A little game theory: does crime pay? (II)

I want to look at the effect of prison terms on the crime problem. The conclusion is that jail terms don't significantly alter criminal behaviour. Here's my model. In view of the conclusion, maybe we should look really critically at the assumptions.

Suppose that a criminal commits, on average, j crimes per unit time (or carries
out j "jobs"). Each crime results in a payoff of S. If caught (with
probability p), the criminal is fined F and spends a time T in jail. If the criminal would lose revenue from legal activities for being in jail, we can take that opportunity cost into account by adding the lost revenue to the fine. So, what is
the optimal crime rate j?


If the criminal commits j crimes per unit time and each time he can be caught
with probability p, this means that he will be caught at a crime jp times
per unit time. The criminal can expect, on average, to be "active" for a time
1/jp before he gets caught. Each time he gets caught, he spends a time T
in jail.


While the criminal is active, he obtains a payoff of jS per unit time. That means he can expect to win an average of S/p for each period that he is active. Note
that this is independent of the crime rate. When he gets caught, he has to
pay a fine F. From the time the criminal starts his activity to the time he
becomes active again after coming out of jail, a time T+1/jp passes.
This means that the expected payoff per unit time is S/p-F
divided by T+1/jp. This is


π=(S-pF)/(pT+1/j)

Conclusions


From the form of the function π it follows that:

• Whether or not crime pays still only depends on S-pF, except that
  now the "fine" includes the (legal) opportunity cost of being in jail. Hovewer, if someone is accruing debt throught their legal acivities, this just means that jail times only make crime more, not less profitable for him. Remember how in the table-top game "monopoly" you can still buy and sell property and collect rent while you're in jail?
  
• when crime pays, the optimal crime rate is "as high as physically possible",
  independently of intensity of law enforcement or legth of jail terms.
  

These conclusions are not unreasonable, but now the question becomes: can someone come up with a model of crime and punishment where the crime rate depends continuously on things like law enforcement pressure, fines or jail times, or is crime essentially a yes-no problem?

What did you expect?

This snippet from Cadena Ser shows just how blatantly illegal the PP's subsidies are:

Asked about the possibility that Spain may appeal or present some kind of defence against these investigations [by the European Commission of illegal subsidies], [Minister of Economy and former European Commissioner] Solbes asked the journalist "on what basis?" (would such a defence be made), making it abundantly clear how hard it is to stop these sanctions, which will include returning the illegal aid and putting Spain's shipyards in an even worse situation than they are currently going through.

Sinking shipyards

The Spanish public shipbuilding company Izar is the result of the 2001 merger of two public companies, the military shipbuilder Bazán, nationalized in 1947, and the civilian shipbuilder AESA, created in 1969 and fully nationalized in 1979. According to the CNT,

On march 14, 2001, the civilian and military shipyards are merged [...]. The reasons were in principle obvious: bankruptcy of AESA and an attempt to finance the civilian shipyards through the military ones.

Spanish shipbuilding had already been reorganized in the 1980's, under the Socialist government of Felipe González, resulting in the loss of 14,000 jobs in the industry.
Labour conflict in the southern shipyards of Sevilla and Puerto Real started again in december 2003. According to a communist newletter, the root of the conflict is the lack of any shipbuilding contracts for 2005, and the decision by the Spanish government not to subsidize the existing contracts (existing EU regulations allow 6% of the contracted price to be subsidized). 4,000 jobs were lost in 2001 in "auxilliary industries", and 11,000 are now in jeopardy due to the lack of contracts.
It turns out that some of the shipyards formerly owned by Bazán, such as Izar-Fene, are barred by law from bidding for civilian contracts. In the case of Izar-Fene, this is a result of some European regulation dating back to 1984, two years before Spain even joined the EU! In November of 2003, the minister of Economy (and now IMF managing director) advocated the creation of a European shipbuilding consortium as the solution to the ills of the Spanish public shipyards. (source)
My conclusion is that the situation in the shipbuilding industry is no different than in mining, and that everything (monopoly, nationalization and subsidies) has been tried already, and both the successive governments and the unions have displayed their characteristic lack of imagination over the past 25 years of industrial reform. There is at least one prosperous private shipbuilder, though, Unión Naval. Indeed, the European Commission reports "complaints" from other Spanish shipyards as part of the reason for their investigation of Izar's subsidies.

¡Señor, qué país!

