# Is Piketty’s “Second Law of Capitalism” fundamental?

on May 28, 2014 at 2:25 pm

Per Krusell and Tony Smith have a new paper on Piketty (pdf), which I take to be reflecting a crystallization of opinion on the theory side.  Here is one excerpt:

There are no errors in the formula Piketty uses, and it is actually consistent with the very earliest formulations of the neoclassical growth model, but it is not consistent with the textbook model as it is generally understood by macroeconomists. An important purpose of this note is precisely to relate Piketty’s theory to the textbook theory. Those of you with standard modern training have probably already noticed the difference between Piketty’s equation and the textbook version that we are used to. In the textbook model, the capital-to-income ratio is not s=g but rather s/(g+ δ), where δ is the rate at which capital depreciates. With the textbook formula, growth approaching zero would increase the capital-output ratio but only very marginally; when growth falls all the way to zero, the denominator would not go to zero but instead would go from, say 0.12—with g around 0.02 and δ = 0.1 as reasonable estimates—to 0.1. As it turns out, however, the two formulas are not inconsistent because Piketty defines his variables, such as income, y, not as the gross income (i.e., GDP) that appears in the textbook model but rather as income, i.e., income net of depreciation. Similarly, the saving rate that appears in the second law is not the gross saving rate as in the textbook model but instead what Piketty calls the “net saving rate”, i.e., the ratio of net saving to net income.

Contrary to what Piketty suggests in his book and papers, this distinction between net and gross variables is quite crucial for his interpretation of the second law when the growth rate falls towards zero. This turns out to be a subtle point, because on an economy’s balanced growth path, for any positive growth rate g, one can map any net saving rate into a gross saving rate, and vice versa, without changing the behavior of capital accumulation. The range of net saving rates constructed from gross saving rates, however, shrinks to zero as g goes to zero: at g = 0, the net saving rate has to be zero no matter what the gross rate is, as long as it is less than 100%. Conversely, if a positive net saving rate is maintained as g goes to zero, the gross rate has to be 100%. Thus, at g = 0, either the net rate is 0 or the gross rate is 100%. As a theory of saving, we maintain that the former is fully plausible whereas the latter is all but plausible.

With the upshot coming just a wee bit later:

…Moreover, whether one uses the textbook assumption of a historically plausible 30% saving rate or an optimizing rate, when growth falls drastically—say, from 2% to 1% or even all the way to zero—then the capital-to-income ratio, the centerpiece of Piketty’s analysis of capitalism, does not explode but rather increases only modestly. In conclusion, at least from the perspective of the theory that we are more used to and find more a priori plausible, the second law of capitalism turns out to be neither alarming nor worrisome, and Piketty’s argument that the capital-to-income ratio is poised to skyrocket does not seem well-founded. [emphasis added by TC]

Krusell and Smith really know their stuff on this topic and their arguments to me seem completely correct.

By the way, here is a Chris Giles follow-up post from The FT, very useful.

1 ummm May 28, 2014 at 2:51 pm

at some point will we have to declare his work an elaborate fraud/hoax? He owes everyone an apology for egregious academic malfeasance

2 Nathan W May 28, 2014 at 3:45 pm

He didn’t answer EVERYTHING at once, and also failed to offer perfect answers to questions which the best and brightest have failed to provide good answers to for quite a few decades now.

Therefore, I reject ALL of his methods, data, conclusions, theories, and generally everything in any way related to anything he touches.

Because he’s not PERFECT 🙂

3 Handle May 28, 2014 at 4:20 pm

I expect to read this exact ‘defense’ repeated over and over by all the usual suspects during the next decade as Piketty’s case is refuted bit by bit.

It’s not about perfectly, it’s about whether the case is solid enough to be compelling. It’s also about whether people will defend anything if it lines up with their policy preferences.

Who will be the first to change their minds on whether Piketty’s case has met the appropriate burden or proof to justify radically redistributive policies? “Well, I use to think taxing the rich was a good idea, but now that I see that Piketty was wrong about depreciation, I guess we ought to let them keep their money after all.”

4 Willitts May 29, 2014 at 1:45 am

It also deftly avoids the fact that Picketty bears the burden of proof.

