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\begin_body

\begin_layout Title
Public Goods
\end_layout

\begin_layout Author
Michael Peters
\end_layout

\begin_layout Date
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
today
\end_layout

\end_inset


\end_layout

\begin_layout Section
Introduction
\end_layout

\begin_layout Standard
In traditional economics, a public good is usually defined as something
 that has two properties - non-excludability and non rivalrousness.
 Non-rivalrousness means that if one person consumes a public good, he or
 she doesn't take it away from anyone else.
 We don't need to compete for public goods - they are there for all to consume.
 Non-excludability means that it is impossible to prevent people from consuming
 the public good once it is produced.
 Almost nothing fits the definition well - maybe clean air.
 Some goods are non-excludable - a fireworks display, loud music, but not
 obviously goods.
 Many public services are at least potentially enjoyed by all, for example
 a bridge across a river, but potentially subject to congestion.
\end_layout

\begin_layout Standard
Economists have long argued that government is needed to provide non-excludable,
 non-rivalrous goods.
 Here is a really cute slide from Benjamin Allen at Harvard University that
 sums up the argument.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename Slide1.png
	scale 60

\end_inset


\end_layout

\begin_layout Standard
In the figure, you can see the selfless cooperators working tirelessly for
 no personal gain to make the world a better place.
 Evil free riders take advantage of this selfless contribution of others
 and waste their time in the pursuit of leisure (I guess that from the picture
 because they are wearing baseball caps) which has no benefit to anyone
 but themselves.
 The policy implication is that a public institution financed by taxation
 is needed to force the free riders to pay.
 If nothing else, this just makes things fairer for the cooperators.
\end_layout

\begin_layout Standard
Economic theory goes a bit further to argue that in the situation described
 above too little of the public good will actually be provided.
 The reason is that the selfless cooperators will actually stop producing
 the public good at a point where the greedy free-riders would be willing
 to pay them to produce more.
\end_layout

\begin_layout Standard
The point of this reading is two fold.
 First, to suggest why the story above is misleading.
 Second, it explains why it is so important to try to get this story right.
\end_layout

\begin_layout Standard
Public goods are often treated as a fringe element of microeconomic theory
 - sometimes relegated to special topics courses.
 In fact, many of the goods that are most important to us are at least non-rival
rous.
 Modern technology makes most them non-excludable.
 The story above suggests that it is those two properties - non-rivalrous
 and non-excludable - are bad things that we need to try to get rid of just
 because they are bad.
 In fact, they are the very properties that give goods value.
 
\end_layout

\begin_layout Standard
A software program is an example, is something that is non-rivalrous and
 non-excludable by nature.
 As in the picture above, it takes effort to produce.
 If you run the same program as I do, I am not hurt by that.
 The program is on a digital file, so it can be freely distributed to everyone.
 Google, Facebook, Twitter are produce enormous amounts of code that produce
 services that most of us seem to love.
 We can't have evil free riders using that software to produce the same
 thing.
 
\end_layout

\begin_layout Standard
Maybe you can see the problem with that argument.
 The first is that all three companies seem to be free-riders.
 They use linux based tools that are open source and free.
 According to a survey of the top 1 million websites in 2015 done by a tech
 analysis company called W3cook (http://www.w3cook.com/), 98% of the servers
 behind those websites were using a Linux based (that is free) operating
 system.
 Your Wordpress website is driven by open source software.
 The html and javascript you use to make up your webpage are free open source
 software.
 At least in the US, the power of the tech sector and silicon valley seems
 to exist because software is non-rivalrous and non-exclusive.
 Those are the properties that make the whole tech industry valuable.
\end_layout

\begin_layout Standard
Secondly, you might see that the simple distinction between cooperators
 and free riders is completely wrong.
 The companies that support open source software do it for their own benefit
 and produce software that is useful to Google, etc.
 At the same time the software that Google produces is offered back to the
 producers of the linux operating system.
 The point being, that all of us are both cooperators and free-riders.
 
\end_layout

\begin_layout Standard
The nice diagram given above, which perfectly summarizes what most economists
 think about non-exclusiveness and non-rivalrousness, really misses the
 point when it tries to describe things that are important to us.
\end_layout

\begin_layout Standard
Perhaps a better way to describe a non-rivalrous good is to say that it
 is a good that is produced at a possibly high fixed cost, but thereafter
 has zero marginal cost.
 For software, one copy and a million copies cost exactly the same thing.
 Music comes on mp3 files which can be costlessly redistributed once they
 are produced.
 Researchers in universities produce data and concepts that are useful to
 a lot of people.
 An idea is non-rivalrous and non-excludable, as are jokes, fashion ideas,
 newspaper articles, etc etc.
\end_layout

\begin_layout Standard
To understand the differences between all these examples, we can use some
 of the basic ideas we have developed so far in consumer theory.
 We'll use the old standby techniques 2 goods and two traders.
 One good we'll just call money, denoted 
\begin_inset Formula $x$
\end_inset

, used for consuming other stuff.
 The other good 
\begin_inset Formula $y$
\end_inset

 will be the one that is non-excludable and non-rivalrous (in other words,
 produced at a cost, but they freely available).
 We'll refer to 
\begin_inset Formula $y$
\end_inset

 as a public good from time to time.
 Consumers have preferences 
\begin_inset Formula $u\left(x,y\right)$
\end_inset

 over these two goods as they always do.
 Consumers can produce more of the public good by giving up money.
\end_layout

\begin_layout Standard
The second idea is that since the public good 
\begin_inset Formula $y$
\end_inset

 is non-rivalrous, if one consumer gives up money to produce more of the
 public good, the other consumer will enjoy this new public good as well.
 This gives us our selfless cooperator, and our selfish free rider.
 The difference is that, as in real life, both consumers play both roles.
 Production of the public good creates a very special type of positive externali
ty in the story that follows.
\end_layout

\begin_layout Standard
Now we can start with a description of something called the 
\emph on
voluntary contribution game
\emph default
, which is a common way to think about how public goods are provided.
 It explains why the amount of the public good in the voluntary contribution
 game is too small (because the outcome is not Pareto optimal: there is
 another outcome that will make both consumers better off).
 
