Taxing versus subsidizing debt under financial frictions

Schabert, Andreas Article — Published Version Taxing versus subsidizing debt under financial frictions Economic Theory Provided in Cooperation with: Springer Nature Suggested Citation: Schabert, Andreas (2024) : Taxing versus subsidizing debt under financial frictions, Economic Theory, ISSN 1432-0479, Springer, Berlin, Heidelberg, Vol. 79, Iss. 4, pp. 1383-1420, https://doi.org/10.1007/s00199-024-01615-3 This Version is available at: https://hdl.handle.net/10419/323259 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Economic Theory (2025) 79:1383–1420 https://doi.org/10.1007/s00199-024-01615-3 RESEARCH ARTICLE Taxing versus subsidizing debt under financial frictions Andreas Schabert1 Received: 3 July 2024 / Accepted: 25 September 2024 / Published online: 22 October 2024 © The Author(s) 2024 Abstract We examine optimal credit market policies in two models with durables/capital as collateral. Pecuniary externalities rationalize ex-ante debt taxes as macroprudential regulation, achieving constrained efficiency. Ex-post debt subsidies can implement first-best by stimulating collateral demand. Due to the same effect, debt subsidies that are constant over time can be superior to debt taxes. Saving subsidies can further enhance efficiency by addressing distributive effects of pecuniary externalities via interest rate reductions. The analysis shows that debt-increasing subsidies can outperform macroprudential regulation, and that constrained inefficiency caused by collateral externalities is insufficient to establish debt taxes as optimal credit market policies. Keywords Financial stability ·Pecuniary externalities ·Collateral constraint · Macroprudential regulation ·Distributive effects JEL Classification E44 ·G18 ·H23 1 Introduction Pecuniary externalities under collateral constraints can lead to financial amplification and crises. The mechanism relies on price-dependent borrowing limits or margin constraints that tighten when asset prices fall. Agents do not internalize the impact of their decisions on asset prices, such that corrective policies can enhance efficiency. Macroprudential regulation, in form of ex-ante debt taxes and capital controls, can restore "constrained efficiency" – defined in the tradition of Stiglitz (1982) – by addressing "overborrowing", as shown by Jeanne and Korinek (2010,2019,2020), Bianchi (2011), Benigno et al. (2016), Korinek and Sandri (2016), Schmitt-Grohe and Uribe (2017), or Korinek (2018). These studies are based on a specific class of models where interest rates are exogenously determined and agents take borrowing limits as given. Thus, neither credit supply nor assets’ collateralizability, which both seem to play a BAndreas Schabert [email protected] 1Center for Macroeconomic Research, University of Cologne, Albertus-Magnus-Platz, 50923 Cologne, Germany 123 1384 A. Schabert central role for the build-up of financial crises (see Geanakoplos 2010, or Justiniano et al. 2019), are taken into account for the analysis of policies aimed to mitigate these crises. Is their neglect irrelevant? This paper shows that endogenous interest rates and collateral premia, i.e. the valuation of assets to serve as collateral, are in fact decisive for optimal credit market policy. We apply two finite horizon models that exclusively contain conventional features; one model is taken from Davila and Korinek (2018).1Lack of commitment induces borrowing to be limited by borrowers’ holdings of durables or capital, serving as collateral. Pecuniary externalities with regard to the collateral price and to the interest rate give rise to "collateral externalities" and "distributive externalities" (see Davila and Korinek 2018). The former are responsible for the main mechanism in the above cited studies, whereas the latter are turned off therein. While state-contingent ex-post credit market interventions can achieve first best, we particularly focus on policies that are less challenging to be implemented than policies that are fully state-contingent. As the main novel contribution, we show that distributive effects and collateral premia are responsible for non-state-contingent credit market subsides that stimulate borrowing to be superior to a debt-reduction policy, in particular, to macroprudential regulation in the form of an ex-ante debt tax.2At large, inefficiencies due to externalities induced by collateral constraints can be most effectively addressed by saving/debt subsidies that reduce interest rates and raise collateral prices. This paper specifically highlights the effectiveness of non-state-contingent policies that promote savings and lower borrowing costs. Real-world examples include home loan saving contracts or tax deductions for mortgage interest payments. The analysis implies that combining these types of policies with ex-ante regulation that limit the build-up of debt, which reflects actual government practices, can principally enhance welfare even further and can replicate state-contingent credit market policies. In a laissez faire equilibrium (of both models), agents do not internalize that the collateral price is too low and the interest rate is too high when borrowing constraints bind in a non-empty set of states. Following Davila and Korinek (2018) classification, we distinguish collateral effects, which refer to uninternalized changes in the price of pledgeable assets affecting the collateral value, from distributive effects, which refer to differential effects of uninternalized interest rate changes on heterogenous agents. In contrast to the above cited studies, where agents take borrowing limits as given, the price of pledgeable assets is positively affected by a collateral premium, i.e. the valuation of assets to serve as collateral.3Corrective policies can address uninternalized changes in the collateral price, leveraging the fact that the collateral price increases with consumption and with agents’ willingness to borrow, which raises the collateral premium. Agents further do not internalize that equilibrium interest rates relate to their consumption/saving choices, which exerts a relevant impact on the equilibrium 1We first develop and analyze a model with durables as collateral under uncertainty. Davila and Korinek (2018) model with capital formation under certainty is subsequently analyzed in Sect.5. 2Throughout the paper, we identify macroprudential regulation with an ex-ante debt tax, following Bianchi and Mendoza (2018): "the macroprudential debt tax [...] is levied in good times when collateral constraints do not bind at date tbut can bind with positive probability at t+1" (p. 591). 3This asset price component is also known as the "collateral value" (see Fostel and Geanakoplos 2008) or the "collateralizability premium" (see Ai et al. 2020). 