It turns out that, between 1997 and 2001, the Spanish government of the PP also illegally subsidized the shipbuilder Izar with €1,500 million which will now have to be returned to the government. Up to 20,000 jobs are now in jeopardy. In Andalucía, for instance, Izar employs 3,400 people and, indirectly, generates 14,000 additional jobs. Here's the European commission's May 12 press release:

Today the European Commission decided that aid provided to the public Spanish shipyards is not in line with EC rules on State aid to shipbuilding. The Commission has established that State holding company Sociedad Estatal de Participaciones Industriales (SEPI), in 1999 and 2000, granted aid worth €500 million to the civil public shipyards that are today all owned by IZAR. The aid took the form of a capital injection, loans and a purchase price above market value. As the loans amounting to € 192.1 million to SEPI were paid back, the sum to be reimbursed will amount to € 308.3 million, plus interest. The Commission concludes that the above amount constitutes further state aid which, after the approval of a final restructuring package, can no longer be approved under the EU shipbuilding aid rules.

Here are the EU's rules for state aid. Since 1997, it is not allowed to subsidize shipbuilding in the EU but, in order to "mitigate the social impact" of this decision, the European Commission allows "accompanying measures" such as funding re-training of workers for other jobs. Here is a good article explaining the situation.
This is all fine and dandy, but so far we're talking about €500, not €1,500. After a little searching, I hit on this news report:

In addition, since May 2003 the Commission is investigating SEPI's capital injections into Izar in an amount close to €1,477 million after the creation of the latter, between 2000 and 2002.
Brussels knows that the Spanish public shipyard company received capital injections of €1,322 million in 2000, €105 million in 2001 and €50 million in 2002.
The Commission doubts that these injections conform to the state aid rules, which only allow subsidies to civilian shipyards which result in economic benefits unattainable through commercial means.
According to the Commission, the Government reported when the aid was awarded that it was intended to help build military ships. Based on this statement, the services of [competition Commissioner] Monti anticipate that the aid may be incompatible with European regulations.

That is, now we're talking almost €2,000 million. What is it with economic conservatives like Aznar and Bush, always talking about market liberalization and small government, and then illegally subsidizing domestic industries? I guess in this respect, too, Aznar is no different from Bush.

Good Lord, what a country!

The decline and fall of Spanish mining

In Spain, mining and Asturias (a province and Autonomous community on the North of the country) are synonimous. According to this site,

In the 1960's the mining industry of Asturias suffered enormous economic losses, to the point that the owners of the mines asked the government to nationalize them. Thus, HUNOSA (a new holding of which the Spanish govenrment owned 77%) was created on March 9, 1967. Started with an initial capital of 3,380 million pesetas, by 1979 HUNOSA had accumulated 65,000 million pesetas in losses. In 1980 the Government agreed with HUNOSA to carry out plans to reduce the weight on mining in Asturian economy. From 1980 to 1990, the number of miners was reduced from 22.000 to 18.000. Because of the mining crisis, Asturias went from being the sixth region of Spain in per capita income in 1955 to twenty-first in 1985, with an unemployment rate above the national average.

Since the 1920's, successive Spanish governments had already helped mining by imposing the obligation to use Spanish coal on all industries. Then the Francoist oligarchy that owned the mines became public employees through HUNOSA, and it was not until 1979-1980, under the first democratic government after Franco's death in 1975, that something was done. This is consistent with other disastrous economic policies of the 1970's, when Franco's govenrments shielded the Spanish people from the oil crisis at huge costs to the state, which ended up blowing up in the face of all Spaniards around 1980. In 1980, the only possible solution to the problem was already a reduction in the number of miners, but for the following 10 years jobs in the industry were reduced only by 9%, probably due to union resistance. The unions had, apparently, won their first collective bargaining contract as late as 1972, when the industry was already on its deathbed. The story contunues:

By 1991, France and Belgium had closed all their pits and Germany only kept open the most productive. Meanwhile in Spain, agreements are signed until 2002, imposing a severe reduction of jobs in the industry, and the closure of the least profitable pits. In 3 years, the number of employees in the industry dropped from 18.000 to 12.000 thanks to early retirement. Even though, at the beginning of the 1990's mining was employing 21,6% of Asturias' workforce.
From 1986, Spain received EEC subsidies due to end in 2002, when all unprofitable mining in the EU must be shut down.

What actually happened is that the European Coal and Steel Community treaty expired in mid-2002, and the EU's general rules on competition came into effect. As a result, subsidies not necessary to counter the effects of dumping by third parties are considered illegal. The
European Commission explains:

This Regulation provides that state aid may be granted for the restructuring of the hard coal industry, taking into account the social and regional aspects of the restructuring as well as the need to maintain, as a precautionary measure, a minimum quantity of indigenous production to guarantee access to reserves.

Apparently, in 2003 Spain continued to subsidize the industry, which now has to return €600 million to the Government. According to Cadena Ser,

It all begins on January 1 2003, when the criteria to award public aid to mining changed. Brussels made the subsidies conditional on prior EU authorization, to enforce the application of the new rules and competition is not distorted. But the People's Party government did not comply with that requirement and awarded the 2003 subsidies without EU approval. In Brussels, nobody moved a finger.

