Kernel density estimate takes values larger than 1

During the research, I used a kernel density estimation and draw the graph.

When I first saw this, I was totally confused. Can density be taking value greater than 1? Isn't density probability? Probability shouldn't take value greater than 1!

Well, my thoughts turned out to be wrong. It's time for reviewing basic statistics courses.

PDF (Probability Density Function) can take value greater than 1. CDF (Cumulative Distribution Function) cannot exceed 1 but PDF can. For example, consider normal density function below.

If standard deviation is very small, then for x=mu, PDF takes very large value.
Even if the height under some value is greater than 1, it does not mean that the probability of having that value is greater than 1. <- This is the source of confusion.

Another thing to note is that PMD (Probability Mass function) for discrete random variable cannot take value greater than 1.

Importance of "Relative" Terms rather than "Absolute" Terms

Economics has long been considered only 'absolute' term. In most standard analysis, utility is determined by the level of own consumption and job matching is occurred according to the level of absolute wage. However, in reality, there are many occasions where relative terms play important role as Kahneman and Tversky pointed out. Depending on the reference point, same absolute term can be interpreted differently. If I have so many wealthy friends, then my income as a graduate student is tiny. However, if most of my friends are still in university, my income as graduate student is considerable. That is, if reference point is decided by people around me then important thing is where I stand, not the amount I have.

I went to a talk by Mounir Karadja today and he gave such an interesting story why Sweden can sustain their social welfare system. According to a 2012 OECD report, the country had the second-highest public social spending as a percentage of its GDP after France, which means that this country has one of the most highly developed welfare states. According to his, one of the reasons is that many of people underestimate their relative income. (Relative income here means where you stand in the country's income distribution.) If they are informed their actual relative position and it was higher than their guess, they tend to support conservative party more. That is, Swedish people's guess about relative income could play a role in their political preferences of supporting welfare systems.

Another interesting piece is by Bertrand, Kastenica, and Pan. They discuss the role of social norm "Husband should earn more than wife" on many economic variables such as marriage, labor market participation, divorce rate and satisfaction in the marriage. Here as well what matters is 'relative' income in the household. Relative income here is defined by men's earning divided by women's earning plus men's earning. If this relative income is less than half, then social norm is violated. In this case, marriage rate is decreased, women's labor market participation is decreased (woman don't want to threat her husband), divorce rate is increased and satisfaction in marriage is lower. I enjoyed reading this paper very much not only because I found it interesting to incorporate the role of social norm in economics but also because the way they thought about income distribution in households (i.e.relative income) was interesting.

In sum, I think relative terms are very important in many economic decisions. Below are some references.

"How Elastic are Preferences for Redistribution? Evidence from Randomized Survey Experiments" March 2013 (I. Kuziemko, M. Norton, S. Stantcheva and E. Saez), Revise and Resubmit, American Economic Review (also released as NBER working paper 18865)

"Inequality at Work: The Effect of Peer Salaries on Job Satisfaction." (David Card, Alexandre Mas, Enrico Moretti, and Emmanuel Saez) NBER Working Paper, No. 16396, September 2010.

"Relative income shocks, beliefs and political preferences: Evidence from a randomized survey experiment" (Mounir Karadja et al.)

Cinderella

Cinderella by Christopher Wheeldon was awesome. I believe that people come to the theater to experience a "fantasy". This revived classical fairy tail was enough to awake dormant fantasy. It was the most impressive full act ballet recently. Appropriate sense of humor and beautiful stage were also great. 

Maria Kochetkova... She is so lovely. I was so happy to see her again... I still vividly remember when I first saw her in San Francisco 5 years ago. Small but strong, soft but has a marked character. Beautiful...<3

Career concern 3 (Dasgupta and Prat 2006)

This is another piece of career concern which has an interesting story. :)
Again, like Morris, this paper focuses on a bad result of career concern. (Remember? Holmstrom paper was more about positive side of career concern.)

In financial economics, there is a theorem called "no trade theorem". (what a fancy name!) Roughly speaking, rational agents will not trade with each other on the basis of differences of information alone. If someone tries to sell a stock on the market, then it signals that that seller has a bad information about that asset. Otherwise, why would he sell? Under this rational expectation, there is no trade occuring. The name of the theorem and logic based on information are very graceful, but this does not match with reality. This theorem relies on lots of assumptions so it's like a well controlled lab like experiment which is useful as a benchmark but not a reflection of reality we face. Our question is then: Why we see huge volume of trade in the stock market? What drives people to trade that much?

Many economists tried to answer this question. One explanation is the existence of "noise traders". They trade not because they have some information but because they are in liquidity needs or personal needs such as hedging. But it's hard explain that huge volume of trade by only considering noise traders. Dasgupta and Prat(2006) maintains that career concern of fund managers can help explain huge trade volume. This explanation makes sense because the proportion of institutional investors is getting bigger and bigger. In 2002, 49.8% of ourstanding corporate equity was held by institutional investors. Fund managers may not have information. But if they don't trade because they don't have information, then it becomes an evidence of lack of ability. Just to show that they are not bad, they trade. Trading without information, they lose in expectation in that period, but they can avoid a situation where they are fired because of their incompetency proven by non trade. This drives excessive trade by fund managers. 

