« Double-dip risks: Update | Main | Neo-classical economics is dead. Sort of. »


Feed You can follow this conversation by subscribing to the comment feed for this post.

Better Nick, better than the vacuous "liquidity as an option to resell" approach.

A further approach would be to try to explain how policies and regulations make assets more or less liquid and when a certain policy is appropriate. The Bond market vs. the Real Estate Market might be a good place to start.... How to deal with Liquidity traps from a time/cost perspective might even be a useful topic for research if it hasn't been already done.

Thoughtful - original - it's great to have you back blogging again Nick!

Arbitragers like used car dealers do two things. As you say, they wait. They also are able to reduce buyers' search costs. Have people talked about information costs as a determinant of the liquidity of an asset?

"Have people talked about information costs as a determinant of the liquidity of an asset?"

yep, I think that is usually taken as the primary determinant. Certainly when Nick was on the "liquidity as the option to resell" thing I was trying to argue that information costs/agency costs (costly state verification) were the primary determinants.

Seems to be taken to be the primary determinant in Gorton's "Slapped by the invisible hand" for example. the best case study in Bernanke-Gertler you're ever gonna read.

Thanks Adam. I feel a bit like the blind men and the elephant. ("Liquidity" is the elephant). I'm sure I haven't captured everything relevant, but I reckon I've grabbed hold of at least one important part of the beast this time. And maybe that's because I started with a concrete example ("go from the concrete to the abstract; from the specific to the general"). But I still think I was on to *something* with that "*value* of the option to re-sell" approach. God knows what though. Gonna just keep trying different angles of approach.

Rick: that would be a next step. But one I'm nowhere near ready to take yet.

My impressions is that this is more or less the standard approach to talking about liquidity (except for the part about the slope; the line has to have a y intercept!). There's another important feature of liquidity, however and that is volume. The quantity you are trying to buy/sell also affects your transaction costs. If you're trying to buy 2 of those Mazdas it will probably be more than twice as expensive as buying 2.

I should add that this means that liquidity is really a transaction cost/price concession surface of two variables, time and trade size.

Thanks Frances!

Car dealers used to reduce search costs, but I'm not sure they do any more. Very few MX6's are sold by dealers. I learned that nearly all seem to be private sales, and the internet makes it easier to search. But dealers do make it easier to coordinate going to inspect and buy the car. (Try buying a car from someone who lives in a Toronto apartment, as I did.)

The other information costs are the Akerlovian "lemons" costs. Dealer reputation can help solve that problem. It's clearly relevant to liquidity. I thought about mentioning it here, but decided to ignore it, so I could concentrate on the matching problem. As Adam says, this is recognised in the literature. The lemon's problem is a subset of costly state verification (where it's cheap for the seller but expensive for the buyer to verify whether or not the car is a lemon).

If you buy a new car, then sell it used, you face a lemons problem (a disproportionate number of sellers will be selling lemons). That's part of why the market price of a new car drops the moment you drive it off the lot. But if you buy used, and sell used, you are both buying and selling at the low, lemons price. So I'm not sure if the lemons problem is a problem at all for people like me (I don't buy new). The lemons problem reduces the liquidity of a bought-new car. But I'm not sure if it reduces the liquidity of a bought-used car. I don't think it does (unless the lemons problem gets proportionately bigger as the car ages).

jsalvati: "My impressions is that this is more or less the standard approach to talking about liquidity (except for the part about the slope; the line has to have a y intercept!)".

But the point I was trying to make in this post is that it's not the intercept -- it's the slope.

(Though I expect someone could come back and say it's both slope and intercept, or rather, the whole curve, since it's unlikely to be linear.)

"There's another important feature of liquidity, however and that is volume."

Yep, agreed. I ignored the volume dimension. It didn't seem very relevant for used cars. One seller, one buyer, one car. (Though it is certainly relevant for other assets). But if I had been trying to buy 2, rather than 1, it would have meant paying more than twice as much, or waiting twice as long, or having to travel twice as far, or something. What this means is that buyers and sellers have dynamic monopsony/monopoly power. They face upward-sloping supply/downward-sloping demand curves. They move the "market price" against themselves. (Except the "market price" doesn't really exist for goods which aren't commodities. The fact that you sold your MX6 Tuesday morning for $1,000 doesn't mean I could have sold my MX6 Tuesday morning for $1,000. Yours is different from mine. Only commodities like financial assets (where each BMO share is identical) have a well-defined market price.