123 Taxing versus subsidizing debt under financial frictions 1385 allocation under different marginal rates of substitution between dates/states (MRSs) of borrowers and lenders (see also Davila and Korinek 2018). Corrective policies can address this externality and reduce the interest rate, which raises borrowers’ consumption and narrows the distance between the MRSs. Apparently, interest rates cannot be reduced when they are assumed to be exogenously determined, and collateral premia do not exist when agents take borrowing limits as given, like in the above cited studies. In this special case, overborrowing prevails and a correction of the collateral price relies on debt reduction via ex-ante debt taxes. To identify optimal credit market policies, we assume that the policy maker acts under full commitment and we apply the Ramsey approach to optimal policy, where the policy problem depends on the set of available instruments.4If state-contingent credit market instruments are available, first best is implementable via a Pigouvian debt subsidy that is introduced ex-post (i.e., in states where collateral constraints bind). The subsidy increases incentives to borrow and thus the willingness to pay for collateral, measured by the collateral premium. An ex-post debt subsidy can thereby raise the collateral price and the borrowing limit such that borrowing can in principle even get unconstrained, which has also been shown by Katagiri et al. (2017). Based on their quantitative analysis, they conclude that the optimal debt subsidy is practically infeasible, given the size and the frequency of required interventions. State-contingency in fact demands policies to be fine-tuned in response to any change in the state of the economy (see e.g. Bianchi and Mendoza 2018). Given that the requirement to accurately track relevant conditions and to timely adjust policy tools can hardly be fulfilled in practice (see e.g. Cochrane 2013), our analysis focusses on Pigouvian policies that are less complex and easier to implement than fully state-contingent policies.5Specifically, we examine ex-ante policies, which are imposed before borrowing constraints might become binding, and constant policies, where the tax/subsidy rate is held constant regardless of the state or period (see columns of Table 1). These policies take the form of taxes/subsidies on borrowing or saving (see rows of Table 1), which are non-equivalent under potentially binding borrowing constraints. Linear preferences of lenders further imply that debt policies do not alter the interest rate and can only address collateral effects, enabling direct comparisons with related studies (see Sect.2). In contrast, saving policies can endogenize interest rates and can thereby address distributive effects of pecuniary externalities (see arrows in Table 1). The first optimal non-state-contingent policy is an ex-ante debt tax. It implements a constrained efficient allocation, as defined in Stiglitz (1982) or Davila et al. (2012). This allocation is chosen by a social planer who determines borrowing and maximizes social welfare subject to budget and borrowing constraints, conditional on maintaining equilibrium price relations under laissez faire. Given that ex-ante debt taxes leave the relevant price relations under laissez faire unchanged, the optimal Ramsey policy in 4This property of the Ramsey approach is also relevant in Benigno et al. (2023), who show that a set of instruments that supports constrained efficiency can also implement an unconstrained allocation. In contrast to Bianchi and Mendoza (2018), who focus on optimal policy under discretion, we abstract from time inconsistency of policy plans. 5An alternative would be to adjust policy instruments with changes in variables that can easily be observed, like gdp or debt (see e.g. Bianchi and Mendoza 2018). In our main model, the latter are however not correlated to the shock that cause borrowing constraints to bind, namely, an unexpected change in income inequality. 123 1386 A. Schabert Table 1 Non-state-contingent Pigouvian policies (numerical examples in bold) Ex-ante Fixed over time Debt 1. tax/subsidy →collateral effects 2. tax/subsidy →collateral effects Saving 3. tax/subsidy →collateral & distributive effects 4. tax/subsidy →collateral & distributive effects both models implements constrained efficiency (like in Davila and Korinek 2018). Concretely, it enhances efficiency by raising the collateral price via a debt reduction that increases the amount of funds available for consumption in states where the borrowing constraint binds. In contrast to ex-ante debt taxes, the other policies under consideration alter the price relations for the interest rate and the collateral price, which would not be possible when agents take borrowing limits as given and interest rates are exogenous. The second non-state-contingent policy is a tax/subsidy on debt that is constant over time and influences borrowing regardless whether the constraint is binding or not. As shown by Bianchi and Mendoza (2018) analysis of optimal debt policy, a policy maker can alleviate currently binding borrowing constraints by an ex-post subsidy and future borrowing constraints by an ex-ante tax. However, a constant debt subsidy, which tends to stimulate borrowing, reduces resources available for consumption when the constraint binds, while it raises agents’ willingness to pay for collateral. It thus combines the inverse effects of an ex-ante debt tax on consumption with the effects of an ex-post debt subsidy on the collateral premium. A constant debt subsidy is superior to a constant debt tax if the effect on the collateral premium dominates. We show that this holds unconditionally in Davila and Korinek (2018) model as well as in the model with durables if the loan-to-value ratio is sufficiently large (including values typically used in quantitative studies). Accordingly, a constant debt tax can be superior to a subsidy in the latter model under smaller loan-to-value ratios, which reduce the effect via the collateral premium. This relates to Bianchi and Mendoza (2018) finding of relatively small welfare gains of constant debt taxes in a model where the loan-tovalue ratio falls in a crisis, and to Bianchi (2011), who reports sizable welfare effects of constant debt taxes for a model without a collateral premium. The remaining two non-state-contingent policies impose taxes/subsidies on lenders. The third policy is an ex-ante saving tax/subsidy, which directly alters the price relation for the equilibrium interest rate. Agents do not internalize the effects of their consumption plans on the interest rate, which cannot be addressed by taxing/subsidizing borrowers, given that the interest rate equals the inverse of the lenders’ discount factor. By subsidizing savings, lenders demand a lower interest rate, which raises borrowers’ current relative to future consumption. The reduction in borrowing costs therefore narrows the distance between the MRS of lenders and borrowers; the latter engaging in precautionary saving under potentially binding borrowing constraints. Increased outstanding debt, however, tends to reduce consumption when the collateral constraint binds and thus to lower the collateral price. Hence, there is a trade-off between the effects on the prices of debt and of collateral. A policy maker decides to subsidize 123 Taxing versus subsidizing debt under financial frictions 1387 saving ex-ante and to reduce the costs of borrowing for potentially constrained agents when distributive effects dominate collateral effects. This is particularly the case in both models when the wealth distribution is sufficiently unequal. The fourth policy is a saving tax/subsidy that is constant over time. In contrast to the ex-ante saving subsidy, it tends to stimulate borrowing as well as consumption before and while the collateral constraint is binding by reducing the interest rate and by raising the collateral premium. It can thereby simultaneously address distributive and collateral effects. To unveil the role of the collateral premium and to reconstruct findings of the studies on macroprudential regulation cited above, we refer to an alternative specification where the borrowing limit is assumed to depend on the aggregate stock of pledgeable assets. For this specification of the borrowing constraint, which is not consistent with the underlying imperfection (i.e. limited commitment), the price that alters the borrowing limit is not affected by the collateral premium, such that debt/saving subsidies can neither implement first best nor address adverse collateral effects. If distributive effects are further disregarded, optimal ex-ante and constant policies are debt taxes, implying that agents overborrow. While the analytical results reveal the main principles, we further provide numerical results for the less stylized model with durables for illustrative purposes. The four optimal policies are (1) an ex-ante debt tax, (2) a constant debt subsidy, (3) an ex-ante saving subsidy, and (4) a constant saving subsidy (see Table 1). Except for the ex-ante debt tax, all policies tend to raise debt before the borrowing constraint binds, and the constant