Department of Economics Microeconomic TheoryUniversity of California, Berkeley Economics 101AMay 2, 2007 Spring 2007Economics 101A: Microeconomic TheorySection Notes for Week 15: Public Goods1 Pareto OptimalityA set of actions (aP1, aP2) is “Pareto optimal” or ”Pareto efficient” if there are is no otherfeasible set of actions (˜a1, ˜a2) such that eitherU1(˜a1, ˜a2) ≥ U1(aP1, aP2) andU2(˜a1, ˜a2) > U2(aP1, aP2)orU1(˜a1, ˜a2) > U1(aP1, aP2) andU2(˜a1, ˜a2) ≥ U2(aP1, aP2)In other words, there is no way to adjust things from a Pareto optimum that makes any oneperson strictly better off while leaving everyone else at least as well off as they already were.An equivalent formulation is that for any feasible set of actions (˜a1, ˜a2)U1(˜a1, ˜a2) > U1(aP1, aP2) ⇒ U2(˜a1, ˜a2) < U2(aP1, aP2)for a Pareto optimum (aP1, aP2). Any set of actions that makes one person strictly better offmust make someone else strictly worse off.2 Solving For Pareto Optima2.1 Guaranteed Minimum UtilityOne way to solve for Pareto optima is to guarantee agents 2 through N a fixed level of utilityk2through kNrespectively and then to maximize agent 1’s utility subject to those N − 1constraints. The set of all possible Pareto optima can be found by adjusting the parametersk2through kN. For example, if there are two agents, a social planner that is trying to findPareto optima would solvemaxa1,a2U1(a1, a2) s.t. U2(a1, a2) ≥ k2the resulting LaGrangian isL(a1, a2, λ; k2) = U1(a1, a2) − λ[k2− U2(a1, a2)]1thus the first order conditions areU1a1(a1, a2) + λU2a1(a1, a2) = 0U1a2(a1, a2) + λU2a2(a1, a2) = 0from these first order conditions one can solve for the actions (aP1(k2), aP2(k2)) which arefunctions of the parameter k2.2.2 Maximizing the Weighted Sum of Individual UtilitiesAnother way to find Pareto optima is to maximize the weighted sum of utilities. Assumethat each agent has utility U1(G, xi) where xiis private consumption and G = g1+ g2where giis agent i’s provision of the public good (like a tax paid). THe price of privateconsumption is pxand the price of the public good is pG. The individual’s budget constraintis pxxi+ pGgi= Ii. Thus a social planner would solvemaxg1,g2λU1(G, x1) + U2(G, x2) =maxg1,g2λU1g1+ g2,I1px−pGpxg1+ U2g1+ g2,I2px−pGpxg2The first order conditions are thusλU1G−pGpxU1x+ U2G= 0λU1G+ U2G−pGpxU2x= 0solving for and equating λ’s givesU1G−pGpxU1xU1G=−U2GpGpxU2x− U2G−U1GU2G= −U1GU2G+pGpxU1GU2x+pGpxU1xU2G−pGpx2U1xU2xpGpxU1xU2x= U1GU2x+ U1xU2GpGpx=U1GU1x+U2GU2x= MRS1G,x+ MRS2G,xwhich is Samuelson’s rule.3 Nash EquilibriumAs opposed to Pareto optima, where a weighted sum of both people’s utility is maximized,a Nash equilibrium involves each agent maximizes her own utility based on the assumptionthat the other agents are behaving in the same manner. To find the Nash Equilibrium, agent1 solvesmaxg1U1g1+ g2,I1px−pGpxg12the first order condition isU1G−pGpxU1x= 0orU1GU1x= MRS1G,x=pGpxin the Nash solution MRS1G,x= MRS2G,x=pGpxand thus GN= gN1+ gN2< gP1+ gP2=