And even commodities like BMO shares don't have a well-defined market price, since if you want to sell a lot of BMO shares quickly, the price will be lower than if you sold a smaller number more slowly. But it's easier to see this with used cars, just because the liquidity problems are bigger.

Nick, this statement: "since if you want to sell a lot of BMO shares quickly, the price will be lower than if you sold a smaller number more slowly" is certainly true and for both buying and selling. That's why Intel paid a premium for McAfee instead of trying to simply buy up all the shares in the open market. Same for BHP and Potash.

However, the statement does not in any way imply "BMO shares don't have a well-defined market price". In fact non-linear pricing is pervasive, see Walmart's volume discounts from its suppliers.

Market makers post two-way tradeable prices with sizes, different trade size means different price. That does not change the fact that the price at which a single share of BMO can be bought at a given point in time (when the market is open) is a publicly known and well defined quantity. The price of a single share has a posted price and the market maker is legally obligated to deal in a single share at the posted bid.

Adam: fair enough. It's just that in simple micro we are so used to thinking of "price" as a scalar, $ per unit, rather than a schedule that relates $ to quantities (or other dimensions).

"Sellers face a two-dimensional trade-off between price and expected-time-to-sell."

I think you are talking about tradeoff between bid-ask spread cost and market timing cost here, which is well-known feature of the stock market (or, any market, in that sense). It is usually translated into tradeoff between market-order and limit-order.

However, your assumption that market timing cost is decreasing function of time seems rather strange. I don't think that assumption is very persuasive.

And, as for the volume matter, it is generally captured in terms of market impact cost.

If you look into market-microstructure literature, you can see that extensive researches have been done on these costs. I suppose you can gain some extra insights from those researches.

himaginary: "However, your assumption that market timing cost is decreasing function of time seems rather strange. I don't think that assumption is very persuasive."

Suppose you had to sell your house: in the next minute; in the next hour; in the next day; in the next week; in the next month; in the next year? How much money do you think you could get for it? Even if house prices generally were constant over time, the longer you had to sell your house, the higher the price you could expect to get. If you waited long enough, you would find a buyer who really really liked your particular house, and would be prepared to pay a lot for it.

I saw some MX6's advertised where the seller quoted a very low price, but said it had to be sold that day, or that week, because the seller had nowhere to park it, or needed the money to pay school fees, or whatever.

I know what "bid-ask spread" means in a market where all trades take place through dealers. It doesn't mean anything for private sales of used cars.

If supply and demand can be curves rather than scalar, why can't liquidity?

It is not enough, I think, to let the slope with respect to time describe liquidity. True, a liquid asset is likely to have a low sensitive to impatience. But the absolute cost still matters, doesn't it? Perhaps not as much as sensitivity to time, but a patience-insensitive commodity on which there is a 10% transaction tax is still less liquid in some sense than one on which there is no tax at all, that can be traded costlessly immediately. Perhaps if we have to choose a scalar measure of liquidity, low time sensitivity is a better measure than absolute cost. But should we choose a scalar measure? If we must, should it be computed from only from this slope, rather than as some weighted average of slope and height?

In deference to jsalvati's comments, I think of market liquidity as a surface whose axes are time, quantity, and cost.

It is an excellent and important point I'd never considered that, while cost is uniformly decreasing in patience (if you assume traders can time optimally), for many illiquid commodities and small quantities, cost can dive beneath the zero.

Steve: Thanks!

I've just read your 2007 post you linked. We are very much on the same page.

Yes, I was pushing it a bit, saying that slope is *the* measure of liquidity. (I think it works quite well for comparing used cars, where quantity is nearly always one, and the transactions tax is the same percentage for all used cars. If I want to say that Mazda MX6's are "less liquid" than Honda Civics, then this slope parameter is the one measure I would focus on). But in a more general case, you are right. If the transactions tax varied across different assets, for example, that would make some more liquid compared to others. Liquidity is the whole curve (with at least 3 dimensions).

I think it's important that the "spread", or cost of a round trip, can be negative. That's why car dealers exist; that's why market-makers exist in stock and bond markets.

But I want to push this point further.