\end_layout

\begin_layout Subsection
The voluntary contribution game
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $f\left(x\right)$
\end_inset

 denote the amount of the public good that can be produced from 
\begin_inset Formula $x$
\end_inset

 units of the private good.
 Suppose the two consumers have utility functions 
\begin_inset Formula $u_{1}\left(x,y\right)$
\end_inset

 and 
\begin_inset Formula $u_{2}\left(x,y\right)$
\end_inset

 respectively.
 Their endowments of the private good are 
\begin_inset Formula $\omega_{1}$
\end_inset

 and 
\begin_inset Formula $\omega_{2}$
\end_inset

.
 The set of points 
\begin_inset Formula $\left\{ \left(x,y\right):y=f\left(\omega_{1}+\omega_{2}-x\right)\right\} $
\end_inset

 is the 
\emph on
production possibilities frontier
\emph default
.
 It looks exactly like the production possibilities frontier that we studied
 before.
\end_layout

\begin_layout Standard
If the first consumer decides to consume 
\begin_inset Formula $x_{1}$
\end_inset

 (and devote the rest of his endowment 
\begin_inset Formula $\omega_{1}$
\end_inset

 to production of the public good) while consumer 
\begin_inset Formula $2$
\end_inset

 decides to consume 
\begin_inset Formula $x_{2}$
\end_inset

 the utilities of each of the consumers are given by
\begin_inset Formula 
\[
u_{1}\left(x_{1},f\left(\omega_{1}+\omega_{2}-x_{1}-x_{2}\right)\right)
\]

\end_inset

 for consumer 
\begin_inset Formula $1$
\end_inset

 and
\begin_inset Formula 
\[
u_{2}\left(x_{2},f\left(\omega_{1}+\omega_{2}-x_{1}-x_{2}\right)\right)
\]

\end_inset

for consumer 
\begin_inset Formula $2$
\end_inset

.
 The important point is that if consumer 
\begin_inset Formula $1$
\end_inset

, say, decides to consume a bit less of the private good and produce a bit
 more of the public good, then consumer 
\begin_inset Formula $2$
\end_inset

 will enjoy the additional public good too without any cost at all - thats
 what non-rivalrous means.
\end_layout

\begin_layout Standard
At this point, we need to make some changes to what we have done before.
 Each of the consumers simply picks the amount of the private good they
 want on their own.
 This is a bit hard to do because the amount that each consumer will choose
 to contribute depends on how much they expect the other consumer to contribute.
 To handle this, we have to replace 
\emph on
Walrasian Equilibrium 
\emph default
with 
\emph on
Nash equilibrium
\emph default
.
 Instead of taking prices to be fixed, each consumer takes the contribution
 of the other consumer to be fixed and chooses the contribution that maximizes
 his utility given this expectation.
 In the Nash equilibrium each consumer must simultaneously choose his consumptio
n of the private good (analogously, his contribution to production of the
 public good) while correctly forecasting what the other player will do.
\end_layout

\begin_layout Standard
Formally, a Nash equilibrium for the voluntary contribution game is a pair
 of private consumptions 
\begin_inset Formula $x_{1}^{\ast}$
\end_inset

 and 
\begin_inset Formula $x_{2}^{\ast}$
\end_inset

 such that
\begin_inset Formula 
\begin{equation}
u_{1}\left(x_{1}^{\ast},f\left(\omega_{1}+\omega_{2}-x_{1}^{\ast}-x_{2}^{\ast}\right)\right)\geq u_{1}\left(x^{\prime},f\left(\omega_{1}+\omega_{2}-x^{\prime}-x_{2}^{\ast}\right)\right)\label{consumer_1}
\end{equation}

\end_inset

 for any alternative contribution 
\begin_inset Formula $x^{\prime}\in\left[0,\omega_{1}\right]$
\end_inset

 and
\begin_inset Formula 
\begin{equation}
u_{2}\left(x_{2}^{\ast},f\left(\omega_{1}+\omega_{2}-x_{1}^{\ast}-x_{2}^{\ast}\right)\right)\geq u_{2}\left(x^{\prime},f\left(\omega_{1}+\omega_{2}-x_{1}^{\ast}-x^{\prime}\right)\right)\label{consumer_2}
\end{equation}

\end_inset

 for any alternative contribution 
\begin_inset Formula $x^{\prime}\in\left[0,\omega_{2}\right]$
\end_inset

.
\end_layout

\begin_layout Standard
One way to view the outcome of this game is given in Figure 
\begin_inset Formula $1$
\end_inset

 where the two consumers choose 
\begin_inset Formula $x_{1}^{\ast}$
\end_inset

 and 
\begin_inset Formula $x_{2}^{\ast}$
\end_inset

.
 The `budget line' that consumer 
\begin_inset Formula $1$
\end_inset

 faces, for example, when consumer 
\begin_inset Formula $2$
\end_inset

 chooses consumption 
\begin_inset Formula $x_{2}^{\ast}$
\end_inset

 is the set of all pairs 
\begin_inset Formula $\left\{ \left(x_{1},y\right):y=f\left(\omega_{1}+\omega_{2}-x_{1}-x_{2}^{\ast}\right)\right\} $
\end_inset

.
 The slope of this is exactly the same as the slope of the production possibilit
ies frontier at the point 
\begin_inset Formula $\left(x_{1}^{\ast}+x_{2}^{\ast},y^{\ast}\right)$
\end_inset

.
 The same is true for consumer 
\begin_inset Formula $2$
\end_inset

.
 So, in the equilibrium of the voluntary contribution game, each consumer
 has the same marginal rate of substitution and the same marginal rate of
 transformation in production.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename public_goods_fig1.eps