"It all begins"??? What an understatement. The story really begins after WWI, when Spain's mining operations failed to be mechanized like their European counterparts, and in the name of "social peace" 80 years of Spanish governments decided to maintain and subsidize a labour-intensive industry. There were 52,000 miners as late as 1958, a testament to everyone's lack of imagination on the issue of how to employ the people of Asturias.
As for the recently ousted PP government, the fact that Spain was unprepared to justify that the mines qualified for aid under the new rules means that Spanish mines were known not to be profitable enough. The PP government, in power since 1996, and the PSOE regional goverment, had up to 6 years (depending on when the new rules were decided on) to evaluate the likelyhood the Spain's mines would qualify for subsidies, and do something about helping Asturias in case the mines had to be shut down. Nothing was done by either of them.
Again according to Cadena Ser, citing "sources in the industry",

[...] for nearly a year and a half, the EU commisioner for Energy allowed the former Ministry of Economy under Rodrigo Rato, close to commissioner Loyola de Palacio in the PP, not to answer Brussels' demands for an explanation of the rules by which Aznar's government was awarding subsidies to mining operations.
But all changed on March 14. That day the PP lost the elections and, barely two weeks later on March 30, Loyola de Palacio sent the Spanish government an ultimatum and started an investigation of the subsidies, since without authorization they might be illegal, incompatible with the common market, and subject to being returned to the EU.

Apparently, De Palacio gave Spain one month, knowing that the transfer of power would barely be completed by then and, for their part, responsible PP officials did nothing to comply. Apparently it was the unions that told the new government about this situation, already after the ultimatum had expired.

Coal-dark future: the EU subsidies to Spanish mining

On may 14, 2004, it became known that the Spanish government illegally subsidized the coal mining industry in the amount of €600 million, which will now have to be returned by the industry. The shady political dealings behind this need not concern us here but, according to radio station Cadena Ser,

The coal mines produce about 12.5 million tonnes of coal each year, for which electric power companies pay—at the international market price—some €750 million, well below the operating costs of the mines.

It is for this reason that Brussels authorizes public subsidies, in order to make up for the difference between the market price and the cost, and that accounts for the €600 million now in question, which is to say, nearly half of the total revenue of the mines. Hence, having to return these subsidies might lead to a collapse of the industry, with 47 companies employing some 14,000 workers and many of them essential in comarcas practically without employment alternatives for their population.

Since the Spanish Constitution defines Spain as a social state and the Draft European Constitution lists solidarity as one of the values on which the EU is founded, it makes sense that the EU and Spain are willing to spend €600 million a year for the sake of preserving the social fabric of the mining comarcas (counties?). To understand the magnitude of the problem, one could estimate the number of affected people at about 100,000 (assume each miner is in a family of four, and double the result). The question that assails me right now is, if Spain and the EU are willing to spend about €6,000 per person per year to help these people, isn't there a better, more imaginative, way to use the money than to subsidize the industry? We are talking almost €43,000 per year for each of the 14,000 employees of the industry!

The problem with subsidies is that they just make the problem worse for the future. If Spain's coal mines are not productive enough to be profitable, the more is extracted from them the less profitable they become, because each new tonne of coal becomes more expensive to extract. The amount of the subsidies must, therefore, increase each year. Presumably, subsidies started when the mines were just short of breaking even, and so have increased to €600 million from a negligible amount. As far back as I can remember, there was talk of industrial reform of the mining sector, and social and labor conflict in the affected areas. Back then, a small subsidy must have seemed a reasonable price to pay for social peace. The price is no longer reasonable, however, when the subsidy accounts for 4/9 of the cost of producing the coal.

Postponing the inevitable closing of the coal mines is only part of the solution. An alternative needs to be provided to the maybe 100,000 people that would have to relocate were the mines to be closed overnight. What can we done with €600 million per year to ease the transition to the unavoidable future when the mines will be no longer in operation, and the mining comarcas deserted?

Challenging the law of supply and demand

According to conventional wisdom an increase in demand for a commodity increases the price it commands in the market, and this in turn leads to an increase in supply, to match the increase in demand. A simple analysis of the optimal supply given a demand curve leads me to believe that it is not impossible, or even implausible, to have the paradoxical situation that an increase in demand at every price can result in a decrease of supply.

