Appendix D: Household Liquidity, Precautionary Saving and Borrowing Constraints | RDP 2021-10: The Rise in Household Liquidity

RDP 2021-10: The Rise in Household Liquidity Appendix D: Household Liquidity, Precautionary Saving and Borrowing Constraints

Gianni La Cava and Lydia Wang

November 2021

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To provide a conceptual framework for the empirical analysis consider the following two-period consumption model. A representative household maximises lifetime utility subject to an intertemporal budget constraint as well as two borrowing and saving constraints. The model is partial equilibrium in nature in that it takes income, housing prices and interest rates as given. To keep things tractable, we abstract from realistic elements such as the life cycle and decisions regarding housing tenure. The household is born with a single asset ( A₀ ), which can be thought of as housing equity, with some fixed fraction of this asset assumed to be initially liquid $(α)$ . The fraction of the asset that is liquid is exogenously given by a technology that allows households to tap the money held in the asset (such as mortgage redraw accounts). The household can choose to either consume or save the liquid fraction of the asset in the first period.

We assume households have log utility in a single consumption good:

V = \ln (c_{1}) + β \ln (c_{2})

where the discount factor is $0 \leq β \leq 1.$

The budget constraint in period 1 is:

c_{1} \leq x_{1} = y_{1} - s + α P A_{0}

where first period consumption ( c₁ ) must be no greater than ‘cash on hand’ ( x₁ ) which is the sum of the first period income endowment ( y₁ ), any borrowing (negative saving) ( –s ) and some fraction of the initial value of the asset $(0 \leq α \leq 1)$ , where the price of the asset is P at the start of the first period and the real volume of the asset is fixed at its initial value ( A₀ ). Cash on hand represents the resources available for consumption and saving to the household, after the realisation of first period income, and can be thought of as the value of liquid assets in the first period.

The budget constraint in period 2 is:

c_{2} \leq y_{2} + R s + (1 - α) P_{1} A_{0}

where the household can consume from its second period income endowment ( y₂ ), any saving, including accumulated interest ( Rs ), and the remaining (illiquid fraction) of the value of the asset, which is equal to P₁ at the start of the second period.

This implies an intertemporal budget constraint:

c_{1} + \frac{c_{2}}{R} \leq y_{1} + \frac{y_{2}}{R} + α P A_{0} + (1 - α) \frac{P_{1}}{R} A_{0}

The household potentially faces a saving constraint:

s \geq \underline{s} = θ P_{1} A_{0}

This can be thought of as a deposit requirement in that the household must hold enough liquid assets to cover some share of the future value of the asset. (This is the flipside of a more standard loan-to-valuation constraint).

The household may also face a borrowing (or equity withdrawal) constraint. Because only a certain fraction of the asset is liquid in the first period, a household may choose to liquidate some of the future value of the asset (which is here known with certainty). We assume that the household is limited to borrow some fraction $(0 \leq γ \leq 1)$ of any increase in the discounted value of the asset $(\frac{P_{1}}{R} - P)$ between period 1 and period 2.

- s \leq - \bar{s} = γ (\frac{P_{1}}{R} - P) A_{0}

We first consider the optimal saving decision of the unconstrained household. Substituting the intertemporal budget constraint into lifetime utility, we find the optimal level of saving:

s^{*} = (\frac{β}{1 + β}) y_{1} - (\frac{1}{(1 + β) R}) y_{2} + (\frac{β}{1 + β}) (α P A_{0}) - \frac{(1 - α)}{1 + β} (\frac{P_{1}}{R}) A_{0}

Note that, due to a wealth effect, household saving is a negative function of any increase in the value of the asset (assuming a positive real interest rate and a positive initial value for the asset):

\frac{δ s^{*}}{δ P_{1}} = - \frac{(1 - α)}{(1 + β) R} A_{0} \leq 0

Note also that, if the asset is fully liquid in the first period $(α = 1)$ , this wealth effect disappears.

We can also consider the optimal saving decision of the household that is constrained by the deposit requirement. This represents the problem facing a potential first home buyer. The saving constraint naturally leads to the prediction that higher future asset prices require greater saving now, and hence a positive correlation between the value of the asset and saving for potential home buyers.

\frac{δ s^{*}}{δ P_{1}} = θ A_{0} \geq 0

Next, we look at the optimal saving of the household that is constrained in terms of borrowing against the future value of their asset through equity withdrawal. This represents the problem facing an owner-occupier that is currently liquidity constrained but has some positive housing equity. If the borrowing constraint is binding, then the optimal level of cash on hand is:

x_{1}^{*} = \bar{x} = y_{1} + γ (\frac{P_{1}}{R} - P) A_{0} + α P A_{0}

Here, higher asset prices are associated with more cash on hand which could be saved or consumed in the first period:

\frac{δ x^{*}}{δ P_{1}} = γ (\frac{1}{R}) A_{0} \geq 0

This implies that equity withdrawal provides a mechanism through which households choose to save in a liquid asset (though households may also choose to consume this equity).

Finally, we introduce uncertainty into the model to highlight the role of precautionary saving. This allows us to consider the problem of the owner-occupier that is worried about becoming liquidity constrained in the future if there is a negative wealth shock. For simplicity, suppose the only source of uncertainty is about the future price of the asset. And consider a mean-preserving spread of the future asset price:

P_{1} = P + ε

where the future asset price consists of a constant mean ( P ) and a stochastic component $(ε)$ . Assume that $E (ε) = 0$ and $V (P_{1}) = σ .$ For expositional purposes, also assume that $β R = 1.$ In this case, the household in the first period faces the following decision:

c_{1}^{*} = {\begin{matrix} E (c_{2}^{*}), & - s > - \bar{s} \\ y_{1} + α P A_{0} - \bar{s}, & - s \leq - \bar{s} \end{matrix}

The first line is the optimality condition when the borrowing constraint does not bind (so the household perfectly smooths consumption). The second line comes from the household budget constraint in period 1 when the borrowing constraint does bind. This condition can be rewritten in compound form:

c_{1}^{*} = \min (E (c_{2}^{*}), y_{1} + α P A_{0} + γ (\frac{E (P_{1})}{R} - P) A_{0})

Given that we have introduced uncertainty in the model, the future asset price enters the equation in expectation. Suppose that uncertainty about the future asset price increases. Very low realisations of the asset price $(ε < 0)$ become more likely, which reduces the household's future income. The borrowing constraint becomes more likely to bind. To avoid this, the household reduces consumption and increases saving in period 1. This precautionary saving effect will be larger for households that hold larger initial (volumes) of the asset.

In effect, when there is uncertainty about future income (e.g. because of increases in interest rates or falls in income), an owner-occupier household may choose to save more today because they are worried about becoming liquidity constrained in the future.