policies induce the largest increases in the collateral price, revealing the relevance of collateral premia. The ex-ante debt tax has the least impact on borrowers’ consumption and leads to the smallest welfare gains relative to laissez faire, which are virtually negligible based on the distance to first best. Saving policies exert relatively large redistributive and social welfare effects via interest rate reductions.6An optimal constant saving subsidy, which leads to the largest welfare gains relative to laissez faire, can thereby reduce welfare losses by about a half compared to first best. By confining our analysis to models employing a linear utility function for lenders, a decision that facilitates the replication of established results on macroprudential regulation, we abstract from distributive effects under debt policies. If lenders’ utility were instead a non-linear function of consumption like borrowers’ utility, interest rates would depend on agents’ endogenous MRSs. An ex-ante debt tax would lower borrowers’ current relative to future consumption, such that lenders’ current consumption would increase relative to future consumption and the period-1 interest rate would unambiguously fall. The reduction in the interest rate would mitigate (but not invert) the debt tax effect on borrowers’ current relative to future consumption. The distributive effects would however demand an increase of borrowers’ current consumption, which is depressed in a laissez faire equilibrium due to precautionary saving. The recommendation regarding an ex-ante debt policy is therefore less clear-cut when considering the relevance of distributive effects under non-linear lenders’ utility. In contrast, saving subsidies can address distributive effects under non-linear lenders’ utility, since they would reduce lenders’ current consumption as well as interest rates. 6Optimal saving policies lead to a redistribution of funds from lenders to borrowers, despite the absence of a redistributive motive of the social planner, owing to quasi-linear preferences. 123 1388 A. Schabert Both effects would induce borrowers to raise consumption relative to lenders, narrowing the distance between their MRSs. This mechanism is in principle relevant whenever borrowing constraints bind with a non-zero probability in a heterogeneous agent economy. The remainder is structured as follows. Section2discusses the related literature. Section3develops the model with durables as collateral under uncertainty. Section4examines optimal policies. Section4.5 presents numerical illustrations. Section 5 presents analytical results for Davila and Korinek (2018) model with endogenous capital formation, where the borrowing constraint binds with certainty. Section 6concludes. 2 Related literature This paper is related to several studies on corrective policies under collateral externalities, like Jeanne and Korinek (2010,2019), Bianchi (2011), Benigno et al. (2016), Korinek and Sandri (2016), Schmitt-Grohe and Uribe (2017), Bianchi and Mendoza (2018), or Korinek (2018). They focus on constrained efficient allocations, as defined in Stiglitz (1982), and macroprudential policies, like debt taxes or capital controls, that are imposed when borrowing constraints are not binding. In contrast to our analysis, these studies apply models where interest rates are exogenously determined and where – except of Bianchi and Mendoza (2018) – agents take borrowing limits as given, implying that there are neither distributive effects nor collateral premia on pledgeable assets. Bianchi and Mendoza (2018) focus on time-consistent policies under discretion, such that commitment to ex-post policies is not possible and first best cannot be implemented. They discuss how the collateral premium principally affects the price of collateral and optimal debt policy. In their quantitative analysis, they report results for macroprudential debt taxes that are imposed when the collateral constraint does not bind and the collateral premium equals zero. They further apply constant debt taxes and find that they lead to relatively small welfare gains or welfare losses, consistent with our results on constant debt policies. Bianchi (2011) finds that a constant debt tax can achieve sizable welfare gains in a model where a collateral premium is nonexistent. In addition to debt taxes, Benigno et al. (2016) analyze policies introduced in other markets, and show that an ex-post tax on non-tradables can raise the collateral price, such that the borrowing constraint does not bind. Bianchi (2016) and Jeanne and Korinek (2020) find welfare gains from ex-post policies in form of debt reliefs or liquidity provisions, which do not implement first best. In addition to these analyses, we examine timeand state-invariant debt/saving subsidies, and show that they can be superior to debt taxes. Our finding that the stimulation of borrowing can enhance social welfare relates to the following studies: Benigno et al. (2013) examine the constrained efficient allocation of an economy where agents take into account that labor supply alters the borrowing limit, which compares to our analysis where agents internalize that borrowing limits depend on their holdings of eligible assets. They show that one should rather reallocate resources between (tradable and non-tradable goods) sectors to raise borrowing limits than subsidize borrowing. In a related model, Arce et al. (2023) show that an ex-ante debt tax is desirable even when ex-post labor market policies are applied and borrowing 123 Taxing versus subsidizing debt under financial frictions 1389 is enhanced in a constrained efficient allocation. Katagiri et al. (2017) apply a variant of Jeanne and Korinek (2010) model where agents internalize collateral services of eligible assets. Like in our models, an ex-post debt subsidy can implement first best by raising the collateral price via the collateral premium such that the borrowing limit is not binding. Their analysis neither examines interest rate effects nor non-statecontingent policies, on which our analysis focusses. Schmitt-Grohe and Uribe (2021) establish the existence of multiplicity in the model examined by Bianchi (2011), giving rise to equilibria with underborrowing due to excessive precautionary savings. For a model with bank intermediation, Chi et al. (2022) show that agents borrow less under laissez faire compared to equilibria with ex-post expansions of bank reserves. Ottonello et al. (2022) shows that constrained inefficiency depends on whether borrowing limits depend on current or future collateral prices, and that debt subsidies can be optimal in the latter case. In contrast to our analysis, none of the above cited studies considers distributive effects. In a seminal paper, Lorenzoni (2008) shows that distributive externalities under financial frictions cause agents to overinvest and to overborrow in an unregulated economy. Davila et al. (2012) show in a model with an endogenous wealth distribution that distributive effects can either lead to overor underaccumulation of capital. Lanteri and Rampini (2023) develop a model of endogenous formation and reallocation of capital. They show that distributive effects of pecuniary externalities with regard to the capital price are larger than collateral externalities, such that a subsidy on new investment enhances efficiency. Both studies do not analyze credit market policies. Davila and Korinek (2018) apply a general framework with capital formation, for which they establish collateral and distributive effects. They show that pecuniary externalities can either cause overor underinvestment, while they emphasize that "collateral externalities generally entail overborrowing" (p. 354). We show for their model that this conclusion holds only if the analysis is restricted to ex-ante debt policies. The above cited studies focus on the analysis of constrained efficient allocations, which can either be derived from a problem of choosing initial allocations or from a Ramsey problem when equilibrium price relations are unaffected by policy instruments (see also Davila and Korinek 2018). In contrast, the solutions to our policy problems differ from this type of constrained efficient allocation when the price relations for the collateral price or for the interest rate are affected by policy. Relatedly, Benigno et al. (2023) re-examine policy instruments used in Benigno et al. (2016), applying the Ramsey approach. Complementary to our analysis of different policy instruments, they show that a set of instruments that can implement a constrained efficient allocation can also be used to implement a superior allocation where borrowing constraints never bind. This possibility relies on the use of taxes/subsidies outside the credit market, while we show that first best is implementable with ex-post credit market policies. 