#LyX 1.6.10 created this file. For more info see http://www.lyx.org/ \lyxformat 345 \begin_document \begin_header \textclass beamer \use_default_options true \language english \inputencoding auto \font_roman default \font_sans default \font_typewriter default \font_default_family default \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \paperfontsize default \spacing single \use_hyperref false \papersize default \use_geometry true \use_amsmath 1 \use_esint 1 \cite_engine basic \use_bibtopic false \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \author "" \end_header \begin_body \begin_layout BeginFrame \end_layout \begin_layout Itemize there are two goods, one public ( \begin_inset Formula $y$ \end_inset ) and one private ( \begin_inset Formula $x$ \end_inset ) \end_layout \begin_layout Itemize there are two consumers \end_layout \begin_layout Itemize the public good is produced using the private good \end_layout \begin_layout Itemize for example, the two consumers belong to a club - they can spend time \begin_inset Formula $x$ \end_inset to organize events at the club (y). \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Itemize the payoffs to the consumers are given by \begin_inset Formula \[ u_{1}\left(x_{1},y\right)\] \end_inset for consumer \begin_inset Formula $1$ \end_inset and \begin_inset Formula \[ u_{2}\left(x_{2},y\right)\] \end_inset for consumer 2 \end_layout \begin_layout Itemize \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $x_{2}$ \end_inset are the time that consumer 1 and 2 respectively get to themselves, while \begin_inset Formula $y$ \end_inset is the total number of events organized by the club - both consumers enjoy all the events at the club equally \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Itemize events are produced when the two consumers spend time organizing them \end_layout \begin_layout Itemize the relationship between the number of events and the time each consumer has to themselves is \begin_inset Formula \[ y=f\left(\omega_{1}+\omega_{2}-x_{1}-x_{2}\right)\] \end_inset where \begin_inset Formula $\omega_{1}$ \end_inset and \begin_inset Formula $\omega_{2}$ \end_inset are the total amount of time that each consumer has to allocate between the two activities \end_layout \begin_layout Itemize there are no rules about volunteering time, each consumer spends whatever time they like organizing \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Itemize this is called the \emph on voluntary contribution game \end_layout \begin_layout Itemize the Nash equilibrium is given by 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 EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Itemize in a Nash equilibrium, each consumer believes that he knows how much time the other consumer is going to volunteer \end_layout \begin_layout Itemize if consumer 1 believes that consumer 2 is going to take \begin_inset Formula $x_{2}$ \end_inset hours for himself, then his or her problem is to maximize \begin_inset Formula \[ u_{1}\left(x_{1},y\right)\] \end_inset subject to \begin_inset Formula \[ y=f\left(\omega_{1}+\omega_{2}-x_{1}-x_{2}\right).\] \end_inset \end_layout \begin_layout Itemize this is a problem you have seen many many times before - so we can draw a picture \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename public_goods_maximization_1.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename public_goods_maximization_2.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename public_goods_maximization_3.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename public_goods_maximization_4.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename public_goods_maximization_5.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename public_goods_maximization_6.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame Find Equilibrium \end_layout \begin_layout Itemize in algebra, let \begin_inset Formula $u_{1}\left(x,y\right)=\alpha\ln\left(x\right)+\left(1-\alpha\right)\ln\left(y\right)$ \end_inset for both players, and suppose that the production function is just \begin_inset Formula $y=\left(\omega_{1}+\omega_{2}-x_{1}-x_{2}\right)$ \end_inset . \end_layout \begin_layout Itemize given his expectation that consumer 2 will contribute \begin_inset Formula $\omega_{2}-x_{2}$ \end_inset to producing the public good, consumer 1 should solve \begin_inset Formula \[ \max_{x_{1}}\alpha\ln\left(x_{1}\right)+\left(1-\alpha\right)\ln\left(\omega_{1}+\omega_{2}-x_{1}-x_{2}\right)\] \end_inset \end_layout \begin_layout Itemize the first order condition is \begin_inset Formula \[ \frac{\alpha}{x_{1}}=\frac{\left(1-\alpha\right)}{\omega_{1}+\omega_{2}-x_{1}-x_{2}}\] \end_inset \end_layout \begin_layout Itemize this gives the simple solution \begin_inset Formula \[ x_{1}=\alpha\left(\omega_{1}+\omega_{2}-x_{2}\right).\] \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Itemize if you write down the same equation for consumer 2 and solve both equations simultaneously for \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $x_{2}$ \end_inset , you will find they are both the same and equal to \begin_inset Formula $\frac{\alpha}{1+\alpha}\left(\omega_{1}+\omega_{2}\right)$ \end_inset \end_layout \begin_layout Itemize the question we want to ask is - if we (as dictators) could choose \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $x_{2}$ \end_inset to be anything at all, would we be happy with the players choices in a Nash equilibrium? or would we want to try to force them to do something else. \end_layout \begin_layout Itemize the algebra doesn't address this question, so lets go back to the diagrams \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Itemize the equation \begin_inset Formula \[ x_{1}=\alpha\left(\omega_{1}+\omega_{2}-x_{2}\right).\] \end_inset is player 1's best reply function \end_layout \begin_layout Itemize we could draw this on a graph \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename best_reply_1.eps \end_inset \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename best_reply_2.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Itemize back to the \begin_inset Quotes eld \end_inset what if we could pick anything \begin_inset Quotes erd \end_inset approach, we could ask what 1 would do if he could pick both \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $x_{2}$ \end_inset \end_layout \begin_layout Itemize the he would solve the problem \begin_inset Formula \[ \max_{x_{1},x_{2}}u_{1}\left(x_{1},f\left(\omega_{1}+\omega_{2}-x_{1}-x_{2}\right)\right)\] \end_inset \end_layout \begin_layout Itemize of course that would make him a lot better off \end_layout \begin_layout Itemize one way to think of the Nash equilibrium is that he does exactly this, but he is constrained to choose \begin_inset Formula $x_{2}$ \end_inset so that it is equal to what he expects \begin_inset Formula $2$ \end_inset to choose \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename best_reply_3.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename public_goods_fig3.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame Patents \end_layout \begin_layout Itemize give player 2 a patent \end_layout \begin_layout Itemize she controls the public good and charges player 1 for all the public good that is produced \end_layout \begin_layout Itemize no competition is allowed \end_layout \begin_layout Itemize if \begin_inset Formula $p$ \end_inset is the (relative) price of \begin_inset Formula $x$ \end_inset , the \begin_inset Formula $\frac{1}{p}$ \end_inset is the relative price of the public good \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename patents_fig_1.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename patents_fig_2.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename patents_fig_3.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename patents_fig_4.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename patents_fig_5.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout Standard \begin_inset Graphics filename patents_fig_6.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename patents_fig_7.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename patents_fig_8.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout BeginFrame \end_layout \begin_layout Standard \begin_inset Graphics filename patents_fig_9.eps \end_inset \end_layout \begin_layout EndFrame \end_layout \end_body \end_document