5 Nathan W May 30, 2014 at 12:34 pm

Which part of the analysis requires which methodological improvement?

6 Z May 28, 2014 at 4:28 pm

That was where everyone started. Now some are trying to make a living proving it.

7 Bill May 28, 2014 at 3:31 pm

So, if I am wealthy, and spend more on myself rather than save as much as Piketty assumes, all will be well.

Goin to the Beach.

8 charlie May 28, 2014 at 5:39 pm

Fixed it for you: “So, if I am wealthy, and spend more on myself rather than save as much as Piketty [implausibly] assumes, [then his pessimistic conclusions do not follow].”

9 Bill May 28, 2014 at 6:14 pm

Charlie, So, do ya think that tax cuts for the top 1% are on the table once again. Maybe some more dividend tax relief (15% isn’t low enough) and we can always lower cap gains too.

10 Joe Teicher May 28, 2014 at 6:36 pm

The dividend tax rate is 23.8% not 15% unless you are poor.

11 Bill May 28, 2014 at 6:46 pm

Joe, You have to be in the 39.6 percent bracket. So everything else must mean you are poor.

12 Bill May 28, 2014 at 6:47 pm

Joe, That referred to the cap gain rate.

13 Alexei Sadeski May 28, 2014 at 9:29 pm

+10% state.

14 Bill May 28, 2014 at 9:56 pm

Alex, most states have rates topping out at 4-6%

15 jpe May 28, 2014 at 11:27 pm

It’s actually 25% with the Pease limitations.

16 Bill May 29, 2014 at 6:49 am

jpe, Pease is a small limitation on certain itemized deductions on AGI greater than \$300k; it is not related to cap gains or the dividend rule.

17 dan1111 May 29, 2014 at 4:18 am

Problems that occur when you assume an argument about the specifics of a proposal to be merely a proxy for a larger ideological battle:

1) It is not necessarily true. Some arguments really are about specifics, not just an attempt to gouge the other side.

2) It leads you to make tiresome straw man arguments.

3) It really, really turns off anyone who disagrees with you ideologically–i.e. the only people who are actually worth debating, since you don’t need to convince anyone on your own side.

18 Nathan W May 28, 2014 at 3:46 pm

I find it unlikely that savings rates will NOT vary with the level of capital accumulation in the longest run, but ceterus parabus this would flow from the assumption that returns to capital are insensitive to the level of capital accumulation.

I’m not sure I can quite grasp the relevance for the conclusions as a whole. Perhaps if returns to capital decline with level of capital accumulation, there will be lower savings, but then wouldn’t that just be a classic Solow situation with some theoretical equilibrium?

Going the next step to tie in that assumption to its expected effects on wealth or inequality is far from obvious to me.

I haven’t read the book yet .. maybe that would clarify some things.

19 Andrew Stegmaier May 28, 2014 at 4:16 pm

I wanted to offer a friendly correction to your post so that future readers don’t get confused:

Your first block of quoted text includes the footnotes in the middle of it. So the text “As it turns out, however, the two formulas are not inconsistent because…” should be followed by “…Piketty deﬁnes his variables, such as income…” not “…Piketty also argues that the return to capital…

By coincidence, the error happens to make syntactic sense, which caused me to read and re-read it before I realized what was happening.

20 Tyler Cowen May 28, 2014 at 4:42 pm

Thanks much, fixed…

21 Carl May 30, 2014 at 1:49 pm

Another friendly correction. I think the capital-to-income ratio by Piketty should read s/g instead of s=g.

22 Willitts May 29, 2014 at 1:47 am

Im sure Picketty made as much syntactic sense.

23 Jan May 28, 2014 at 5:25 pm

Is there anyone whose interpretation of Piketty does not totally align with their previous position on inequality?

24 F. Lynx Pardinus May 28, 2014 at 5:28 pm

No, especially the guy upthread who is confidently and unironically predicting how Piketty will be viewed in a decade.

25 Jan May 28, 2014 at 5:57 pm

And I include myself here, though I don’t have the econ chops to understand some of the more technical arguments being exchanged.