In this paper's 2-period model, payment from the investor to the manager is linear function of return(ax+b if x is return). In other words, there is a fixed fee b, and proportional rate a. If a and b are not too big, investors delegate investment to fund managers. Here good fund managers perfectly observe true state of the world and act efficiently, so bad fund manager's behavior becomes a problem. Bad fund managers don't want to be fired in the next period so have an incentive to take a gamble in the first period. (They don't gamble in second period because there is no third period.) If their gamble succeeds (say, 50% probability), they can be retained by the investor. It might fail but at least this is better than do nothing and got fired, if cost of gamble(a) is not too expensive. So if a is not too big, then bad managers take a risk which has negative expected return not to show that they lack information. Thus, parameters a and b are important to predict fund manager's behavior. For example, if b=0 then bad manager has no incentive to trade in the first period by taking dangerous negative expected return gamble.

In addtion to that, signal structure, proportion of good fund managers, existence of contingent contracts matter in predicting the fund manager's behavior. In this paper, there exists only two periods so how career concerns affect fund manager's behavior in the long run (dynamic settings) is an interesting extension. In dynamic setting, it is likely that their true ability is revealed at some point as in Holmstrom. So I guessed if we are lucky, we might have efficient information transmission in the dynamic model. But it seems like we are not lucky enough. Dasgupta and Prat(2008, JET) studied career concern in dynamic setting finance market. I didn't read whole paper, but the conclusion of the paper is that financial market cannot be informationally efficient even in infinite horizon if there is reputational concern.

Reference:
"Financial Equilibrium with Career Concerns" (Amil Dasgupta and Andrea Prat), Theoretical Economics, forthcoming, Volume 1, Issue 1, March 2006.
"Information aggregation in financial markets with career concerns," (Amil Dasgupta and Andrea Prat), Journal of Economic Theory, 143(1): 83-113, November 2008.

What makes Europe and China different?: Finding answers from different patriarchy (Wolf 05)

*It was written as a report in Topics in Gender and Economics. 


The Hajnal line, which named after a Hungarian-English scholar John Hajnal, is a border that links Saint Petersburg, Russia and Trieste, Italy. West and east of this line show two very different marriage patterns: "European pattern" and "non-European pattern". Former is characterized by a higher age at marriage and a high proportion of people who never marry. Latter is completely opposite. It is characterized by a younger age at marriage and a low probability of not getting married. Moreover, this line not only divides marital patterns but also divides demographic and kinship regimes. But then why these two different parts show different types of marriage, demographic and kindship regimes? Wolf tries to find the answer to this question from the form of patriarchy.

As we have two parts of the world in terms of marriage and other demographic regimes, we have two according forms of patriarchy to explain the difference. One is ``property patriarchy", which is appropriate to explain Europe and the other is ``state patriarchy", which is for China (non-European). He assumes that parents want to control their children because children are useful resources. Under this assumption, patriarchy is a means of controlling their children. Under property patriarchy, parents try to control their children through their property. Under state patriarchy, parents use authority of state to control their children. In European countries, especially before absolutist state, the only way the parents could control their children was to give gifts or threatened disinheritance. This means that if parents didn't have enough fortune, they had no way to control their children. Parental authority was not high in Europe. Since parents didn't have control over children, it was hard for them to use their children as a labor force. This being the case, parents had no reason to invest in their children. Thus, marriage was formed only after individual has enough material basis to maintain family and is based on individual's consent rather than parent's arrangements. This is why the average marriage age was pretty high. Parents, lacking control over children, hired servants as a labor force. Since western countries had better contract enforcement, it was easier for the parents to hire people than to use their children as a labor force. Through this mechanism, European countries west of Hajnal line showed distinctive characteristics: late marriage, frequent celibacy, low fertility, low mortality, one married couple per household, a close link between marriage and entry into household headship, and servanthood as a common stage in the life cycle. 

In contrast to property patriarchy, state patriarchy is authorized by the state. In China, filial piety was emphasized and if children did not respect their parents state punished them harshly. Relying on this state authority, parents could gain strong control over their children. Thus, they used their children as a labor force. In societies with advanced agriculture, people needed plenty of labor force. Since every single family member acted as a labor, there was an incentive for parents to make their children marry early and give birth to children as many as possible. Parents preferred boy to girl because boy usually could work more and better. Statistics in China show that brotherless women were more likely to have children before marriage. This is because parents without boy made their daughter prostitute because in that way, they could make money. In sum, in countries where authorized state enabled parents exercise control over their children, parents tried to make profit through their children. This mechanism characterizes the eastern part of Hajnal line by early and nearly universal marriage, high fertility, high mortality, complex households containing two or more married couples, a long delay between marriage and household headship, and a tendency for people to employ their children at home rather than to let them out as servants.

