I will sketch a simple model in which the distribution of wealth gets more unequal over time, how the equilibrium real interest rate falls over time, eventually leading to a zero nominal interest rate, and unemployment. I will then show that an increase in the money supply can increase employment, despite zero nominal interest rates.

Let me give you the intuition first.

People differ in how patient they are, along a continuum. In the initial equilibrium, the more patient save and the less patient dissave. Over time, the patient accumulate assets and the impatient reduce their assets. Eventually the impatient become borrowing-constrained.

The initial equilibrium real rate of interest depends on the average degree of patience of the whole population. But when the impatient become borrowing-constrained, the equilibrium real rate of interest depends on the average degree of patience of that subset of the population which is not borrowing-constrained. That subset excludes the less patient, which means the equilibrium real interest rate falls over time as more and more people become borrowing-constrained. Eventually the equilibrium real interest rate falls so much that the nominal interest rate hits the zero lower bound.

There are two assets: money and bonds. Money is a medium of exchange; bonds are not. When the nominal interest rate hits zero, the most patient continue to save, but can only save in the form of money. The least patient dissave, and slowly reduce their holdings of money.

Since the most patient now hold money purely as a store of wealth, at the margin, the average desired velocity of circulation of money across the population falls over time. As velocity falls over time, aggregate demand falls over time, and unemployment rises over time.

Unexpected deflation would prevent unemployment by allowing the real money stock to rise over time. But fully anticipated deflation might worsen unemployment by increasing the real interest rate above equilibrium. A permanent increase in the money supply would prevent unemployment for a long time, provided some of the money were given to the borrowing-constrained.

Here's a sketch of the model.

Agents are identical, except in their rate of time preference (patience). There is a continuous distribution of rates of time preference F(d). Otherwise agents are identical. They consume, and supply labour inelastically.

Labour produces consumption services. No investment or storage.

Let's start with a barter version of the model.

There is one asset: government bonds. Agents are unable to issue private bonds, so an agent who runs out of bonds is borrowing-constrained.

Each agent has a consumption-Euler equation (personal IS curve): C0 = C(C1,d-r). Where r is the real rate of interest.

The market rate of interest adjusts until aggregate savings equals zero. The more patient agents will be saving, and the less patient will be dissaving. So the stock of bonds flows from the less patient (r<d) to the more patient (r>d). Eventually the least patient person runs out of bonds, becomes borrowing constrained, and stops dissaving. Aggregate savings therefore rises, and so the rate of interest falls, and continues to fall over time as more and more people become borrowing-constrained. Eventually, the most patient person in the whole population will hold all the bonds, and everyone else is borrowing-constrained.

Now introduce money, as a second asset. Money pays no interest, but is a medium of exchange. Each agent has a cost of having a high personal velocity of circulation (low real balances relative to nominal expenditure). The marginal cost falls to zero when velocity gets below some minimum level (call it 1), at which point he is satiated in monetary services.

Some agents own bonds. They can separate their savings decision (consumption vs. savings) from their portfolio allocation decision (money vs. bonds).

Each agent who owns bonds has a personal LM curve, which defines velocity as a function of the nominal rate of interest. The LM curve has the normal slope, but suddenly goes horizontal at a zero nominal rate of interest.

But other agents do not own bonds, and are borrowing-constrained. Their savings decisions cannot be separated from their money demand decisions. If the real rate of interest is below their rate of time-preference, they will dissave, and slowly reduce their holdings of money over time. But as their money holdings get smaller, their level of dissaving will fall, so eventually their savings drops to zero, and their money holdings stop falling when they hit some lower bound (whether at a zero or positive level of money depends on their preferences).

The aggregate stocks of bonds and money stay constant over time.

At the beginning of time, the bonds and money are divided equally between all agents. Then play begins.

Over time, the patient agents save, the impatient agents dissave, and bonds flow from the impatient to the patient.

Eventually, the least patient agent runs out of bonds, his savings rate slowly falls, as his stock of money falls, and then he stops dissaving when his holdings of money hit the lower bound. As more and more agents hit the lower bound, the real rate of interest falls over time. Eventually the real rate of interest falls so low that the nominal rate of interest hits zero.

When the nominal rate of interest hits zero, and the real rate of interest can fall no further, desired aggregate savings at full employment is positive. Unemployment, initially at zero, now begins to rise. The more patient, who are saving, accumulate both bonds and money, past the point of satiation of money holdings.

After some time, there are four sets of agents. 1) The most patient are still saving, and hold bonds and money, and are past satiation in money (they hold "idle hoards"). 2)The slightly less patient are dissaving, and holding bonds and money. 3)The slightly less patient still are dissaving, and holding only money. 4) The least patient have stopped dissaving, and hold only money (the lower bound).

Unemployment would cause the price level to fall, if we assume flexible prices. A lower price level would help eliminate unemployment, by increasing the real value of all agents' holdings of money. The fourth group of agents would now join the third group, and would start dissaving again. (And the third group might dissave more). This will reduce aggregate savings, and help reduce unemployment. But expected deflation would increase the real interest rate, cause an increase in savings, and increase unemployment. The second effect would dominate (assume it doesn't and you get a contradiction, which would take me too long to prove).

So let's just assume the price level stays fixed once unemployment starts to appear.

Then the central bank decides to do something about unemployment.

An open market operation will not help (I think). But a helicopter increase in the money supply will help. The fourth group of agents will start dissaving again. The third group of agents will dissave more. So aggregate savings falls.

If the helicopter can increase the supply of money by a big enough amount to get to full employment, then the price level can start to rise if extra money is added. So the policy prescription is one big helicopter drop, to get to full employment, then steadily increase the money supply over time to make inflation positive, and allow the real rate to drop below zero.

Well, the model works, sort of. But I am not happy with it. A helicopter drop of bonds could have the same initial effect, since the borrowing-constrained could sell the bonds for some of the money held in idle hoards. The bond market (and that's the only asset market) is perfectly liquid, in that bonds can be swapped for money at zero cost. Put it another way: it's the wealth effect, as much as any liquidity effect, that's doing most of the work (I think).

But, the model does show that monetary policy could work, even at zero nominal interest rates. So I'm going to post it, even though it's not really doing what I want it to do.