please answer question 1,2,3,4
Consider a game played by three agents: a parent (P), and two children (1 and 2). Suppose that each player simultaneously takes some action (ai0, i=P,1,2) that affects the earnings of every family member (yj0,j=P,1,2). In particular, earnings are given as follows: yP=fP(aP,a1,a2)=6+aP6aP26a126a22y1=f1(aP,a1,a2)=2+a16aP26a126a22y2=f2(aP,a1,a2)=2+a26aP26a126a22 In other words, by increasing ai, party i can increase his or her own earnings, but imposes a negative externality on the earnings of others. Unless stated otherwise, assume that the utility of each party is just equal to that party's income, and that actions (ai) do not affect utility directly. 1. (5 points) Assuming that a central planner can make lump sum redistributions to achieve any desired distribution of resources across the three parties, what are the socially optimal actions? What is the resulting total income for the family unit? 2. (5 points) Assume that the parties independently and simultaneously choose their actions. Do players have dominant strategies? If so, what are they? What is the Nash equilibrium? Is it unique? What is the total income of the family unit in equilibrium? 3. (5 points) Imagine that the government can impose a tax/subsidy scheme to address the externality. What is th optimal corrective tax (known as a Pigouvian tax)? Verify your answer by recalculating the equilibrium. 4. (10 points) Now imagine that the parent, P, cares about both children. In particular, suppose that the parent's utility function is u(cP,c1,c2)=lncP+21lnc1+21lnc2 where ci is the consumption of party i. Consumption may differ from income because the parent is allowed to make monetary transfers: cP= yPt1t2, and ci=yi+ti for i=1,2. To keep the analysis simple, assume that the parent can make positive or negative transfers (i.e. there is no non-negativity constraint). However, only the parent can make transfers. The game unfolds as follows: first, all parties simultaneously choose actions ai; second, the parent chooses transfers, having observed actions and realized incomes. Solve for the subgame perfect equilibrium of this game. (Hint: there's an easy way to do this, and a hard way. Think about how the consumption of each party varies with income on the continuation path. What does that tell you about each party's objectives in stage 1 ?) Consider a game played by three agents: a parent (P), and two children (1 and 2). Suppose that each player simultaneously takes some action (ai0, i=P,1,2) that affects the earnings of every family member (yj0,j=P,1,2). In particular, earnings are given as follows: yP=fP(aP,a1,a2)=6+aP6aP26a126a22y1=f1(aP,a1,a2)=2+a16aP26a126a22y2=f2(aP,a1,a2)=2+a26aP26a126a22 In other words, by increasing ai, party i can increase his or her own earnings, but imposes a negative externality on the earnings of others. Unless stated otherwise, assume that the utility of each party is just equal to that party's income, and that actions (ai) do not affect utility directly. 1. (5 points) Assuming that a central planner can make lump sum redistributions to achieve any desired distribution of resources across the three parties, what are the socially optimal actions? What is the resulting total income for the family unit? 2. (5 points) Assume that the parties independently and simultaneously choose their actions. Do players have dominant strategies? If so, what are they? What is the Nash equilibrium? Is it unique? What is the total income of the family unit in equilibrium? 3. (5 points) Imagine that the government can impose a tax/subsidy scheme to address the externality. What is th optimal corrective tax (known as a Pigouvian tax)? Verify your answer by recalculating the equilibrium. 4. (10 points) Now imagine that the parent, P, cares about both children. In particular, suppose that the parent's utility function is u(cP,c1,c2)=lncP+21lnc1+21lnc2 where ci is the consumption of party i. Consumption may differ from income because the parent is allowed to make monetary transfers: cP= yPt1t2, and ci=yi+ti for i=1,2. To keep the analysis simple, assume that the parent can make positive or negative transfers (i.e. there is no non-negativity constraint). However, only the parent can make transfers. The game unfolds as follows: first, all parties simultaneously choose actions ai; second, the parent chooses transfers, having observed actions and realized incomes. Solve for the subgame perfect equilibrium of this game. (Hint: there's an easy way to do this, and a hard way. Think about how the consumption of each party varies with income on the continuation path. What does that tell you about each party's objectives in stage 1 ?)