My mind is not quite clear on this point, but let me take a flying leap anyway: absent things like taxes, the *average* "spread" across all buyers and sellers (including dealers) *must* by logic be zero. Money doesn't just disappear into thin air. If one person (the impatient one) buys "high" and sells "low", then someone else (the patient one) must be taking the other side of the trade, and selling "high" and buying "low".

If I am right on this point (and I think I must be right, if only I could think about it and state it clearly), then (absent things like taxes) the average height of the curve is always zero, so we can't use the height of the curve to give us any measure of liquidity at all. It must be slope, not height. Height must be zero, for any asset, on average (absent things like taxes). Only slope remains.

Do you see what I'm trying to say? There's no such thing as "spread", on average, across all buyers and sellers. Private traders see spread as a cost; dealers see spread as a benefit. Spread nets to zero. In my case, as a private buyer of a used car from a private seller, in a market where there are no dealers, I was forced to see that "spread" doesn't exist.

Steve: just in case it wasn't obvious from my above comment, your comment, and your post, have been very useful to me in thinking this through.

The other information costs are the Akerlovian "lemons" costs. Dealer reputation can help solve that problem. It's clearly relevant to liquidity. I thought about mentioning it here, but decided to ignore it, so I could concentrate on the matching problem. As Adam says, this is recognised in the literature. The lemon's problem is a subset of costly state verification (where it's cheap for the seller but expensive for the buyer to verify whether or not the car is a lemon).

A plug is in order here for Phil Edmonston, author of the Lemon Aid series of car reviews. Edmonston, btw, was the first NDP MP elected in Quebec.

I wrote to Phil (now retired and living in Florida) concerning a Mazda 626 I once owned whose transmission blew at 95 k (it had a 100k /5 yr warranty - mine blew at 6 yrs). He advised me to sue Mazda and they would probably quickly settle(the tranny problems occured when Ford took over Mazda in the earlier 90's) but I couldn't be bothered. My repairs were about $2500.

An aside,slightly O/T - but a valuable private sector resource - Lemon Aid - worth a plug.

JVFM: Was it a 1993 or later 4 cylinder 626 with an automatic transmission? Notorious problem. Much discussed on 626.net. The solution is to install a tranny oil cooler.

1996 four banger with A/T- you're right - I think I cooked it travelling in the southern US - I paid for a lifetime warranty at Aamco, but didn't go in for their yearly annual checkups ( a condition of the warranty - so when it fried again at 150k I junked it).

When I wrote to Phil, it was not a notorious problem then. He now lists the problem - perhaps the notoriety soon followed as you describe.

"Even if house prices generally were constant over time"

I think that's a big if. In general, house prices decline over time because of depreciation. So there is a possibility that you can never sell your house at desired price, or possibility that you have to lower your selling price. These should be counted as market timing cost (or non-execution cost, if not executed), as they do in implementation shortfall method.

"I know what "bid-ask spread" means in a market where all trades take place through dealers. It doesn't mean anything for private sales of used cars."

But the dealer system is not the only way trading system is implemented. For example, the Tokyo Stock Exchange adopts auction system, or order-driven system. So, in a sense, TSE may be more similar to private sales of used cars than it is to NYSE. And, of course, "bid-ask spread" does have meaning in TSE.


"I saw some MX6's advertised where the seller quoted a very low price, but said it had to be sold that day, or that week, because the seller had nowhere to park it, or needed the money to pay school fees, or whatever."

That price lowering is exactly the component of bid-ask spread. It is said that bid-ask spread reflects "price of immediacy". (One of other components is said to be "price of information asymmetry".)

himaginary: generally, in a growing economy, the house itself depreciates (wear and tear), but the land it is built on appreciates (richer people and a bigger population are willing to pay more for prime locations).

But it doesn't affect my point. Make any assumption you like about the trend of house prices. My point is that the longer you have to sell your house, the better the price you can get, *relative to trend*. (Of course, this presumably asymptotes to some upper limit.) Try to sell a house in one day, and you will probably have to sell it at a "firesale price".

I've been thinking more about "spreads", in the light of your comment, and Steve's. Yes, it's true, that on the TSE for example, there's a bid-ask spread for BMO shares at any second. But if the spread is positive, there are no trades taking place. If bid is $40 and ask is $41, no shares are traded. If I instruct my broker to buy "at the market", he implicitly translates that into "bid $41", and buys shares for me at $41 (assuming quantity is sufficient). So for a split second the bid and ask are both $41, so the spread is zero, the trade takes place, and the spread opens up again.