\end_inset


\end_layout

\begin_layout Standard
With private goods, this is exactly what you want.
 Recall that, in the Edgeworth box, both consumers' indifference curves
 were tangent and the common slope of their indifference curves was equal
 to the slope of the production possibilities frontier.
 With public goods, this is not the outcome that you want.
\end_layout

\begin_layout Standard
You might be wondering what happened to the Edgeworth Box.
 Recall that in the Edgeworth box, one point could be used to represent
 the outcome for both consumers.
 The reason is that increasing the amount of good 
\begin_inset Formula $x$
\end_inset

 that 
\begin_inset Formula $1$
\end_inset

 consumes automatically lowers the amount that 2 consumes.
 Good 
\begin_inset Formula $x$
\end_inset

 is rivalrous.
 In the usual story the same is true of good 
\begin_inset Formula $y$
\end_inset

.
 Here, however, good 
\begin_inset Formula $y$
\end_inset

 is non-rivalrous - if 1 consumes more good 
\begin_inset Formula $y$
\end_inset

, then so does 2.
\end_layout

\begin_layout Standard
So lets modify the approach to account for this.
 For consumer 1, the indifference curve is just the collection of 
\begin_inset Formula $\left(x,y\right)$
\end_inset

 pairs that satisfy
\begin_inset Formula 
\[
u_{1}\left(x,y\right)=K.
\]

\end_inset

The graphs of these curves look no different than they do in any other problem.
 Higher indifference curves (further away from the origin) represent higher
 values of 
\begin_inset Formula $K$
\end_inset

 in the equation above, and correspondingly higher payoffs to consumers.
 
\end_layout

\begin_layout Standard
The leap we are going to make here is that when we choose a bundle 
\begin_inset Formula $\left(x,y\right)$
\end_inset

 for consumer 1, this bundle induces a corresponding bundle for consumer
 2 given by
\begin_inset Formula 
\[
\left(\omega_{1}+\omega_{2}-f^{-1}\left(y\right)-x,y\right).
\]

\end_inset

In words, we figure out how much money is required to produce 
\begin_inset Formula $y$
\end_inset

, that is the 
\begin_inset Formula $f^{-1}\left(y\right)$
\end_inset

, then subtract that from the total endowment 
\begin_inset Formula $\omega_{1}+\omega_{2}$
\end_inset

 to get the total amount of money left over after producing 
\begin_inset Formula $y$
\end_inset

 units of software.
 Subtract the 
\begin_inset Formula $x$
\end_inset

 we want to give to consumer 1, then give the rest to consumer 2.
 The function 
\begin_inset Formula $f^{-1}\left(y\right)$
\end_inset

 is sometimes referred to as the cost function for the public good.
\end_layout

\begin_layout Standard
Given this logic, we could define an indifference curve for 
\emph on
consumer 2 
\emph default
by finding all the consumption bundles for 
\emph on
consumer 1 
\emph default
that induce the same payoff for player 2.
 Formally, an indifference curve for player 2 is the collection of all bundles
 for player 1 that satisfy
\begin_inset Formula 
\[
u_{2}\left(\omega_{1}+\omega_{2}-f^{-1}\left(y\right)-x,y\right)=K.
\]

\end_inset


\emph on
 
\end_layout

\begin_layout Standard
To figure out what these indifference curves look like is a bit daunting.
 Holding 
\begin_inset Formula $y$
\end_inset

 constant, the less money consumer 1 has, the more is left over for consumer
 2.
 In this sense, player 2's indifference curves represent higher payoffs
 for 2 the closer they are to the origin.
 
\end_layout

\begin_layout Standard
For the rest, we'll rely on the idea that the public good provides diminishing
 marginal utility to consumer 2.
 What that means is that the more of the public good that is being produced,
 the less valuable an additional unit of the public good will be.
 Fix the consumption of the private good of consumer 1 at some value 
\begin_inset Formula $x_{0}$
\end_inset

 and travel up the vertical line through 
\begin_inset Formula $x_{0}$
\end_inset

.
 Remember that changing the public good in this case means that all the
 cost of producing the public good is borne by consumer 2 since the amount
 of money consumer 1 has is being held constant at 
\begin_inset Formula $x_{0}$
\end_inset

.
 Initially as you increase the public good, this makes consumer 2 better
 off since the public good is very valuable to her when there isn't much
 of it.
 Eventually the thrill will wear off, and as the production of the public
 good gets higher and higher, consumer 2's payoff will begin to decline.
 What that means is that consumer 2's indifference curves will typically
 cross any vertical line twice.
\end_layout

\begin_layout Standard
Of course, there will be one indifference curve that is just tangent to
 the vertical line.
 This one is illustrated in the picture below.
 
\end_layout

\begin_layout Standard
In a Nash equilibrium, consumer 2 will choose how much of the public good
 to produce assuming that the consumption of the public and private good
 by consumer 1 are fixed.
 What that means is that consumer 2 will choose a level of the public good
 at which her indifference curve as we have just described it is vertical.
\end_layout

\begin_layout Standard
The indifference curves should look like backward C's as in the following
 diagram, where I have superimposed consumer 2's indifference curve into
 the original diagram depicting the Nash equilibrium of the voluntary contributi
on game.
 Let me explain why.
 At the point where the indifference curve is just tangent to the vertical
 line, consumer 2 could increase her production of software 
\begin_inset Formula $y$
\end_inset

 and travel further up the vertical line.
 If she travels directly upward, it means that she is holding consumer 1's
 holdings of money constant.
 In other words, if she increases production, she pays all the costs herself.
 At the tangency, she doesn't want to do this - the marginal benefit she
 gets from increasing production is exactly equal to the marginal cost.
 Formally
\begin_inset Formula 
\[
\frac{\partial u_{2}\left(\omega_{1}+\omega_{2}-f^{-1}\left(y^{\ast}\right)-x_{1}^{\ast},y^{\ast}\right)}{\partial x}\frac{1}{f^{\prime}\left(y^{\ast}\right)}=\frac{\partial u_{2}\left(\omega_{1}+\omega_{2}-f^{-1}\left(y^{\ast}\right)-x_{1}^{\ast},y^{\ast}\right)}{\partial y}.
\]