26 carlospln May 28, 2014 at 7:14 pm

You don’t need to.

There is no such thing as ‘econ chops’.

“Krusell and Smith really know their stuff on this topic and their arguments to me seem completely correct”

‘completely correct’?

‘Irony stands mute’ Jozef Stalin

27 C May 28, 2014 at 7:57 pm

No. And that’s because 95% of those who comment on this book haven’t read it. They’ve read a review from a person whose political outlook they share.

If you’d like another example of this, check out all of our climate change deniers. The deaf lead the blind.

28 Chip May 28, 2014 at 10:10 pm

Surveys show skeptics of human-led climate change are more knowledgeable of the issue than proponents.

That you use the denier construct – as if skeptics deny that climate changes – suggests you could brush up on the subject as well.

29 Jan May 28, 2014 at 10:23 pm

I am not surprised by this. I would expect that skeptics of conventional wisdom in general have a bit better understanding of basic science. However, we shouldn’t take that to mean that skeptics are more often correct, especially when a large majority of experts in a particular field have reached general consensus on an issue.

30 Chip May 28, 2014 at 11:29 pm

And the consensus is that CO2 is a greenhouse gas and that it’s production by humans contributes to warming.

After that, the consensus breaks down. This is often lost on people who trumpet the 97% or other hazy references to authority.

The consensus is also that we do not know the human signal in warming, and that uncertainty has increased since the failure to predict the recent pause.

The consensus fractures further when we consider the scale of warming, it’s effects, the cost of mitigation etc.

The climate is much more complex than economics, and we see on this blog daily how much uncertainty plagues the dismal science.

So why the unblinking, table-pounding confidence on climate?

31 Jan May 29, 2014 at 8:23 am

Nah, consensus is pretty clear that awg is happening and that we could do something to slow it if we wanted to.

32 Jan May 29, 2014 at 8:24 am

AGW

33 C May 29, 2014 at 6:10 am

Citation needed, and which proponents? All opinions aren’t equal.

34 Chip May 29, 2014 at 7:08 am

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1871503&http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1871503

That’s from 2011. People who still today insist man is the primary cause of temperature change are going to be even less informed in light of the continued divergence of empirical data from predictive models.

35 derek May 28, 2014 at 10:41 pm

Some guy writes a book, and you assume that what he says is true?

36 C May 29, 2014 at 6:34 am

Some guy writes a review, and you assume that what he says is true?

37 Urstoff May 28, 2014 at 9:43 pm

Of course not because most of us haven’t read it, and those who have don’t have the prior knowledge required to be able to fully evaluate it.

38 mpowell May 29, 2014 at 2:31 pm

I would consider myself to be an example. I probably line up somewhere around Delong and Krugman politically and I think this book is an unfortunate distraction. Frankly, I’m not sure why those two guys have been so enthusiastic about it. The data may be valid and therefore useful for something, but it certainly doesn’t support any of the general conclusions reached by the book. Either regarding policy or the nature of capitalism. Those arguments are simply wrong.

39 dearieme May 28, 2014 at 5:50 pm

“The economics profession has been considerably more forgiving than the FT of the constructed data in Prof Piketty’s datasets.” And some people still claim that Economics is a Science. Mind you, plenty of Climate Scientists are out to prove them right.

40 dearieme May 28, 2014 at 6:01 pm

Aha! ” Prof Piketty’s choice to migrate from data based on estate tax records to data from the Survey of Consumer Finances and showed this in the bottom line in the following chart.” He changed his thermometers. Climate Science! Climate Science! Climate Science!

41 Mike V May 28, 2014 at 5:51 pm

Now I know why Piketty didn’t like MIT and all those geeky theorem-provers. It just gets in the way of being a rock star economist.

42 rayward May 28, 2014 at 6:46 pm

We’ve moved back to the second phase, highly technical criticisms, which is probably better than the third phase, focused on the data. The second phase has the distinct advantage of self-promotion, not to mention obfuscation, as the esoterica dulls everyone but the faithful.