3 A model with incomplete markets and limited commitment In this section, we develop a finite horizon model with durables in fixed supply. Section5presents Davila and Korinek (2018) model with capital formation, which is 123 1390 A. Schabert slightly more stylized (without uncertainty and without discounting).7There exist two imperfections in both models: Only non state-contingent debt is available and agents are not able to commit to debt repayment. The latter leads to the key financial friction, i.e. a borrowing constraint with the borrower’s asset serving as collateral. 3.1 Details There are two mass-one groups {b,l}with infinitely many agents, who live for three periods t=1,2,3. In each period t, a household i∈{b,l}derives utility from consumption of a non-durable good, ci,t, and a durable good (or housing), di,t,as given by the function ui,t=u(ci,t,di,t). Agents maximize their expected lifetime utility, E3 t=1βt−1u(ci,t,di,t), where uis strictly increasing and concave, Edenotes an expectations operator conditional on information in period 1, and β∈(0,1)is a discount factor. In each period, agents receive a potentially random endowment yi,tof non-durable goods and they exhibit an initial endowment of durables di,0. Agents can borrow and lend only in terms of non state-contingent one-period bonds bi,t, which are issued at the price 1/rt. The budget constraint of an agent ifor period tis given by ci,t+qt(di,t−di,t−1)+(1−τi,t)bi,t/rt=bi,t−1+yi,t+Ti,t,(1) where τi,tdenotes distortionary taxes/subsidies on debt/saving. Specifically, we consider Pigouvian-type fiscal interventions, where budgetary effects of taxes/subsidies are (ex-post) neutralized in a non-distortionary way: Ti,t=−τi,tbi,t/rt,(2) which is not internalized by agents. There is no uncertainty in the periods 1 and 3, where total endowment with non-durables is equally distributed: yb,1=yl,1=y/2. Agents b (l) start with negative (positive) initial net financial wealth bb,0<0(bl,0>0) and will be called borrowers (lenders). In period 2, endowments are randomly determined and can either take the same values as in period 1 (state L) or can be unequally distributed (state H). Specifically, both states are equally likely and endowment of borrowers in state H(with Higher inequality) is yb,2=y/(1+δH), where δH>1. We assume that agents cannot commit to repay debt and that debt can be renegotiated after issuance in the same period. Borrowers can make a take-it-or-leave-it offer to reduce the value of debt. If a lender rejects the offer, she/he can seize a fraction γof the borrower’s durable goods, which she/he can sell at the market price qt.Offersare therefore accepted when the repayment value of debt at least equals the current value of seizable assets. Without loss of generality, we assume that default and renegotiation never happen in equilibrium. When debt is issued, the amount of debt −bi,tis therefore 7While the analysis of their model includes an additional (capital investment) tax/subsidy, the results on borrowing and saving taxes/subsidies correspond to the result for the model with durables. 123 Taxing versus subsidizing debt under financial frictions 1397 4.1 An ex-ante Pigouvian tax on debt We first consider the case, where a tax/subsidy on debt might be introduced in period 1, whereas no policy instrument is applied in period 2. Ex-ante debt taxes, which correspond to capital controls in open economies, have already been examined in several related studies (see Davila and Korinek 2018, or Erten et al. 2021, for an overview), establishing that they can implement a constrained efficient allocation as defined by Stiglitz (1982). Following Bianchi and Mendoza (2018), we will refer to an ex-ante debt tax as macroprudential regulation. Under such a policy, borrowers’ optimality condition (5) changes to 1−τb,1c−1 b,1/r1=βEc−1 b,2.(17) In equilibrium, condition (17) and the optimal lending choice 1/r1=βimply 1−τb,1=cb,1Ec−1 b,2. By taxing debt in period 1, τb,1>0, agents can be induced to borrow less, which tends to raise cb,2relative to cb,1and the durables price q2 via (12).12 Given that borrowers do not internalize the adverse effect of period-1borrowing on the durables/collateral price and thus the borrowing limit in period 2, a policy maker can enhance efficiency by addressing collateral effects of pecuniary externalities with an ex-ante debt tax. This mechanism is well-established in the literature on macroprudential regulation and capital controls, and has led to the notion of “overborrowing”. Proposition 2 Suppose that the policy maker can apply a Pigouvian tax/subsidy on debt before the borrowing constraint might be binding. Then, the optimal allocation is constrained efficient and associated with a tax on debt, satisfying τb,1=cb,1γd(1−βγ)Eμtb1 2·χc≥0,(18) where μtb1 2≥0denotes the multiplier on the borrowing constraint of the policy problem. Proof See Appendix.  The optimal ex-ante debt tax described in Proposition 2implements the “constrained efficient allocation”, which is chosen by a social planer respecting budget and borrowing constraints and allowing markets for durables and non-durables to clear in a competitive way (see Stiglitz 1982, or Davila et al. 2012). Concretely, a constrained efficient allocation is chosen by a social planer who determines borrowing and maximizes social welfare Wsubject to budget and borrowing constraints, while taking the competitive equilibrium relations for interest rates (10) and the durables price (12) under laissez faire into account. The Ramsey optimal ex-ante debt tax leads to the same outcome, since it leaves the pricing Eqs. (10) and (12) unaffected.13 In contrast, 12 This positive effect of higher net worth of borrowers on the durables/collateral price corresponds to the effect in Davila and Korinek (2018) imposed by their condition 1. 13 Benigno et al. (2016) and Davila and Korinek (2018) also show that this approach can be equivalent to a Ramsey optimal policy where the policy maker chooses taxes ex-ante or on first period allocations. 