# RDP 2019-06: The Effect of Mortgage Debt on Consumer Spending: Evidence from Household-level Data Appendix A: Debt Overhang Model

Below we set out a highly stylised two-period partial equilibrium model to illustrate the channels through which higher debt levels might affect spending, and hence explain a debt overhang mechanism. The household lives in an endowment economy, faces no uncertainty and has a lifetime utility function given by:

$V=U( C 1 )+βU( C 2 )$

where the discount factor is $0\le \beta \le 1$. For simplicity, assume that the utility function is given in logarithmic form.

$V=ln( C 1 )+βln( C 2 )$

The household has a budget constraint in the first period given by:

$C 1 = Y 1 + L 1 −δ D 0$

where the household consumes (C1) in the first period out of their current income (Y1) and new borrowing (L1) less repayments on debt (D0) that they inherit from period 0. The household is assumed to repay a fixed fraction $\left(0<\delta <1\right)$ of the initial debt, and this fraction is exogenously given. The household is also assumed to be endowed with an asset from period 0 (A0), which can serve as collateral in period 1 but which cannot be liquidated until period 2.

The budget constraint in the second period is:

$C 2 = Y 2 −( 1+r ) D 1 +( 1+r ) A 0$

where the real interest rate on debt per period is r. We assume that the stock of debt follows a law of motion:

$D 1 = L 1 +( 1−δ ) D 0$

where the total debt at the end of period 1 is equal to any borrowing during the period plus any outstanding initial debt that the household was born with less the repayment made in the period. The repayment of past debt includes both an interest and principal component. A household that makes a larger principal payment than required (or ‘prepayment’) will have more liquidity at its disposal in the future than a similar household that makes a smaller (or no) prepayment. When δ = 1 this corresponds to the case of one-period debt.

The household's lifetime budget constraint is:

$C 1 + C 2 ( 1+r ) = Y 1 + Y 2 ( 1+r ) + A 0 − D 0$

This says that the household can consume out of the present discounted value of lifetime income plus any net wealth that they are born with.

However, when taking on debt, the household potentially faces a borrowing constraint:

$D 1 ≤ D ¯ = φ ¯ D A 0$

where the maximum LVR in any period is given by $0\le {\overline{\phi }}_{D}\le 1$. This states that the total debt for the household is limited by the value of its gross assets (collateral). Given the law of motion for debt, this implies that the household is potentially constrained in the amount it can borrow in period 1. Moreover, the maximum amount of new borrowing depends on the household's previous debt and any prepayments:

$L 1 = D 1 −( 1−δ ) D 0 L 1 ≤ L ¯ = φ ¯ D A 0 −( 1−δ ) D 0 =[ φ ¯ D −( 1−δ ) γ 0 ] A 0$

where ${\gamma }_{0}\le {\overline{\phi }}_{D}$ is the household's leverage from period 0. Accordingly, if the household was born with the maximum amount of debt ${D}_{0}=\overline{D}={\overline{\phi }}_{D}{A}_{0}$ , it can at most borrow the amount it repaid, that is ${L}_{1}\le \delta {\overline{\phi }}_{D}{A}_{0}$ .

The household may also face a liquidity (or cash flow) constraint in period 1:

$D 1 ≤ D ¯ ¯ =θ ( Y 1 −r D 0 ) 1−δ$

This condition is essentially a debt servicing requirement imposed by mortgage lenders. It states that the household's repayments in period 2 (when it fully repays the loan) $\left(\left(1-\delta \right){D}_{1}\right)$ must be less than some fraction $\left(0\le \theta \le 1\right)$ of its current disposable income, which is equal to its endowment less the interest payments on the initial debt (Y1 −rD0). Using the law of motion for debt, this condition again implies a constraint on new borrowing in the first period:

$L 1 ≤ L ¯ ¯ =θ ( Y 1 −δ D 0 ) 1−δ −( 1−δ ) D 0$

To see how this works, consider a household that was born with debt and no prepayment $\left(\delta =0\right)$. This household will have more disposable income (or cash flow) in period 1, but will be required to make a larger repayment in period 2. This will mean the constraint is more likely to bind in period 1. Another household that fully prepaid the loan $\left(\delta =1\right)$ will have less cash flow in period 1, but also no repayments in period 2, and hence a greater stock of liquid resources during that final period. For this household, the liquidity constraint will never bind.