43 Turpentine May 28, 2014 at 8:46 pm

That’s it, I’m done. The way this whole debate has evolved is ridiculous. Krugman and co. are having a field day screaming “I told you so!” two seconds after having finished reading the book. TC, Mankiw and co. appear to spend their every waking hour linking to every single person criticizing the book, having had a particularly orgasmic field day the moment the FT piece came online.

Seriously. To start with, the Piketty plots all these guys are arguing over never were that much of a smoking gun in the first place. They show interesting trends, but I never had any doubt that basic sampling errors around them must be pretty big to start with. That’s fine. So how about Krugman, Cowen and Mankiw’s genius minds start working on trying to make these confidence intervals narrower instead of making mood-affiliation fools of themselves? That’s how progress is made. That’s how we move forward. I would expect Tyler to rise above such pettiness.

Actually, Cowen’s attitude in all this has to be particularly laughable, given how proud he is of his 2001 posts on “mood affiliation”. This whole Piketty episode has had the unfortunate side effect of making me much more doubtful of the wisdom of Tyler, and raising my cynicism towards econ blogs. It’s sad.

44 carlospln May 28, 2014 at 9:56 pm

1) Economics is not a science

2) The Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne is not a ‘Nobel Prize’

3) Cowen is a useful idiot running interference for Ray Lopez and the 1%.

45 charlie May 28, 2014 at 10:05 pm

Tyler’s position on this book is not that hard to understand and not that embarrassing.

As it was coming out, Tyler promoted the book as a provocative leftist take on the topic that was likely to be pretty obscure and overlooked outside of academic specialists; he was, I think, agnostic about its claims. Then, it became the leading coffee-table book of educated people in coastal US cities and was widely heralded as “proving” with “data” that the knee-jerk class warfare of those Occupy fools was right all along. At that point, the book needed to seriously be knocked down a peg.

46 mpowell May 29, 2014 at 2:36 pm

As someone whose political views are substantially to the left of Cowen’s (though not leftist by any means) I would agree with this. The reaction to this book has gotten completely out of hand, but I would say the left is to blame in this case much more than the right. And that’s a really weird thing for me to write.

47 derek May 28, 2014 at 10:46 pm

The best criticism of Piketty that I’ve heard is from a Canadian economist who ran for the NDP. He says that inequality is a given in a growing and thriving economy, and if you have a growing and thriving economy you can afford all the nice things that the left wants. He disagrees with Piketty’s conclusions saying that over 50% tax is confiscatory and will constrain growth.

So how does that comment fit into your nice tidy little slots about what people think?

I tend to agree with him.

48 Ben Hughes May 29, 2014 at 12:39 am

From the comments I can see this entire discussion has devolved into all accusing each other of mood affiliation and confirmation bias.

It’s getting in the way. The fallacer’s fallacy is real. Can we focus on the actual arguments and data, as they stand or fall on their own, and not the motives of those making arguments?

49 Willitts May 29, 2014 at 2:05 am

True, but not surprising. Given that models are abstractions and data are drawn from the unobserved population, there is seldom ironclad proof of anything, especially in a social science. The differing viewpoints reveal where one’s values focus.

I can confidently say that most people care about the outcome of an appellate decision rather than how it was reached. To attorneys, the latter is far more important. However, many believe justices decide what they want and work backwards. In that case, the reasoning is less persuasive albeit still important to the law. Few people are moved to change their minds about the outcome by virtue of the dicta.

50 dan1111 May 29, 2014 at 4:47 am

@Ben, I agree to a certain extent (and I admit that I am guilty of the same). But doesn’t your comment make you the worst offender by your own definition? 🙂

51 Willitts May 28, 2014 at 11:37 pm

How extraordinary is it to have such an undertaking avoid the peer review process by way of a book?

Did Piketty base his book on prior peer reviewed articles?

52 Colin May 29, 2014 at 11:08 am

Not all that strange (see, for example, Shiller), and yes, for the most part.

53 Ray Lopez May 28, 2014 at 11:53 pm

TC gets credit for highlighting what Piketty is saying, that if growth is near zero, then you get bad things like the rich getting richer sometime in the future due to both his historical data and his ‘law’.

But this passage is less strong than it could be ( my emphasis in bold ):

” Piketty’s argument that the capital-to-income ratio is poised to skyrocket does not seem well-founded.