Does Hajnal line still exist today? I would say no. Many eastern countries are westernized after 19th century as they adopted western legal system and mode of thought. Moreover, those countries are industrialized, so the need of hands in household is reduced. As a result, parents don't have such a control over children even in China and children are not valuable as before. Thus marriage pattern these days seems to be converging to what have been called "European pattern". In other words, convergence of patriarchy due to the change in society results in convergence of marriage and demographic regimes. Late marriage, low fertility, low mortality, and nuclear family are now common in both west and east of Hajnal line. 

Reference:

Europe and China: Two kinds of patriarchy by Arthur P. Wolf 

(This is a chapter of the book "Marriage and the family in Eurasia: Perspectives on the Hajnal hypothesis.)

What Men Want: Then and Now (New York Times)

 

Interesting...

 

What about women? What women want?

I guess for women, they want pretty same thing then and now.

 

http://www.nytimes.com/imagepages/2012/02/12/opinion/sunday/12-coontz-gfx.html

How to View Marriage in the Eyes of Economics: Comparing Becker (1973) and Edlund (2013)

*It was written as a report in Topics in Gender and Economics.


Marriage, a socially or ritually recognized union or legal contract between spouses, is a universal phenomenon across different cultures even though the forms are various (monogamy, polygamy, same-sex marriage, etc.) Gary Becker (University of Chicago) first started to analyze marriage, which seems to have no connection with economics at first site, in the framework of economics and kicked off family economics. Like other economics model, agents maximize utility and decide whether to marry or not by weighing costs and benefits of marriage. Marriage market, which is filled with many single men and women, are in equilibrium state. With this model, we can induce many implications related to marriage such as how sorting occurs and how women decides labor force participation, etc. However, since this was the very first economics paper which analyzed marriage, there were also some unsatisfactory predictions. For example, Becker predicted negative sorting in terms of wages but empirically, we observe positive sorting for wages (i.e. wealthy man marries wealthy woman). Edlund (2013) also adopts economic model to analyze marriage but she viewed marriage in different aspect. She interpreted marriage as men's means of obtaining children. This view helps to explain the phenomenon which Becker couldn't explain clearly. Thus it is worth comparing those two influential papers in detail.

 

First of all, the definitions of marriage in both papers are different. Becker defined marriage as sharing the same household. But Edlund defined marriage as a contract over parental rights where women sell and men buy. This view of Edlund stems from women's absolute advantage in bearing and giving birth to a child. This ability is exclusive to women. Compared to Edlund, Becker emphasized comparative advantage of men and women. With marriage, men and women specilize in labor market or household chores according to their comparative advantages as if countries concentrate on production with comparative advantage and trade. Becker thought gains of marriage are mainly from this specializing. But Becker also recognized that children is very important factor for marriage. He thought that children are one of the important gains from marriage so reflected this in his model by assuming that time of men and women are not perfect substitute. In terms of having a child, men and women cannot be perfect substitutes and complements are rather appropriate. In sum, Edlund emphasized innate differences of men and women and Becker put emphasis on comparative differences of men and women.

Those papers importantly address the issue of sorting in marriage market. How men and women are matched according to their many characteristics is interesting and important question. According to Becker, postive sorting occurs for most characteristics such as physical capital, education, intelligence, height, race, etc. But for earnings and traits of men and women that are close substitues, negative sorting occurs. The reason for negative sorting is also related to specilization. By matching man who earns well and woman who earns not well or man who earns not well and woman who earns well, we can maximize total output. This is because if someone with low earning ability specializes in household production, forgone earning is minimized. The logic is very clear cut. But the problem is that this does not match with the data. Edlund's model overcomes this pitfall of Becker model. As mentioned above, in her model, men pay for marriage because they obtain parental rights through marriage. In this case, negative sorting is unlikely. Men tend to marry down then the remainings are low quality men and high quality women. In this case, it is unlikely for them to be matched. High quality man did not marry high quality woman because she was too expensive for him. This being the case, it is likely that low quality cannot pay that expensive price neither. If low quality man's outside option to be single is too bad, then he might be willing to pay for that expensive price. So negative sorting can happen but is very unlikely.

Another advantage of Edlund's model is that it can explain why non marital cohabitation occurs. Since Becker's model defined marriage as sharing same household, there was actually no difference between legal marriage and non marital cohabitation. I guess at that time, non marital cohabitation was less common than these days, so he overlooked the difference of marriage and non marital cohabitation. In Edlund's model, non marital cohabitation also gives men rights for children, but this rights are more restricted than the rights coming from marriage. Non marital cohabitation is likely to involve low quality men because they pay less for non marital cohabitation. Since we have rising non marital cohabitation and children born outside of marriage, explaining why this happens and how this affects overall marriage market is important question. One of merits of Edlund's model is that she incorporated non marital cohabitation in the model and opened the door for analyzing young people's matching trend.