Chapter 8, Unintended Consequences of Policy Interventions Video Solutions, Lectures on Microeconomics: The Big Questions Approach

Problem 1

Modify the routing problem by assuming that any type- $i$ agent's cost of taking any route $x^i \in X^i$ is
$$
\sum_{l \in x^i}\left(c_l\left(t_l\right)+\tau_l\left(t_l\right)\right)
$$
where $c_l$ is differentiable, and $\tau_l\left(t_l\right)$ is a congestion tax that the agent pays the government for the right to pass through link $l$. The utilitarian traffic flow is unchanged because, in the utilitarian social welfare function, which now includes the government's payoff $\sum_{l \in \mathcal{L}} t_l \tau_l\left(t_l\right)$, all tax transfers cancel out, and, so, the utilitarian social welfare function itself is unchanged. By contrast, Nashequilibrium traffic flow is affected by the transfers.
1. Design a tax structure $\left(\tau_l\right)_{l \in \mathcal{L}}$ such that, in SM2LA, the Nash-equilibrium routing coincides with the utilitarian one.
2. Repeat the exercise in part 1 for SM2P with highway 101 added.
3. Repeat the exercise in part 1 for the general routing problem.

Carson Merrill

Numerade Educator

Problem 2

Consider congestion example SM2LA. Modify the example by assuming that fraction $\alpha \in[0,1]$ of agents are Kantian, and fraction $1-\alpha$ are Nash. Each agent's action is the probability with which he takes the freeway, with the probability of taking the side streets being complementary. Each agent is an expected-utility maximizer (i.e., expected-cost-oftime-in-traffic minimizer). In a Kantian-Nash equilibrium, each Nash agent chooses his action taking all others' actions as given; whereas each Kantian agent chooses his action assuming that every Kantian agent would choose the same action as him and taking as given only the actions of the Nash agents.
1. Find a Kantian-Nash equilibrium in which any two Kantian agents take the same action, and any two Nash agents take the same action. Assume that the law of large numbers holds: if, say, each Kantian agent chooses the freeway with probability $p$, then exactly measure pa of Kantian agents will end up on the freeway.
2. How does the equilibrium you have identified in part 1 depend on $\alpha$ ?
3. Is the equilibrium outcome you have identified in part 1 Pareto efficient?

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Problem 3

Find all equilibria for the friendship graphs in Figures 8.3a and $8.3 \mathrm{~b}$ and interpret your findings in the context of Theorem 8.4. If necessary, modify the example in Figure 8.3 to illustrate the aspects of Theorem 8.4 that are missing from that example.

Timothy Telesca

Numerade Educator