\end_inset


\end_layout

\begin_layout Standard
Then if you travel up the vertical line a bit above 
\begin_inset Formula $\left(x_{1}^{\ast},y^{\ast}\right)$
\end_inset

, then consumer 2 is strictly worse off.
 How to restore her payoff? You have to give her more money, which means
 reducing the amount of money you leave for consumer 1.
 That means that you have to travel left of the vertical line to restore
 2's payoff.
 The indifference curve must lie to the left of the vertical line at every
 point except at the tangency.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename public_goods_fig5.eps

\end_inset


\end_layout

\begin_layout Standard
Since the point 
\begin_inset Formula $\left(x_{1}^{\ast},y^{\ast}\right)$
\end_inset

 is supposed to coincide with the Nash equilibrium in the voluntary contribution
 game, consumer 1's indifference curve has the same slope as the production
 possibilities frontier at the point where 
\begin_inset Formula $y^{\ast}$
\end_inset

 is produced.
 In other words, it is downward sloping, not vertical.
 The lens between the curves represents a situation in which both consumers
 could be made better off.
\end_layout

\begin_layout Standard
We draw the conclusion that the Nash equilibrium of the voluntary contribution
 game is not pareto optimal.
 Remember what I have said about 'pareto optimal' - it has nothing to do
 with optimal.
 An outcome where consumer 2 gets whatever she likes, while consumer 1 gets
 nothing is pareto optimal.
 Second, remember the lesson of Google, Facebook, Twitter, all your social
 media products - they come from an equilibrium that is not pareto optimal.
 It seems unlikely you would be too upset about the products that are produced
 in that market, so maybe something that isn't pareto optimal really isn't
 so bad.
\end_layout

\begin_layout Standard
Notice why it isn't so bad.
 All consumer are producers of the public good - they are cooperators and
 free riders at the same time - they help each other.
 Many important products have another property that is relevant here.
 In the story above, if consumer 1 produces more software, decreasing returns
 means that it is more expensive for consumer 2 to produce an 
\emph on
extra 
\emph default
unit of software than it was before.
 Software isn't like that, if consumer 1 produces more software, it becomes
 cheaper for producer 2 to produce new software.
 If you let me write more software, you will begin to want to write software
 that was too hard or expensive to produce before.
 Music is also like that, the more that is produced the easier it is to
 produce.
 Research is like that, jokes are like that, newpaper articles are like
 that, and many more.
 That probably explains why goods that are non-rivalrous and non-exclusive
 are so often produced in such abundance - despite the fact that the correspondi
ng equilibrium outcome isn't pareto optimal.
 
\end_layout

\begin_layout Standard
We won't develop this formally because we only have two goods, but we can
 go much further.
 Benefiting from the efforts of another (what we started out calling free
 riding) allows consumers to devote their resources to other activities,
 these other activities may involve production of public goods.
 Probably every organization on earth is based on this principle.
 When one person volunteers to do some work on something, the others in
 the organization don't free ride on this, they use the time that has been
 freed to do other useful things.
 Most organizations don't work perfectly, nonetheless many of them work.
 
\end_layout

\begin_layout Section
Intellectual Property
\end_layout

\begin_layout Standard
The reason it is so important to think about public goods and the narrative
 of cooperators and free riders is because of the awful policies it has
 spawned.
 The voluntary contribution game provides a sheen of logic to a narrative
 which can then be easily misused.
 The outcome in the voluntary contribution game is not pareto optimal, even
 though it looks like all the other problems you encounter in intermediate
 microeconomics.
 The reason appears to be that when a consumer produces the public good,
 other people can use it.
 Other people can't use your car because it is your property.
 Therefore we should turn public goods into property - in other words, try
 to make a non-exclusive good into an exclusive one.
 Then things would be pareto optimal again as they are when all goods are
 private goods.
\end_layout

\begin_layout Standard
There are a bunch of laws that people now associate with intellectual property
 - copyright law, trademarks and patents.
 They seem on the surface to be associated with things that people thought
 up, so they must be intellectual - therefore innovative.
 The story then goes that we need laws to protect intellectual property
 to promote innovation.
 There is a nice article by Richard Stallman at http://www.gnu.org/philosophy/not-
ipr.en.html which explains in a pretty simple way how these laws relate to
 one another, and how none of them have anything to do with intellectual
 property or promoting innovation.
\end_layout

\begin_layout Standard
You can read about US copyright and patent laws in a lovely (free) book
 called 
\begin_inset Quotes eld
\end_inset

Against Intellectual Monopoly
\begin_inset Quotes erd
\end_inset

 by David Levine and Michele Boldrin (http://www.dklevine.com/general/intellectual
/againstfinal.htm).
 The book is full of historical anecdotes and simple economic models to
 explain what is going on.
 The gist of their argument is that copyright and patent laws in the US
 are designed to give corporations levers that they can use to prevent entry
 and suppress competition.
 Evidence that these laws encourage creation of public goods is basically
 non-existent, what evidence there is suggests the opposite.
 This is important for Canada since the US almost always insists that countries
 adopt US copyright and patent law before they will engage in trade negotiation.
\end_layout

\begin_layout Standard
So lets continue with the example of software patents, and try to explain
 the effect that they have using our second year economic theory.
\end_layout

\begin_layout Standard
The way patents work is that one of the two consumers applies for a patent,
 and is then given a monopoly over production of the patented product, in
 this case our software.
 For example, Mircosoft has a couple of strange patents - one is a patent
 for double clicking on an icon to launch an application, awarded in 2004.
 Another is patent for using the page up and page down buttons on a keyboard
 to shift the content of a page up or down (2008).
 These operations are basic to just about all software.
 Consistent with the patent, anyone writing software would then have to
 pay microsoft a fee to use those procedures in their software.
 