Except for taxes, and broker's commissions, the actual spreads at which trades take place are always zero. The only reason we don't see zero spreads is because as soon as the spread goes to zero trade instantly takes place and eliminates either the bid or the ask (or both). But prior to the market opening, you do see zero spreads. If there were a 1 minute delay on trades, you would see zero spreads much more frequently. All actual trades are made at zero spreads.

I know this is a weird way of looking at it.

I'm not sure the claim that "the actual spreads at which trades take place are always zero" makes sense Nick.

Just because I execute on the ask doesn't mean my bid is the market bid. It just means that I value the BMO share at some amount greater than $41. If I value the share at $45 then I've just captured a surplus not different from a consumer's surplus.

But in normal(perfectly competitive market) price theory even though most consumers would have paid more than the market price and thus captured a surplus does not lead us to conclude the market price is undefined or that all of their bids are the market bid. The market price is the bid of the marginal bidder, the one who got no surplus.

The same should apply here, just because you bid $41 and dealt doesn't make your bid the marginal bid, even for an instant. After all, somebody else could in principle be selling to the market maker at $40 at exactly the same instant as you are buying. The marginal bid is really, more or less by construction, the market maker's bid and that is still $40.

This is from Steve Waldman by email (for some reason he had difficulty posting his comment):

Nick -- Sorry for the slow response. With apologies to Alan Greenspan,
if anything I've said seems useful, you've probably misunderstood me.

A couple of comments: It's an interesting point that, in a sense, the
average spread must equal to zero, because the liquidity demanders
(impatient buyers and sellers) transfer their spread precisely to
liquidity providers (more patient buyers willing to hold "inventory"
-- a position, positive or negative -- for a while).

I think that whether this is "true" has a lot to do with how one
defines cash flows, whether one attributes them to liquidity or to
something else. The spread paid by an impatient trader allows that
trader to satisfy her preferences immediately and with certainty. More
patient traders must react to the demands of the impatient (if the
impatient are to be satisfied at all), implying that for some period
of time, they may be left holding a portfolio that is not what they
deem optimal. Obviously, my thinking on this is informed by financial
asset portfolios, but it works with cars too. Say you have a car, and
I have a great need for a car now. I approach you on the street and
offer you three times the value of your vehicle, in cash, now. You are
left stranded -- the financial benefit you receive as a liquidity
provider has a real cost! -- but deeem the compensation, the spread,
to be more than sufficient and so take the deal. Conventionally, the
surplus I pay over the car's "fair value" would be accounted as a
liquidity-related cost, "the price of immediacy", while the same
spread as you receive it would be accounted as compensation for your
inconvenience, as a payment. Obviously, the aggregate flows sum to
zero, but it might (or might not) remain useful to refer to only the
expense as "liquidity" related and the income as something else.

What about in financial markets, where liquidity providers buy and
sell positions routinely, and no one is stranded on a streetcorner?
It's really the same story. Liquidity providers try to earn income by
responding to others' demand for immediacy, but in doing so, they are
forced to deviate from their optimal portfolio. If you want to sell a
share of BMO "at market", to satisfy that I must end up with more than
one share of BMO than I wanted. (There could be a "natural cross",
where simultaneously you want one less and I want one more share of
BMO, in which case we might both agree to trade at "fair price".
Assets that trade in high volume often have low apparent spreads in
part because the probability of a "coincidence of wants" is fairly
high, and often one can trade within those apparent spreads.) If you
sell a share of BMO at market, we account for the cost you pay over
its contemporaneous fair value as the price of immediacy.
("Contemporaneous fair value" is conventionally guessed as the
midpoint of the listed spread, but where within the spread it really
lies is unknowable.) But we don't call the money you pay the negative
price of indifference to immediacy (though we could). We call it
compensation for bearing the risk of holding a position you would not
desire until such time as you can liquidate economically.

