The Instability of Dynamic Consumer Credit Pricing

I am attempting to come up with a model demonstrating that dynamic risk-based pricing of consumer credit could well be unsustainable. The foundation of my thinking is a 27 year old paper by Sargent and Wallace entitled "Some Unpleasant Monetarist Arithmetic". In that paper, SandW show that when real interest rates exceed economic growth in the presence of deficit spending, tight monetary policy is not sustainable. I have only a sketch so far.

Suppose consumer credit markets price debt using some sort of Modigliani-Miller formulation: as the leverage ratios of households rise, the price the pay for debt rises. We can think of the leverage ratio as the value of household assets (including human capital) divided by assets less debt outstanding, or equity. For households operating at the margin, human capital may contain all assets.

Interest charges in each period is a function of the household's leverage ratio in the previous period. So

r(t) = rf + LR(t-1)(rh -rf)

LR(t)= LR(t-1) + f(Min payment - r(t-1))

Where r(t) is the interest rate on consumer credit at time t, rf is a rick free rate, rh is the return on assets for an unleveraged household, and LR(t-1) is the leverage ratio at time t-1.

Suppose a household is on a path where it is amortizing its debt. Then the cost of credit falls over time, which accelerates the point at which a household becomes debt-free.

But now suppose that household receives a negative permanant shock: a job loss, a health problem or a divorce. In the short tun, it can make a minimum payment on a credit card balance, but not enough to pay interest due, so the balance gets bigger. This raises the cost of interest in the following period, which in turn means that the minimum payment pays even a smaller share of interest costs. The leverage ratio increases.

Under these circumstances, the amount owed is increasing as rapidly as the discount rate, and so a traversality condition (where in the limit, the present value of debt goes to zero) is violated. This implies (I think) that some sort of interest rate ceiling might improve stability. Thoughts are welcome.