\end_layout

\begin_layout Standard
You can see from these two examples, that patents (at least software patents)
 have nothing to do with innovation.
 In Microsoft's defense, they do need a portfolio of patents that they can
 use to defend themselves against other large companies who also hold patents
 - often for no better reason that to litigate a competitor out of existence
 (see Apple's litigation against other smartphone makers https://en.wikipedia.org/
wiki/Apple_Inc._v._Samsung_Electronics_Co).
\end_layout

\begin_layout Standard
To make things simple here, we'll just assume that people have to pay microsoft
 to produce their software for them.
 That really isn't any different than having them pay a fee to microsoft
 whenever they write their own software.
\end_layout

\begin_layout Standard
It might seem strange that microsoft could get a patent for a technique
 people have been using forever.
 It is important to realize that the US (and now Canada) has what is called
 a 'First to Patent' law, which means that whoever files a patent on something
 first gets the patent whether they developed the idea or not.
 
\end_layout

\begin_layout Standard
Lets just assume that consumer 2 wins this race.
 Consumer 2 now owns the public good.
 She can 
\emph on
exlude 
\emph default
consumer 1 and charge whatever price she likes for access to it.
 Suppose she sets the price 
\begin_inset Formula $p$
\end_inset

 for the public good.
\end_layout

\begin_layout Standard
Normally we do all our stuff using the price of good 
\begin_inset Formula $x$
\end_inset

 and letting the price of good 1 be normalized to 1.
 We'll switch that here, but this shouldn't create too much of a problem.
 The public good 
\begin_inset Formula $y$
\end_inset

 has a price 
\begin_inset Formula $p$
\end_inset

 , while the private good has price 1.
 Given the price set by consumer 2, consumer 1 now finds the best bundle
 of public and private goods he can afford.
 This is given by the point where his indifference curve is tangent to the
 budget line that passes through the point 
\begin_inset Formula $\left(\omega_{1},0\right)$
\end_inset

.
 Why care? The reason is that consumer 1 will no longer be able to produce
 the public good on his own (of if he does, he will have to pay consumer
 2 a fee because she owns the public good).
 So the price that consumer 2 sets will change consumer 1's desire to have
 the public good.
 If consumer 2 sets a price too high, consumer 1 won't want any.
\end_layout

\begin_layout Standard
What this means is that the patent doesn't by itself give consumer 2 any
 incentive to produce more of the public good - production depends on price,
 which is chosen by consumer 2.
 What the patent does give her is a device that she can use to extract money
 from consumer 1.
\end_layout

\begin_layout Standard
The figure below depicts what consumer 1 would want for three different
 prices.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename public_goods_fig6.eps

\end_inset


\end_layout

\begin_layout Standard
The steepest budget line is the one that has the lowest price for the public
 good, the flattest budget line has the highest price.
 Since 2 now has the 'intellectual' monopoly, she can set any price that
 she likes.
 She will pick the price that maximizes her payoff.
 
\end_layout

\begin_layout Standard
It would seem quite daunting to find this price, since the change in price
 will change consumer 1's consumption of the public good, which will in
 turn effect how much consumer 2 needs to produce.
 
\end_layout

\begin_layout Standard
Fortunately we have just the graphic device we need figure out what consumer
 2 will do, since we just figured out what her indifference curves looked
 like in the space depicting consumer 1's consumption of the private and
 public good.
 She will pick the price that put her on the highest indifference curve
 consistent with consumer 1 maximizing subject to his budget constraint.
 You'll notice in the curve above that there is a smooth line that connects
 all the tangency points for different prices.
 This curve is sometimes referred to as the offer curve.
 The picture that follows shows what happens when consumer 2 chooses the
 point on this offer curve that is tangent to her own indifference curve.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename public_goods_fig7.eps

\end_inset


\end_layout

\begin_layout Standard
The indifference curve for consumer 2 is now tangent to consumer 1's offer
 curve so consumer 2 is getting the best payoff she can get.
 Consumer 1 is getting the best payoff he can afford.
 Notice that because of the way the offer curve is drawn the two consumers
 indifference curves won't be tangent.
 Both consumers would be better off if more of the public good were produced.
 So the patent solution just produces another inefficient outcome.
 This is why proponents of patents will never talk about how well patents
 work.
 Instead they focus on how bad the outcome is likely to be without them
 - we went over that - not really so bad.
\end_layout

\begin_layout Standard
We can continue with graphic reasoning a bit more, but then we'll need to
 switch to algebra.
 The following figure reproduces the outcome for consumer 1 in the equilibrium
 of the voluntary contribution game.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename public_goods_fig9.eps

\end_inset


\end_layout

\begin_layout Standard
Recall that the budget line for consumer 1 with patents starts at his or
 her endowment point 
\begin_inset Formula $\left(0,\omega_{1}\right)$
\end_inset

.
 If consumer 2 sets a price for software which allows consumer 1 to buy
 the outcome he enjoyed in the volunary contribution game, then the budget
 line consumer 1 faces would be given by the red line in the picture.
 Since this budget line always lies below the production possibilities curve
 that consumer 1 faces in the volutary contribution game, it will be steeper
 than consumer 1's indifference curve at his allocation in the equilibrium
 of that game.
\end_layout

\begin_layout Standard
So far, this is completely working.
 If the price of the software were set so that consumer 1 could reproduce
 the outcome in the volunary contribution game, then consumer 1 would be
 able to purchase the additional software that he wants from consumer 2.
 Consumer 2 would in turn be willing to produce it because she is making
 more than enough money to compensate her for her additional cost.
\end_layout

\begin_layout Standard
What patents get wrong is what they do next - they give consumer 2 control
 over price.
 Most economics students understand at some level that markets don't work
 when some market participant can control the price.
 In this example, it is easy to see why this is the case.
 The patent holder now has a new device for earning money that has nothing
 at all to do with having to produce software.
 It would first occur to her than since she controls price, she can actually
 get the same level of the public good 
\begin_inset Formula $y^{\ast}$
\end_inset

 as in the voluntary contribution game, while ending up with a lot more
 money for herself.
 