Krusell and Smith really know their stuff on this topic and their arguments to me seem completely correct. ”

I see mood-affiliation. Authors are saying ‘does not seem ‘, which is equivocal, while TC is saying, unequivocally, ‘completely correct’.

54 Yancey Ward May 29, 2014 at 1:18 am

Jeez, in all your highlighting you might have spotted the second instance of “seem/s”.

55 Willitts May 29, 2014 at 1:55 am

It was, after all, on a separate line thanks to the gratuitous use of a three syllable word.

In Ray’s defense, “seems completely correct” is a rather odd phrase. If by “seems” he means “appears to be,” then I suppose it is OK. But “completely” seems out of place with “seems.”

56 dan1111 May 29, 2014 at 2:08 am

In Ray’s non-defense, he is claiming that Tyler is exaggerating the original argument, when all Tyler did was quote it and say “I agree”. Whether “seems” is what it seems seems to be completely irrelevant, seemingly.

57 dan1111 May 29, 2014 at 2:04 am

This is hilarious! Tyler said “seem”, too. And even if he hadn’t, a statement of full agreement with someone else’s argument by definition cannot be stronger than the original argument.

Person 1: There is a 50% chance of rain today.

Person 2: Person 1 is completely correct.

By your argument, person 2 believes there is a 100% chance of rain.

58 Brian Donohue May 29, 2014 at 9:37 pm

It’s been a rough week for Ray. Stumbles on a couple threads, and having his cover blown here as TC’s true puppet master (take that, Kochs!)

Perhaps his smoke show Filipina can massage his bulging Adonis physique and rub his throbbing 140 IQ temples while they frolic atop his bags of 1%er cash and make it all better.

59 wiki May 29, 2014 at 4:07 am

Here’s my summary of the whole Piketty story:

Distinguished economist with good work on detailing cross national inequality in recent decades based on tax data writes a book on the subject. He goes beyond his caveats to draw parallels between recent changes and historical inequality. He invokes macroeconomic arguments about the rate of growth of r vs g. He shows that it was bad in the 19th century and implies that this was related to 19th century inequality. It says it’s bad now and he thinks it’s likely to get to 19th century levels. He implies that we will have 19th century badness tied to inequality. He propose various remedies which are mostly unenforceable.

But it seems most of his historical leaps are suspect. He ignored the existing literature by other economists on 19th century inequality. He does literally nothing to link 19th century inequality to r and g. Moreover, the 19th century was an unusually good time for workers, the rise of the welfare state, progressive reforms, consumption, innovation, and overall prosperity. His theories about r and g are speculative and critiqued even by supporters of his such as liberal economist Solow. Other theoretical analyses are harsher and go poorly for him. Most of the other conjectures he makes not directly related to his empirical work are equally suspect. There is even less of a link between his theoretical analysis — which now seems weak or flat out wrong — and his prescriptions. And there might even more caveats than even the journal suspected about his core data. The most uncontroversial parts about inequality and its increase in recent decades have been known independent of Piketty’s work.

Other than that, How did you like the play Mrs. Lincoln?