The household chooses consumption in each period to maximise its lifetime utility subject to the lifetime budget constraint, the borrowing constraint and the liquidity constraint.

For the unconstrained household, the consumption function is:

$C 1 * = 1 ( 1+β ) ( Y 1 + Y 2 1+r + A 0 − D 0 )$

And the sensitivity of spending to outstanding debt for the unconstrained household is:

(A1) $∂ C 1 * ∂ D 0 = −1 ( 1+β )$

This shows that there is a negative wealth (or income) effect of debt on spending for the unconstrained household.

For a borrowing-constrained household, the consumption function is:

$C 1 * = Y 1 + L ¯ −δ D 0 = Y 1 + φ ¯ D A 0 −( 1−δ ) D 0 −δ D 0$

And the sensitivity of spending to outstanding debt for this household is:

(A2) $∂ C 1 * ∂ D 0 =−( 1−δ )−δ=−1≤− 1 ( 1+β )$

When $\beta >0$ and the borrowing constraint binds, any increase in debt reduces consumption by more than the wealth effect. This works through a reduced ability to borrow $\left(1-\delta \right)$ and lower disposable income to meet the required repayments $\left(\delta \right)$.

For a liquidity-constrained household, the consumption function is:

$C 1 * = Y 1 + L ¯ ¯ −δ D 0 = Y 1 +θ ( Y 1 −r D 0 ) 1−δ − D 0$

And the sensitivity of spending to outstanding debt for this household is:

(A3) $∂ C 1 * ∂ D 0 =−( 1+ θr 1−δ )≤−1$

This condition holds as long as the real interest rate is positive (r > 0). In effect, there is a negative wealth (or income) effect of debt on spending for the constrained households, but also additional borrowing and liquidity effects. So the effect of debt on spending is larger for constrained households than for unconstrained households. This implies that a debt overhang effect will be stronger for households that face binding financing constraints (Eggertsson and Krugman 2012).

Next, we consider the role of uncertainty in driving the debt overhang effect (King 1994; Albuquerque and Krustev 2015). For simplicity, suppose the only source of uncertainty is about the interest rate to be paid on the debt as the household enters the second period.[25] And consider a mean-preserving spread of the interest rate:

$r= r ¯ +ε$

where the interest rate consists of a constant mean $\left(\overline{r}\right)$ and a stochastic component $\left(\epsilon \right)$. Assume that $E\left(\epsilon \right)=0\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}V\left(r\right)=\sigma$. For expositional purposes, also assume that $\beta r=1$. In this case, the household in the first period faces the following decision:

$C 1 * ={ E( C 2 * ), D 1 < D ¯ ¯ Y 1 + L ¯ −δ D 0 , D 1 ≥ D ¯ ¯$

where the first line is just the optimality condition when the constraint does not bind, while the second line comes from the household budget constraint in period 1 when the constraint does bind. This condition can be rewritten in compound form:

$C 1 * =min( C 2 * , Y 1 + L ¯ ¯ −δ D 0 )=min( C 2 * , Y 1 ( 1+ θ 1−δ )− θE( r ) D 0 1−δ − D 0 )$

Given that we have introduced uncertainty in the model, note that the interest rate enters the equation in expectation. Now suppose that uncertainty about interest rates increases. Very high realisations of interest rates $\left(\epsilon >0\right)$ become more likely, which reduces the household's expected disposable income. As a result this makes the cash flow constraint more likely to bind in the future and reduces the value of $\overline{\overline{L}}={Y}_{1}\left(1+\frac{\theta }{1-\delta }\right)-\frac{\theta E\left(r\right){D}_{0}}{1-\delta }-{D}_{0}$. To avoid this, the household reduces consumption in period 1. This precautionary saving effect will be larger for households with more initial debt.

In effect, when there is uncertainty about the ability to meet future mortgage repayments due to an increase in uncertainty about future income or interest rates, the household may choose to consume less today even though they are not currently liquidity constrained, but because they are worried about becoming constrained in the future.

## Footnote

The mechanism would apply equally to uncertainty about the future income of the household. [25]