\end_layout

\begin_layout Standard
In our formalism, we can use consumer 1's offer curve to understand this.
 Since the budget line that consumer 1 faces when the price is set in such
 a way that he can choose the equilibrium outcome in the voluntary contribution
 game is steeper than his indifference curve at that point, it means that
 consumer 1's offer curve cuts the horizontal line through 
\begin_inset Formula $y^{\ast}$
\end_inset

 at a point to the left of consumer 1's original equilibrium allocation.
 The offer curve should look like the solid blue line in the figure above.
 So consumer 2 can raise the price of software until the budget line faced
 by consumer 1 looks like the green line in the next figure.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename public_goods_fig10.eps

\end_inset


\end_layout

\begin_layout Standard
If you recall that the definition of the offer curve is all the points at
 which consumer 1's indifference curve is tangent to a line that runs from
 that point back to the endowment point, consumer 1 will voluntarily choose
 
\begin_inset Formula $y^{\ast}$
\end_inset

 units of software when the price is set to the green line is the budget
 line.
 By allowing consumer to control of the price, the patent basically creates
 a subsidy to the patent holder.
 
\end_layout

\begin_layout Standard
Consumer 2 might not be satisfied with this subsidy.
 She has really embarked at this point on a new venture - surplus extraction.
 Software is somewhat secondary at this point and she receives her reward
 by manipulating price.
 It is theoretically possible that she might want to raise output of software,
 but as in the figures above, this isn't her main objective, it is to find
 the place where her indifference curve is tangent to consumer 1's indifference
 curve.
\end_layout

\begin_layout Standard
Whether she does or not can be determined by travelling along the dashed
 horizontal line to the left of the equilibrium point in the voluntary contribut
ion game until you reach the offer curve, given by the solid blue line in
 the figure above.
 Since consumer 2 can achieve any point on the offer curve that she wants,
 what she does will depend on what her indifference curve looks like through
 that point.
 Generally, the tangency with the offer curve may involve either more or
 less software than in the voluntary contribution game.
 So patents are as likely from a theoretical perspective to lower output
 of software as they are to raise it.
\end_layout

\begin_layout Standard
Whether software patents work well or not is not a theoretical issue - it
 all depends on preferences and costs.
 You might think that this means that theory has nothing to say about whether
 patents are good or bad.
 Yet the message of the theory is unambigous - having a blanket policy where
 everything is patented (or otherwise protected as 'intellectual property')
 is going to do a lot of harm.
 
\end_layout

\begin_layout Standard
We can illustrate this with a couple of familiar examples.
 We'll start with one where patents act exclusively as a tax who proceeds
 are transferred to the patent holder, then follow up with an example in
 which patents can be beneficial to everyone (but more often than not just
 act as a way of tranferring income to patent holders).
 In each case we'll follow the assumptions above, but assume that each consumer
 has money income 
\begin_inset Formula $\omega$
\end_inset

 to begin.
 We'll also assume that each unit of software can be produced for one unit
 of income.
 You should work out for yourself what implications those assumptions have
 for the diagrams above.
 All we'll change in the following two examples are consumer preferences.
\end_layout

\begin_layout Subsection*
Quasi-linear preferences
\end_layout

\begin_layout Standard
Assume that consumer preferences are given by 
\begin_inset Formula $u\left(x,y\right)=x+\ln\left(y\right)$
\end_inset

, where as above 
\begin_inset Formula $x$
\end_inset

 is money income, 
\begin_inset Formula $y$
\end_inset

 is software.
 In words preferences are quasi-linear in income, a very common assumption
 in economics.
 In the voluntary contribution game, each consumer's best reply is determined
 by maximizing 
\begin_inset Formula 
\[
x+\ln\left(2\omega-x-x_{2}\right)
\]

\end_inset

 where 
\begin_inset Formula $x_{2}$
\end_inset

 is the amount of money the other player retains for herself.
 The solution is given by solving
\begin_inset Formula 
\[
1=\frac{1}{2\omega-x-x_{2}},
\]

\end_inset

 so in the symmetric equilibrium of the voluntary contribution game
\begin_inset Formula 
\[
x=\omega-\frac{1}{2}.
\]

\end_inset

This gives output of software in the voluntary contribution game as 
\begin_inset Formula $1$
\end_inset

.
\end_layout

\begin_layout Standard
If consumer 2 is given a patent and sets the price 
\begin_inset Formula $p$
\end_inset

 for software, consumer 1 will maximize
\begin_inset Formula 
\[
\omega-py+\ln\left(y\right)
\]

\end_inset

 which, as you know, has solution 
\begin_inset Formula $y=\frac{1}{p}$
\end_inset

.
 We can write consumer 2's payoff for any price 
\begin_inset Formula $p$
\end_inset

 as
\begin_inset Formula 
\[
\omega-\frac{1}{p}+1+\ln\left(\frac{1}{p}\right).
\]

\end_inset

Consumer 2 starts with 
\begin_inset Formula $\omega$
\end_inset

, but has to produce 
\begin_inset Formula $\frac{1}{p}$
\end_inset

 units of software as requested by consumer 1.
 This costs 
\begin_inset Formula $\frac{1}{p}$
\end_inset

.
 She then receives 
\begin_inset Formula $p$
\end_inset

 times 
\begin_inset Formula $\frac{1}{p}$
\end_inset

 dollars of revenue from consumer 1, which gives her 
\begin_inset Formula $\omega-\frac{1}{p}+1$
\end_inset

 dollars for herself, and 
\begin_inset Formula $\frac{1}{p}$
\end_inset

 units of software.
 It is straightforward that consumer 2 will choose the price 
\begin_inset Formula $p=1$
\end_inset

, so that the same amount of software is produced with patents as is produced
 in the voluntary contribution game.
 