123 1398 A. Schabert we will examine polices in the subsequent sections that alter (10) and (12), such that the prices q2,1/r1, and 1/r2can be altered by policy in more direct ways. Under alternative credit market polices, competitive equilibrium allocations can thereby be implemented that are superior to the constrained efficient allocation under the ex-ante debt tax. 4.2 A constant Pigouvian tax/subsidy on debt In this model, state contingency cannot simply be induced by cyclicality of policy instruments. We therefore consider that the debt tax/subsidy τbcan neither be made contingent on specific periods nor on the state of the economy, i.e. on the distribution of agents’ endowment, such that the debt tax/subsidy is constant and equally imposed in the periods t=1 and t=2. In this case, the tax/subsidy has ex-ante and ex-post effects relative to the state of the economy where the borrowing constraint might be binding. The borrowers’ optimality conditions (5) and (7) then change to (1−τb)c−1 b,1/r1=βEc−1 b,2,(19) (1−τb)c−1 b,2/r2=β+μb,2,(20) where 1/r1=1/r2=β. Condition (19) and (20) imply that the multiplier on the collateral constraint satisfies μb,2=βc−1 b,2cb,1Ec−1 b,2−1, which differs from laissez faire μb,2=(c−1 b,2−1)β. The collateral premium ξ2=γμ b,2and condition (9) then lead to the following price relation: q2=vd(d)(1+β) c−1 b,21−βγcb,1Ec−1 b,2+βγ ,(21) which simplifies in state Lto q2=cb,2vd(d)(1+β). The durables price q2tends to be higher under a larger collateral premium ξ2(see 9), while a constant debt tax τb>0 tends to reduce the multiplier μb,2and thus ξ2(see 20). Due to this effect on q2and the negative effect of the debt tax on non-durables consumption cb,1relative to cb,2 (see 19), the price q2is here characterized by a positive relation to cb,1in equilibrium (see 21). A constant debt tax tends to induce agents to borrow and to consume less in period 1 relative to period 2 (as in the case of the ex-ante tax), but also tends to reduce borrowing and consumption in period 2 when the borrowing constraint might be binding (see 20). Due to a lower collateral premium, a debt tax can induce a reduction in the durables price and in the borrowing limit in period 2. It might therefore be preferable to apply a subsidy rather than a tax on debt. These two effects of debt policies due to borrowing constraints that bind in the current period and in the subsequent period correspond to those discussed in Bianchi and Mendoza (2018) for an optimal state-contingent policy under discretion. In contrast to a policy maker under discretion, who can influence expectations about future policy makers’ choices only via endogenous state variables, 123 Taxing versus subsidizing debt under financial frictions 1399 a policy maker under commitment fully accounts for agents conditioning their expectations on the policy choices. For the policy problem under commitment, the following proposition reveals when a debt subsidy (or tax) is preferable14: Proposition 3 Suppose that the policy maker can apply a constant Pigouvian tax/subsidy on debt in the periods 1 and 2. Then, the optimal allocation is associated with a tax/subsidy rate on debt satisfying τb=cb,1γdEμtb 2χc−χξ·2cb,1 cb,2 +βγβ c2 b,2,(22) where μtb 2≥0denotes the multiplier on the borrowing constraint of the policy problem, and the rate τbis negative if cb,1 cb,2(H)>1−β2γ 2βγ . Proof See Appendix.  As revealed by Proposition 3, the collateral effects given on the RHS of (22)imply that an optimal non-contingent debt policy can either be a tax (τb>0) or a subsidy (τb<0). The reason is that a constant debt subsidy tends to raise q2via the collateral premium on durables (see χξ), similar to an ex-post debt subsidy (see Sect.3.2.2). At the same time, a constant debt subsidy tends to reduce cb,2and thus q2via increased debt (see χc), which are the inverse effects of an ex-ante debt tax. If the impact on the collateral premium summarized by the positive term in the curly brackets in (22) dominates the latter effect, a debt subsidy is optimal, τb≤0. The inequality at the end of the proposition, reveals that this holds if the ratio of period-1-consumption to period-2-consumption in state H,cb,1/cb,2(H), is sufficiently large. Recall that this ratio is equal to one under first best (see 13), while it exceeds one under laissez faire when the collateral constraint binds (see 11). The inequality is therefore satisfied when the allocation under the constant tax/subsidy is (still) characterized by a binding collateral constraint, inducing the ratio cb,1/cb,2(H)to exceed one, and the threshold 1−β2γ 2βγ is smaller or equal to one. The latter is in fact the case when the liquidation value of collateral γis sufficiently large, i.e. if γ≥1/β2+2β, which can in principle be satisfied by empirically plausible loan-to-value-ratios (e.g. γ=0.8 ). For smaller loanto-value-ratios, a constant debt tax can be superior to a debt subsidy. Relatedly, Bianchi and Mendoza (2018) find relatively small welfare gains of an optimized constant debt tax in an infinite-horizon model where the loan-to-value ratio falls in a crisis, which tends to reduce the beneficial effects of a debt subsidy via the collateral premium. Apparently, the term in the curly brackets in (22) is equal to zero if there were no collateral premium, like under a borrowing constraint that does not depend on the individual stock of durables (see 16). In this case, the RHS of (22) would be strictly positive, such that the optimal constant policy imposed on borrowers would be a debt tax. This corresponds to the welfare-enhancing constant debt tax in Bianchi (2011), where a collateral premium is non-existent. 14 Bianchi and Mendoza (2018) do not analytically or quantitatively identify the conditions under which a debt subsidy is optimal. 123 1400 A. Schabert 4.3 An ex-ante Pigouvian tax/subsidy on saving We now consider a tax/subsidy on saving as a closely related policy instrument, which is however imposed on lenders. Given that borrowers and lenders structurally differ with regard to preferences and constraints, the impact of a tax/subsidy on saving will in general not be equivalent to the impact of a tax/subsidy on debt. Specifically, the analysis will reveal that distributive effects of pecuniary externalities play an important role for the policy maker’s choice under a saving policy, which directly alters the interest rate. Notably, the absence of a redistributive motive for the social planner under quasi-linear preferences (see Sect.3.2) implies that redistributive effects of interest rate changes exclusively stem from the mitigation of pecuniary externalities. In contrast, the interest rate was exogenous under the linear utility function of lenders (see Sects.4.1 and 4.2) as long as only borrowers were taxed. Under an ex-ante tax/subsidy on saving, the interest rate in period 1 can directly be altered by policy, as shown by the lenders’ optimal saving decision (1−τl,1)/r1=β. (23) Combining (23) with the borrowers’ optimality condition (5), gives 1/(1−τl,1)= cb,1Ec−1 b,2, implying that borrowers’s period-1 non-durables consumption cb,1tends to decrease relative to cb,2with a saving tax, τl,1<0. Given that the interest rate now becomes endogenous via the policy maker’s optimal choice, the relevant price relation is given by r1=c−1 b,1/βEc−1 b,2,(24) while the collateral price satisfies the laissez faire price relation (12), like under the ex-ante debt tax. An ex-ante tax/subsidy on saving can indirectly alter the borrowing limit via the effect of cb,2on the collateral price similar to the ex-ante debt tax, while it can additionally affect the interest rate in a direct way via (23). The social planer can utilize the latter effect and lower the interest rate to address distributive effects of pecuniary externalities. In fact, the distributive effects call for a subsidy on saving and the collateral effects for a tax on saving. The sign of the optimal tax/subsidy rate therefore depends on the relative magnitudes of both effects. Proposition 4 Suppose that the policy maker can apply a Pigouvian tax/subsidy on saving before the borrowing constraint might be binding. Then, the optimal allocation is associated with a tax/subsidy rate on saving satisfying τl,1=−bb,1r1E(μtl1 2)E∂φb 1 ∂cb,1−r1E∂φb 1 ∂cb,2   ≥0 −r1βγd (1−βγ)−1E(μtl1 2χc)   ≥0 ,(25) 123 Taxing versus subsidizing debt under financial frictions 1401 where ∂φb 1/∂cb,1>0,∂φb 1/∂cb,2<0, and μtl1 2≥0denotes the multiplier on the borrowing constraint of the policy problem and φb 1=β(cb,1/cb,2)the stochastic discount factor. Proof See Appendix.  