In a previous post, I argued that assets became less liquid in the last couple of months, and that the fall in liquidity might have been a major cause of the fall in asset values. So, why did liquidity fall?

There's a literature based on a model of informed traders, uninformed traders, and "noise" traders (better called "liquidity traders"). Robert Waldmann knows that literature better than I do, and has a post, similar to mine, in which he also offers an explanation of the fall in liquidity.

Here's my own very simple model of liquidity. It explains why a small initial shock to asset values could have a much larger effect on equilibrium asset values by making assets less liquid.

There are 100 people living in a country, and 100 cars, and each person owns one car. 80 of the cars are good cars, worth $100. The other 20 cars are lemons, worth only $50 because of higher repair bills, so the average car is worth $90. You can't tell a good car from a lemon just by looking, but after owning a car for a year you know if it's a lemon.

Each year a random 20 people are forced to emigrate, to be replaced by 20 immigrants. Each year there is a market, where the emigrants sell their cars to the immigrants. If the market could be restricted to emigrants, each car would sell for $90. Unfortunately, nobody can tell the emigrants apart from those staying at home. Stay-at-homes who own lemons would want to sell their cars, worth $50, for $90. And the immigrants know this.

In equilibrium, all lemons-owners come to the market to sell their lemon, hoping to replace it with a good car from an emigrant. We have 16 good cars from the emigrants, 4 lemons from the emigrants, and 16 lemons from the stay-at-homes, all offered for sale. So the average value of those 36 cars is $2600/36=$72.22, and that's the price they all sell at.

The emigrants are getting ripped off, since their cars are worth $90 on average, but they can only sell them for $72.22. Their cars are illiquid assets. They wouldn't sell them at that price unless they had to.

The 100 cars in the country seems to be worth $7,222 at mark-to-market accounting values, even though we know the stock of cars is really worth $9,000.

Now suppose we look at an alternative universe, where there are no lemons, and all 100 cars are worth $100. The stock of cars is worth $10,000 in that alternative universe, whether we use mark-to-market or "true value" accounting. And emigrants could sell their cars for exactly what they are worth; cars would be perfectly liquid.

Now imagine we used to live in the alternative universe, with no lemons, but suddenly switched to living in the universe where 20% of the cars are lemons. The true value of the stock of cars falls from $10,000 to $9,000 (a 10% decline), but the market value of the cars falls to $7,222 (a 27.8% decline).

I think that's part of what happened in the financial crisis, but it's not the whole story. It's not just that some cars became lemons. And it's not just an accounting illusion, based on market prices in an unrepresentative market. The decline in liquidity caused a decrease in the true value of cars. Let's complicate the model a little.

Suppose each person has a choice between buying a car and catching the bus, and chooses whichever is cheaper. The bus costs $10 per year, they have a rate of time preference of 10% per year, so will be prepared to buy a known-good car at $100, if they know they can sell it at the same $100 when they emigrate.

Now we switch to the universe where 20% of the cars are lemons. Lemons cost $5 per year in maintenance (good cars cost nothing). Someone who knew for certain he would never emigrate would be willing to pay $100 for a good car, $50 for a lemon, and $90 for the average car, rather than catch the bus. But if he knows he has a 20% chance of emigrating per year, and will suffer a capital loss of x% on the car if he emigrates and is forced to sell it, he will have a discount rate of 10% + 0.2x%. The 10% is the rate of time preference; the 0.2x% is the liquidity premium. The higher discount rate reduces the amount he would pay for a car.

If someone with a 20% chance of emigrating were to pay $90 for an average new car, he would face a capital loss of $17.78 if he sold it at $72.22. He would rather catch the bus than buy a new car at $90. So $90 cannot be the amount he would pay. And then neither is $72.22 the equilibrium price on the market.

We have to solve for all prices simultaneously.

UPDATE 14/3/2009: Kevin Quinn has spotted a math mistake in what follows. I switched a + for a - in the appendix. The appendix should read:

"Let G be the value of a known-good car

Let B be the value of a known-bad car

Let A be the value of an average car in the country

Let S be the market price of cars

A = 0.8G + 0.2B

S = (16G + 20B)/36

10%G + (not minus) 20%(G-S) = $10 (comparing cost of owning good car to annual cost of bus)

10%B + (S-B) + $5 = $10 (comparing cost of owning bad car, including $5 maintenance, to annual cost of bus)"

Kevin's solutions (and they seemed correct when I checked them) are:

G=91.54, B = 83.92, S = 87.32 and A = 90

This means that the loss of liquidity does not affect the average fundamental value of cars (I had thought it did).

In would need to change the model to make emigration endogenous to get the results I wanted. If owners of good cars decided to forego profitable opportunities to emigrate, because the profits would be less than the capital loss from selling a good car at less than it is worth, then the illiquidity of cars would cause real losses in aggregate, and the average fundamental value of cars would fall.

Kudos to Kevin, for actually checking the model.

END OF UPDATE.

If I have the arithmetic right (don't trust me), someone would be willing to pay $74.58 for a good car (with a 20% chance of having to sell it next year for $61.86), $51.69 for a lemon (which he will definitely sell next year for $61.86), and so $70 for an average car (with a 20% chance of being a lemon), rather than catch the bus. (And if it costs $90 to produce a new car, Tobin's Q is now less than one, and there will be no investment in new cars.)

Let's recap. Originally all cars were good, and worth $100, and perfectly liquid assets. Then 20% of the cars turned out to be bad. With an unchanged discount rate, based on pure time preference, the fundamental value of the average car dropped to $90, but the market value, based on a market in which mostly lemons were traded, appears to be $72.22. But the discount rate will not stay unchanged, because a liquidity premium is added to the time preference premium. The fundamental value of the average car in the stock, the amount a person would rationally pay for one of the 100 cars chosen at random, rather than catch the bus, has fallen to $70. And the market value of a car, on a market where mostly lemons are traded, has fallen to $61.86.

If people didn't care about liquidity, the average car would be worth 10% less. If people do care about liquidity, mark-to-market accounting shows a 38% drop in value. This overstates the drop in value, but the 30% drop in true value when people care about liquidity is nevertheless real.

That's my parable of the financial crisis, and why the loss in asset value was much bigger than the shock which caused it. And it doesn't really matter if the shock was merely that we realised that 20% of the assets had been lemons all along. The change in perceptions caused the real value to change.

Appendix (for anyone who wants to check my arithmetic)

Let G be the value of a known-good car

Let B be the value of a known-bad car

Let A be the value of an average car in the country

Let S be the market price of cars

A = 0.8G + 0.2B

S = (16G + 20B)/36

10%G - 20%(G-S) = $10 (comparing cost of owning good car to annual cost of bus)

10%B + (S-B) + $5 = $10 (comparing cost of owning bad car, including $5 maintenance, to annual cost of bus)

Awesome! Very well explained, and it sounds right to this non-economist ...

Posted by: Phil | December 09, 2008 at 10:46 PM

Thanks Phil! Out of curiosity, what is your (non-economist) background?

Here's my wild guess: you're an engineer? (Engineers are comfortable with modelling).

Posted by: Nick Rowe | December 11, 2008 at 07:34 PM