\end_layout

\begin_layout Standard
What changes in the solution with patents is the distribution of income.
 Consumer 2 ends up with payoff 
\begin_inset Formula $\omega+\ln\left(1\right)=\omega$
\end_inset

 while consumer 1 ends up with 
\begin_inset Formula $\omega-1$
\end_inset

.
 In other words, the patent simply allows player 2 to charge player 1 for
 all the software, whereas in the voluntary contribution game they would
 have split the cost.
\end_layout

\begin_layout Subsection*
Cobb-Douglas preferences 
\end_layout

\begin_layout Standard
Things work a bit better for patents when consumers have Cobb-Douglas preference
s.
 Under the right conditions, they can actually benefit both consumers, though,
 for the most part, their main role is still as a redistribution device.
 When preferences are Cobb-Douglas we have 
\begin_inset Formula 
\[
u\left(x,y\right)=x^{\alpha}y^{1-\alpha}.
\]

\end_inset

Consumer 
\begin_inset Formula $1$
\end_inset

 maximizes
\begin_inset Formula 
\[
x^{\alpha}\left(2\omega-x-x_{2}\right)^{1-\alpha}
\]

\end_inset

where 
\begin_inset Formula $x_{2}$
\end_inset

 is the amount of income consumer 1 expects consumer 2 to retain for herself
 (which makes her contribution to the production of software 
\begin_inset Formula $\omega-x_{2}$
\end_inset

).
 This is done subject to the constraint that 
\begin_inset Formula $x\le\omega$
\end_inset

.
 
\end_layout

\begin_layout Standard
The first order condition gives
\begin_inset Formula 
\[
x=\alpha\left(2\omega-x_{2}\right)
\]

\end_inset

which has a solution between 
\begin_inset Formula $0$
\end_inset

 and 
\begin_inset Formula $\omega$
\end_inset

 provided 
\begin_inset Formula $2\alpha$
\end_inset

 is less than 1 and 
\begin_inset Formula $x_{2}<\omega$
\end_inset

.
 The symmetric equilibrium for the voluntary contribution game is then 
\begin_inset Formula 
\[
x=\frac{2\alpha}{1+\alpha}\omega
\]

\end_inset

 for each of the two players.
 The total output of software would then be
\begin_inset Formula 
\begin{equation}
2\omega-\frac{4\alpha}{1+\alpha}\omega=2\omega\frac{\left(1-\alpha\right)}{1+\alpha}.\label{voluntary-y}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
On the other hand, if 2 has the patent, then as you know, consumer 1 will
 keep the fraction 
\begin_inset Formula $\alpha$
\end_inset

 of his money income and choose to buy 
\begin_inset Formula $\frac{\left(1-\alpha\right)\omega}{p}$
\end_inset

 units of the public good.
 Since 
\begin_inset Formula $\frac{2}{1+\alpha}\alpha\omega$
\end_inset

 is obviously strictly larger than 
\begin_inset Formula $\alpha\omega$
\end_inset

 you can see that consumer 1 will have a lot less income with patents.
 
\end_layout

\begin_layout Standard
So when 2 charges 
\begin_inset Formula $p$
\end_inset

 for the public good, her payoff is
\begin_inset Formula 
\[
\left(\omega\left(2-\alpha\right)-\frac{\left(1-\alpha\right)\omega}{p}\right)^{\alpha}\left(\frac{\left(1-\alpha\right)\omega}{p}\right)^{\left(1-\alpha\right)}.
\]

\end_inset

To figure out whether 1 will end up with more of the public good, we have
 to figure out what price 2 will choose.
 The price that maximizes the expression above will also maximize
\begin_inset Formula 
\[
\alpha\ln\left(\omega\left(2-\alpha\right)-\frac{\left(1-\alpha\right)\omega}{p}\right)+\left(1-\alpha\right)\ln\left(\frac{\left(1-\alpha\right)\omega}{p}\right)
\]

\end_inset

which is a little simpler.
 Differentiate it to get
\begin_inset Formula 
\[
\frac{\alpha}{\omega\left(2-\alpha\right)-\frac{\left(1-\alpha\right)\omega}{p}}\frac{\left(1-\alpha\right)\omega}{p^{2}}-\frac{1-\alpha}{p}.
\]

\end_inset

This means that the price 2 will set satisfies
\begin_inset Formula 
\[
\alpha\omega=p\left(\omega\left(2-\alpha\right)-\frac{\left(1-\alpha\right)\omega}{p}\right).
\]

\end_inset

This has solution.
 
\begin_inset Formula 
\[
p=\frac{1}{2-\alpha}.
\]

\end_inset


\end_layout

\begin_layout Standard
Now we can compare the two outcomes.
 Since consumer 1 determines the output of software, the total amount of
 software produced with be
\begin_inset Formula 
\[
\frac{\left(1-\alpha\right)\omega}{p}=
\]

\end_inset

Now lets use the diagram above to figure out whether 2 will want more of
 less software than is created in the voluntary contribution game.
 We can decompose her price adjustment into two parts.
 The first thing we could imagine is that she takes the subsidy and raises
 the price to the point where 1 will want the same level of the public good
 as he did in the equilibrium of the voluntary contribution game.
 With Cobb-Douglas preferences, the consumer will always want to keep 
\begin_inset Formula $\alpha\omega$
\end_inset

 of his income.
 So we want to find a price that make consumer 1 purchase the level of output
 in the voluntary contribution game.
 From (
\begin_inset CommandInset ref
LatexCommand ref
reference "voluntary-y"

\end_inset

) this is 
\begin_inset Formula $2\omega\frac{\left(1-\alpha\right)}{1+\alpha}$
\end_inset

