This section presents a model of borrower demand and lending behavior in the presence of both traditional mortgages and aggressive lending instruments in the context of a competitive real estate market. The quantity of housing services consumed by each household is fixed and exogenous to our model. The borrower can rent or purchase the housing services for each period. The evolution of wealth for the borrower is given by

- (1)

where *t* denotes the time period, *W*_{t} denotes the total wealth at time *t*, *Y*_{t} is the stochastic income at time *t*, δ denotes the rent payment, *r* is the nonstochastic interest rate and *P*_{t} is the equilibrium price of housing. Each period the agent chooses to rent or purchase the housing services in order to maximize the expected utility of terminal wealth

- (2)

where γ denotes the risk-aversion parameter and *T* denotes the final period. If at any point in time the wealth of the agent becomes zero or negative, then the agent is in default. Note that this is the total wealth of the agent, not just equity in the home. This is consistent with the lack of ruthless default as discussed in Quercia and Stegman (1992), Pavlov (2001) and Deng, Pavlov and Yang (2005).

If the agent defaults, his or her wealth resets to a small amount above zero and the agent's credit score, *C*, goes down to 500.^{3} Each period the agent maintains wealth above zero, his or her credit score increases by 30, to a maximum of 850. Therefore, in addition to maximizing expected utility of terminal wealth, the agent considers the probability that she or he has to default in the future, as well as the negative consequences of default, namely, the inability to purchase a home in the future until the credit score improves.

Lenders require a minimum credit score to fund a mortgage. This constraint is analogous to the wealth and loan-to-value (LTV) constraints. These three constrains are conceptually similar because they all can eliminate a particular set of agents from becoming homeowners regardless of their optimal choice. For numerical tractability in what follows, we only consider the credit score constraint. Adding the LTV constraint directly would require modeling consumption of nonhousing services, which is beyond the scope of this article. For evidence of the importance of the credit score constraint, as well as wealth and income constraints, see Firestone, Calem and Wachter (forthcoming). The credit score represents the constraint in our model, and thus the rent-versus-buy decision, and the resulting equilibrium price of ownership, is solved through constrained optimization. When minimum credit score requirements are lowered through aggressive lending practices, the entire group of borrowers with credit scores above the new constraint, but below the original one, would then be considered unconstrained and thus able to purchase housing at the prevailing equilibrium price.

This mechanism is also the source of endogenous cycles in the economy. Incomes are stochastic, and real estate prices respond to changes in income. On top of this response, if lenders relax and tighten the minimum credit score requirements procyclically and/or if they reprice their products procyclically, we get a magnified real estate cycle above and beyond what could be justified by shifts in incomes.

Our model further assumes that the stock of owner-occupied homes is constant and rental properties cannot be converted to be owner occupied. In reality, these two assumptions would not hold perfectly. However, we justify their use in our model by the fact that when lending rates fall the demand for both rental and owner-occupied housing increases either through increased household formation and/or through increased demand for second homes. We do not explicitly model these effects here.

#### Solution Methodology

The main mathematical complication of the above model lies in the ability of the agent to predict the future price distribution of property prices and choose whether to rent or buy given this future price distribution. In our model agents account for the fact that incomes fluctuate and, therefore, prices fluctuate. However, the mistake borrowers make is that they do not foresee if, when and by how much lenders will withdraw credit. Because an explicit solution for the future real estate price distribution is not available, we employ a version of the Longstaff and Schwartz (2001) least-squares simulation approach. First we generate simulation paths for future personal income. We assume future income follows a zero-mean Brownian motion of the form

- (3)

where *σ*_{Y} denotes the volatility of income. We start by assigning random wealth levels for each simulation path and time period. We then assume that the terminal real estate price level is half of the terminal wealth. We justify this last assumption by appealing to the stylized fact that at retirement people tend to spend half of their total wealth, including present value of expected future retirement payments, on housing, and the other half they use for consumption. We have solved our model for various other levels of final prices, and while the level of real estate prices change, the comparative statics we report below remain unchanged. Another way of stating this assumption is that the rate of substitution between housing services and other consumption at retirement is constant and certain. While the exact rate of substitution does not matter, the fact that it is constant is important to our model. Absent this assumption, the rate of substitution between housing services and other consumption becomes an additional source of uncertainty, which needs to be explicitly modeled as a stochastic process. This would, in turn, add an additional stochastic state variable to our model, thus greatly increasing the computational demands of the solution.

