A penalty function approach to occasionally binding credit constraints

A substantial part of the literature using dynamic stochastic general equilibrium (DSGE) models features financial frictions in the form of credit constraints. In this concept some agents (entrepreneurs or households) are limited in their borrowing capacity by the amount of collateral that they can provide to the lender. The constraint is assumed to be eternally binding, which facilitates the model solution as standard perturbation techniques can be applied. A number of papers used this approach to model frictions in the housing market. However, while conceptually and computationally attractive, the eternally binding constraint (EBC) setup suffers from a major shortcoming. The permanent nature of collateral constraints generates strong, short-lived and symmetric reactions of macroeconomic variables to shocks. This means in particular that the EBC modeling strategy does not allow to distinguish between “normal” and “stress” periods. This model feature seems inconsistent with empirical evidence, which shows that residential investment, housing stock, change in mortgage loans and house price inflation are all skewed downwards, i.e. left tail events are relatively more frequent. This suggests either that shocks affecting the housing market are asymmetric, or that it responds to symmetric shocks in a skewed fashion. In this paper we consider a model in which asymmetries emerge endogenously from constraints facing the agents. The discussion presented above suggests that the collateral constraints should not be applied in a permanently and symmetrically binding fashion. A preferred specification would feature constraints that do not matter under normal circum-stances (from the modeling perspective: in the vicinity of the steady state), but become binding occasionally, i.e. during episodes of unfavorable economic conditions (e.g. after a series of negative macroeconomic or financial shocks). The idea of occasionally binding constraints (OBC) is not new. However, given their highly non-linear nature, they should ideally be solved with global methods. Due to the curse of dimensionality, however, these can be applied only to relatively small models with a limited number of state variables. In spite of the progress achieved in the area of global solution techniques in recent years, such methods are still out of range for models of the size used for practical policymaking, i.e. featuring a number of real and nominal rigidities. Adding these frictions seems indispensable when the models are to be applied for instance for analyzing business cycle consequences of macroprudential policies. For such models, local solution methods are still the only feasible option. For these reasons, we thoroughly investigate a potentially attractive shortcut to approximate occasionally binding constraints, i.e. the so-called barrier or penalty function method. This approach essentially consists in converting inequality constraints into equality constraints, making the use of standard perturbation techniques possible. To this end, we construct a DSGE model with a standard set of rigidities and collateral constraints, except that the latter are introduced in the form of a smooth penalty function. We parametrize the model in such a way that the constraint does not play an important role close to the steady state, but becomes binding when the economy is hit by sufficiently large negative shocks. Next, we investigate the main features of the model both under per-fect foresight and in a stochastic setting using its local approximations of various orders. Our main findings are as follows. First, the introduction of occasionally binding con-straints via the penalty function approach allows to generate asymmetric and non-linear reactions of the economy to shocks. Second, this feature can be also reproduced for local approximations, though only for orders higher than two. Third, and less optimistic, stochastic simulations for 2nd, 3rd and 4th order approximations suffer from serious stability problems that make them inapplicable in practice. Approximations of order higher than four are, on the other hand, prohibitively expensive in terms of storing and computing power for medium-sized business cycle models. All in all, while being practical for non-stochastic models, the penalty function approach unfortunately fails to fulfill our expectations in a stochastic environment. This makes it an attractive way of introducing financial frictions into deterministic models like GEM or EAGLE. However, a fully-fledged application in a realistic stochastic framework seems currently out of range.