The condition for the optimal ex-ante tax/subsidy rate (25) in Proposition 4reveals that the sign of the tax/subsidy rate depends on two opposing effects: The first term (in curly brackets) on the RHS is strictly positive and summarizes the distributive effects induced by the borrowing constraint that is binding with a positive probability (see Assumption 2).15 These effects, which would be inexistent without pecuniary externalities (see terms in the square brackets), call for a saving subsidy, τl,1>0, inducing a lower interest rate. Due to the higher debt price 1/r1, borrowers can increase their consumption of non-durables in period-1 relative to period 2 compared to laissez faire (see 11). The second term (in curly brackets) on the RHS is also strictly positive and summarizes the collateral effects, which can be addressed by reducing borrowing via a saving tax, τl,1<0, that tends to reduce the supply of debt (like an ex-ante debt tax tends to reduce the demand for debt, see Proposition 2). Evidently, the policy maker applies a saving subsidy, τl,1>0, when collateral effects are dominated by distributive effects, which is more likely for higher levels of debt −bb,1(see 25); the latter primarily depending on the exogenously given initial debt level −bb,0. 4.4 A constant Pigouvian tax/subsidy on saving Now suppose that the tax/subsidy on saving can neither be made contingent on particular periods nor on the state of the economy, such that the tax/subsidy rate is equally imposed in the periods t=1 and t=2. This policy regime would even be nonequivalent to a constant tax/subsidy on debt if all agents were ex-ante identical, because of the asymmetry of agents’ problems in period 2 induced by the borrowing constraint. The lenders’ optimality conditions are then given by (1−τl)/r=β, where r1=r2=r,(26) instead of (10), implying that the constant saving tax/subsidy alters the interest rate in both periods, 1 and 2. These interest rate effects of the constant saving tax/subsidy further affect the borrowing decisions in period 1 and 2 c−1 b,1/(1−τl)=E[c−1 b,2],(27) c−1 b,2β/(1−τl)=β+μb,2.(28) The conditions (27) and (28) indicate that a constant saving subsidy τl>0 tends to raise borrowers’ non-durable consumption in period 1 and 2. Simultaneously, it alters the valuation of the borrowing constraint, measured by the multiplier on the borrowing 15 Notably, the multiplier μtl1 2now depends on the difference between the marginal utilities of borrowers and lenders (see proof of Proposition 4). 123 1402 A. Schabert constraint μb,2and thereby the collateral premium ξ2. Combining (27) and (28), gives μb,2=c−1 b,2cb,1βE[c−1 b,2]−β, which can be used to substitute out the multiplier μb,2 in (6). Then, the durables price relation differs from the laissez faire version (12) and satisfies (21), like under the constant debt tax/subsidy. With these changes in the price relations for durables and debt, the policy maker can use a constant tax/subsidy on saving to simultaneously address collateral effects via the durables price q2as well as distributive effects via the interest rate r. Proposition 5 Suppose that the policy maker can apply a constant Pigouvian tax/subsidy on saving in the periods 1 and 2. Then, the optimal allocation is associated with a tax/subsidy rate on saving satisfying τl=+, with (29) =β−bb,1rβEμtl 2 φbE∂φb ∂cb,1−rE∂φb ∂cb,2 −Ebb,2 μtl 2 φb∂φb ∂cb,1 −r∂φb ∂cb,2≥0 =βγdEμtl 2χξ·1+2rcb,1 cb,2γβc−2 b,2−rχc, and ∂φb/∂cb,1>0,∂φb/∂cb,2<0, and μtl 2≥0denotes the multiplier on the borrowing constraint of the policy problem and φb=β(cb,1/cb,2)the stochastic discount factor. The term is positive if cb,1 cb,2(H)>1−βγr−1 2βγ . Proof See Appendix.  According to Proposition 5, distributive effects, which are summarized by the term  in (29), can be addressed by a constant saving subsidy, which relates to the findings in Proposition 4. Compared to an ex-ante saving subsidy, a constant saving subsidy additionally reduces the interest rate in period 2, where the borrowing constraint might be binding. This additional effect is captured by the last term in the square brackets of . In contrast to the terms referring to the distributive effect, the sign of the term , which summarizes the collateral effects, is ambiguous and depends on the effects on the collateral premium summarized in the curly brackets in . Given that a higher willingness to borrow increases the valuation of collateral, the collateral effects also call for a saving subsidy if the impact on the collateral premium is sufficiently large. Otherwise, the term is negative and calls for a saving tax, which tends to increase consumption by reducing debt (see χc). For >0, the inequality cb,1 cb,2(H)>1−βγr−1 2βγ has to hold, which differs from the corresponding condition for the constant debt subsidy by r−1replacing β(see Proposition 3). As in Sect. 4.2, this inequality is likely to be satisfied when borrowing remains constrained even under the optimal policy (such that cb,1exceeds cb,2(H)) and for sufficiently high loan-to-value ratios γ, which strengthen the effect via the collateral premium. If the borrowing constraint were however independent of individual holdings of durables (see 16), such that there would be no collateral premium effect on q2,the 123 Taxing versus subsidizing debt under financial frictions 1403 Fig. 1 Instruments and prices (benchmark values γ=0.8andδH=1.1) term would be strictly negative. Yet, even in this case the policy maker would apply a saving subsidy if the distributive effects dominate; the latter being more likely under higher initial debt levels −bb,0. 4.5 Prices, allocation and welfare We now aim at illustrating the impact of corrective policies on prices, the allocation and social welfare, and the possibility to improve on the (constrained efficient) allocation implemented by an ex-ante debt tax via alternative non state-contingent policies. We introduce a functional form for v(db,t):v(db,t)=κlog db,t. We further assign values to model parameters that induce the – admittedly stylized – model to generate meaningful values of targeted variables. Specifically, we normalize yand set it equal to 2, and further set d/y=1.5, bb,0/y=−0.25, κ=0.1, and β=0.9, leading to a housing-to-GDP ratio, debt-to-income ratios, interest rates, and tax/subsidy rates within reasonable ranges. The benchmark values for the inequality measure δHand for the share of seizable collateral γare 1.1 and 0.8, respectively; the latter relating to commonly applied loan-to-value ratios. We then examine the sensitivity of the effects by altering the tightness of the borrowing constraint γand income inequality δH. The solutions for the equilibrium objects under the four non-state-contingent policies summarized in Table 1and under laissez faire are presented in the Figs.1,2,3.The first row in all Figures refers to a variation in γ, where an increase in γreduces the tightness of the borrowing constraint and thereby the strength of the financial friction, which de-emphasizes the collateral effect. The second row in all Figures refers to a variation in δH, where an increase in δHincreases the inequality of agents’ nondurables endowment in state Hin period 2 and thereby the relevance of the financial friction as well as of distributive effects. 123 1404 A. Schabert Fig. 2 Debt and consumption (benchmark values γ=0.8andδH=1.1) Fig. 3 Social welfare (benchmark values γ=0.8andδH=1.1) The first column of Fig.1shows the tax and subsidy rates under all five regimes. The laissez faire case (black dotted lines) exhibits zero tax/subsidy rates. The first policy regime (solid black lines with crosses) is the optimal ex-ante tax on debt τb,1>0 (see Proposition 2), which decreases with γand increases with δH. The second policy regime (red dashed lines with crosses) is the optimal constant subsidy on debt τb<0 (see Proposition 3). The third (blue solid lines with circles) and the fourth regime (green dashed lines with circles) are the optimal ex-ante and the optimal constant saving subsidy, τl,1>0 and τl>0, as characterized in Propositions 4and 5. The second 123 Taxing versus subsidizing debt under financial frictions 1405 column shows the durables (collateral) price in period 2, which is slightly increased compared to laissez faire under the ex-ante debt tax. The constant debt subsidy, which tends to raise borrowing in both periods 1 and 2 (see Fig.2), also leads to higher durables prices due to its impact on the collateral value. In contrast, the ex-ante saving subsidy, which raises debt in period 1 and reduces non-durables consumption cb,2 in period 2 (see Fig.2), leads