internal funds have been depleted. In fact, it shows that borrowers trade off future wealth for asset purchases, as if they were financing the last unit of their asset holdings using the maximum allowed debt. This imply that agents prioritize using internal funds ands use as little debt as possible. Proof of Theorem 1: From the definitions of compensated elasticities we have that, xζ x,p ∂x ∂x ∂ρ ∂x xζ x,ρ = = = = p ∂p ∂ρ ∂p ∂ρ ρ where

p ρ

⇒

ζ x,p =

p ζ x,ρ ρ

corresponds to the maximum or marginal leverage used in home purchases.

Proof of Theorem 2: We have that S ζ x,p

ζ x,p

<

1 − f (1 + r M ) p ζ x,π + f (1 + τ) p¯ f ζ x,p

First note that the RHS is strictly positive since 0 < f < 1, which is the case for a representative borrower. In addition, the second term is non-negative given Assumption 1. Substituting the expression for the equilibrium effect on the asset price, equation (16), we get that, d pτ p¯ f − = dτ 1 + rM

(1− f ) p¯ 1+r M

+

1+

p ζ x,π 1+τ ζ x,p S ζ x,p ζ x,p

+

−

S p¯ f ζ x,p 1+r M ζ x,p

px ζ x,y y ζ x,p

>0

It follows that

 d pτ p¯ f  dV(τ) = −λ − x<0 dτ dτ 1 + rM That is, an increase in borrowing subsidies reduces the agent indirect utility. On the other hand, an increase in borrowing subsidy will have an effect of the same magnitude as an increase in borrowing subsidies but with the opposite sign, so dV(τ) − <0 dτ which is the effect on agent’s indirect utility of an increase in borrowing taxes. Proof of Theorem 4: We have that S ζ x,p

ζ x,p

<

(1 − f )(r M − τ) (r M − τ) ζ x,π + f (1 + r M ) (1 + τ) f ζ x,p

Note that the RHS is strictly positive since 0 < f < 1, which is the case for a representative borrower. In addition, the second term is non-negative given Assumption 1.

35

Substituting the expression for the equilibrium effect on the asset price, equation (23), we get that, f S τ (1 − f )ζ x,p + π1τ ζ x,π − 1−π τ ζ x,p d pτ 1 τ dπ (1 − π ) >0 −p f = τ) τ )x (1−π (1−π 1 S dτ dτ ζ x,y 1 + r M pτ ζ x,p + p ζ x,p + y τ

It follows that

 dV(τ) d pτ dπτ  = −λ (1 − πτ ) − pτ f x<0 dτ dτ dτ That is, an increase in borrowing subsidies reduces the agent indirect utility. On the other hand, an increase in borrowing taxes will have an effect of the same magnitude as an increase in borrowing subsidies but with the opposite sign, thus an increase in borrowing taxes will benefit borrowers.

36