.
 So we want a price such that 
\begin_inset Formula 
\[
2\omega\frac{\left(1-\alpha\right)}{1+\alpha}=\frac{\left(1-\alpha\right)\omega}{p}
\]

\end_inset

or
\begin_inset Formula 
\[
p=\frac{1+\alpha}{2}.
\]

\end_inset

It isn't too hard to show that 
\begin_inset Formula $\frac{1+\alpha}{2}>\frac{1}{2-\alpha}$
\end_inset

 for all 
\begin_inset Formula $0<\alpha<1$
\end_inset

, so consumer 2 will set a lower price for software than the price that
 would have supported the same output of software as in the voluntary contributi
on game.
\end_layout

\begin_layout Standard
We could have figured this out more directly.
 Once consumer 2 gets a patent, she automatically receives a transfer of
 income from consumer 1.
 However, the size of the transfer doesn't change as 2 varies the price
 because of the properties of Cobb Douglas preferences.
 So if consumer 1 has Cobb Douglas preferences, the only way consumer 2
 can increase her payoff is by using the transfer to pay for more software
 for herself.
\end_layout

\begin_layout Standard
More generally, if the consumer has more reasonable preferences for which
 the proportion of income that she spends on software rises as software
 prices rise, then consumer 2 will be able to extract even more income from
 consumer 1 by raising software prices.
\end_layout

\begin_layout Standard
The fact that consumer 2 produces more software looks good, but it doesn't
 mean that consumer 1 is any better off than she would have been without
 the patent.
 
\end_layout

\begin_layout Standard
If consumer 2 has the patent, then consumer 1 has payoff
\begin_inset Formula 
\[
\left(\alpha\omega\right)^{\alpha}\left(\omega\left(1-\alpha\right)\left(2-\alpha\right)\right)^{1-\alpha}=
\]

\end_inset


\begin_inset Formula 
\[
\omega\left(2-\alpha\right)\left(\frac{1}{2-\alpha}\right)^{\alpha}\left(\frac{\alpha}{\left(1-\alpha\right)}\right)^{\alpha}\left(1-\alpha\right).
\]

\end_inset

In the voluntary contribution game, consumer 1 has payoff 
\begin_inset Formula 
\[
\left(\frac{2\alpha\omega}{1+\alpha}\right)^{\alpha}\left(2\omega\frac{1-\alpha}{1+\alpha}\right)^{1-\alpha}=
\]

\end_inset


\begin_inset Formula 
\[
\frac{2}{1+\alpha}\omega\left(\frac{\alpha}{1-\alpha}\right)^{\alpha}\left(1-\alpha\right)
\]

\end_inset

To compare the payoffs, we need to compare 
\begin_inset Formula $\omega\left(2-\alpha\right)\left(\frac{1}{2-\alpha}\right)^{\alpha}$
\end_inset

 and 
\begin_inset Formula $\frac{2}{1+\alpha}\omega$
\end_inset

.
 We can't really figure this out analytically, so we'll turn to the computer.
\end_layout

\begin_layout Standard
The next picture shows a plot of the payoffs of consumer 1 as they vary
 with his propensity for keeping is own income, 
\begin_inset Formula $\alpha$
\end_inset

, when 
\begin_inset Formula $\omega=2$
\end_inset

.
 This calculation, and the draft are drawn using wxMaxima, which is a free
 computer algebra program.
 If you want to try it for yourself, I left the code for the calculation
 at the top of the diagram.
\end_layout

\begin_layout Standard
Recall, that with CD preferences, 
\begin_inset Formula $\alpha$
\end_inset

 is the weight that the consumer gives to retained income 
\begin_inset Formula $x$
\end_inset

 - if 
\begin_inset Formula $\alpha$
\end_inset

 is small, the consumer desperately wants software and is willing to pay
 a lot for it.
 If 
\begin_inset Formula $\alpha$
\end_inset

 is large, the consumer isn't so interested in software.
 This is consistent with the 
\emph on
demand 
\emph default
for software 
\begin_inset Formula $\frac{1-\alpha}{p}$
\end_inset

 - at any price, the consumer will buy less software the larger is 
\begin_inset Formula $\alpha$
\end_inset

.
 The red curve in the figure below represents the function 
\begin_inset Formula $\frac{2}{1+\alpha}\omega$
\end_inset

 which is the factor associated with the equilibrium of the voluntary contributi
on game.
 The blue figure represents the value of the factor associated with patents.
 The implication of the fact that the red curve lies above the blue curve
 is that consumer 1 is never better off than he was in the equilibrium of
 the voluntary contribution game.
 He gets more software, but the price of the software is much more expensive
 to him than the software he would have produced on his own.
 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\left(\alpha\omega\right)^{\alpha}\left(\frac{\left(1-\alpha\right)\omega}{p}\right)=\left(\alpha\omega\right)^{\alpha}\left(\frac{2\left(1-\alpha\right)\omega}{1+\alpha}\right)^{\left(1-\alpha\right)}
\]

\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename maxima_plot.png
	scale 40

\end_inset


\end_layout

\begin_layout Standard
We conclude that though the patent does result in more software being produced,
 its primary purpose is still to transfer income to the patent holder -
 essentially a tax.
\end_layout

\begin_layout Subsection*
Problems to work on.
\end_layout

\begin_layout Standard
One way in which the model describe above differs from open source software
 is that, with software, public good production tends to involve increasing
 returns - the more software other people write, the more software you can
 write for the same amount of money.
 Try to carry out the analysis in the section with quasi-linear software
 above when the production function for public goods is given by 
\begin_inset Formula 
\[
y=x^{2}
\]

\end_inset

 where 
\begin_inset Formula $y$
\end_inset

 is output of the public good and 
\begin_inset Formula $x$
\end_inset

 is money spent on developing the public good.
 Go as far as you can, but at the very least, verify for yourself that when
 production is subject to increasing returns, a lot more software will be
 produced in the voluntary contribution game.
\end_layout

\end_body
\end_document