to lower durables prices q2. Simultaneously, it reduces the interest rate in period 1 below its laissez faire value (see third column), such that borrowing funds requires issuance of less debt bb,1. The constant saving subsidy leads to the most pronounced increase in the durables price q2. It further leads to a reduction in the interest rate r1in period 1 that is larger than under the ex-ante saving subsidy and it equally reduces the interest rate r2in period 2. Figure 2further shows that all three subsidies raise debt −bb,1and lead to higher levels of non-durables consumption in period 1 compared to laissez faire, which in contrast decreases under the ex-ante debt tax. The constant debt subsidy and the ex-ante saving subsidy reduce consumption cb,2due to a higher debt burden in period 2. The opposite result for cb,2is induced by the ex-ante debt tax and by the constant saving subsidy, which lowers borrowing costs in both periods 1 and 2. Figure3presents the welfare effects of the policy regimes. The welfare measure is based on W(see 4) and expressed in terms of equivalents of borrowers’ non-durables consumption in period 1. The first column shows welfare effects of the four policy regimes relative to the laissez faire case. Evidently, the debt policies (ex-ante debt tax and constant debt subsidy) lead to much smaller welfare gains than the saving subsidies. This result is simply due to the fact that the former policies can – by construction – not address distributive effects by changes in the interest rate.16 In contrast, saving policies induce substantial interest rate reductions compared to laissez faire, which leads to a redistribution of resources in favor of borrowers (see also Fig.4). The second column of Fig.3zooms in into the welfare effects of the debt policies, revealing that the ex-ante debt tax leads to the smallest welfare gains under the benchmark parameter values. It further shows that the ex-ante debt tax can principally be superior to the constant debt subsidy for tighter borrowing constraints, i.e. for lower loan-to-value ratios γ(see also Proposition 3), which reduce the positive impact of the constant debt subsidy on the collateral price via the collateral premium, ξ2=γμ b,2. A larger income inequality δHreduces borrowers’ period-2 consumption cb,2under laissez faire (see Fig.2), lowering the collateral price according to (12). The welfare gains of ex-ante debt taxes, which raise cb,2via a debt reduction, therefore increase monotonically with δH(see Fig.3). Correspondingly, the welfare loss of an ex-ante debt subsidy, which lowers cb,2, would increase with δH. The total welfare effect of a constant debt subsidy does however further depend on the effect on the collateral premium, which increases with the multiplier on the collateral constraint, given by μb,2=c−1 b,2cb,1βEc−1 b,2−β. When income gets more unequal, the adverse effect of the debt subsidy on cb,2(see Fig.2) dominates the positive effect on the collateral premium, such that the total welfare gain falls (see Fig.3). The constant debt subsidy can therefore be outperformed by the ex-ante debt tax for high δHvalues. The last 16 This would in principle be possible under alternative specifications of lenders’ utility, for example, logarithmic utility, which is neglected here to keep the exposition transparent using polar cases. 123 1406 A. Schabert Fig. 4 Welfare of borrowers and lenders (benchmark values γ=0.8andδH=1.1) column of Fig.3presents welfare losses compared to first best. The values for laissez fare and the ex-ante debt tax are virtually identical, indicating that the total welfare gains of an ex-ante debt tax are negligible relative to first best. In contrast, the constant saving subsidy can substantially reduce the welfare loss in a competitive equilibrium compared to first best. For the benchmark values, it reduces the welfare loss by about a half. Finally, Fig. 4confirms the existence of redistributive effects under saving policies. Borrowers gain and lenders lose compared to laissez faire due to lower interest rates. The first and the second column of Fig.4show that these redistributive effects monotonically increase with the tightness of the collateral constraint (lower γ) and the inequality of income (higher δH). Likewise, these welfare effects increase with initial debt −bb,0(not shown). In contrast, debt policies solely affect borrowers’ welfare via intertemporal substitution, which is revealed in the last column of Fig.4, showing the same effects as for aggregate welfare (see second column of Fig.3). 5 A model with capital formation To assess the robustness of our findings and to facilitate comparisons, we further apply a model with endogenous capital formation, like Bianchi and Mendoza (2018) or Davila and Korinek (2018). Concretely, we use Davila and Korinek (2018) model applied for collateral externalities and replicate their results on an ex-ante debt policy that implements the constrained efficient allocation. In addition, we examine the other three policy regimes given in Table 1, like in the analyses of the previous model (see Sect.4). There is no uncertainty and there are no durable consumption goods in this economy. Agents’ lifetime utility satisfies ul=cl,1+cl,2+cl,3and ub=log cb,1+log cb,2+cb,3, which accords to Assumption 1without durables (v=0) and implies no discounting 123 Taxing versus subsidizing debt under financial frictions 1413 λtl1 b,1=r1βEλtl1 b,2,(49) βλtl1 b,2=βc−1 b,2−1−λtl1 b,1bb,1 ∂φb 1 ∂cb,2 +βμtl1 2γ∂q2 ∂cb,2 d,(50) μtl1 2=βλtl1 b,2≥0.(51) Applying expectations and substituting out the multipliers λtl1 b,1and λtl1 b,2in (48)-(50), gives r1 1+bb,1E∂φb 1/∂cb,1 1+r1bb,1E∂φb 1/∂cb,2=c−1 b,1−1 β(Ec−1 b,2−1)+βγdEμtl1 2∂q2/∂cb,2. Combining the latter with (23) and (24) and using ∂q2(cb,2) ∂cb,2=(1−βγ)χc(see 12), leads to the following condition for the ex-ante tax/subsidy rate on saving τl,1=−bb,1r1Eμtl1 2E∂φb 1/∂cb,1−r1E∂φb 1/∂cb,2 −r1βγdEμtl1 2(1−βγ)χc, where ∂φb 1/∂cb,1=β/cb,2>0 and ∂φb 1/∂cb,2=−β(cb,1c−2 b,2)<0. Combining (48), (50), and (51), shows that μtl1 2satisfies Eμtl1 2=r−1 1(c−1 b,1−1)/(1+ bb,1E[∂φb 1/∂cb,1])≥0.  Proof of Proposition 5For the policy maker’s primal problem under commitment in Lagrangian form, we define φb 1(cb,1,cb,2)=βcb,1/cb,2and φd 2(cb,1,cb,2)= vd(d)(1+β) c−1 b,2−(cb,1c−2 b,2−1)βγ , and use the goods market clearing conditions to rewrite the welfare function, for convenience: L=E{log cb,1+v(d)+(y−cb,1)+βlog cb,2+v(d)+y−cb,2 +β2[y+v(d)]+λtl b,1bb,0+yb,1−cb,1−bb,1φb 1(cb,1,cb,2) +βλtl b,2bb,1+yb,2−cb,2−bb,2φb 1(cb,1,cb,2) +βμtl 2[γφd 2(cb,1,cb,2)d+bb,2]}, leading to the first order conditions λtl b,11+bb,1E∂φb 1/∂cb,1 (52) =c−1 b,1−1−βEλtl b,2bb,2∂φb 1/∂cb,1+βEμtl 2γd∂φd 2/∂cb,1, βλtl b,21+bb,2∂φb 1/∂cb,2 (53) =βc−1 b,2−1−λtl b,1bb,1∂φb 1/∂cb,2+βμtl 2γd∂φd 2/∂cb,2, 123 1414 A. Schabert λtl b,1=rβEλtl b,2,(54) μtl 2=φbλtl b,2≥0,(55) where we used Eφb 1(cb,1,cb,2)=1/r. Taking expectations and substituting out the multipliers λtl b,1and λtl b,2in (52)–(54), leads to Eμtl 2/φb 11+bb,1E∂φb 1/∂cb,1−Eμtl 2/φb 11+rbb,1E∂φb 1/∂cb,2 =1 rβc−1 b,1−1−Ec−1 b,2−1−1 rEμtl 2/φb 1bb,2∂φb 1/∂cb,1 +1 rγdEμtl 2∂φd 2/∂cb,1+Eμtl 2/φb 1bb,2∂φb 1/∂cb,2 −γdEμtl 2∂φd 2/∂cb,2, and by applying (26), to the following condition for the constant tax/subsidy rate τl=−bb,1rβEμtl 2/φb 1E∂φb 1/∂cb,1−rE∂φb 1/∂cb,2 +βE−bb,2μtl 2/φb 1(∂φb/∂cb,1)−r(∂φb 1/∂cb,2)+, (56) where ∂φb 1/∂cb,1>0, ∂φb 1/∂cb,2<0, and =βγdE[μtl 2∂φd 2/∂cb,1−r∂φd 2/ ∂cb,2].Thetermon the RHS of (56) can by using ∂φd 2 ∂cb,1=χξγβ c2 b,2 and ∂φd 2 ∂cb,2= χc−χξ2γβcb,1 c3 b,2 be rewritten as =βγdEμtl 2χξ γβ c2 b,21+2rcb,1 cb,2−rχc,(57) Further applying χξ=c2 b,2χc(see 9) to rewrite (57)as=βγdrE[μtl 2(χc{γβ(r−1+ 2cb,1c−1 b,2)−1})], shows that ≥0if cb,1 cb,2(H)>1−βγr−1 2βγ . Proof of Proposition 6Consider the economy with capital formation. Under a Pigouvian debt subsidy in period 2, the capital price satisfies qk=A3[c−1 b,2(1−φ)+ φ+τb,2·φc−1 b,2]−1. Thus, the collateral constraint is slack under the first best allocation, −bfb b,2≤φqkkfb, if the subsidy rate satisfies τb,2≤A3kb,2[−bfb b,2]−1−φ−1, where we used cfb b,1=cfb b,2=1 and kfb =(A2+A3)/α and bfb b,2is given by bfb b,2=yb,2+yb,1−2−(A2+A3)2/(α2)+A2(A2+A3)/α. Proof of Proposition 7Consider the economy with capital formation. In equilibrium, where capital is entirely held by borrowers, the budget constraints can be written as cb,1+αk2/2+bb,1/r1=yb,1,cb,2+bb,2/r2=yb,2+bb,1+A2k,cb,3= yb,3+bb,2+A3k,cl,1+bl,1/r1=yl,1,cl,2+bl,2/r2=yl,2+bl,1, and cl,3=yl,3+bl,2. 