All that said, I think you are absolutely right that in, for example,
my liquidity surfaces, over time the "spread" cost to an optimal timer
should go negative, not approach zero as I've graphed things. Because
what we call "liquidity" is always the spread relative to fair price,
which is a moving target. Assuming risk-averse agents, if a patient
trader cannot earn money (pay a negative spread) from the mechanics of
buying and selling, they should be unwilling to bear the risk of
fluctuations in fundamental value that are normalized out of my

Re the curiosity that spreads are always zero when assets are traded,
that's right, but i think unhelpful. A limit order book basically
describes a supply/demand diagram with its left-hand side truncated.
If one cumulates the bids, one can generate a demand curve, where
quantity demanded falls to approximately zero below the equilibrium
price. If one cumulates the asks, quantity supplied rises from a
approximately zero above the equilibrium price. So the diagram looks
like a K, if we place the vertical axis at quantity 0. If we let
assets be continuous, so that minimum bid and ask sizes needn't be
discrete, "approximately zero" should be zero, and there should be no
"spread" at all. Then what happens if I suddenly want to buy a full
share of BMO, right now, at "market"? There will be a rightward shift
in the existing demand curve by one unit, with a vertical line drawn
upward from the offset demand curve. The equilibrium price will
change, ephemerally, by an amount determined by the slope of the
supply curve. I will pay a premium over the prior equilibrium price.
If I put in a market order to sell, the supply curve shifts right with
a downward vertical line from the previous zero point, and I realize
less than the equilibrium price from the sale, depending on the slope
of the demand curve.

The observable spread is then a scalar summary of the pre-order supply
and demand curves. It captures some sense of what the equilibrium
price would ephemerally become if a market order to buy or to sell
were to be entered. It's a linear approximation of the cost of a round
trip, valid for small orders under the assumption that neither the
equilibrium price nor the slope of the reactive supply/demand curves
are altered by the order. (These assumptions may be approximately true
for small orders of high-turnover assets, though they are distinctly
false in general.) You can read the spread off of our idealized supply
and demand diagram as just the vertical distance between supply and
demand curves that meet at zero when quantity is set to one (if that
is the quantity bid and asked).

When a market order for a discrete number of units is placed, the
vertical distance between the two curves at the quantity ordered goes
to zero, and there is no spread. But we are interested in observable
spreads before the actual transaction, because they predict something
about what will happen in future trades, assuming there is no
fundamental news but there are occasional traders who demand immediate

The biggest problem with all this stuff is the assumption that we can
cleanly separate liquidity motivated costs, wherein a person buys or
sells an asset for idiosyncratic reasons that shouldn't affect the
equilibrium price, from costs due to changes in the equilibrium price.
In the end, all we observe is a series of trading prices and
quantities that bounces around. It's important to remember that if the
equilibrium price was fixed and certain, liquidity providers would
compete the spread to zero: both the supply and demand curves would
approach be horizontal (ignoring fixed transaction costs). That we
observe spreads and variable liquidity costs at all is due to the
possibility of price moves, the risk and uncertainty associated with
holding assets that must be compensated for when a trader demands
immediacy. The real spread is then related to the degree of
competition to provide liquidity for a particular market and the
degree of uncertainty surrounding its equilibrium price. "Liquidity"
related transaction costs and "solvency" related valuation changes are
hopelessly entangled together, and spreads are up-front compensation
for changes in the equilibrium price that (in aggregate) liquidity
demanders know something about but liquidity suppliers do not. When
people observe large spreads and claim a "liquidity" problem, they are
implicitly admitting a great deal of uncertainty about the value of
the thing in question (assuming there aren't barriers to competitive
liquidity provision). When we talk about "liquidity" costs, we are
really running an unlikely thought experiment where we imagine the
equilibrium price at time t is known, and we want to know how much a
trader would demand to trade at time t given the possibility that the
equilibrium price will change over a holding period sufficient for
cheap liquidation. But in fact the equilibrium price is never actually
known, and apparent spread costs can actually be steps along the path
to a new equilibrium (and therefore not costs at all, just fair-or- even-better-than-fair prices paid).

Blah! This feels like nonstop digression on my part. (I've spent a
fair amount of time working on the microstructure of liquidity, so
it's easy for me to not shut up about this.)

I hope I have disabused you of any notion that my blatherings might be

The above is from Steve Waldman.

with respect to Steve's paragraph that begins "Re the curiosity that spreads are always zero when assets are traded, that's right..."

I agree, for the most part, with the description of the limit order market but think he's still missing something. As I said above just because you're willing to execute at the $41 ask price doesn't make your bid the marginal bid.