At each time period between *T* – 1 and 2, going backwards, we regress the future price on each path on the income and wealth level on that path. In our base model we utilize a regression of the form:

- (4)

We then use the estimated regression for each period to derive a distribution of prices for period *t* + 1 conditional on the state variables at time *t*.

The conditional future distribution of real estate prices allows us to determine the current price, *P*_{t}, that numerically equates the expected utility of renting (which is certain) to the expected utility of buying (which involves price risk). In this model the total demand for rental and owner-occupied housing does not change, nor does the supply of each type of housing. Instead, the price of owner-occupied housing changes to make sure that the utility of renting and owning is the same, thus ensuring that the total demand and supply of the two types of hosing are equated.

Once we have the price *P*_{t}, which equates the utilities of rent and own, we can solve for the current level of wealth, *W*_{t}, using the evolution of wealth given in Equation (1). Given these new levels of current wealth, *W*_{t}, we repeat the regression estimation (Equation (4)) and recompute current wealth levels until we reach a fixed point for which current wealth levels do not change anymore.

We then repeat the above-described algorithm until time 2. At the first time period we have only one level of income and wealth, so we do not estimate regression Equation (4) but rather use the prices in period 2 to compute the price in period 1 that equates the expected utilities of renting and buying.

We then go forward through our simulation and set the credit score to 500 for any path for which the wealth level falls below zero. We also increase the credit score by 30 for every period the agent maintains wealth above zero. If the wealth level does fall below zero, it is reset to a small positive amount. For numerical tractability we cannot set the wealth level exactly at zero. The agent is then not allowed to purchase on that path until the agent's credit score improves above the predetermined minimum. In our solutions we do not reset the price in that period if some paths result in negative wealth. Nonetheless, credit scores do impact prices because they impact the ability of the agent to purchase real estate on that path until the agent's credit score improves.

By following the above methodology, the initial wealth level is different on each path. We then alter the final period wealth and repeat the entire procedure until the initial wealth level on all simulation paths equals the original wealth level, set to 100 in our case. Once this is achieved, we have a solution of the model in which wealth levels are consistent on each path and for each time period and prices on each path and time period are set to equate the utility of renting and owning.

We provide a step-by-step description of our solution algorithm in the Appendix.

#### Model Solution

Table 1 reports the base parameters we use in the numerical solution. In addition to the parameters mentioned above, we set the number of time periods to 10, each one representing roughly 3 to 5 years of the agent's life, and we set the number of simulations to 10,000. While we would have liked to increase the number of simulation paths, the above procedure is computationally very demanding, and increasing the simulation paths greatly increases the time to find a solution.

Table 1▪. Base model parameters. Variable | Base Level |
---|

*Risk-aversion*, γ | 5 |

*Rent yield*, δ | 5% |

*Mortgage interest rate, r* | 5% |

*Volatility of income* | 30% |

*Credit score after default* | 500 |

*Credit score improvement if no default* | 50 |

*Minimum credit score to borrow* | 700 |

*Number of time periods* | 10 |

*Number of simulations* | 10,000 |

Figure 1 reports the equilibrium real estate price at time 1 as a function of the lending rate for minimum credit score requirement of 600 and 700. In both cases, higher borrower cost relative to the cost of renting results in lower prices today. In other words, with high borrowing costs, agents require high expected future price appreciation to make the decision to buy.

Furthermore, the high minimum credit score requirement of 700 places a potential future burden on homeowners as it increases the penalty if they are forced into default in the future. The penalty is increased because it would take longer for an agent to recover his or her credit score and borrow again.

Importantly, the minimum credit score requirement has a relatively larger impact for very low interest rates because the penalty of not being able to borrow in the future is larger when homeownership is relatively more attractive.

Figure 2 focuses on the effect of minimum credit score requirement on initial prices for three levels of the borrowing rate: 3%, 4% and 5%. The higher the minimum credit score requirement, the more reluctant are agents to borrow and own. This is particularly true if interest rates are low relative to the cost of renting, in which case the penalty of being unable to buy real estate is significant and very restrictive.

The overall conclusion of the above analysis is that eased lending terms, either in the form of low borrowing costs, low credit score requirements or both have a positive impact on the real estate markets and push prices higher.