Describing demand
Let d(p) be the average amount of the commodity that will be sold (per unit time) at the price p. Note that, even if the commodity is sold in indivisible units, this average is a continuous variable. One can safely assume that this demand function is monotonically decreasing and, generically although not necessarily, strictly decreasing. The reason is that, at a lower price, one is at least as likely to purchase the commodity (and at least as often) as at a higher price. The demand function can therefore be inverted, to give the supply
function p(s), which is the price that the commodity commands in the market if it is supplied (and consumed) at the rate s.

Optimizing supply
For the supplier, the problem is to determine the rate at which to supply the commodity so as to maximize his profit. The revenue that can be expected from selling the commodity at the rate s is sp(s). Let c(s) be the cost of bringing the commodity to market at the rate s. The form of the cost function is, interestingly, not important for our current analysis.
The profit derived from bringing the commodity to market at the rate s is, therefore, P(s)=sp(s)-c(s). The commodity will
be supplied at the rate s₀ maximizing the profit. This satisfies P'(s₀)=0 and, generically, P"(s₀)<0.

The response of the supply to changes in the demand
It is well-known from the mean-value theory of phase transitions that s₀ does not necessarily depend continuously on small changes in the profit function P(s). For instance, it is possible that a small change in the unit cost leads to a large change in the optimal supply. This behaviour is not generic, though.

In any case, suppose that the demand changes by δp(s), with a concommitant change δs₀ in the optimal supply. Then, P'(s₀)=0 implies
P"(s₀)δs₀+[s₀δp(s₀)]'=0.
In other words, since P"(s₀)<0, the optimal supply s₀ increases if, and only if, sδp(s) is an increasing function of s at s=s₀. This makes sense, since this function represents
the increase in expected revenue for a given supply level s. It is also worth noting that this result does not depend on the specific form of the profit function or any of its components, only on the variation of the revenue function.
Moreover, p(s) is itself everywhere decreasing. It is therefore likely that δp(s) will also be decreasing, and not impossible that sδp(s) will be decreasing at least for some range of values of s.

Conclusion
The conclusion of this analysis is that, if the demand changes in such a way that the expected revenue grows much faster at larger prices than the current market price than at smaller prices, the price of the commodity will increase in response to the increase in demand (as expected) while at the same time supply will contract. This is true even if the demand increases across the board!

Mixed economy and money supply

In an earlier post, I wrote:

It seems that the role of the government in a mixed economy is both that of judge and party: the government judges the quality of competition by ensuring that no single agent can affect prices, but in order to promote public goods it must have the ability to itself affect the prices of public goods favourably. For this the government must have access to a substantial fraction of the available money, and the only way to obtain it is by taxation. Since the government regulates the market, it makes sense that economic activity is taxed (income tax, sales tax, etc). Poll taxes make less sense because they tax entities whether or not they take part in economic exchanges in the market.

Since then I have come across some interesting ideas on monetary reform along the following lines.

Currently, money is created by being lent at interest (both by the central bank lending to financial institutions and the latter lending to individuals or firms). Public spending can never be fully funded by taxes, and so the government must incur debt (by issuing bonds).

Alternatively, the government would create the money necessary to cover any budget shortfall, and then spend it (not lend it) into the economy. One of the problems with this approach to the money supply is that, according to conventional wisdom, increasing the amount of available money increases inflation.

But, apparently, modern macroeconomics is based on neutrality of money, namely, that the amount of money available and the interest rates have no impact on the economy! Then there should be no problem in eliminating taxation altogether and just printing all the money that the government needs to get things done.
What is going on?

A little game theory: does crime pay?

I think I'll jump ahead of Joel and write a little about the conclusions of our conversation today on game theory.
The question, arising from a class at UCR, is: does crime pay? Here's our analysis.

In the simple model introduced in that class, the criminal has a choice between committing or not committing a certain crime (for a benefit S). He has a probability p of being caught, for a penalty F. The payoff matrix is

payoff	crime	no crime
caught (p)	S-F	0
not caught (1-p)	S	0
expected payoff	S-pF	0

Conclusion
It pays off to commit the crime as long as the probability p of getting caught is less than S/F. In particular, unless the penalty is larger than the benefit obtained from the crime, no amount of law enforcement can make it disadvantageous (on average) to commit the crime. This is certainly not the conclusion that the UCR professor wanted to reach, which involved some argument about decreasing profit leading to less crime but no discussion of threshold values of the probability of getting caught.

Possible extensions include:

• Make the utility of the crime or the penalty depend on the current utility of the criminal. For instance: if the benefits and penalties are monetary, what happens if the utility of money is nonlinear? How does risk-prone versus risk-averse utility affect the analysis? Are poor people more likely than wealthy people to commit the crime? What if the criminal is so poor that they can't pay the fine?
  
• Consider replacing the one-shot crime with a probability of committing the crime per unit time (say, the number of "jobs" per year).