It is perfectly possible, particularly in electronic markets with computers executing trades, for someone else to be executing a sell at the market maker's bid of $40 at exactly the same instant. This does not mean that for that instant there are two market prices, it means that at that instant the spread is not zero despite the fact that you're executing a trade.

The "market price" is the marginal bid, not the highest bid. That will, by construction, be the market maker's bid.

So the cost is a function of the time required/allowed to complete the transaction. But couldn't this be reversed? The time required to find a buyer/seller is a function of the price? I think this might be visible in the housing market, where time on the market might be related to the percent of the asking price that the house actually sells for.

rharris: other things equal (like the location and size of the house, the state of the housing market, etc.), the longer you are prepared to wait to sell your house, the higher the price you will eventually get (relative to the trend of house prices, and allowing that there is a limit, etc.). And this relation can (usually, subject to technical qualifications [implicit function theorem blah blah]) be "inverted" (as economists would say), so that the higher the price you insist on for your house, the longer you will have to wait to sell it (and it could be forever).

But there's a separate point, that I think you are making. Sometimes sellers misjudge the state of the market, or the quality of their particular house in the eyes of potential buyers. Some think their house will sell more quickly/at a higher price than it in fact will. (The curve relating price to time-to-sell is lower than they think it is). So when they get no offers after a few months, they revise downwards their expectations, and accordingly revise downwards their asking price.

(And other sellers misjudge the market in the opposite direction, and are surprised at how quickly their house sells.)

So, if we see a house that stays unsold on the market for an especially long time, it might be due to (a combination of) one of (at least) three possible reasons:

1. The house is very "different" from most houses, so only a small number of buyers will value it very highly. The curve is steep, rather than flat, so it pays to wait for the right buyer to come along. It's an especially illiquid house. It's a Mazda MX6, not a Honda Civic.

2. The seller is very patient, in no hurry to sell.

3. The seller misjudged the market, and had overly-optimistic expectations.

4. Or it could just be bad luck. Buyers are "lumpy" individuals, so new buyers don't always come onto the market in a smooth steady stream of one per week. Some weeks there will be 2 or 3 new buyers; other weeks there will be none.

Adam and Steve: I'm still thinking.

Topic for future post: Why do houses sit on the market for longer in a falling market? (I think that stylised fact is true).

1. Is it simply because sellers are irrational and hate to sell their house for less than they paid for it (or could have sold it for last year)? Sticky Price Hypothesis. Bygones are not bygones.

2. Is it simply because sellers are slow to adjust their expectations about the state of the market? Adaptive expectations. Rational expectations with imperfect information and signal processing.

3. Or might it be because a fall in demand (a smaller flow of new buyers) somehow shifts the slope of the price/time-to-sell curve, makes houses less liquid, and makes it rational for sellers to wait longer on average?

Damn yes, I like that topic! It brings together all my pet themes: sticky prices, rational/adaptive expectations, imperfectly competitive macro, business cycle theory, and liquidity! (And I must be able to squeeze used cars in there somehow!)

I think the relationship and difference between liquidity and price discovery and volatility should be argued more carefully to proceed with this line of argument.

If price is more volatile, the chance your ask/bid order being executed increases, even your ask/bid price is distant from the current price. So, from your viewpoint, the average time of trade execution is shortened, i.e. liquidity increased, as volatility increased. However, in general, high volatility means less liquidity, because it means that there is insufficient number of traders in the market, and/or that the market cannot absorb incoming new information very well.

As for liquidity and price discovery, Maureen O’Hara once put it, "Markets provide liquidity and price discovery. These two concepts are related, but they are not the same." I think liquidity is something that facilitates price discovery, but not the price discovery itself.

" But if I had been trying to buy 2, rather than 1, it would have meant paying more than twice as much, or waiting twice as long, or having to travel twice as far, or something."

Since your search is simultaneous, you wouldn't have to wait twice as long. (You don't start the second search after the first one is over, you do both at once.) A similar argument applies to distance as to time. :)

Sorry, Nick. I take that back. Suppose that the probability of finding a suitable car is constant over time. Then after any time period in which the search has failed, it is like starting over. Also, we may assume that finding one suitable car does not perceptibly alter the probability of finding another. Then twice the time or distance is right, on average.

The comments to this entry are closed.

Search this site

  • Google

Blog powered by Typepad