123 Taxing versus subsidizing debt under financial frictions 1415 The social welfare function (4) can thus for β=1/r1=1/r2=1 be rewritten as W=log cb,1+yl,1+log cb,2+yl,2+yb,3+bb,2+A3k+yl,3. To establish the claims made in the first part of the proposition, consider that the policy maker introduces an investment tax/subsidy τk,1and an ex-ante debt tax/subsidy τb,1, which are fully compensated (ex-post) by type-specific lump-sum transfers (like 2). The borrowers’ optimality conditions then satisfy (1−τb,1)=cb,1/cb,2,(58) (1−τk,1)αk1/cb,1=1/cb,2A2+qk.(59) The primal policy problem of the policy maker is identical to the problem of a social planer who determines period-1-borrowing as well as the capital investment decision and maximizes social welfare Wsubject to budget and borrowing constraints taking the equilibrium price relation (31) under laissez faire into account, leading to a constrained efficient allocation. The problem can be summarized as max Ww.r.t. cb,1,cb,2,bb,1,bb,2,and ksubject to cb,1+αk2/2+bb,1=yb,1, cb,2+bb,2=yb,2+bb,1+A2k, and bb,2+φqk(cb,2)k≥0, where qk(cb,2)satisfies (31) and thus ∂qk/∂cb,2>0. The Lagrangian can be written as L=log cb,1+yl,1+log cb,2+yl,2+yb,3+bb,2+A3k+yl,3 +λt1 1yb,1−cb,1−αk2/2−bb,1+λt1 2yb,2+bb,1+A2k−cb,2−bb,2 +μtb1 2bb,2+φqk(cb,2)k, leading to the first order conditions for cb,1,cb,2,bb,1,bb,2,and k λt1 1=1/cb,1,λt1 2=(1/cb,2)+μtb1 2φk∂qk/∂cb,2,λ t1 1=λt1 2,(60) μtb1 2=λt1 2−1≥0,(61) λt1 1αk=A3+λt1 2A2+μtb1 2φqk(cb,2). (62) Substituting out the multipliers λt1 1and λt1 2using the three conditions in (60), gives 1/cb,1=1/cb,2+μtb1 2φk∂qk/∂cb,2.Using(58) to substitute out 1/cb,2in the latter, leads to the following condition for the ex-ante debt tax/subsidy rate τb,1: τb,1=μtb1 2cb,1φk∂qk/∂cb,2≥0, where μtb1 2=(c−1 b,1−1)≥0. Further substituting out the multipliers with λt1 1= λt1 2=1/cb,1and μtb1 2=1/cb,1−1in(62), gives αk1/cb,1=A3+1/cb,1A2+ 1/cb,1−1φqk. Rewriting it with the capital trading decision qk(1/cb,2)=A3+ κb,2φqkas (1+τk,1)αk1/cb,1=1/cb,2A2+A3+1/cb,2−1φqkand combining with (59), implies that the investment tax/subsidy rate satisfies τk,1=− 1/cb,1−1/cb,2A2+φqk A3+1/cb,1A2+μtb1 2φqk. 123 1416 A. Schabert Since 1/cb,1=1/cb,2+μtb1 2φk∂qk/∂cb,2implies 1/cb,1≥1/cb,2, the policy maker subsidizes capital τk,1≤0iffA2+φqk≥0. This establishes the claims made in the first part of the proposition. For the second part of the proposition, we consider a constant debt tax/subsidy and an investment tax/subsidy, which are fully compensated (ex-post) by lump-sum transfers. Agents’ borrowing and investment decisions then satisfy (1−τb)/cb,1=1/cb,2,(63) (1−τb)/cb,2=1+κb,2,(64) and (59). Substituting out κb,2in the capital trading condition qk(1/cb,2)=A3+ κb,2φqkwith (64) and then the tax/subsidy rate τbwith (63), gives the price relation qk=A3cb,2 1−cb,1/cb,2φ+cb,2φ, (65) implying that qkrelates to cb,1and cb,2by ∂qk/∂cb,1=φA3c2 1φcb,1−cb,2−φc2 1−2 >0 and ∂qk/∂cb,2=1−2φcb,1/cb,2∂qk/∂cb,1. The Lagrangian of the policy maker’s problem can be written as L=log cb,1+yl,1+log cb,2+yl,2+yb,3+bb,2+A3k+yl,3 +λtb 1yb,1−cb,1−αk2/2−bb,1 +λtb 2yb,2+bb,1+A2k−cb,2−bb,2+μtb 2[bb,2+φqk(cb,1,cb,2)k], where qk(cb,1,cb,2)satisfies (65). The first order conditions for cb,1,cb,2,bb,1,bb,2, and kare λtb 1=1/cb,1+μtb 2φk∂qk/∂cb,1,λ tb 2=1/cb,2+μtb 2φk∂qk/∂cb,2,λ tb 1=λtb 2, (66) μtb 2=λtb 2−1≥0,(67) λtb 1αk=A3+λtb 2A2+μtb 2φqk(cb,1,cb,2). (68) Substituting out the multipliers λtb 1and λtb 2using the first three conditions in (66), 1/cb,1+μtb 2φk∂qk/∂cb,1=1/cb,2+μtb 2φk∂qk/∂cb,2, and substituting out 1/cb,2with (63), gives the following condition for the debt tax/subsidy rate τb: τb=μtb 2cb,1φk[(∂qk/∂cb,2)−(∂qk/∂cb,1)]. Using that the capital price qksatisfies ∂qk/∂cb,2=1−2φcb,1/cb,2∂qk/∂cb,1and ∂qk/∂cb,1>0 (see 65), the latter can be rewritten as τb=−μtb 2cb,1φk2φcb,1/cb,2∂qk/∂cb,1≤0. For the third part of the proposition, we consider an ex-ante tax/subsidy on saving and an investment tax/subsidy, which are fully compensated (ex-post) by lump-sum 123 Taxing versus subsidizing debt under financial frictions 1417 transfers (see 2). Agents’ saving and investment decisions then satisfy (1−τl,1)/r1=1,(69) and (59). Given that (69) endogenizes the interest rate for the policy maker, 1/r1= cb,1/cb,2is a relevant restriction to the policy problem. Using the resource constraints to substitute out cl,t, the Lagrangian of the policy maker’s problem can be written as L=log cb,1+(y1−cb,1)+log cb,2+y2−cb,2+[y3+A3k] +λtl1 1yb,1−cb,1−αk2/2−bb,1(1/r1) +λtl1 2yb,2+bb,1+A2k−cb,2−bb,2+μtl1 2bb,2+φqk(cb,2)k, whereweusedyt=yb,t+yl,t, and qkand 1/r1satisfy (31) and 1/r1=cb,1/cb,2, respectively. The first order conditions for cb,1,cb,2,bb,1,bb,2,and kare given by λtl1 11+bb,1∂(1/r1)/∂cb,1=1/cb,1−1, (70) λtl1 1bb,1∂(1/r1)/∂cb,2+λtl1 2=(1/cb,2)−1+μtl1 2φk∂qk/∂cb,2,(71) λtl1 1(1/r1)=λtl1 2≥0, (72) μtl1 2=λtl1 2,(73) and (62). Substituting out the multipliers λtl1 1and λtl1 2in (70)–(72), leads to r1 (1+bb,1∂(1/r1)/∂cb,1) 1+r1bb,1∂(1/r1)/∂cb,2=1/cb,1−1 (1/cb,2)−1+μtl1 2φk∂qk/∂cb,2 . Further using (69)aswellas1/r1=cb,1/cb,2and rearranging terms, gives τl,1=−bb,1r1μtl1 2(∂ (1/r1)/∂cb,1)−r1∂(1/r1)/∂cb,2 −r1μtl1 2φk∂qk/∂cb,2,(74) where ∂(1/r1)/∂cb,1>0 and ∂(1/r1)/∂cb,2<0. For the fourth part of the proposition, we consider a constant tax/subsidy on saving and an investment tax/subsidy, which are fully compensated (ex-post) by lump-sum transfers (see 2). Agents’ saving and investment decisions then satisfy (1−τl)/r=1,(75) where r=r1=r2, and (59). Substituting out the interest rates in agents’ borrowing decisions with (75), c−1 b,1/(1−τl)=1/cb,2and c−1 b,2/(1−τl)=1+κb,2, and combining the latter to cb,1/cb,2=cb,21+κb,2, implies that qksatisfies the price relation (65). Proceeding as above, the Lagrangian of the policy maker’s problem can be written as 123 1418 A. Schabert L=log cb,1+(y1−cb,1)+log cb,2+y2−cb,2+[y3+A3k] +λtl 1yb,1−cb,1−αk2/2−bb,1(1/r) +λtl 2yb,2+bb,1+A2k−cb,2−bb,2(1/r)+μtl 2bb,2+φqkcb,1,cb,2k, where qkand 1/rsatisfy (65) and 1/r=cb,1/cb,2, respectively, leading to the following first order conditions for cb,1,cb,2,bb,1,bb,2,and k λtl 1(1+bb,1∂(1/r)/∂cb,1)+λtl 2bb,2∂(1/r)/∂cb,1 =1/cb,1−1+μtl 2φk∂qk/∂cb,1, (76) λtl 1bb,1∂(1/r1)/∂cb,2+λtl 21+bb,2∂(1/r)/∂cb,2 =(1/cb,2)−1+μtl 2φk∂qk/∂cb,2,(77) λtl 1(1/r)=λtl 2, (78) μtl 2=λtl 2(1/r)≥0,(79) and (68). Substituting out λtl 1and λtl 2in (76)-(78) and taking differences, leads to rμtl 2bb,1∂(1/r)/∂cb,1+μtl 2bb,2∂(1/r)/∂cb,1 −rrμtl 2bb,1∂(1/r)/∂cb,2+rμtl 2bb,2∂(1/r)/∂cb,2 =r−11/cb,1−1−(1/cb,2)−1+r−1μtl 2φk∂qk/∂cb,1 −μtl 2φk∂qk/∂cb,2,(80) where ∂(1/r)/∂cb,1>0 and ∂(1/r)/∂cb,2<0. Combining (80) with c−1 b,1/r= 1/cb,2and (1−τl)=r, leads to the following condition for the tax/subsidy rate τl=rμtl 2bb,1+μtl 2bb,2∂(1/r)/∂cb,2−(μtl 2bb,1+1 rμtl 2bb,2)∂(1/r)/∂cb,1 +1 rμtl 2φk∂qk/∂cb,1[2φ+1−r]/r,(81) whereweused∂qk/∂cb,2=1−2φcb,1/cb,2∂qk/∂cb,1to derive the last term in (81).  Acknowledgements The author is grateful to Felix Bierbrauer, Emanuel Hansen, and Joost Roettger for helpful comments and suggestions, as well as to Anton Korinek for insighful comments on several details of the analysis. Funding Open Access funding enabled and organized by Projekt DEAL. Financial support was received from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2126/1 – 390838866. 123 Taxing versus subsidizing debt under financial frictions 1419 Declarations Conflict of interest The author has no conflict of interest to declare that are relevant to the content of this article. 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