60 Robert Ryan May 31, 2014 at 4:48 am

The first half of the paper is correct but astoundingly trivial ……and the second is pure gibberish becasue the author claims stuff that his own graphs/data dont show (clearly its all just placerholder filler right now….for example the author says there is no relationship between gross and net savings but the vague and uncited Figure 4 shows almost a perfect linear relationship between the two!!!…. and there is no data backing any iof the author’s claims yet)…………………….. This is a rough draft papr so I am critiquing it as an academic reviewer would…… Sure, we can all agree that Picketty needs some sort of baseline number to keep savings from being 100% if growth is zero. But what should it be? And Why? A real simple way to also argue this is that savings of 100% means people all die…and savings of 99% means that MOST people die…. so we could just put in a constant to keep some # of people alive. Let us start with the assumption of a marginal propensity to consume (Keynesian) to ensure there is a basic floor to consumption and a associated ceiling to savings. Easy. Not rocket science, and not enough of an observation worthy of a top journal publication. If they dont want to use a consumption variable, fine, use a depreciation variable that represents capital consumption (that is, the turnover of inputs to capital that require human management and prevent everyone from being incomeless…but in an accounting identity perspective, that number is probably roughly equal to the MPC, so let’s use the clearer demand side llustration for the moment). If the authors wanted a real journal pub ouf of this paper they need to show WHAT that floor is or to show importantly WHY. A paper complainig about Picketty’s lack fo seleciton of a consumption floor is NOT an academic paper, it must go beyond complaining. As for why: A minimum level of consumption that depends on savers’ personal wealth distribution matters a lot. the authors should start there. How walth is distributed would add another equation, but it would procide the consumption floor that, in turn, would provide a limit on Picketty’s dilemma without violating hte spirit of Picketty’s argument. This added variable takes away the absurdity of outrageously high savings! An equation for extreme wealth distribution shows that concentrated wealth– super rich savers who dominate the economy– have a tiny need for consumption; if the rich dont care if others die in social darwinistic manner, then savings rates can be absurdly high, like >99% in the short run. Picketty, in a sense, ASSUMES that wealth is highly concentrated already, and that may be fair but is also unfortunately TOUTOLOGICAL. Easily fixed. That wealth is highly concentrated and savings is explosive in the short run is entirely plausible if we permit wealth distribution variables to affect the denominator s/(g + cf), were cf= consumption floor according to a wealth distribution). If all the wealth is in a few hands, then we could have an economic genocide if growth approaches zero, and the supposedly “absurd” 100% savings rate means capitalism and its adherents all died, or quit that particular economic regme by opting out of that economy. If all the wealth is evenly distributed, however, this should indeeed be absurd, and we should instead expect savings to be constant REGARDLESS of growth as it is in the neoclassical assumptions. Thus, evenly distributed savings creates a long run constant savings rate regardless of growth and is a HIDDEN SOCIALIST assumption of neoclassical theory. For mainstream neoclassical theory to make any sense, we must assume distributed savings, wealth, capital, etc.!!! Sheesh. This stuff is rather obvious, isnt it???

61 Robert Ryan May 31, 2014 at 4:57 am

Aside from that, wiki’s comment here is silly– it is something someone says when they have nothing useful to add to the conversation themselves. Picketty’s work is not nearly THAT flawed. IN fact his work is well cited and jives with prior literature, so wiki is entirely wrong on tha tpart. The only major problems of Picketty’s work is that he repeats grossly simplifying assumptions in order to make his models elegant and his points easy to follow. Of course it is to caorse grained. As I just demonstrated, it can be easily amended with more fine grained analysis, but what I stated is just scratching the surface. The questuon is wether or not anyone can FOLLOW it anyhow. Do people really want to fix the arugment with good faith effort to carefully prune the details, or are they just looking to reaffirm their own gross simplifications? 98% of those out there who follow and comment on economics really cannot keep up with analyses so coarse-graied and so they prefer to throw mud at each other’s simplistic generalities instead of contributing meaningfully to each other’s work. This is part of the reason why most academcis stay away from the public: we tend to carefully prune each other’s work with good faith in an attempt to collaboratively reign in a better explanation of reality, while the public throws mud around viciously for no good purpose. Public intellectuals and academic (usually small school or niche subarea of research) radical mouthpieces give the false impresion that these debates are really intensely polar. The authors of this paper are merely correcting a little minor, even trivial FLAW, not overturning his whole WORK. Problem easily solved.

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63 Unlearningecon June 2, 2014 at 7:12 am

“Is Piketty’s “Second Law of Capitalism” fundamental?”

“First, it is important to be clear that the second fundamental law of capitalism, β = s / g, is applicable only if certain crucial assumptions are satisfied. First, this is an asymptotic law, meaning that it is valid only in the long run: if a country saves a proportion s of its income indefinitely, and if the rate of growth of its national income is g permanently, then its capital/income ratio will tend closer and closer to β = s / g and stabilize at that level. This won’t happen in a day, however: if a country saves a proportion s of its income for only a few years, it will not be enough to achieve a capital/income ratio of β = s / g.”

Hmmm…

64 Robert Ryan June 3, 2014 at 4:06 am

Exactly.

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