It's an interesting paper (pdf). It's a very standard New Keynesian macro model with one twist. It's a twist worth doing. There are two types of people: the impatient, who borrow from; the patient. And there's an exogenous limit to the debt the impatient are allowed to accumulate.
It's a math model, of course. I'm not going to stick literally to the assumptions of the model, because many of those assumptions were made to make the math simple, and aren't really central. Here's my reading.
1. For a given level of current and expected future income, the equilibrium real rate of interest will drop. That's because some people and firms who want to borrow won't be able to, or won't be able to borrow as much. The demand for loanable funds from qualified borrowers falls, and so does the equilibrium rate of interest. Equivalently, for a given level of expected future income and real interest rate, there is a fall in the level of current income at which desired savings equals desired investment (ignoring the desires of those who want to borrow and spend but can't). Whichever way you think about the IS curve, it shifts down, or it shifts left. Same thing.
2. The marginal propensity to spend out of current income increases. A change in current income that leaves future income unchanged has little effect on permanent income, and so will normally have little effect on the demand for goods. But borrowing-constrained people and firms will have a marginal propensity to spend out of current income of one. So anything that causes current income to change will have a bigger multiplier effect.
3. A change in the real rate of interest will have a smaller direct effect on the demand for goods. That's because the borrowing-constrained people and firms want to borrow and spend more if the rate of interest falls, but are unable to.
4. The IS curve may become either flatter or steeper, depending on whether 2 or 3 above dominates. A less interest-elastic demand for goods means a steeper IS curve. But a bigger multiplier means a flatter IS curve. With an exogenous borrowing constraint, a given fall in the rate of interest will have a smaller direct effect on increasing demand, but that direct effect will have a bigger multiplier effect. So the total effect of a fall in the rate of interest on demand for goods could be either bigger or smaller.
Now suppose an exogenous shock causes the borrowing constraint to bind more tightly. This causes the IS curve to shift left, and makes the natural rate of interest strongly negative. Since the nominal rate of interest cannot fall below zero, and since the expected rate of inflation isn't high enough, the economy goes into a recession. Any fall in the current price level increases the real value of existing nominal debt, and makes the borrowing constraint bind more tightly still. (That's the Fisher debt-deflation effect).
Monetary policy can only work if it can increase the expected future price level, and thereby increase expected inflation and lower the real rate of interest. Fiscal policy can work because a cut in current taxes will raise the current disposable income of borrowing-constrained people and firms, causing them to increase their demand for goods, even if they know that future taxes will be increased. In effect, the government is acting as lender to the borrowing-constrained, and relaxing that constraint.
One main problem with the model is the exogeneity of the level of real debt at which the borrowing constraint bites. I can think of three main channels in which making the borrowing constraint endogenous might change the results of the model:
B1. The amount that people and firms are allowed to borrow depends on their income. If a recession causes a decline in current and expected future income, this could tighten the borrowing constraints still further, deepening the recession. In effect, we get a borrowing-constraint multiplier.
B2. The amount that people and firms are allowed to borrow depends on the rate of interest. A fall in the rate of interest will lower the interest payments on a given debt, and would loosen borrowing constraints and help the economy escape the recession.
B3. A cut in current taxes implies an increase in expected future taxes, which reduces expected future disposable income, and reduces the ability of debtors to service debt in future, which may tighten the borrowing constraint. So fiscal policy may still fail, because the increased government lending may be fully offset by an equivalent tightening of the borrowing constraint. Ricardian Equivalence may still hold.
Let me add a quasi-monetarist gloss to the model: The recession is not caused by an excess of desired savings over investment. Instead, when lenders are unable to find borrowers they think are safe, they choose to save in the form of money instead. It is the excess demand for the medium of exchange that causes the recession. In a barter economy, those who desired to lend but who were unable to find borrowers they think are safe would be forced to consume or invest their income themselves.
Let me add a more monetarist objection (I'm trying to guess what Scott Sumner would say): It was the Fed, by having too tight a monetary policy, so that expected future nominal income fell, that caused the borrowing constraint to tighten. It wasn't exogenous at all.
And I would add that monetary policy might actually be more powerful in this sort of model than it would be in a similar New Keynesian model with a liquidity trap but without a borrowing constraint. The higher marginal propensity to spend out of current income is higher, so the multiplier effect of anything that gets spending started will be higher. Plus, more importantly, the borrowing constraint creates its own multiplier, because any increase in current or expected future income relaxes the borrowing constraint which creates additional demand and income.